Proof and Problem Solving at University Level, January 2013
ANNIE SELDEN2013, The Mathematics Enthusiast, Special Issue: International Perspectives on Problem Solving Research in Mathematics Education (Manuel Santos-Trigo & Luis Moreno-Armella, Guest Editors), Vol. 10, 1&2 (January 2013), 303-334.
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This paper is concerned with undergraduate and graduate students’ problem solving as they encounter it in attempting to prove theorems, mainly to satisfy their professors in their courses, but also as they conduct original research for theses and dissertations. We take Schoenfeld’s (1985) view of problem, namely, a mathematical task is a problem for an individual if that person does not already know a method of solution for that task. Thus, a given task may be a problem for one individual, who does not already know a solution method for that task, or it may be an exercise for an individual who already knows a procedure or an algorithm for solving that task.
Related papers
Proof and Problem Solving at University Level
The Mathematics Enthusiast, 2013
This paper will be concerned with undergraduate and graduate students' problem solving as they encounter it in attempting to prove theorems, mainly to satisfy their professors in their courses, but also as they conduct original research for theses and dissertations. We take Schoenfeld's (1985) view of problem, namely, a mathematical task is a problem for an individual if that person does not already know a method of solution for that task. Thus, a given task may be a problem for one individual, who does not already know a solution method for that task, or it may be an exercise for an individual who already knows a procedure or an algorithm for solving that task. A Continuum of Tasks from Very Routine to Very Non-routine While what is a problem depends on what a solver knows, it is possible for most mathematics teachers to judge what is difficult for most students in a given class. Thus, we see mathematical tasks for a given class, such as a calculus class, on a continuum from those that are very routine to those that are genuinely difficult problems (Selden, Selden, Hauk, & Mason, 2000). At one end, there are very routine problems which mimic sample worked problems found in textbooks or lectures, except for minor changes in wording, notation, coefficients, constants, or functions that are incidental to the way the problems are solved. Such problems are often referred to as exercises (and might not be considered to be problems at all in the problem-solving literature). The vast majority of exercises in calculus textbooks are of this nature. Lithner (2004) distinguished three possible solution strategies for typical calculus textbook exercises: identification of similarities (IS), local plausible reasoning (LPR), and global plausible reasoning
1 Operating on and Understanding of Problem Situations in Proving
2015
Proving in secondary mathematics often appears as a solving process of proof problems. This paper presented a model of proving processes which is based on one conception of mathematical problem solving. The aim of this paper is to investigate students ’ proving processes to check the validity of this model and get insights into students proving processes from the viewpoint of this model. For this aim, two proving processes in each of which a pair of 9th grade students tried to solve a plane geometry problem using a dynamic geometry software will be analyzed qualitatively. The analysis will illustrate the interactions between the students ’ understanding of the problem situation and their explorations and the importance of deepening solvers ’ understanding in proving process. It will also show that operating on the problem situation and getting in touch with it is useful throughout proving processes and that solvers ’ interpretations are influenced by their understanding of the probl...
Mathematical and Non-mathematicalUniversity Students' Proving Difficulties, PME-NA, October 2011.
Proceedings of the 33rd Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education, Reno, Nevada (pp. 675-683). , 2011
This paper discusses university students’ mathematical and non-mathematical proving difficulties. A total of forty-one difficulties have been observed and organized into nine categories. Of these difficulties, twenty-seven are briefly described below. These observations come from several years of teaching an experimental proving course to beginning graduate and advanced undergraduate mathematics students and from teaching an experimental voluntary proving supplement to an undergraduate real analysis course. We believe that discussing and categorizing these difficulties will lead to a greater understanding of students’ thinking with regard to proof and to future research.
Teaching Advanced Students to Construct Proofs, longer version. Shorter version appeared in O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepulveda (Eds.), Proceedings of the Joint Meeting of PME 32 and PME-NA XXX (p. I-305), 2008.
2008
In this paper, we describe preliminary results arising from the development of a modified R. L. Moore Method course devoted entirely to helping advanced university mathematics students improve their proving abilities. The paper describes the course and why it might be needed. We also discuss kinds and aspects of proofs in a way that may be useful in gauging student progress, and in particular, introduce the idea of the formalrhetorical and problem-centered parts of a proof. We go on to propose a theoretical perspective suggesting that much of proving depends on procedural knowledge in the form of small habits of mind, or behavioral schemas, and give three examples.
The Proof Construction Process for Undergraduate and Beginning Graduate Students, an NSF Proposal, written 2009.
The Project Summary was as follows: Intellectual Merit. This proposed contextual research – empirical research project will use a new theoretical perspective on the learning of proof construction. We view much of the proof construction process as a sequence of mental (or physical) actions in response to situations that arise in the partially completed process. Similar situations tend to yield similar actions that can become, more or less, permanently linked with those situations. We regard the smallest of such linked, automated situation-action pairs as persistent mental structures, and call them behavioral schemas. One such behavioral schema, that we have observed and that students often take a long time to develop, links theorems of the form “For all numbers x, P(x)” [the situation] with writing into the proof “Let x be a number,” meaning x is fixed but arbitrary [the action]. The idea of behavioral schemas is similar to ideas that some social psychologists have been investigating about the automaticity of much of everyday life, except that we are reifying the linking of a situation to an action into a mental object (a behavioral schema) that can be altered with the aid of a teacher. We have developed a six-point theoretical sketch of the genesis and enactment of behavioral schemas (given in the proposal). One point is that behavioral schemas are acquired through practice. That is, to acquire a beneficial schema a person should actually carry out the appropriate action correctly a number of times – not just understand its appropriateness. Changing a detrimental behavioral schema, which may have been tacitly acquired, requires similar, perhaps longer, practice. This research project will produce immediately usable outcomes. We will find and categorize, ways to promote beneficial behavioral proving schemas, as well as uncover and find ways to change detrimental ones. We will also find and categorize kinds and aspects of proofs at a level of detail that will be practical for teachers. The overarching research method is the design experiment. As with most design experiments, our intent is to develop theories about the process of learning (to construct proofs), and about the teaching and teaching materials that support learning. The research results, however, are intended to be paradigmatic, that is, to apply more widely to situations and students other than our own. We are proposing six interconnected subprojects: (1) a design experiment studying the development of a “proofs” course for advanced undergraduate and beginning graduate mathematics students; (2) a design experiment developing a “proving skills” supplement for undergraduate real analysis; (3) a design experiment developing a nontraditional transition-to-proof course; (4) conducting and studying sustained one-on-one tutoring; (5) using tablet computers to record and observe students independently constructing proofs; and (6) developing a new kind of electronic book/course notes that is adjustable by teachers to suit differing classes and specific content. Broader impact. It is common knowledge, among many who teach beginning mathematics graduate students in the U.S., that significantly many such students have difficulty constructing proofs and that this difficulty sometimes persists for a long time. Sometimes the inability to prove theorems causes students to fail because content course assessments are based primarily on that ability. The remaining students, who cannot significantly improve their proving skills, are likely to leave mathematics voluntarily for some other field. A similar problem occurs at the upper undergraduate level for students, such as preservice secondary mathematics teachers, in proof-based content courses such as abstract algebra and real analysis. This is a time in their studies when some undergraduates realize (perhaps wrongly) that mathematics is going to require a skill that they cannot do well and that may seem impossible to learn. Such students are likely to start seriously considering a non-mathematical career. The current “solution” for graduate students is often to put them in “remedial” courses, such as undergraduate real analysis or abstract algebra, but by and large, this attempted “solution” is not successful. Perhaps this is because such undergraduate content courses concentrate on extending students’ knowledge of mathematical topics, and can rarely devote much class time to improving proving skills. Because beginning mathematics graduate students are the survivors of a long sequence of younger students turning away from mathematics, sometimes referred to as “the leaky pipeline,” they represent a considerable national investment. A Challenge of Numbers noted that the attrition rate from mathematics from ninth grade on, is roughly 50% per year, and the attrition rate for Latinos and African Americans is significantly larger. This is a waste of the nation’s resources and does not bode well for the future of the U.S. STEM workforce. The information to be provided by this project is meant to solve both the above graduate and undergraduate problems.
Errors and Misconceptions in College Level Theorem Proving. Tennessee Technological University Mathematics Department Technical Report. No. 2003-3
Online Submission, 2003
A Theoretical Perspective for Proof Construction, Powerpoint Presentation, 18th Annual Conference on Research in Undergraduate Mathematics Education, February 2015
2015
This theoretical paper suggests a perspective for understanding university students' proof construction based on the ideas of conceptual and procedural knowledge, explicit and implicit learning, behavioral schemas, automaticity, working memory, consciousness, and System 1 and System 2 cognition. In particular, we will discuss proving actions, such as the construction of proof frameworks that could be automated, thereby reducing the burden on working memory and enabling university students to devote more resources to the truly hard parts of proofs.
Errors and Misconceptions in College Level Theorem Proving, Cornell Misconceptions Conference Proceedings, July 1987
Proceedings of the Second International Seminar on Misconceptions and Educational Strategies in Science and Mathematics (Joseph D. Novak, Ed.), Vol. III, Cornell University, July 1987, 457-470.
The Development of Proofs in Analytical Mathematics for Undergraduate Students
Journal of Physics: Conference Series
Proofs in analytical mathematics are essential parts of mathematics, difficult to learn because its underlying concepts are not visible. This research consists of problems involving logic and proofs. In this study, a short overview was provided on how proofs in analytical mathematics were used by university students. From the results obtained, excellent students obtained better scores compared to average and poor students. The research instruments used in this study consisted of two parts: test and interview. In this way, analysis of students' actual performances can be obtained. The result of this study showed that the less able students have fragile conceptual and cognitive linkages but the more able students use their strong conceptual linkages to produce effective solutions
Understanding and Transforming University Students’ Learning of the Proof Construction Process in Mathematics, an NSF Proposal, written 2010.
The project summary for this grant proposal was as follows: This empirical research proposal is in the STEM learning strand. It will produce transformational, fundamental knowledge about: How can advanced university students learn, and be taught, to construct mathematical proofs autonomously? The emphasis will be on what students are capable of, in contrast to the literature’s focus on their difficulties. The proposal is also partly in the cognitive underpinnings strand because it will develop a new theoretical view of how the mind controls its own deductive reasoning. The above question (and more tractable subquestions) will be answered through a naturalistic study (Lincoln & Guba, 1985) using a generative/integrative analysis (Clement, 2000) of data arising from an unusual experimental “proofs” course. This course is for beginning graduate and some advanced undergraduate mathematics students who need help with proving. Its sole purpose is to provide that help, and classes consist very largely of students presenting their proofs and receiving critiques and advice. The course is consistent with a constructivist point of view, in that the teachers attempt to help students reflect on, and learn from, their own proof writing experiences. It is also somewhat Vygotskian (1978) in that the teachers represent to the students how the mathematics community writes proofs. That is, they are instruments in the cultural mediation of community norms and practices. Data will be collected mainly in four ways: (1) videoing and analyzing all class meetings; (2) videoing extensive one-on-one tutoring of students constructing proofs; (3) task-based interviews of former students; and (4) for the first time, using tablet computers to record and observe students independently constructing proofs. The theoretical framework used in a data analysis of a naturalistic study emerges from the data itself. From pilot project data a framework has already started emerging including: (1) kinds of proofs – needed to keep track of student progress; and (2) aspects of a proof – mathematicians fail to prove a theorem because of its “problem-oriented aspect”, but students (mainly) fail to prove a theorem because they cannot write its “formal-rhetorical aspect” (which is teachable). The pilot project data have also yielded a view of much of the proof construction process as a sequence of mental (or physical) actions in response to situations that arise in the partially completed proof construction. Similar situations tend to yield similar actions that can become permanently linked with those situations. The smallest of such linked, automated situation-action pairs are regarded as persistent mental structures, called behavioral schemas. An example of a behavioral schema, that students often take a long time to develop, links theorems of the form “For all numbers x, P(x)” [the situation] with writing into the proof “Let x be a number,” meaning x is fixed but arbitrary [the action]. Such schemas are stored in procedural memory and can be activated without requiring conscious recollection, thus reducing the demands on working memory. The idea of behavioral schemas in this project is not derived from other schema theories in psychology. However, it is reminiscent of social psychologists’ view of automaticity in everyday life, except that the linking of situations to actions is reified into mental objects. Intellectual merit: (1) The idea of a course and a teaching experiment devoted to teaching proving at this level is novel, and the project will produce much detailed information on what students can do, rather than focusing mainly on their difficulties. (2) One of the proposed data collection methods is new and will provide insights into students’ actions and thinking when alone. (3) The emerging framework will allow one to see that an aspect of a proof that blocks a mathematician’s progress is different in nature from the aspect that blocks most students. Thus teaching students to mimic mathematicians may not help. (4) The emerging view of behavioral schemas should prove very helpful in teaching, and elucidate how procedural knowledge can guide the use of conceptual knowledge. Such schemas may also illustrate the utility of converting System 2 cognition to System 1 cognition (Stanovich & West, 2000). Broader impact: (1)This project will provide information for designing courses for teaching students to prove theorems on a variety of topics and for students at a variety of levels – beginning graduate, mid-level undergraduate, and even high school. (2) It is common knowledge among teachers of beginning graduate students that significantly many need help constructing proofs, and some, who do not improve, drop out of mathematics. The experimental course in this project could be conveniently adapted to other universities to alleviate this situation. (3) When students first learn to construct proofs, they often find the experience empowering, and raise their career expectations and their interest in mathematics. This can be particularly helpful for minority populations whose expectations are not high.
The Mathematics Enthusiast
ISSN 1551-3440
(formerly The Montana Mathematics Enthusiast)
VOL. 10, Nos1&2, January 2013, pp.1-506
Editor-in-Chief
Bharath Sriraman, The University of Montana
Associate Editors:
Lyn D. English, Queensland University of Technology, Australia
Simon Goodchild, University of Agder, Norway
Brian Greer, Portland State University, USA
Luis Moreno-Armella, Cinvestav-IPN, México
International Editorial Advisory Board
Mehdi Alaeiyan, Iran University of Science and Technology, Iran
Miriam Amit, Ben-Gurion University of the Negev, Israel
Ziya Argun, Gazi University, Turkey
Ahmet Arikan, Gazi University, Turkey
Hyman Bass, University of Michigan
Raymond Bjuland, University of Stavanger, Norway
Morten Blomhøj, Roskilde University, Denmark
Constantinos Christou, University of Cyprus, Cyprus
Helen Doerr, Syracuse University, USA
Paul Ernest, University of Exeter, UK
Viktor Freiman, Université de Moncton, Canada
Fulvia Furinghetti, Università di Genova, Italy
Anne Birgitte Fyhn, Universitetet i Tromsø, Norway
Marja van den Heuvel-Panhuizen, University of Utrecht, The Netherlands
Laura Jacobsen, Radford University, USA
Kristina Juter, Kristianstad University, Sweden
Gabriele Kaiser, University of Hamburg, Germany
Yasemin Kýymaz, Ahi Evran Üniversitesi, Turkey
Tinne Hoff Kjeldsen, Roskilde University, Denmark
Jean-Baptiste Lagrange, IUFM-Reims, France
Kyeonghwa Lee, Seoul National University
Stephen Lerman, London South Bank University, UK
Frank Lester, Indiana University, USA
Richard Lesh, Indiana University, USA
Claus Michelsen, University of Southern Denmark, Odense
Reidar Mosvold, University of Stavanger
Nicholas Mousoulides, University of Cyprus, Cyprus
Norma Presmeg, Tenessee, USA
Gudbjorg Palsdottir,University of Iceland
João Pedro da Ponte, University of Lisbon, Portugal
Demetra Pitta Pantazi, University of Cyprus, Cyprus
Linda Sheffield, Northern Kentucky University, USA
Olof Bjorg Steinthorsdottir, University of Northern Iowa
Günter Törner, University of Duisburg-Essen, Germany
Renuka Vithal, University of KwaZulu-Natal, South Africa
Dirk Wessels, UNISA, South Africa
Nurit Zehavi, The Weizmann Institute of Science, Rehovot, Israel
AIMS AND SCOPE
The Mathematics Enthusiast (TME) is an eclectic internationally circulated peer reviewed journal
which focuses on mathematics content, mathematics education research, innovation,
interdisciplinary issues and pedagogy.
The journal is published on a print-on-demand basis by Information Age Publishing and the
electronic version is hosted by the Department of Mathematical Sciences- The University of
Montana. The journal supports the professional association PMENA [Psychology of
Mathematics Education- North America] through special issues on various research topics. TME
strives to promote equity internationally by adopting an open access policy, as well as allowing
authors to retain full copyright of their scholarship contingent on the journals’ publication ethics
guidelines: http://www.math.umt.edu/TMME/TME_Publication_Ethics.pdf The journal is
published tri-annually.
Articles appearing in the journal address issues related to mathematical thinking, teaching and
learning at all levels. The focus includes specific mathematics content and advances in that area
accessible to readers, as well as political, social and cultural issues related to mathematics
education. Journal articles cover a wide spectrum of topics such as mathematics content
(including advanced mathematics), educational studies related to mathematics, and reports of
innovative pedagogical practices with the hope of stimulating dialogue between pre-service and
practicing teachers, university educators and mathematicians. The journal is interested in research
based articles as well as historical, philosophical, political, cross-cultural and systems perspectives
on mathematics content, its teaching and learning. The journal also includes a monograph series
on special topics of interest to the community of readers The journal is accessed from 110+
countries and its readers include students of mathematics, future and practicing teachers,
mathematicians, cognitive psychologists, critical theorists, mathematics/science educators,
historians and philosophers of mathematics and science as well as those who pursue mathematics
recreationally. The editorial board reflects this diversity. The journal exists to create a forum for
argumentative and critical positions on mathematics education, and especially welcomes articles
which challenge commonly held assumptions about the nature and purpose of mathematics and
mathematics education. Reactions or commentaries on previously published articles are
welcomed. Manuscripts are to be submitted in electronic format to the editor in APA style. The
typical time period from submission to publication is 8-12 months.
Please visit the journal website at http://www.math.umt.edu/TMME/.
Indexing Information
Australian Education Index ;
EBSCO Products (Academic Search Complete);
EDNA;
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PsycINFO (the APA Index);
MathDI/MathEDUC (FiZ Karlsruhe);
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Bombacaceae Ceiba Tree in Bangalore © 2013 Bharath Sriraman
THE MATHEMATICS ENTHUSIAST
(formerly The Montana Mathematics Enthusiast)
ISSN 1551-3440
Vol.10, Nos.1&2, 2013, pp.1-506
International Perspectives on Problem Solving Research in Mathematics Education
Guest Edited by Manuel Santos-Trigo & Luis Moreno-Armella
-1. Editorial: Why (yet) another issue on problem solving?
Bharath Sriraman (USA)………………………………………………………………..….pp.1-2
0. Introduction to International perspectives on problem solving research in mathematics education
Luis Moreno-Armella & Manuel Santos-Trigo (Mexico)……………………….……….…pp.3-8
1. Reflections on Problem Solving Theory and Practice
Alan H. Schoenfeld (USA)……………………………………………………………......pp.9-34
2. Problem Solving in the Primary School (K-2)
Richard Lesh (USA), Lyn English (Australia), Chanda Riggs & Serife Sevis (USA)
……………………………………..……………………………………………..…….pp.35-60
3. Prospective Teachers’ Interactive Visualization and Affect in Mathematical Problem-Solving
Inés Mª Gómez-Chacón (Spain)………………………………………...……………....pp.61-86
4. Young Children Investigating Advanced Mathematical Concepts With Haptic Technologies:
Future Design Perspectives
Stephen Hegedus (USA)………………………………..…………...…………………pp.87-108
5. Cognitive processes developed by students when solving mathematical problems within
technological environments
Fernando Barrera-Mora & Aarón Reyes-Rodríguez (Mexico)…………………....……pp.109-136
6. Problem Solving and its elements in forming Proof
Joanna Mamona-Downs & Martin Downs (Greece)…………………………….……pp.137-162
7. Becoming Aware Of Mathematical Gaps In New Curricular Materials: A Resource-Based
Analysis Of Teaching Practice
José Guzman (Mexico)& Carolyn Kieran (Canada)...........................................………….pp.163-190
8. Mathematical Problem Solving in Training Elementary Teachers from a Semiotic Logical
Approach
Martín Socas & Josefa Hernández (Spain)…………………..…………….…………..pp.191-218
9. Developing problem solving experiences in practical action projects
François Pluvinage (Mexico)……………………………………………...…………..pp.219-244
10. Thoughts About Research On Mathematical Problem- Solving Instruction
Frank K. Lester, Jr. (USA)………………………………………………………...….pp.245-278
11. Framing the use of computational technology in problem solving approaches
Manuel Santos-Trigo (Mexico) & Matías Camacho Machín (Spain)…………………..pp.279-302
12. Proof and Problem Solving at University Level
Annie Selden & John Selden (USA)……………………………….………………….pp.303-334
13. Cognition and Affect in Mathematics Problem Solving with Prospective Teachers
Lorenzo J. Blanco, Eloisa Guerrero Barona &
Ana Caballero Carrasco (Spain)……………………………………………………....pp.335-364
14. Developing the art of seeing the easy when solving problems
Alfinio Flores & Jaclyn Braker (USA)……………………………………….…….….pp.365-378
15. Two-step arithmetic word problems
Enrique Castro-Martínez & Antonio Frías-Zorilla (Spain)……………….…………...pp.379-406
16. Trajectory of a problem: a study in Teacher Training
Alain Kuzniak, Bernard Parzysz & Laurent Vivier (France)………………..……..…..pp.407-440
17. Primary school teacher’s noticing of students’ mathematical thinking in problem solving
Ceneida Fernández, Salvador Llinares & Julia Valls (Spain)………………………….pp. 441-468
18. A Proposal for a Problem-Driven Mathematics Curriculum Framework
Judith S. Zawojewski, Marta T. Magiera & Richard Lesh (USA)…………………...….pp.469-506
TME, vol10, nos.1&2, p .1
Editorial: Why (yet) another issue on Problem Solving?
Bharath Sriraman
This is the 10th volume of The Mathematics Enthusiast, consisting of 500+ pages in 18 articles
that give reflections, directions and the state of the art of mathematical problem solving as it
relates to the field of mathematics education. This impressive collection compiled and guest
edited by Manuel Santos-Trigo and Luis Moreno-Armella contains a treasure trove of
scholarship from both the pioneers of this area of research (Alan Schoenfeld, Richard Lesh,
Frank Lester, among others) as well as reports on new areas of study from Mexico, France and
Spain. Two of the articles (Mamona-Downs & Downs, Selden& Selden) discuss the connections
between problem solving and proof, and one piece (Flores & Braker) explores an interesting
open-ended problem. There are many themes in this double issue- For instance those interested
in advances in problem solving as a result of new technologies such as haptic devices will find
articles (e.g., Hegedus) that report on cutting edge investigations. Others interested in cognition
and learning trajectories as a result of problem solving practices will find articles that cater to
this particular taste. The reflections by forerunners such as Alan Schoenfeld and Frank Lester are
well worth reading for anyone that wants to catch up with developments in problem solving in
the last 40 years.
Mathematics education (in the U.S) has been victimized as not having “really” progressed in
terms of experimental research by the National Mathematics Advisory Panel (see Greer, 2008),
which prescribed algebra as a panacea to cure our students mathematical ills. As noted in an
earlier survey (English & Sriraman, 2010) and numerous articles in this double issue, there have
been tremendous advances in the area of problem solving which unfortunately did not translate
into curricular or “test-score” gains as measured by the testing industry. Problem solving as
implemented in schools in the 90’s also became a fad caught in the pendulum swing of
mathematics education reform. Polya style heuristics that capture the nuances of real
mathematical thinking became didactically transposed and dogmatized by the textbook industry
into prescriptive condition-action rules or flowcharts (Lesh & Sriraman, 2010). Several articles
in this double issue revisit Polya style heuristics and capture its real essence. Some provide
existence “proofs” of the mathematical thinking young children are capable of when presented
with semi structured open-ended problems in conditions that foster novelty (see Lesh, English,
Riggs & Sevis). This should offer the community hope that problem solving research and well
documented empirically validated skill sets can be promoted and made relevant for the new
generation of school children, particularly in an age where thinking across the disciplines is
necessary in many professions. Hopefully this answers the question posed in the title of the
editorial.
The journal imposed an 18 month embargo on submissions (which will end on 04/2014) to clear
up the backlog of articles as well as make room for special issues in the works. In 2011, the
journal received the honor of being selected by National Science Foundation's Math and Science
Partnership (MSP) program committee to assemble and publish a set of papers over the next two
years to expand avenues for more MSP projects to share what they are learning about
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos.1&2, pp. 1-2
2013©The Author(s) & Dept. of Mathematical Sciences-The University of Montana
Sriraman
mathematics and science education through an internationally recognized peer-reviewed journal
that is widely available. Over the next two years some special issues, starting with Vol10, no3
[July 2013] will feature articles reporting on MSP projects. These projects include large
partnerships targeting science and/or mathematics teaching and learning in specific grade bands
or disciplinary areas, institute partnerships focusing on developing teacher leadership,
partnership incubator (or “Start”) projects focusing on learning about institutional partnership
development.
Another important change to be noted is that TME now allows authors to retain full copyright of
their work as opposed to transferring it to publishing entities that use our work to generate profit
(Sriraman, 2012). Indeed the journal now exists as an independent entity, with open access, as
well as supports professional associations like PMENA and other grass roots research groups by
providing a peer reviewed outlet for ongoing research. Vol.11, no.3 [July 2014] will feature
articles synthesizing 5+ years of research within the PMENA working group on Pre-service
Elementary Mathematics Teacher Content Knowledge. This topic is particularly poignant to me
since the first issue of this journal (vol1, no1, 04/2004) was the result of four idealistic
elementary school teachers believing in the mission of this journal and writing about their
attempts to reconcile the mathematics content they were learning in a mathematics for
elementary school teachers course with existing mathematics education research found in
practitioner’s journals as well as standards imposed by institutions framing policy.
I am thankful to the community for supporting the mission and the existence of this journal. Ten
years ago, I dared to dream and imagine the possibilities of and for this journal. Time and
dedicated work have allowed it to flourish in myriad uncharted directions and benefit many
people. I wish the editors, authors and readers of The Mathematics Enthusiast a Happy 2013Unlike the doomsday soothsayers predictions things continue to exist! To that end for T(i)ME to
continue to exist (pun intended), I ask for your continued support…
Bangalore, India
Jan 4, 2013.
References
English, L., & Sriraman, B (2010) Problem solving for the 21st Century. In B. Sriraman & L.
English (Eds.) Theories of Mathematics Education : Seeking New Frontiers. Springer, pp.
263-285.
Greer, B. (2008). Critical Notice on the National Mathematics Advisory Panel Report. The
Montana Mathematics Enthusiast, 5(2&3),365-428
Lesh, R., & Sriraman, B. (2010). Reconceptualizing mathematics education as a design science.
In B.Sriraman & L. English (Eds). Theories of Mathematics Education: Seeking New
Frontiers. Springer, pp. 123-146.
Sriraman, B. (2012). Dogmatism and the Knowledge Industry: More accurately assessing the
work of gifted scholars. In D. Ambrose, R. Sternberg, B. Sriraman (Eds) Confronting
Dogmatism in Gifted Education. Routledge, Taylor and Francis, pp.115-127.
TME, vol10, nos.1&2, p .3
Introduction to International Perspectives on Problem Solving Research
in Mathematics Education
Luis Moreno Armella1 & Manuel Santos-Trigo2 (Mexico)
Any field of research and innovation must be exposed to revisions, criticisms and to an
intense scrutiny not only to discuss the state of the art but, hopefully, to identify
prospective changes and new areas of study and exploration as well.
Problem Solving has been such an area, with a prominent place in mathematics education
and whose contributions continuously appear in conference proceedings, handbooks,
journals, books and, more recently, in digital endeavors. Problem Solving involves an
approach that fosters reflection and delving into mathematical ideas to explain
individuals’ cognitive behaviors within social media. Here, we argue that ideas do not
live by themselves isolated from the semantic networks that sustain the life of cognition:
meaning. These networks constitute a key ingredient for developing understanding and
structural perspective of concepts through problems. In the long term, (and maybe not
that long) these networks provide integration of knowledge that learners need to construct
and integrate in order to gain a wide perspective.
Problem Solving drives developments through research programs, curriculum design,
teachers’ mathematical education, and mathematical instruction at the level of the
classroom. Taking this and more into due account, and in order to identify current trends
in mathematical problem solving and to foster a further exchange of ideas within our
1
2
[email protected]]
[email protected]
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos.1&2, pp. 3-8
2013©The Author(s) & Dept. of Mathematical Sciences-The University of Montana
Moreno-Armella & Santos-Trigo
community, we discussed the present project with Bharath Sriraman, the editor of The
Mathematics Enthusiast, who generously accepted to devote a special issue of the Journal
to Mathematical Problem Solving. We invited colleagues, who have made significant
contributions to this field, to contribute to the special issue. Previously, we had identified
some lines of development to eventually frame their contributions. Of course this was not
meant as a restriction to their freedom; rather to orient their possible directions. Some
questions were posed and discussed to help us identify possible themes to consider in the
volume. Thus, we tried to answer: What are the current trends in problem solving
research and what are the main results that influence teachers’ practices and curricula
design? In addition, with the significant development of digital tools and environments,
we are in need to understand to what extent the present research agenda is being or need
to be transformed by their influence. This touches, for instance, deep epistemic issues
concerning the nature of valid mathematical reasoning and results in mathematics in the
classroom. We have to take into account that mediation tools are not neutral, neither from
the cognitive nor from the epistemic viewpoint. That the knowledge students generate
and/or appropriate, is intertwined with these tools. However, we cannot forget that a
school culture always leaves significant marks on students’ and teachers’ values. Artigue
(2005, p. 246) states that ‘‘these [previous] values were established, through history, in
environments poor in technology, and they have only slowly come to terms with the
evolution of mathematical practice linked to techno- logical evolution.’’ Thus, the school
culture requires the gradual re-orientation of its practices to gain access to new habits of
mind and to the new environments resulting from a serious presence of digital
technologies.
TME, vol10, nos.1&2, p .5
We consider that how we understand the learning of mathematics through a problemsolving approach is deeply related, today, with the presence of the mediation tools that
students will find and use in and outside the classroom.
We have shared with the invited authors a list of themes; we would like to mention some
of them we consider particularly relevant.
Mathematical Problem Solving Foundations. Any domain of study needs to make
explicit tenets and principles that support and justify its academic agenda. As we
mentioned before, we are interested in documenting the extent to which theoretical and
pragmatic frameworks are helpful to explain the problem solvers’ development of new
mathematical knowledge. Besides taking into account the role of meditational tools as the
foundation of a research program, we need to consider as well the contrasts with a
modeling approach to problem solving.
It is relevant to investigate how the presence of digital tools has transformed the agenda
of problem solving approaches, which, in its early stages, has developed within a culture
of paper and pencil mathematics. Of course, the lines of reasoning supported with and
within, the new expressive media reflects what we have, before, termed the cognitive and
epistemological consequences of the digital tools.
Mathematical Problem Solving and International Students’ Mathematical
Assessment. Currently, results from international assessments like PISA or TIMSS are
used to compare or contrast students’ mathematical competences among different
Moreno-Armella & Santos-Trigo
countries. In general, the media use the results to talk about the success or failure of
national educational systems in science, mathematics, and language. Thus, it becomes
important to discuss issues regarding the role of problem solving activities in the
students’ development of competences associated with those types of assessments. Some
questions to discuss in this section involve:
a) How are PISA and TIMSS goals and ways to assess students’
mathematical achievement related to mathematical problem solving? Is the
PISA framework consistent with frameworks used in mathematical
problem solving?
b) What makes a good task or problem foster and evaluate the students’
mathematical thinking? The role of routine and non-routine problems in
problem solving approaches.
c) How can a problem be used for teaching and evaluating the students’
comprehension of mathematical concepts?
d) To what extent should we expect students to pose and solve their own
problems?
e) To what extent international assessments like PISA or TIMSS actually
evaluate problem-solving competences, including those that demand the
use of computational technology?
Mathematical Problem Solving and Curriculum Frameworks. A distinguishing
feature of some current curriculum proposals is that they are structured to enhance
problem-solving activities through all school levels. However, there is a need to discuss
TME, vol10, nos.1&2, p .7
what those proposals entail and should include in terms of contents and mathematical
processes. Thus, relevant questions to discuss in this section involve the structure and
organization of a curriculum centered in problem solving activities. Besides, we need to
identify fundamental ideas and processes that are central to foster students´ appropriation
of mathematical knowledge and the ways digital media can be incorporated within the
eventual proposals. Needless to say, the assessments conundrum will be lurking turning
the corner.
As a consequence, the presence of digital technologies in education calls us to address
this fundamental issue that curricular structures eventually will be inhabited by these
technologies. It has already happened in the past: the technology of writing and the
technology of positional notation of numbers are two of the milestones in the history of
semiotic representations with a living impact on education.
Future developments of Mathematical Problem Solving. It is widely recognized that
students should develop abilities, mathematical resources, and ways of thinking that help
them formulate and solve not only school problems but also situations that they encounter
outside the institutional settings. In this context, it becomes important for students to
examine and explore phenomena in which they have the opportunity to examine
information embedded in a variety of contexts in order to formulate, explore, and
formulate meaningful mathematical questions. Thus, we will be in need to research the
extent to which students can transfer problem-solving experiences learnt within the
school context to new situations. This will of course, demands from the students the
Moreno-Armella & Santos-Trigo
abilities to move across the semantic field of a mathematical notion. This is far from
being a trivial activity.
For instance how could students through problem solving phases that involve gathering
data, modeling activities, find solutions and provide interpretations?
We received a positive response to our invitation letter from the authors and their
contributions often address several issues discussed above. We hope that readers will find
the contents of these two special issues useful to reflect on and extend their views about
problem solving and we invite all to continue the discussion directly with the authors and
other members of the problem solving community.
TME, vol10, nos.1&2, p. 9
Reflections on Problem Solving Theory and Practice
1
Alan H. Schoenfeld
University of California, Berkeley, CA, USA
Abstract: In this article, the author reflects on the current state of mathematical problem
solving, both in theory and in instruction. The impact of the book Mathematical Problem
solving (Schoenfeld, 1985) is also discussed, along with implications of problem solving
today with the advent of 21st century technologies.
Keywords: Mathematical problem solving; Mathematics teaching; Mathematical
learning
Introduction
My book Mathematical Problem Solving (Schoenfeld, 1985), which I shall refer
to as MPS) was published more than 25 years ago. MPS, which was fundamentally
concerned with research and theory, had been developed in dialectic with a course in
problem solving at the university level. The book provided a theoretical rationale for the
course, and evidence that it worked; the course was an existence proof that, with the
“right” kinds of instruction, students could become more effective problem solvers. The
book-plus-course addressed a series of theoretical and pragmatic questions, some of
which they answered, some of which they suggested answers to, and some of which they
left unaddressed. Either directly or by logical extension the ideas in the book had the
potential for significant curricular impact, if the “lessons” in them were taken seriously.
The question is, what has been the fate of the ideas that the book and the course
embodied? Which ideas survived, which flourished? Which evolved in unpredictable
1
[email protected]
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos.1&2, pp.9-34
2013©The Author(s) & Dept. of Mathematical Sciences-The University of Montana
Schoenfeld
ways, which withered with unfulfilled promise? I am grateful to the editors for the
opportunity to reflect on the past and to think about future opportunities.
I begin by describing what, in my opinion, were the achievements, failures, and
potential of that early work (which, of course, built upon and reflected the state of the
field in 1985). This is followed by a characterization some of the main outcomes of the
evolution of problem solving research and development. There is, of course, a huge
literature on problem solving. It is impossible to do justice to that literature, and my
comments will be selective. My most general comments are based, in part, on the volume
Problem solving around the world – Summing up the state of the art (Törner, Schoenfeld,
& Reiss, 2008). That volume provides a recent overview of theory and practice (and to
some degree, curricular politics) in a wide variety of nations. This article will update my
article in that volume (Schoenfeld, 2008), characterizing recent and potentially significant
events in the U.S.
Problem Solving as of 1985 – a retrospective view
In theoretical terms, what MPS offered in 1985 was a framework for the analysis
of the success of failure problem solving attempts, in mathematics and hypothetically in
all problem solving domains. “Problem solving” at its most general was defined as trying
to achieve some outcome, when there was no known method (for the individual trying to
achieve that outcome) to achieve it. That is, complexity or difficulty alone did not make a
task a problem; solving a system of 100 linear equations in 100 unknowns without the
use of technology might be a real challenge for me, but it is not a problem in the sense
that I know how to go about getting an answer, even if it might take me a very long time
and I agonize over the computations.
TME, vol10, nos.1&2, p .11
The core theoretical argument in MPS, elaborated slightly in Schoenfeld (1992),
was that the following four categories of problem solving activity are necessary and
sufficient for the analysis of the success or failure of someone’s problem solving attempt:
a) The individual’s knowledge;
b) The individual’s use of problem solving strategies, known as heuristic
strategies;
c) The individual’s monitoring and self-regulation (an aspect of metacognition);
and
d) The individual’s belief systems (about him- or herself, about mathematics,
about problem solving) and their origins in the students’ mathematical
experiences.
Regarding (a), little needs to be said; one’s mathematical knowledge is clearly a
major determiner of one’s mathematical success or failure. Regarding (b): In 1985 I
singled out heuristic strategies for special attention, because my major intuition when I
began doing research on problem solving was that, with the right kinds of help, students
could learn to employ the heuristic problem solving strategies described by Pólya
(1945/57, 1954, 1962,65/81). Regarding (c): research over the course of the 1970s and
early 1980s had revealed that how well problem solvers “managed” the resources at their
disposal was a fundamental factor in their success or failure. When working complex
problems, effective problem solvers monitored how well they were making progress, and
persevered or changed direction accordingly. Unsuccessful problem solvers tended to
choose a solution path quickly and then persevere at it, despite making little or no
progress (see, e.g., Brown, 1987; Garofalo & Lester, 1985). Finally, regarding (d): by the
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time that MPS was published, many counterproductive student beliefs, and their origins,
had been documented. For example, students whose entire mathematical experience
consisted of working exercises that could be solved in just a few minutes came to believe
that “all problems can be solved in five minutes or less,” and ceased working on
problems that they might have been able to solve had they persevered.
By these categories of behavior being “necessary and sufficient” for the analysis
of problem solving success or failure, I meant that:
They were necessary in the sense that if an analysis of a problem solving failed to
examine all four categories, it might miss the cause. It was easy to provide
examples of problem solving attempts for which each of the four categories above
was the primary cause of success or failure.
They were sufficient in that (I posited that) no additional categories of behavior
were necessary – that the root cause of success or failure would be found in
categories (a) through (d) above.
In MPS I claimed that the framework described above applied for all of
mathematical problem solving; I had ample evidence and experience to suggest that that
would be the case. I conjectured, on the base of accumulated evidence in other fields, that
the framework applied to all problem solving domains, broadly construed. If you take
problem solving in any of the sciences, there was a face value case for the framework.
The relevant knowledge and strategies would be different in each domain – knowledge
and heuristic strategies are different in physics or chemistry than in mathematics – but it
was easy to se that the framework fit. But the potential application was broader. Consider
writing, for example. Someone who sits down to write an essay, for example, is engaging
TME, vol10, nos.1&2, p .13
in a problem solving task – the task being to create a text that conveys certain
information, or sways the opinion, of a particular audience. Various kinds of knowledge
are relevant, both factual and in terms of text production. Writers use heuristic strategies
for outlining, using topic sentences, etc. They can profit from monitoring and selfregulation; or they can lose track of their audience or argument, thus wasting time
producing text that will ultimately be discarded. Finally, beliefs are critically important:
the writer who believes that writing simply consists of writing down what you think will
produce very different text from the writer who believes that crafting text is a challenging
art requiring significant thought and multiple edits.
In sum, MPS offered a framework for analyzing the success or failure of problem
solving, potentially in all problem solving domains. At the same time, the work reported
in MPS had significant theoretical limitations. My analyses of problem solving all took
place in the lab: one or two individuals sat down to work on problems that I had chosen.
In various ways, this represented very significant constraints on their problem solving,
and thus on my analyses. First, they were given the tasks. In most real-world problem
solving, the tasks emerge in practice and have a history or context of some sort. Second,
the goals were pre-determined (the students were to solve my problem) and the problems
themselves were fixed. In problem solving “au naturel,” goals and the problems
themselves often change or emerge in interaction. Third, the timescale was relatively
short. Fourth, social interactions were minimal. Fifth and most important, MPS offered a
framework, highlighting what was important to examine in order to explain success or
failure. What MPS did not offer was a theory of problem solving – a characterization that
allowed one to explain how and why people made the choices they did, while in the midst
Schoenfeld
of problem solving. All of these were limitations I wished to overcome. My ultimate
theoretical goal has been to provide a theoretical explanation that characterizes, line by
line, every decision made by a problem solver while working on a problem (trying to
achieve one or more complex goals) in knowledge intensive, highly social, goal-oriented
activities. In 1985 that goal was far beyond what the field could do.
Let me now turn to issues of practice. First and foremost, MPS was an existence
proof, at multiple levels. At the macro level, the book provided evidence that my problem
solving courses really worked – that my students became much more effective problem
solvers, being able to solve more and more difficult unfamiliar problems after the course
than before. At a finer level of grain size, examining students’ work after the course
showed that it was indeed possible for the students to master a range of problem solving
heuristics; that they could become more effective at monitoring and self-regulation; and
that on the basis of their experiences in the course, students were able to evolve much
more productive beliefs about themselves and mathematics. At a yet finer level of grain
size, MPS offered a methodological blueprint for developing problem solving instruction.
The challenge in 1975, when I began my problem solving work, was that heuristic
strategies “resonated” – when mathematicians read Pólya’s books his descriptions of
problem solving strategies felt right – but, it had not yet been possible to teach students to
use such strategies effectively. A major realization was that Pólya’s descriptions of the
strategies were too broad: “Try to solve an easier related problem” sounds like a sensible
strategy, for example, but it turns out that, depending on the original problem, there are at
least a dozen different ways to create easier related problems. Each of these is a strategy
in itself; so that Pólya’s name for any particular strategy was in fact a label that identified
TME, vol10, nos.1&2, p .15
a family of strategies. Once I understood this, I could “take apart” a family by identifying
the main strategies that fell under its umbrella. I could teach each of those particular
strategies (e.g., solving problems that had integer parameters by looking at what
happened for n = 1, 2, 3 , 4 . . . ; looking at lower-dimensional versions of complex
problems; etc.), and when the students had learned each of these, they had mastery of the
family of strategies that Pólya had named. What that meant was that understanding and
teaching Pólya’s strategies was no longer a theoretical challenge, but an empirical one.
One could imagine a purely empirical, pragmatic program: take the main heuristic
strategies identified by Pólya; consider each as a family of strategies and decompose
them into their constituent parts; and work out a straightforward instructional program
that enabled students to learn each of the constituent strategies. In this way, it should be
possible to make problem solving accessible to all students. I hoped that some such work
would take place.
A quarter-century later . . .
Issues of theory
Here there is good news, both in terms of what has been achieved and how the
theoretical horizon has expanded. As noted above, the major challenge with regard to
problem solving was to build a theory of problem solving, rather than a framework for
examining it. More broadly, the challenge was to build a theory of goal-oriented decision
making in complex, knowledge-intensive, highly social domains. Mathematical problem
solving or problem solving in any content area, is an example. The goal is to solve the
problem; knowledge (including knowledge of various strategies) is required; and,
depending on the context, the problem solving activities may be more or less socially
Schoenfeld
engaged. Mathematics (or other) teaching is another, much more complex activity. The
goals here are to help students learn mathematics. Achieving those goals calls for a huge
amount of knowledge and strategy, and for deploying that knowledge amidst dynamically
changing circumstances: when a student suddenly reveals a major misconception, for
example, or it becomes apparent that the class does not have a good grip on something
that the teacher thought they understood, the current “game plan” has to be revised on the
spot and something else put in its place. In fact, if you can model decision making during
teaching, it is straightforward to model decision making in other complex knowledgeintensive domains such as medical practice, electronic trouble-shooting, and more.
By “model” I mean the following. One needs to specify a theoretical architecture
that says what matters, and say how decision making takes place within that architecture.
Then, given any instance of such decision making (e.g., problem solving or teaching),
one should be able to identify the things that matter in that instance, and show how the
decision making took place in a principled way (that is, through a structured model
consistent with the theoretical architecture), using only the constructs in the theoretical
architecture to build and run the model. By way of crude analogy, think of Newton’s
theory of gravity as providing a theoretical architecture (the inverse square law) for
characterizing the motion of a set of objects. For each object (say the planets in our solar
system, plus the sun) the some parameters need to be specified: mass, position , direction,
velocity. The model of the solar system is given these data for time T, and the theory is
used to specify these parameters for time T+1. The theory, then, is general; each model
(whether of our solar system or some other galactic system) is a specific instantiation of
the theory. The quality of any particular model is judged by how well the behavior of the
TME, vol10, nos.1&2, p .17
objects represented in the model corresponds to the behavior of the objects being
represented. (A model of the solar system had better produce motion that looks like the
motion of the planets in our solar system!) The quality of the theory is judged by its
accuracy and it scope – what is the range of the situations for which it can generate
accurate models? (A theory that only modeled two-body gravitational systems wouldn’t
be very exciting.)
Twenty-five years after MPS was published, my new book How We Think
(Schoenfeld, 2010) builds on the earlier work and lays out the structure of a general
theory of in-the-moment decision making. The architecture it describes is
straightforward: what one needs for a theoretical account of someone’s decisions while
that person is engaged in a familiar goal-oriented activity such as problem solving,
teaching, or medical practice is a thorough description of:
a) The goals the individual is trying to achieve;
b) The individual’s knowledge (and more broadly, the resources at his or her
disposal);
c) The individual’s beliefs and orientations (about himself and the domain in
which he or she is working); and
d) The individual’s decision-making mechanism.
These categories represent the natural evolution of the categories in the 1985
framework. Regarding (a), depicting the goals is necessary in that the theory describes a
much broader spectrum of behavior than problem solving. Depending on context, one’s
highest priority goal may be, for example: to solve a problem; to make sure that one’s
students understand a particular body of mathematics; or to diagnose a patient
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appropriately and set him or her on a path toward recovery. Regarding (b), the role of
knowledge is still central, of course: what one can achieve depends in fundamental ways
on what one knows. In my current theoretical view I fold access to and implementation
of heuristic strategies into the category of knowledge. I always viewed problem solving
strategies as a form of knowledge, of course – but, in the problem solving work I was
trying to validate their importance and utility, so they were separated out for special
attention. In addition, I add “resources” into the category of “what the individual has to
work with”: the approach one takes to a problem may vary substantially depending on,
for example, whether one has access to computational tools on a computer. Regarding
(c), beliefs still play the same central role in shaping what the individual perceives and
prioritizes as in my earlier work. I have chosen to use the word orientations (including
preferences, values, tastes, etc.) as a more encompassing term than beliefs because, for
example, choices of what to purchase for dinner and how to cook it, while modelable in
terms of the architecture I specify, aren’t necessarily a matter of beliefs.
Regarding (d), the decision making mechanism in the theory is implemented in
two ways. If circumstances are familiar – that is, one is collecting homework or going
over familiar content in class – people use various mechanisms described in the
psychological literature (scripts, frames, schemata, etc.) that essentially say what to do
next. If circumstances suddenly vary from the predictable – e.g., a student makes a
comment indicating a serious misconception, an explanation obviously doesn’t work –
then it is possible to model the individual’s decision making using a form of subjective
expected utility. (The various options that might be used are evaluated in light of their
perceived value to the person being modeled, and the higher a valuation an option
TME, vol10, nos.1&2, p .19
receives the more likely the option is to be chosen.) Monitoring and self-regulation,
which were a separate category in MPS, still play a centrally important role – but here
they are placed as a major component of decision making.
To my mind How We Think has roughly the same status today that MPS had in
1985. The book offers a number of very detailed case studies, showing how a wide range
of mathematics teaching can be modeled, and an argument suggesting (by virtue of the
breadth of “coverage” in the cases) that the model applies to all teaching. Then, there are
suggestions that the theory should suffice to describe goal-oriented decision making in all
knowledge-intensive fields. This is a heuristic argument similar to the argument I made in
1985, that the problem solving framework I explicated for mathematics should apply to
all problem solving disciplines. Time will tell if the theory holds up.
While How We think brings to fruition one theoretical line of inquiry, it also
opens up a number of others – lines of investigation that I think will be fruitful over the
coming decades. These may or may not strike the reader as falling under the banner of
“problem solving” – but, they should, if the question is, what do we need to know about
thinking, teaching, and learning environments to help students become more effective
mathematical thinkers and problem solvers? (I will revisit this question directly when we
turn to practical issues.)
My work to date has examined problem solving through the lens of the individual,
at any point in time. That is, the question has been, how and why does the individual go
about making decisions in the service of some (problem solving) goals, given what he or
she knows? These are serious limitations. First, the focus on what is happening in the
moment ignores questions of learning and development. The person who has worked on,
Schoenfeld
and solved, a problem, is not the same person who began working on it. He or she
approaches the next problem knowing more than before. So, one question is, how can
issues of learning and development be incorporated into a theory of decision making?
This is a deep theoretical question, which may not have immediate practical applications
– but, if we can trace typical developmental trajectories with regard to students’ (properly
supported) ability to engage in problem solving, this might help shape curriculum
development. More generally, if our goal is to theorize cognition and problem solving,
such issues need to be addressed.
Second, individuals do not work, or learn, in a vacuum. As will be seen below,
characterizing productive learning environments – and the norms and interactions that
typify them – is an essential endeavor, if we are to improve instruction. But learning
environments are highly interactive, and the ideas that individuals construct are often
built and refined in collaboration with others. At minimum, a theory of learning and
cognition that explains how ideas grow and are shared in interaction is critical. There is
much to be done on the theoretical front.
Issues of practice
Here, the question is whether one wishes to view the metaphorical glass as being
half empty or half full. There is reason to be disheartened, and reason to be encouraged.
And there is work to be done.
On the one hand, there are ways in which we could and should be much further
along in curricular development (and the research that would undergird it) than we are.
As explained above, there was an implicit blueprint for progress in MPS: the methods I
described for decomposing heuristic strategies into families of more fine-grained
TME, vol10, nos.1&2, p .21
techniques, and finding out how much instruction was necessary for those techniques to
become learnable, were well enough characterized for others to implement them. That is,
25 years ago it was theoretically possible to begin a straightforward program of
development that would result in successful instruction on a wide range of problem
solving strategies. “All” that was needed was a huge amount of work! That work did not
get done. There are systemic reasons for this, which Hugh Burkhardt and I (Burkhardt &
Schoenfeld, 2003) have explored. University reward systems work against this kind of
work. There is no theoretical “glory” in working through such pragmatic issues, either for
the individual or in terms of promotion decisions at research universities. Making
significant progress at the curricular level calls for a team of people, and university
reward structures are stacked against that as well – our system tends to reward individual
achievements, and to give less credit for collaborative work. Perhaps for those reasons,
perhaps because there are fads and fashions in educational research (as in all fields), an
area that I considered to be fertile ground for practical development went unexplored. I
think that’s a shame.
At the same time, some good things have happened in K-12 education. A global
summary of developments can be found in Törner, Schoenfeld, & Reiss, 2008. Here I
will summarize the optimistic view regarding the past 25 years in the U.S., and then point
to the fact that we are at a turning point, where much hangs in the balance.
In contrast to some nations where a ministry of education or its equivalent makes
curricular decisions that are implemented nationwide, the U.S. has had what is best
described as a “loosely coupled” system. For almost all of its history, each of the states
had its own educational system, which was responsible for setting statewide standards.
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Historically, textbook decisions have been local – each of the roughly 15,000 school
districts in the U.S. could choose is own textbooks. Until the past decade, few states had
statewide assessments, so there was little pressure to “teach to the test.” There were
homogenizing factors, of course. There were a small number of textbook publishers, so
textbook choice, though theoretically unconstrained, was limited in practice; and, most
school districts aimed at preparing their college-intending students for the (essentially
universal) college calculus course, so the goal state was clearly established. By tradition,
grades K-8 focused on arithmetic and then pre-algebra; algebra I was taken in 9th grade,
plane geometry in 10th, algebra II and possibly trigonometry in 11th, and pre-calculus in
12th. Some students accelerated through calculus in high school; many students dropped
out of the pipeline altogether. (The generally accepted figure in the 1980s was that each
year, some 50% of the students at each grade level in secondary school failed to take the
next year’s mathematics course.) There was huge variation in the courses students took,
but “traditional” instruction focused mostly on conceptual understanding and mastery of
skills and procedures. There was a negligible amount of “problem solving,” by any real
definition.
In 1989 the National Council of Teachers of Mathematics produced the
Curriculum and Evaluation Standards for School Mathematics. The volume was intended
for teachers, and had few references; but the authors knew the problem solving research,
and it showed. For the first time in a major policy document, there was a significant
emphasis on the processes of doing mathematics: The first four standards at every grade
level focused on problem solving, reasoning, communication using mathematics, and
connections within and outside of mathematics. The U.S. National Science Foundation,
TME, vol10, nos.1&2, p .23
recognizing that commercial publishers would not build such textbook series on their
own, issued a request for proposals for the creation of “Standards-based” texts. In each of
these texts, the authors elaborated their own vision of what it meant to learn according to
the Standards. This variety was a good thing: different visions of a richer mathematics,
focused on problem solving and reasoning, began to emerge. It is hard to get precise
figures, but some estimates are that 20-25% of the K-12 textbooks in use today are
Standards-based. Given the vagaries of the “loosely coupled” educational system in the
U.S., that’s a non-trivial impact for research ideas! (Of course it took 25 years, and the
ideas don’t necessarily reflect, and may sometimes be contradictory to, the views of the
original researchers. But that’s the way the system works.)
So, there has been curricular progress in K-12 mathematics in the U.S., if not as
much as one would like. Recent political events mean that the progress will either be
accelerated or blocked, in the near future. As part of an attempt to improve mathematics
instruction
called
the
“Rate
to
the
Top”
initiative
(see
http://www2.ed.gov/programs/racetothetop/index.html), the federal government offered
fiscal incentives to collections of states that produced high quality standards and plans for
assuring that students reached them. Given the short time frame to apply for funding, the
U.S. National (State) Governors Association and the Council of Chief State School
Officers supported an effort to construct a set of standards for mathematics, known as the
Common Core State Standards (see http://www.corestandards.org/, from which the
Common Core State Standards for Mathematics (CCSSM) can be downloaded.)
As of this writing, 44 of the 50 states have committed to the Common Core
initiative, meaning that they will replace their current state standards with CCSSM. By
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federal statute, they will need to use assessments (tests) that are deemed consistent with
the CCSSM in order to measure student progress toward the goals of CCSSM. Two
national consortia have been funded to produce assessments consistent with the CCSSM:
PARCC (the Partnership for Assessment of Readiness for College and Careers; see
http://www.achieve.org/PARCC) and the SMARTER Balanced Assessment consortium
(see http://www.k12.wa.us/smarter/). By the time this article appears, both consortia will
have published their “specs” for assessments consistent with CCSSM. Simply put, those
assessments (tests) will shape the mathematics experiences of the students in the states
that have committed to the Common Core State Standards initiative. As we know, testing
– especially high-stakes testing – determines the foci of classroom instruction. The
CCSSM place significant emphasis on what they call mathematical practices, claiming
that people who are mathematically proficient:
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
If the tests produced by the consortia provide students with opportunities to
demonstrate such mathematical habits of mind, the tests will serve as a lever for moving
the K-12 system in productive directions. But, if they consist largely of short answer
TME, vol10, nos.1&2, p .25
questions aimed at determining students’ mastery of facts and procedures, they will serve
impede the kind of progress we have been making over the past 25 years.
In sum, progress in K-12 has been slow but steady; it may get a boost or a setback
in the immediate future, depending on the high-stakes tests that the two assessment
consortia adopt. But there has been significant progress. I wish I could say the same
about collegiate mathematics over the same time period. For a while calculus reform
flourished, but it seems to have stabilized and become “same old, same old.” There have
been glimmers of excitement surrounding innovations in linear algebra and differential
equations (stimulated in some part by technology), but not so much that the general
zeitgeist of collegiate mathematics instruction is noticeably different from what it was
when I was a math major. And that was a long time ago!
Rethinking “problem solving”
I got my “start” in problem solving and I still think that, in some ways, it deserves
to be called “the heart of mathematics” (Halmos, 1980). More broadly, there is a view of
mathematics as the “science of patterns” (Steen, 1988). What I like about this framing is
that one thinks of science as consisting of systematic explorations – and mathematics as
we practice it certainly has that character. This broad a framing includes problem posing
as well as problem solving, and a certain form of empiricism, which was made explicit in
Pólya’s (1954) title, “patterns of plausible inference.” In doing mathematics we explore;
we seek systematicity; we make conjectures; and, we use problem solving techniques in
the service of making and verifying those conjectures. Yet more broadly, we engage in
Schoenfeld
“thinking mathematically” – a title used by John Mason, Leone Burton, and Kaye Stacey
(1982, 2010) and which I wish I’d thought of. Here I want to push things one step further.
At heart, doing mathematics – whether pure or applied – is about sense-making.
We observe an object, or a relationship, or a phenomenon, and we ask: What properties
must it have? How do we know? Do all objects that look like this have the same
property? Just what does it mean to “look like this”? Are there different ways to
understand this? With that mindset, simple objects or observations become the starting
points for explorations, some of which become unexpectedly rich and interesting. Third
graders observe that every time they add two odd numbers the sum is even. Must it
always be so? How would one know? We observe that some numbers can be factored,
others can’t. How many of the unfactorable kind are there? How can I measure the height
of a tree without climbing it? How many different crayons do I need to color a map, so
that every pair of countries that share a border have different colors?
What I strive to do in my problem solving courses is to introduce my students to
the idea that mathematics is about the systematic exploration and investigation of
mathematical objects. Elsewhere (see, e.g., Schoenfeld, 1989; see also Arcavi, Kessel,
Meira & Smith, 1998; English & Sriraman, 2010) I have described our first-week
discussions of the magic square. We start with the 3 x 3, which my students solve easily.
But that is just a start. Did we have to get an answer by trial and error, or are there
reasons that even numbers go in the corners, and that 5 goes in the center? Is the solution
unique (modulo symmetry), or are there distinct solutions? Having finished with the
original 3 x 3, we ask: what if I had nine other integers? Say 2 through 10, or the odd
numbers from 1 through 17, or any arithmetic sequence? Can we find a magic square for
TME, vol10, nos.1&2, p .27
which the sum of each row, column, and diagonal is 87? How about 88? We observe that
if we multiply all the cells in a magic square by a constant, we get a magic square; if we
add a constant to each cell, we get a magic square. Thus, we can generate infinitely many
3 x 3 magic squares. But can we generate all the 3 x 3 magic squares this way?
The reason for this discussion is that I want to introduce my students to what it
means to do mathematics. I want them to understand that mathematics isn’t just about
mastering facts and procedures, but that it’s also about asking questions (problem posing,
if you will) and then pursuing the answers in reasoned ways. The problem solving
strategies are tools for sorting things out, seeing what makes the mathematical objects
and relationships “tick.” So yes, we are solving problems, but as part of a larger sensemaking enterprise. That, in part, is why attending to my students’ beliefs is so important a
part of the course. Having been “trained” by their prior experience to understand
(believe) that doing mathematics means “mastering” content selected and organized by
others; that all problems can be solved in short order, usually by the techniques the
teacher has presented within the past week; that proofs have nothing to do with
discovery; and so on2, my students needed to be “untrained” or “retrained” by their
experiences in my course. Thus I give them extended opportunities to make observations
and conjectures, and provide them with the tools that enable them to experience the doing
of mathematics as a sense-making activity.
My question is, can’t we approach all mathematics teaching this way? I believe
that all of K-12 mathematics, and a good deal of collegiate mathematics, can be seen as a
set of sensible answers to a set of sensible questions. Given the pace at which K-12
2
See Schoenfeld (1992) for a list of counterproductive beliefs that students typically
develop.
Schoenfeld
mathematics proceeds, I am sure that this could be done without any formal loss of
content. What would be gained is that students would experience mathematics as an
exciting sense-making domain, which is the way we see it as mathematicians. If K-12
students truly experienced mathematics that way, I’m willing to be that similarly oriented
collegiate instruction could build on well-established habits of mind, and proceed much
more effectively than it currently does. (Despite the fact that the students in my problem
solving courses through the years could be labeled “the best and the brightest” – it’s a
non-trivial achievement to get into Berkeley – my feeling has always been that my
problem solving courses have been remedial in a significant way. The vast majority of
the students who entered those courses were unaware of basic mathematical problem
solving strategies, and, as a function of their experience, did not view mathematics as a
domain that they could make sense of.)
Rethinking Research on Classroom Environments
A comment made by one of the advisory board members of one of my projects
(“Classroom Practices that Lead to Student Proficiency with Word Problems in Algebra”,
NSF grant DRL-0909815), struck me as particularly interesting. Megan Franke (2011)
noted that, of the various classroom variables she had looked at, the one that seemed to
have the strongest impact on student learning was the amount of time students spent
explaining their ideas. This resonates, not only with the discussion of sense-making
above, but with an emerging body of research focusing on the character of classroom
environments that support the kinds of rich student engagement and thinking that one
would like. Engle, for example (Engle, in press; Engle and Conant, 2002) has developed
TME, vol10, nos.1&2, p .29
a “productive disciplinary engagement framework.” Reviewing the best-known examples
of rich learning environments in mathematics and science, Engle concludes that the most
powerful learning environments all include aspects of:
Problematizing – students participate in the act of framing meaningful
questions, which the class explores.
Agency and authority – students are empowered to seek information, distill it,
craft arguments, and explain them.
Disciplinary accountability – students learn what it is to make claims and
arguments that are consistent with disciplinary norms.
Resources – when tools or information is needed, the students have access to
them.
For an elaboration of these ideas with examples drawn from my problem solving
courses, see Schoenfeld (2012). Gresalfi, Martin, Hand, & Greeno (2009) offer a
framework (see Fig. 1) that indicates ways on which classroom participation structures
can lead to differential outcomes in terms of student agency, argumentation, and
accountability.
Schoenfeld
Fig. 1. A model of how competence gets constructed in the classroom.
From Gresalfi, Martin, Hand, & Greeno (2009), p. 54, with permission.
This kind of framework can be useful, as we seek to understand both how to craft
classrooms more focused on sense making and to document their effects.
Rethinking Technology (a.k.a. entering the 21st century)
When I took my first statistics class, all of the examples were “cooked.” This was
before the days of widespread access to calculators and computers, so everything I did
had to be hand-computable. As a result, the variation of every distribution I worked with
was a perfect square! The presence of computational technology should have radicalized
the ways in which our students can engage with statistics, in that no data analysis is now
an obstacle. Students should be able to ask their own questions and gather their own data.
Yet, few students have this experience. In Schoenfeld (2012) I give the example of how,
sitting at my desk in early June and watching it rain, I wondered whether this was
TME, vol10, nos.1&2, p .31
atypical. The San Francisco Bay Area is supposed to have a “dry” season, and it seemed
that we had gotten more rain than we should have. Using Google I was quickly able to
find data regarding annual monthly rainfall and recent rainfall, at which point I could do
some simple statistical analyses to verify that this year’s June rainfall was anomalously
high. (In fact, it went on to set a record.) From my perspective, I was clearly doing
mathematics. My question is, where do today’s students learn to gather such information
and to operate on it? Asking questions, seeking data, building models, and drawing
inferences should be everyday experiences for our students.
Similarly, the presence of computational tools – whether symbolic calculators,
graphers, or Wolfram Alpha (see http://www.wolframalpha.com/) – has the potential to
radically reshape the knowledge to which students have access in mathematics
classrooms, and the ways they can operate on it. Pure mathematics can become an
empirical art for students in ways that it was not, even for mathematicians, until recently.
Where are students learning to harness these skills – not for the sake of learning to be
fluent with technology, but as means to mathematical ends? There have been some
inroads along these lines, for example with dynamic geometry software, but for the most
part these positive examples are the exceptions that probe the rule.
Concluding Comments
I thank to the editors for the opportunity to think about the current state of
mathematical problem solving, both in theory and in instruction. I became a mathematics
educator many years ago because of my love for mathematics and my wish to share it
with students, who were typically deprived of the pleasures that I consistently
experienced as a mathematician. Problem solving provided a way into the joys of doing
Schoenfeld
mathematics and the pleasures of discovery. I firmly believe that problem solving – or a
broader conception of mathematics as sense making – still can do so, and I hope to see us
make progress along those lines.
References
Arcavi, A., Kessel, C., Meira, L., & Smith, J. P. (1998). Teaching mathematical problem
solving: An analysis of an emergent classroom community. In A. H. Schoenfeld, J. J.
Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education (Vol. III)
(pp. 1–70). Washington, DC: Conference Board of the Mathematical Sciences.
Brown, A. (1987). Metacognition, executive control, relf-regulation, and other more
mysterious mechanisms. In F. Reiner & R. Kluwe (Eds.), Metacognition, motivation,
and understanding (pp. 65-116). Hillsdale, NJ: Erlbaum.
Burkhardt, H., & Schoenfeld, A. H. (2003). Improving educational research: toward a
more useful, more influential, and better funded enterprise. Educational Researcher
32(9), 3-14.
Common Core State Standards for Mathematics. (2011). Available at the website of the
Common Core State Standards Initiative, http://www.corestandards.org/.
Engle, R. A. (in press). The productive disciplinary engagement framework: Origins, key
concepts, and continuing developments. In D. Y. Dai (Ed.), Design research on
learning and thinking in educational settings: Enhancing intellectual growth and
functioning. London: Taylor & Francis.
Engle, R. A., & Conant, F. R. (2002). Guiding principles for fostering productive
disciplinary engagement: Explaining an emergent argument in a community of
learners
classroom.
Cognition
and
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20(4),
399-483
English, L., & Sriraman, B. (2010). Problem solving for the 21st century. In B. Sriraman
& L. English (Eds). Theories of Mathematics Education: Seeking New Frontiers
(pp.263-290), Springer Science and Business.
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Franke, M. (June 5, 2011). Personal communication.
Garofalo, J., & Lester, F. (1985).
Metacognition, cognitive monitoring, and
mathematical performance. Journal for Research in Mathematics Education,16(3),
163-176.
Gresalfi, M., Martin, T., Hand, V. & Greeno, J. G. (2009). Constructing competence: an
analysis of student participation in the activity systems of mathematics classrooms.
Educational Studies in Mathematics, 70, 49–70
Halmos, P. (1980). The heart of mathematics. American Mathematical Monthly, 87, 519524.
Mason, J., Burton, L., & Stacey, K. (1982, 2010) Thinking mathematically. First edition,
1982, New York: Addison Wesley. Second edition, 2010, Harlow, England: Prentice
Hall.
Pólya, G. (1945; 2nd edition, 1957). How to solve it. Princeton: Princeton University
Press.
Pólya, G. (1954). Mathematics and plausible reasoning (Volume 1, Induction and
analogy in mathematics; Volume 2, Patterns of plausible inference). Princeton:
Princeton University Press.
Pólya, G. (1962,1965/1981). Mathematical discovery (Volume 1, 1962; Volume 2,
1965). Princeton: Princeton University Press. Combined paperback edition, 1981.
New York: Wiley.
Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic Press.
Schoenfeld, A. H. Teaching mathematical thinking and problem solving. In L. B.
Resnick and L. Klopfer (Eds.), Toward the thinking curriculum: Current cognitive
research (1989 Yearbook of the Association for Supervision and Curriculum
Development), pp. 83-103.
Washington, DC: Association for Supervision and
Curriculum Development 1989.
Schoenfeld, A. H. (2008). Problem Solving in The United States, 1970-2008: Research
and Theory, Practice and Politics. In G. Törner, A. H. Schoenfeld, & K. Reiss (Eds.),
Problem solving around the world – Summing up the state of the art. Special issue of
the Zentralblatt für Didaktik der Mathematik: Issue 1, 2008.
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Schoenfeld, A. H. (2010). How we think: A theory of human decision-making with
educational applications. New York: Routledge.
Schoenfeld, A. H. (2012). Problematizing the didactical triangle. ZDM, The International
Journal of Mathematics Education, 44, 5, 587-599.
Steen, L. A. (29 April 1988). The science of patterns. Science, 240, 611-616.
Törner, G., Schoenfeld, A. H., & Reiss, K. (Eds.). (2008) Problem solving around the
world – Summing up the state of the art. Special issue of the Zentralblatt für Didaktik
der Mathematik, 39(5-6). Issue 1, 2008.
TME, vol10, nos.1&2, p.35
Problem Solving in the Primary School (K-2)
Richard Lesh1
Lyn English
Indiana University
Queensland University of Technology
Chanda Riggs
Serife Sevis
Sharon Elementary School
Indiana University
Lesh: “Do you really think your children can do this?”
Riggs: “So far, nobody has taught them yet about what they can’t do.”
Abstract: This article focuses on problem solving activities in a first grade classroom in a
typical small community and school in Indiana. But, the teacher and the activities in this
class were not at all typical of what goes on in most comparable classrooms; and, the
issues that will be addressed are relevant and important for students from kindergarten
through college. Can children really solve problems that involve concepts (or skills) that
they have not yet been taught? Can children really create important mathematical
concepts on their own – without a lot of guidance from teachers? What is the relationship
between problem solving abilities and the mastery of skills that are widely regarded as
being “prerequisites” to such tasks? Can primary school children (whose toolkits of
skills are limited) engage productively in authentic simulations of “real life” problem
solving situations? Can three-person teams of primary school children really work
together collaboratively, and remain intensely engaged, on problem solving activities that
require more than an hour to complete? Are the kinds of learning and problem solving
experiences that are recommended (for example) in the USA’s Common Core State
Curriculum Standards really representative of the kind that even young children
encounter beyond school in the 21st century? … This article offers an existence proof
showing why our answers to these questions are: Yes. Yes. Yes. Yes. Yes. Yes. And: No.
… Even though the evidence we present is only intended to demonstrate what’s possible,
not what’s likely to occur under any circumstances, there is no reason to expect that the
things that our children accomplished could not be accomplished by average ability
children in other schools and classrooms.
Keywords: Common core standards; elementary mathematics education; problem
solving in elementary school;
1
[email protected]
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos.1&2, pp.35-60
2013©The Author(s) & Dept of Mathematical Sciences-The University of Montana
Lesh, English, Riggs & Sevis
Can Children Solve Problems involving Concepts they have not been Taught?
Most people’s ordinary experiences are sufficient to convince them about the
truth of two important assumptions about learning and problem solving.
First, the kinds of things that students can learn, and the kinds of problems
that they can solve, tend to be strongly influenced by the things they already
know and are able to do. So, the accompanying “common sense assumption”
is that these prerequisites must be mastered before students are expected to
learn relevant new ideas, or solve relevant new types of problems.
And
consequently, learning is viewed as a long step-by-step process in which
prerequisites are checked off one at a time.
Second, concepts and abilities do not go from unknown to mastered in a single
step. They develop! And, so do associated abilities. In fact, especially for the
most important “big ideas” in the K-12 curriculum, development typically
occurs over time periods of several years, and along a variety of dimensions –
such as concrete-abstract, intuition-formalization, situated-decontextualized,
specific-general, or increasing representational fluency, or increasing
connectedness to other important concepts or abilities. So, in situations which
are meaningful and familiar to students, rapid developments often occur for
clusters of related concepts and abilities. And, in these contexts, students’
ways of thinking often integrate ideas and abilities associated with a variety of
textbook topic areas – so that the resulting knowledge and abilities are
organized around experiences as much as around abstractions.
TME, vol10, nos.1&2, p.37
For readers who are familiar with Vygotsky’s zones of proximal development, the
title of this section poses a question that is clearly naïve. Learning does not occur in this
all-or-nothing manner.
For example, in a series of projects known collectively as The
Rational Number Project (RNP, 2011), it is well known that the “difficulty level” of a
given task can be changed by years – simply by changing the context or the
representational media in which problems are posed (e.g., written symbols, written
language, diagrams or graphs, concrete models, or experience-based metaphors).
Consequently, when students encounter a problem in which some type of mathematical
thinking is needed, all of the relevant concepts and abilities can be expected to be at some
intermediate stages of development – not completely unknown, yet not completely
understood – regardless of whether these concepts or abilities have been formally taught.
In fact, for researchers who have investigated what it means to “understand” the
most powerful and important ideas in the elementary school curriculum, it has become
clear that most of the “big ideas” that underlie the K-12 curriculum begin to develop in
early years– in topic areas ranging from rational numbers and proportional reasoning
(RNP, 2011), to measurement and geometry (e.g., Krutetskii, 1976), to statistics and
probability (e.g., Zieffler, Garfield, delMas, & Reading, 2008), to early ideas in algebra
(English, in press; Thompson, 1996) or calculus. In fact, in each of these domains of
mathematical thinking, many important understandings typically begin to develop even in
the primary grades (K-2). Such observations are reminiscent of Bruner’s claim, long ago,
that: Any child can be taught any concept at any time – if the concept is presented in a
form that is developmentally appropriate (Bruner, 1960). Of course, the “if clause” in
this quote is very significant. That is, in order for remarkable developments to occur,
Lesh, English, Riggs & Sevis
relevant concept and abilities need to be accessible in the forms that are developmentally
appropriate.
For the problems that will be described in this chapter, the two primary tests of
developmentally appropriateness are: (a) Do the children try to make sense of the
problem using their own “real life” experiences – instead of simply trying to do what they
believe that some authority (such as the teacher) considers to be correct (even if it doesn’t
make sense to them)? (b) When the children are aware of several different ways of
thinking about a given problem, are they themselves able to assess the strengths and
weaknesses of these alternatives – without asking their teacher or some other authority?
When these two criteria are satisfied, children are able to go from “first-draft of thinking”
to “Nth-draft of thinking” without interventions from an outside authority.
When referring to “real life” sense-making abilities, it is important to emphasize
that we are not assuming that a first grader’s “real life” interpretations of experiences are
the same as an adult’s one. For example, for first graders, children’s stories often engage
their sense making abilities more than situations that an adult might consider to be a “real
life” situation. So, for the problems that we’ll describe in this article, the tasks were
presented in the context of stories such as Two Headed Stickbugs, The Proper Hop (for
Beauregard the Frog), Fussy Rug Bugs, Isabelle Talks, The Royal Scepters, or Tubby the
Train (see Figure 1) – most of which appeared first in Scott Foresman’s longest running
kindergarten book - written by Lesh & Nibbelink (1978).
For our purposes in this article, some other important of “real life” characteristics
that we tried to build into our problems include the following. (a) The product that the
children are challenged to produce often is not just a “short answer” to a pre-
TM
ME, vol10, noos.1&2, p.339
mathematized qu
uestion. It is
i a sharable and reusaable tool or artifact thatt needs to bbe
poweerful (in the given situattions) and allso sharable (with other people) andd reusable (iin
situattions beyond
d the given situation). (b) The chilldren need tto know whho needs theeir
produ
uct - and wh
hy? Otherwisse, they won
n’t be able too assess the sstrengths and weaknessees
of altternative pro
oducts; and, they won’t be
b able to juudge: is a 3-ssecond answ
wer sufficiennt;
or, iss a 3-minutee or 30-min
nute needed
d? (c) Produuct developm
ment is likeely to requirre
severral cycles off testing and
d revising – similar
s
to firrst, second, or third draafts of writteen
descrriptions of drawn
d
pictu
ures. (d) Prroducts willl often needd to integraate ideas annd
proceedures drawn
n from a varriety of textb
book topic aareas. One rreason for thhat this is truue
is beccause “real lift”
l
problem
m often invo
olves partly cconflicting cconstraints aand trade-off
ffs
(e.g., low cost bu
ut high qualitty, simple bu
ut powerful) .
Lesh, English, Riggs & Sevis
Figure 1. The First Pages from Six Stories
Figure 1 shows the six contexts that were used for the problems which will be
described in this article. Then, Figure 2 briefly describes the tasks that accompany each
of these stories. For each task, the children worked in groups of three; the work spaces
TM
ME, vol10, noos.1&2, p.441
usuallly were at least the sizee of large po
oster boards;; and, the prroducts usuaally needed tto
include letters to
o “someone else” descriibing how thhey could ddo and whatt our studentts
writing aspeect of the tassks was empphasized parrtly because the school iin
did. This letter-w
h we worked
d was focusiing on writin
ng abilities. But, it alsoo was used bbecause, to bbe
which
consiistent with our
o published
d principles for designinng model-elicciting activitties (Lesh, eet.
al., 2000),
2
it is important
i
fo
or the produ
ucts to be m
more than sinngle solutionns to isolateed
probllems. Solutions to our problems also needed too be sharablle and reusaable. So, onne
straig
ghtforward way
w to achieeve this goaal was to askk children too write letteers describinng
tools or artifacts that
t can be used
u
by otheers.
Two-H
Headed Stickbu
ugs
The Propper Hop
Two different
d
sized “stickbugs”
“
weree used – one maade
with po
opsicle sticks, an
nd the other mad
de with meter-lo
ong
strips of
o wood. In a warm-up
w
activity, the children wo
ork
in grou
ups of three, and
a
use meter-long “stickbug” to
measurre as many distaances as possiblee in the playgrou
und
of their school. (note: On boring plaaygrounds, mark
kers
can bee placed using highway markeers or signs.) The
T
teacherr should record these distances by
b drawing arro
ows
and po
oints on a posterr-sized photograp
ph (or drawing)) of
the sch
hool yard. Theen, the next day
y, children shou
uld
again work
w
in groups of
o three, and use the teacher’s no
otes
and po
opsicle-sized stickbugs to creatte miniature scaalemodelss of their playgro
ound.
The greenn dots shown bbelow are lily ppads. The
lines betw
ween dots indicate “proper hopps” (which
must be hhorizontal or verrtical hops to addjacent lily
pads). Thhe red, yellow, and orange dotss are where
three of Beauregard’s friends live. And, the
Beauregard
children’ss task is to find a place where B
(x) shouldd live so that thhe sum of the distances to
his three ffriends’ houses iis as small as posssible.
Fussy Rugbugs
R
What is the largest number
n
of rugb
bugs that can live
oup
totally inside the area shown below? (Give each gro
enough
h post-it stickiees to cover thee area; and, give
g
approx
ximately one-thiird of the stick
kies to each chiild.
Either square or roun
nd stickies can be used. But, in
c
the stickiees should (a) be completely inside
either case,
the areea indicated, (b) not overlap, and
d (c) fit togetherr as
closely
y as possible.
Isabel Tallks
The “treees” shown bellow should be unevenly
distributedd. They repressent apple treess, with big
juicy applles that Isabel looves to eat whille standing
in their shhade. The goal is to build a cloosed fence
which enccloses the largeest number of aapple trees.
The fencee is a loop of sodda straws or cofffee stirrers
– strung oon a closed loop of string.
Lesh, E
English, Rigggs & Sevis
The Ro
oyal Scepters
Severaal different temp
plates for gingerrbread houses (like
the on
ne shown below
w) can be dow
wnloaded from the
interneet – and the sizess of the pieces caan be marked usiing
small “toothpick” ru
ulers. Then, diffferent groups of
childreen should cut caardboard pieces to make parts for
severall houses – e.g., with one group specializing in
roofs, and
a other groups specializing in
n side walls, or end
e
walls. The goal is forr the pieces to fiit together for eaach
house.
New Traccks for Tubby
Tubby the Traain
For the T
Problem, the train tracck
pieces loook like the onnes
shown herre. The goal is to
make a trrack for Tubby sso
that:
1. As maany as possible of
the pieeces are used uup
– so tthat none will bbe
wastedd.
2. The ttrack makes as
many closed loops as
possibble – so that manny
differeent animal houses
can bbe put inside of
loops.
3. The ppieces of track ffit
togethher smoothly – sso
will be no bumpps
there w
and sccreeches.
4. The ttrack should nnot
have aany dead ends.
Notice thaat all of the traiin track pieces aare marked
on hexagoonal shapes so thhat it is as clear aas possible
when the ttracks fit togetheer well.
Figurre 2. Tasks for
f Six of thee Stories
Can Children Create Imporrtant Matheematical Cooncepts on T
Their Own??
mine the title of this section, two oppposing answ
wers to this qquestion seem
m
If we exam
vious - depending on ho
ow the questtion is interppreted. If thhe question is
to be equally obv
C children
n invent the most
m fundam
mental conceepts in algebbra, geometryy,
intended to ask: Can
TME, vol10, nos.1&2, p.43
or calculus?, then the answer clearly must be: No! It took years for some of history’s
most brilliant mathematicians to invent these concepts. So, average ability children
cannot be expected to do such things during single class period? But, if the question is
asking: Can children use numbers to describe mathematically interesting situations in the
mathematical “objects” involve more than simple counts of discrete objects (i.e., cardinal
numbers), then one of the main points of this paper is that the answer to this question is:
Yes!
For example, the six problems that we describe in this article involve using
numbers to describe locations (coordinates, or ordinal numbers), lengths or distances (or
other types of measurable quantities), signed quantities (negative numbers), directed
quantities (vectors), actions (operators, transformations, functions), changing quantities
(rates or intensive quantities), or accumulating quantities (calculus). In particular, for the
six stories described here:
Children’s responses to the Stickbug Problem often use numbers to describe
lengths, distances, and sometimes even coordinates – if the “map” is thought
of as a simple kind of grid.
Children’s responses to the Proper Hop Problem often use numbers to
describe locations, actions (hops), number patterns, or quantities that have
both a magnitude and a direction.
Children’s responses to the Fussy Rugbugs Problem often use numbers to
describe areas or dimensions (concerning how “rugs” are aligned within
shapes).
Children’s responses to the Isabel Talks Problem often use numbers to
describe relationships between areas and perimeters, and even negative
Lesh, English, Riggs & Sevis
numbers (because when borders are rearranged to include some new “trees”
and other “trees” tend to be lost).
Children’s responses to the Royal Scepters Problem often use numbers to
describe scaling-up, proportions, ratios, lengths, distances, shapes (e.g.,
rectangles, triangles), and sometimes angles or areas .
Children’s responses to the Tubby the Train Problem often use numbers to
describe lengths, angles, and negative quantities (which occur pieces of tracks
are inserted or deleted in order to eliminate dead ends, or in order to enlarge or
shrink enclosed areas).
Of course, from a child-eye view, the preceding situations are not about ordinal
numbers, coordinates, signed numbers, vectors, operators – or areas, volumes, or
densities. To the children, they are simply contexts in which numbers are used to
describe things such as: hops, measuring sticks, sticky post-it notes, straws, or paths.
Nonetheless, because the tasks require children to externalize their thinking in forms that
are visible to the students themselves (as well as teachers and researchers), the seeds are
apparent for many of the most important “big ideas” that span the entire K-12
mathematics curriculum.
In general, what research based on models & modeling perspectives (Lesh &
Doerr, 2003) shows that, if children clearly recognize the need for a specific kind of
mathematical description, diagram, artifact, or tool, and if the children themselves are
able to assess strengths and weaknesses of alternative ways of thinking, then remarkably
young children are often able to produce impressively powerful, reusable, and shareable
tools and artifacts in which the mathematical “objects” being described involve far more
TME, vol10, nos.1&2, p.45
than simple counts. However, even though children are able to generate such descriptions
without guidance from adults, this claim does not imply that there is no role for teachers.
For example, even if children succeed in developing a powerful, sharable, and reusable
artifact or tool in response to a problem, they usually lack powerful ways to visualize
underlying constructs, and they are not often aware of strengths and weaknesses of
alternative ways of thinking. Furthermore, because their results often integrate concepts
and procedures drawn from a variety of textbook topic areas, they usually have not
unpacked these ideas-or, expressed them using elegant language and notations.
Can Teams of Primary School Children Work Collaboratively, and Remain
Intensely Engaged, on Problem Solving Activities that Require an Hour to
Complete?
Lesh: How long do you think primary school children are able to work on these kinds of
tasks? And, what is it about such activities that stimulate sustained work from children?
Riggs:
In general, the children worked on one modeling activity for two or three
consecutive days for an hour or more each day. The fourth day was reserved for sharing
explanations of their modeling to their classmates. Due to the cooperative nature of the
activities, complemented by children's engagement in problem solving, the children were
highly motivated and often requested additional time to devote to the task. Through
sharing, children learned to appreciate diversity in problem solving. I believe that
introducing concepts through interesting children's stories gives the children a purpose
for their learning; this purpose is what stimulates them to complete the task no matter the
amount of time required or how challenging it seemed. The children viewed learning as
Lesh, English, Riggs & Sevis
something they wanted to do instead of something they were required to do; modeling
activities provide that motivation. The activities were designed to open and close within a
week. One reason for this policy was because class time is precious. These stories
served as “chunks” that children could use to organize ideas and skills related to a
central “big idea”. If these “chunks” got too large, the children would lose sight of the
"big idea". Memorable stories also help children remember what they have learned. The
children continued to think about the "big idea" after class - and after we moved to other
topics. Weeks after they had finished activities directly associated with one of our
stories, they often referred back – saying: This is like Stickbugs, or Beauregard, or Tubby
the Train. Then, they would use concepts and abilities that they had developed during
those tasks. … So again, several smaller stories are better than one big story.
Lesh: How much and what kind of guidance did you need to provide in order for
children to be successful for these tasks?
Riggs: When the children work in groups, they tend to persevere when they otherwise
might have given up. But also, in every one of our activities, children worked together to
build some concrete tools or artifacts – such as pathways, fences, villages, maps, or
scaled-up houses. So, as long as they clearly understood what was needed and why, and
as long as they were able to test their thinking without asking me “Am I done?” or “Is
this right?”, they were able to move from first-draft thinking to second and third-draft
thinking without much guidance from me.
Self-assessment is important because, in complex activities, if children need to
wait for their teacher’s approval at each step, then things move too slowly, and young
TM
ME, vol10, noos.1&2, p.447
childrren lose intterest. Also
o, when chiildren expreess their thiinking in foorms that arre
visiblle, it’s easier for teacherrs to wisely pinpoint whhen and whaat kind of heelp is neededd.
Usuaally, the kind
d of help thatt is most effeective involvves reflectingg more than guiding. Foor
exam
mple: interventions that tend
t
to be most
m
helpful are questions like: Whho needs thiss?
Why do they neeed it? What do
d they need
d to do? Doees everybodyy agree? … They are noot
comm
mands like: Think
T
of a similar
s
probllem that youu have solveed. Or: Everrybody listenns
to Allice. … Th
he goal is fo
or children to
t iterativelyy express, teest, and revvise their ow
wn
ways of thinking
g. The goal is not for th
hem to supeerficially adoopt the teachher’s ways oof
think
king.
Acting
g Out Proper Hops in Suga
ar Swamp
Record
ding Information about Pro
oper Hops
Figurre 3. The Pro
oper Hops Prroblem
Three P
Paths in Sugarr Swamp
Record
ding Patterns aabout Proper Hops
Lesh, E
English, Rigggs & Sevis
A Rug
gbug Village – Getting Starrted
Maximiizing Rugbug Homes in an Area
An Ex
xample of a Ty
ypical “Letterr” for the Rug
gbug Problem))
Figurre 4. The Ru
ugbug Probleem
As the “lletter” in th
he Figure 4 shows, 1st graders’ lettters cannott be expecteed
comm
municate obj
bjectively to another peerson. But, in this part
rticular studyy, the schoool
where we were working
w
had
d made a sch
hool-wide coommitment to focus on writing. Soo,
the op
pportunity was
w too good
d to miss. For
F our purpooses, the maain point of ttrying to craaft
letterrs was not to
o press the ch
hildren for writing
w
excelllence. The main purposse was (a) foor
the children to understand
u
th
hat “someon
ne else” wannted to use the informaation, tool, oor
artifaact that they produced, an
nd (b) to em
mphasize the tools and arrtifacts needeed a be usefuul
to oth
her people to use. In oth
her words, we
w wanted tto make it aas clear as poossible to thhe
childrren that theiir proceduree (or tool) neeed to be shharable and rreusable. … On the onne
TM
ME, vol10, noos.1&2, p.449
hand,, the solutions that child
dren produce in such sittuations are highly situaated forms oof
know
wledge. On the
t other han
nd, sharable and reusablee solutions aalso are transsferrable.
Makin
ng a Fence to Enclose the Most
M Trees
A Fence for Isabell
Figurre 5. Apples for Isabelle Problem
Measu
uring with Stickbug Rulers
Eliminaating “Dead E
Ends” for Tubb
by
Figurre 6. Measurring with Royal Scepterss & Laying T
Tracks for Tuubby the Traain
Lesh, English, Riggs & Sevis
What is the Relationship between Problem Solving & the Development of
“Prerequisite” Concepts & Skills?
Lesh: This project was not an experiment that treated your children like guinea pigs in a
laboratory. It was simply a joint effort that you and our research team decided to
provide the best kind of learning experiences for your children. Yet you, like most
teachers, administrators, and schools on these days, are being held accountable for
learning gains which are measured by standardized tests which (I believe) don’t measure
much beyond low-level skills. So, even though we didn’t have any experimental “control
group”, how do you think your students will perform, compared to others, on
standardized tests that are relevant to you and others in your school?
Riggs: I believe that my students will perform as well, if not better, on standardized
assessments after using the model eliciting activities.
Given that the children learn to
problem solve in ways that make sense to them, and they can see their results from the
models created, the model eliciting activities provided a knowledge base where
information can be retrieved and applied as needed. The students' ability to apply what
they had learned became evident when they would remember the "big idea" weeks after
we had finished the activity, and when they would apply it to situations in their own lives.
One example: Three weeks after completing The Proper Hop, a student stated that living
in an apartment complex is like living in Sugar Swamp - there are a lot of lily pads. After
helping Beauregard to find the best lily pad closest to his friends, this student understood
why her Mom didn't want her to walk all the way over to the other side of the complex to
visit a friend. It was too far away; it was like Beauregard hopping 20 hops. She said
that her Mom allowed her to go next door to visit a friend; for Beauregard, it would only
TME, vol10, nos.1&2, p.51
be one or two hops.
This student also wished she could pick the location of her
apartment to be close to her friends - just like she helped Beauregard find his home in
The Proper Hop.
Ever since the seminal work of William Brownell (1970), it has been known that,
even if we only care about skill-level knowledge, “varied practice” is far more effective
than “routine practice” (or drills that are repeated again and again). Brownell identified
three kinds of varied practice. The first type involves mixed activities in which attention
shifts among several skills – rather than emphasizing just one. This is effective partly
because “understanding” involves more than knowing how to do something; it also
involves knowing when to do it. The second type of varied practice involves practicing
skills in a full range of situations in which they are intended to be useful.
This is
effective partly because useful skills need to be flexible, not rigid. And, the third type of
varied practice involves using skills during complex activities – similar to the way
excellent chefs not only know how to use each of the tools sold in chef’s catalogues, but
they also know how to orchestrate the use of these tools during the development of
complex meals.
Can Primary School Children Engage Productively in Authentic Simulations of
“Real Lift” Problem Solving Situations?
According to the models & modeling perspectives that underlie our work (Lesh &
Doerr, 2003), we reject the notion that children learn, or learn to be effective problem
solvers, by first learning concepts and skills, and then learning to use them in meaningful
“real life situations.” By far the most important characteristic of the models & modeling
Lesh, English, Riggs & Sevis
perspectives that distinguish our work from traditional research on problem solving is the
recognition that – regardless of whether investigations focus on decision making by
medical doctors, business managers, chess players, or others in real life decision-makers in virtually every field where learning scientists have investigated differences between
ordinary and exceptionally productive people, it has become clear that exceptionally
productive people not only do things differently, but they also see (or interpret) things
differently. Furthermore, when problem solvers interpret situations they don’t simply
engage models that are completely mathematical or logical in nature.
Their
interpretations also tend to include feelings, values, dispositions, and a variety of
metacognitive functions. But, instead of mastering these other higher-order functions
separately, and then attaching them to mathematical models, research on models and
modeling shows that they develop as integral parts of the relevant interpretation systems
(Lesh, Carmona & Moore, 2010).
Traditionally, problem solving has been characterized as a process of (a)
getting from givens to goals when the path is not obvious, and (b) putting
together previously learned concepts, facts, and skills in some new (to the
problem solver) way to solve problems at hand. But, when attention shifts
toward models & modeling, problematic situations are goal directed activities
in which adaptations need to be made in existing ways of thinking about
givens, goals, and possible solution steps. So, modeling is treated as a way of
creating mathematics (Lesh & Caylor, 2007); and, modeling and concept
development are expected to be highly interdependent and mutually
supportive activities – especially for young children.
TME, vol10, nos.1&2, p.53
Traditionally, problem solving strategies and metacognitive functions have
been specified as lists of condition-action rules – and have been thought of as
providing answers to the question: What should I do when I’m stuck (i.e.
when I am not aware of any productive ways of thinking about the problem at
hand).
But, when attention shifts toward models & modeling, the goal of
metacognitive processes is to help problem solvers develop beyond their
current ways of interpreting the situations, rather than helping them identify
“next steps” within current ways of thinking.
Traditionally, problem solving in mathematics education has focused on
individual students working without tools on textbook word problems. But,
because research on models and modeling tends to focus on simulations of
“real life” situations, problem solvers often are diverse teams of students each
of whom are likely to have access to a variety of specialized technical tools
and resources. So, capabilities that become important include: modularization,
communication, explanation, and documentation - as well as planning,
monitoring, and assessment – all of which tend to be overlooked in the
traditional mathematics education problem solving literature; and, all of which
emphasize modern socio-cultural perspectives on learning.
Because model development activities are, above all, research sites for directly
observing the development of interpretation systems that involve some of the most
important aspects of what it means to “understand” many of the most important concepts
and “big ideas” in mathematics education, research on models and modeling has led to
new views about: (a) how the modeling cycles that students go through during one 60-
Lesh, English, Riggs & Sevis
minutes model-eliciting activity often are remarkably similar to developmental sequences
that Piagetian psychologists have identified during timespans of several years based on
normal everyday experiences, (b) how average ability students often develop (locally)
through several Piagetian stages during single 60-minutes problem solving episodes, (c)
how students’ final-draft solutions often embody mathematical thinking that is far more
sophisticated than traditional curriculum materials ever dared to suggest they could be
taught, (d) how student solutions which are expressed in the form of sharable and
reusable tools often enable students to exhibit extraordinary abilities to remember and
transfer their tools to new situations, (e) how the processes that enable students to move
from one model to another seldom look anything like currently touted “learning
trajectories” which describe learning and problem solving using the metaphor of a point
moving along a path, (f) how the tools and underlying models which students produce in
“real life” model development often integrate concepts and abilities associated with a
variety of textbook topic areas, (g) how students’ early interpretations often involved
collections of partial interpretations – which tend to be both poorly differentiated and
poorly integrated, (h) how later interpretations tend to notice patterns of information,
rather than the kind of pieces of information that tend to dominate earlier interpretations,
(i) how model development tends to involve gradually sorting out and integrating several
earlier interpretations, (j) how model development often occurs along a variety of
interacting dimensions – such as concrete-abstract, intuition-formalization, specificgeneral, global-analytic, and so on, (k) how the origins for final interpretations often can
be traced back to several conceptual grandparents, and (l) how final models tend to
include not only systems of logical/mathematical “objects”, relations, operations, and
TME, vol10, nos.1&2, p.55
patterns, but they also usually included dispositions, feelings, and a variety of relevant
metacognitive functions.
Are the Learning & Problem Solving Experiences Recommended (for example) in
the USA’s Common Core State Curriculum Standards Representative of Those
Children Encounter beyond School in the 21st Century?
For mathematics in the primary school (K-2), the main themes of the CCSC
Standards are clear. One of its laudable overall goals is to focus on deeper “conceptual”
treatments of fewer standards.
Another is to emphasize research-based learning
progressions about how students’ mathematical knowledge, skill, and understanding
develop over time. And, another is to treat mathematical understanding and procedural
skill as being equally important.
What do the CCSC Standards mean by focusing on deep treatments of a small
number of “big ideas”?
They say: Mathematics experiences in early
childhood settings should concentrate on (1) number (which includes whole
number, operations, and relations) and (2) geometry, spatial relations, and
measurement, with more mathematics learning time devoted to number than
to other topics.
What does mathematical understanding look like? They say: One hallmark of
mathematical understanding is the ability to justify, in a way appropriate to
the student’s mathematical maturity, why a particular mathematical statement
is true or where a mathematical rule comes from.
Lesh, English, Riggs & Sevis
Modeling with mathematics is mentioned in only one small paragraph in these
standards. And, what do the CCSC Standards mean by “modeling with
mathematics”? They say: Mathematically proficient students can apply the
mathematics they know to solve problems arising in everyday life, society, and
the workplace.
The goal of describing and comparing measurable attributes is mentioned in
precisely one sentence in the CCSC Standards for the primary grades. But,
this sentence is overwhelmed with statements and examples focusing on
number operations, and on counts of discrete objects in sets.
The preceding prejudiced portray of a view of mathematics, learning, and
modeling that is extremely different than the one described briefly in this article. The
CCSC preoccupation with counts is not focused. It is narrow. And, it is not at all
consistent with the kinds of situations that even young children encounter where numbers
and arithmetic outside their school classrooms. Similarly, the CCSC’s notion of what it
means to “understand” important concepts and processes completely overlooks the
development of powerful sense-making systems - that is, models for describing
(quantifying, dimensionalizing, coordinatizing, or in general: mathematizing) situations
in forms so that the concepts and procedures that they profess to emphasize will be useful
beyond mathematics classrooms (Lesh & Sriraman, 2010; Lesh, Sriraman & English,
2013).
Similarly, the notion of modeling in the CCSC as “applying mathematics that they
know to solve problems arising in everyday life” is not at all what we have described in
this paper – where 1st grade children learned to actively develop impressively
TME, vol10, nos.1&2, p.57
sophisticated descriptions of meaningful situations – similar to those that occur beyond
school classrooms.
And finally, the CCSC’s notion of “research progressions”
completely ignores the large literature on situated cognition – where knowledge is
recognized as being organized around mathematically rich experiences (like our stories)
as much as around the kind of decontextualized abstractions that the CCSC Standards
continues to emphasize in the examples and detailed descriptions of curriculum goals that
are given. Why is this oversight so important? One reason is because most “learning
progressions” of the type that the CCSC appears to have in mind envision long strings of
prerequisites as being necessary to “master” before children can proceed to more
important milestones. So, learning is thought of as a long and arduous process – which
looks nothing like the rapid local developments that we describe in this article.
Certainly “real life” situations where number and arithmetic concepts are useful
involve many kinds of mathematical “objects” including beyond counts.
Examples
include locations, actions, weights, likelihoods, and so on. But, unlike the word problems
that fill K-12 textbooks, which can be characterized as situations described by a single
rule (or function) going in one direction. “Real life” situations often involve several
“actors” or several functions – so that feedback loops and 2nd-order effects are important,
and where issues such as maximization, minimization, or stabilization occur regularly.
For example, in the story-based problems that we have emphasized here, most of them
involved several interacting arithmetic operations, as well as issues such as minimization
or maximization.
Most of all, this article is intended to portray mathematical model development as
an important aspect of mathematical “understanding” that is unabashedly optimistic about
Lesh, English, Riggs & Sevis
the level of mathematical thinking that is accessible – even to primary school children,
and to students of average-ability as measured on standardized achievement tests.
References
Brownell, W. A. (1970). Psychological considerations in the learning and teaching of
arithmetic. In J. K. Bidwell & R. G. Clason (Eds.), Readings in the history of
mathematics education (pp. 504–530). Washington DC: National Council of Teachers
of Mathematics. (Reprinted from Tenth yearbook of the National Council of Teachers
of Mathematics: The teaching of arithmetic, pp. 1–31, 1935. New York: Bureau of
Publication, Teachers College, Columbia University.)
Bruner, J. (1960). The process of education. New York: Vantage Books.
English, L. D. (In press). Reconceptualising statistical learning in the early years through
data modeling. In L. D. English & J. Mulligan (Eds.), Reconceptualising early
mathematics learning. Dordrecht: Springer.
Krutetskii, V. (1976). The psychology of mathematical abilities in school children.
Chicago, IL: University of Chicago Press.
Lesh, R. & Doerr, H. (2003) Beyond constructivism: A Models & Modeling Perspective
on Mathematics Teaching, Learning, and Problems Solving. Hillsdale, NJ: Lawrence
Erlbaum Associates.
Lesh, R., & Caylor, B. (2007). Introduction to the special issue: Modeling as application
versus modeling as a way to create mathematics. International Journal of computers
for mathematical Learning, 12(3), 173-194.
Lesh, R., & Nibbelink, W. (1978). Mathematics around us: Kindergarten. Glenview,
IL: Scott Foresman & Co.
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Lesh, R., Carmona, G. & Moore, T. (2010) Six sigma learning gains and long term
retention of understanding and attitudes related to models & modeling.
Mediterranean Journal for Research in Mathematics Education, 9(1), 19-54.
Lesh, R., Kelly, A., Hoover, M., Hole, B., & Post, T. (2000) Principles for developing
thought-revealing activities. In A. Kelly & R. Lesh (Eds.) (2000) The Handbook of
Research Design in Mathematics and Science Education. Hillsdale, NJ: Lawrence
Erlbaum Associates.
Lesh, R., & Sriraman, B. (2010). Reconceptualizing mathematics education as a design
science. In B.Sriraman L. English (Eds). Theories of Mathematics Education: Seeking
New Frontiers. Monograph 1 of Advances in Mathematics Education, Springer
Science, Berlin/Heidelberg, pp. 123-146.
Lesh, R., Sriraman, V., & English, L. (2013). Theories of learning mathematics. In
Encyclopedia of Mathematics Education, Springer Reference Works. Retrieved from
http://www.springerreference.com/docs/html/chapterdbid/313336.html.
The Rational Number Project. (2011). A Five Volume Series containing more than 100
publications from the Rational Number Project – spanning the years from 1979-2010.
University of Minnesota Press.
Thompson, P. W. (1996). Imagery and the development of mathematical reasoning. In L.
P. Steffe, P. Nesher, P. Cobb, G. A. Goldin, & B. Greer (Eds.), Theories of
mathematical learning (pp. 267-285). Mahwah, NJ: Erlbaum.
Zieffler, A., Garfield, J., delMas, R., and Reading, C. (2008). A framework to support
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Lesh, English, Riggs & Sevis
TME, vol10, nos.1&2, p. 61
Prospective Teachers’ Interactive Visualization and Affect in
Mathematical Problem-Solving
Inés Mª Gómez-Chacón1
Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid
Abstract: Research on technology-assisted teaching and learning has identified several
families of factors that contribute to the effective integration of such tools. Focusing on
one such family, affective factors, this article reports on a qualitative study of 30
prospective secondary school mathematics teachers designed to acquire insight into the
affect associated with the visualization of geometric loci using GeoGebra. Affect as a
representational system was the approach adopted to gain insight into how the use of
dynamic geometry applications impacted students’ affective pathways. The data suggests
that affect is related to motivation through goals and self-concept. Basic instrumental
knowledge and the application of modeling to generate interactive images, along with the
use of analogical visualization, played a role in local affect and prospective teachers’ use
of visualization.
Keywords: problem-solving strategies, visual thinking, interactive learning, drawing,
diagrams, teacher training, visual representations, reasoning, GeoGebra.
1. Experimental conditions and research questions addressed
At present, the predominant lines of research on problem-solving aim to identify
underlying assumptions and critical issues, and raise questions about the acquisition of
problem-solving strategies, metacognition, and beliefs and dispositions associated with
problem-solvers’ affect and development (Schoenfeld, 1992; Lester and Kehle, 2003).
Problem-solving expertise is assumed to evolve multi-dimensionally (mathematically,
metacognitively, affectively) and involve the holistic co-development of content,
problem-solving strategies, higher-order thinking and affect, all to varying degrees
(English & Sriraman, 2010). This expertise must, however, be set in a specific context.
1
[email protected]
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos.1&2, pp.61-86
2013©The Author(s) & Dept. of Mathematical Sciences-The University of Montana
Gómez-Chacón
Future research should therefore address the question of how prospective teachers’
expertise can be holistically developed.
The research described here was conducted with a group of 30 Spanish
mathematics undergraduates. These future teachers took courses in advanced
mathematics in differential and Riemannian geometry, but worked very little with the
classical geometry they would later be teaching. They were accustomed to solving
mathematical problems with specific software, mainly in areas such as symbolic
calculation or dynamic geometry, but were not necessarily prepared to use these tools as
future teachers. Research on teaching in technological contexts (Tapan, 2006) has shown
that students are un- or ill-acquainted with mathematics teaching, i.e., they are unaware
of how to convey mathematical notions in classroom environments and find it difficult to
use software in learning situations. Hence the need to specifically include the classroom
use of software in teacher training.
This paper addresses certain understudied areas in problem-solving such as
visualization and affect, with a view to developing discipline awareness and integrating
crucial elements for mathematics education in teacher training. As defined by Mason
(1998), teachers’ professional development is regarded here as development of attention
and awareness. The teacher’s role is to create conditions in which students’ attention
shifts to events and facts of which they were previously unaware. Viewed in those terms,
teaching itself can be seen as a path for personal development.
The main aim of this essay is to explain that in a dynamic geometry environment,
visualization is related to the viewer’s affective state. The construction and use of
imagery of any kind in mathematical problem-solving constitute a research challenge
TME, vol9, nos.1&2, p.63
because of the difficulty of identifying these processes in the individual. The visual
imagery used in mathematics is often personal in nature, related not only to conceptual
knowledge and belief systems, but laden with affect (Goldin, 2000; Gómez-Chacón,
2000b; Presmeg, 1997). These very personal aspects are what may enhance or constrain
mathematical problem-solving (Aspinwall, Shaw, and Presmeg, 1997; Presmeg, 1997),
however, and as such should be analyzed, especially in technological contexts.
Gianquinto (2007) and Rodd (2010) contend that visualization is “epistemic and
emotional”. Giaquinto suggests that visual experience and imagining can trigger beliefforming dispositions leading to the acquisition of geometrical beliefs that constitute
knowledge. According to Rodd (2010), the nature of belief-forming dispositions is not
confined to perception, but incorporates the results of affect (or emotion-perception
relationships). Hence, the belief-forming dispositions that underlie geometric
visualization are affect-laden.
The present study on teaching geometric loci using GeoGebra forms part of a
broader project involving the design, development and implementation of multimedia
learning scenarios for mathematics students and teachers2. The solution of geometric
locus problems using GeoGebra was chosen as the object of study because a review of
the literature revealed that very little research has been conducted on teaching that aspect
of geometry. A recent paper (Botana, 2002) on computational geometry reviewed current
approaches to the generation of geometric loci with dynamic geometry systems and
compared computerized algebraic systems to dynamic symbolic objects. However, it did
not address the educational add-ons needed by teachers. Several authors have compared
2
Complutense University of Madrid Research Vice-Presidency Projects PIMCD-UCM-463-2007, PIMCDUCM-200-2009; and PIMCD-UCM-115-2010
Gómez-Chacón
the visual (and sometimes misleading) solutions generated by dynamic geometry systems
to the exact solutions obtained using symbolic computational tools (Botana, Abánades
and Escribano, 2011). The approximate solution problem affects all dynamic geometry
systems, due to the numerical nature of the calculations performed. The GeoGebra team
has been working on improving this feature as part of the GSoC3 project. In the
meantime, however, external tools must be used to obtain accurate solutions4.
This article specifically explores the role of technological environments in the
development of students’ competence as geometricians and future teachers. More
precisely, it focuses on the relationship between technology and visual thinking in
problem-solving, seeking to build an understanding about the affect (emotions, values
and beliefs) associated with visualization processes in geometric loci using GeoGebra.
The questions posed are: how does affect impact visual thinking through dynamic
geometry software (GeoGebra)? and how does interactive visualization impact affect in
learning mathematics? The difficulties encountered in training students to build strategic
knowledge for the classroom use of technology, which weaken personal problem-solving,
are also explained.
The rest of the paper is organized as follows. A description of the scientific theory
underlying the research is followed by a presentation of the training and research
methodology used. A subsequent section discusses the results of all the analyses,
including tentative answers to the questions formulated above. A final section addresses
the preliminary conclusions of the study and suggestions for future research.
2. Theoretical considerations
3
http://www.geogebra.org/trac/wiki/Gsoc2010
4
http://nash.sip.ucm.es/LAD/LADucation4ggb/
TME, vol9, nos.1&2, p.65
Different theoretical approaches to the analysis of visualization and representation
have been adopted in mathematics education research. In this study the analysis of the
psychological (cognitive and affective) processes involved in working with (internal and
external) representations when reasoning and solving problems requires a holistic
definition of the term visualization. Arcavi’s proposal (Arcavi, 2003: 217) has
consequently been adopted: “the ability, the process and the product of creation,
interpretation, use of and reflection upon pictures, images, diagrams, in our minds, on
paper or with technological tools, with the purpose of depicting and communicating
information, thinking about and developing previously unknown ideas and advancing
understandings”.
Analysis of those two complementary elements, image typology and use of
visualization, was conducted as per Presmeg (2006) and Guzmán (2002). In Presmeg’s
approach, images are described both as functional distinctions between types of imagery
and as products (concrete imagery (“picture in the mind”), kinesthetic imagery, dynamic
imagery, memory images of formula, pattern imagery). In Guzman they are categorized
from the standpoint of conceptualization, the use of visualization as a reference and its
role in mathematization, and the heuristic function of images in problem-solving
(isomorphic visualization, homeomorphic visualization, analogical visualization and
diagrammatic visualization5). This final category was the basis adopted in this paper for
addressing the handling of tools in problem-solving and research and the precise
5
Isomorphic visualization: the objects may correspond ”exactly” to the representations. Homeomorphic
visualization: inter-relationships among some of the elements afford an acceptable simulation of the
relationships between abstract objects They serve as a guide for the imagination. Analogical visualization:
the objects at hand are replaced by that are analogously inter-related. Modeling process. Diagrammatic
visualization: mental objects and their inter-relationships in connection with aspects of interest are merely
represented by diagrams that constitute a useful aid to thinking processes. (Guzmán, 2002).
Gómez-Chacón
distinction between the iconic and heuristic function of images (Duval, 1999; Souto and
Gómez-Chacón, 2011) to analyze students’ performance. The heuristic function was
found to be related to visual methods (Presmeg, 1985) and cognitive aspects as part of
visualization: the effect of basic knowledge, the processes involved in reasoning
mediated by geometrical and spatial concepts and the role of imagery based on analogical
visualization that connects two domains of experience and helps in the modeling process.
The reference framework used to study affective processes has been described by
a number of authors (DeBellis and Goldin, 1997 & 2006; Goldin, 2000; Gómez-Chacón,
2000 and 2011), who suggest that local affect and meta-affect (affect about affect) are
also intricately involved in mathematical thinking. Goldin (2000: 211) contends that
affect has a representational function and that the affective pathway exchanges
information with cognitive systems. According to Goldin, the potential for affective
pathways are at least in part built into the individual. Both these claims were
substantiated by the present data. For these reasons, while social and cultural conditions
are discussed, the focus is on the individual and any local or global affect evinced in
mathematical problem-solving in the classroom or by interviewees. This aspect of
students’ problem-solving was researched in terms of the model summarized in Figure 1
and used in prior studies (Goldin, 2000: 213; Gómez-Chacón, 2000b: 109-130; Presmeg
and Banderas-Cañas, 2001: 292), but adapted to technological environments.
Affective pathway 1 (enabling problem-solving): curiosity →puzzlement→ bewilderment
→encouragement→ pleasure →elation →satisfaction →global structures of affect
(specific representational schemata, general self-concept structures, values and beliefs)
Affective pathway 2 (constraining or hindering problem-solving): curiosity → puzzlement
→ bewilderment → frustration → anxiety → fear/despair → global structures of affect
(general self-concept structures, hate and rejection of mathematics and technology-aided
mathematics)
TME, vol9, nos.1&2, p.67
Figure 1. Emotions and meta-affective aspects in problem-solving
This idealized model illustrates how local affect might influence the heuristic
applied by a problem solver. This model was used in individual case studies because it
provides insight into how visual processes, emotions and cognitive strategies interact. It
also helps detect mental blocks and emotional instability where confusion and perceived
threat are significant, generating high anxiety levels, and therefore conditioning visual
thinking and attitudes. Here, emotions are not mere concomitants of cognition, but are
intertwined with and inseparable from it. Most importantly, they are bound up with the
individual’s self-image and self-concept and the global affective dimension where
purpose, beliefs and goals have a substantial impact.
3. Training and the research methodology used
The qualitative research methodology used consisted of observation during
participation in student training and output analysis sessions as well as semi-structured
interviews (video-recording). The procedure used in data collection was student problemsolving, along with two questionnaires: one on beliefs and emotions about visual
reasoning and the other on emotions and technology (one was filled in at the beginning of
the study and the other after each problem was solved). All screen and audio activity on
the students’ computers was recorded with CamStudio software, with which video-based
information on problem-solving with GeoGebra could be generated. Consequently, at
least four data sources were available for each student.
Six non-routine geometric locus problems were posed, to be solved using
GeoGebra during the training session. Most of the problems were posed on an analytical
register (Table 1: for a fuller description see Gómez-Chacón and Escribano, 2011).
Gómez-Chacón
Finding the solutions to the problems called for following a sequence of visualization,
technical, deductive and analytical steps.
Table 1: Geometric locus problems
PROBLEM
Problem 1: find the equation for the
locus formed by the barycenter of a
triangle ABC, where A = (0, 4), B = (4,
0) and C is a point on circle x² + y² + 4x
= 0.
Problem 2: assume a variable line r that
cuts through the origin O. Take point P
to be the point where line r intersects
with line Y=3. Draw line AP from point
A = (3,0), and the line perpendicular to
AP, s. Find the locus of the intersection
points Q between lines r and s, when r
is shifted.
Problem 3: assume a triangle ABC and
a point P. Project P on the sides of the
triangle: Q1, Q2, Q3. Are Q1, Q2 and
Q3 on the same line? Define the locus
for points P when Q1, Q2 and Q3 are
aligned.
Problem 4: the top of a 5-meter ladder
rests against a vertical wall, and the
bottom on the ground. Define the locus
generated by midpoint M of the ladder
when it slips and falls to the ground.
Define the locus for any other point on
the ladder.
Problem 5: find the locus of points such
that the ratio of their distances to points
A = (2, -3) and B = (3, -2) is 5/3.
Identify the geometric object formed.
Problem 6: find the equation for the
locus of point P such that the sum of the
distances to the axes equals the square
of the distance to the origin. Identify the
geometric object formed.
Level: basic
Geometric locus: the wording of the
problem determines the steps to be
followed.
Level: medium
Geometric locus: in this problem, the
difficulty is to correctly define a
variable line. That done, the rest is
fairly straightforward. The instructions
for using GeoGebra are stated explicitly
in the problem.
Level: medium – advanced
Geometric locus: the locus cannot be
drawn with the “locus” tool in
GeoGebra, because it is non-parametric.
There is no mover point.
Level: medium – advanced
Geometric locus: the problem does not
give explicit instructions on the steps to
follow. The situation is realistic and
readily understood, but translation to
GeoGebra is not obvious. An ancillary
object is needed.
Level: Advanced
Geometric locus: the problem is simple
using paper and pencil. The difficulty
lies in expressing “distance” in
GeoGebra.
Level: Advanced
Geometric locus: the problem is simple
using paper and pencil. The difficulty
lies in expressing “distance” in
GeoGebra.
TME, vol9, nos.1&2, p.69
Geometric locus training was conducted in three two-hour sessions. Prior to the
exercise, the students attended several sessions on how to use GeoGebra software, and
were asked to solve problems involving geometric constructions.
In the two first sessions, the students were required to solve the problems
individually in accordance with a proposed problem-solving procedure that included the
steps involved, an explanation of the difficulties that might arise, and a comparison of
paper and pencil and computer approaches to solving the problems. Students were also
asked to describe and record their emotions, feelings and mental blocks when solving
problems.
The third session was devoted to common approaches and the difficulties arising
when endeavouring to solve the problems. A preliminary analysis of the results from the
preceding sessions was available during this session.
The problem-solving results required a more thorough study of the subjects’
cognitive and instrumental understanding of geometric loci. This was achieved with
semi-structured interviews conducted with nine group volunteers. The interviews were
divided into two parts: task-based questions about the problems, asking respondents to
explain their methodologies and a series of questions designed to elicit emotions, visual
and analytical reasoning, and visualization and instrumental difficulties.
A model questionnaire proposed by Di Martino and Zan (2003) was adapted for
this study to identify subjects’ belief systems regarding visualization and computers to
study their global affect and determine whether the same belief can elicit different
emotions from different individuals. In this study, students were asked to give their
opinion of a belief and choose the emotion (like/ dislike) they associated with it, e.g.:
Gómez-Chacón
Table 2: Example of items of student questionnaire on beliefs and emotions
‐ Visual reasoning is central to mathematical problem solving.
‐ Visual reasoning is not central to mathematical problem solving.
Give reasons and examples. How do you feel about having to use problem
representations or visual imagery?
I like it.
I don’t like it.
I’m indifferent.
…..Explain the reasons for your feelings.
A second questionnaire, drawn up specifically for the present study, was
completed at the end of each problem. The main questions were:
Table 3: Student questionnaire on the interaction between cognition and affect
Please answer the following questions after solving the problem:
1. Was this problem easy or difficult? Why?
2. What did you find most difficult?
3. Do you usually use drawings when you solve problems? When?
4. Were you able to visualize the problem without a drawing?
5. Describe your emotional reactions, your feelings and specify whether you got stuck
when doing the problem with pencil and paper or with a computer.
6. If you had to describe the pathway of your emotional reactions to solving the problem,
which of these routes describes you best? If you do not identify with either, please
describe your own pathway.
Affective pathway 1 (enabling problem-solving): curiosity →puzzlement→ bewilderment
→encouragement→ pleasure →elation →satisfaction →global structures of affect
(specific representational schemata, general self-concept structures, values and beliefs).
Affective pathway 2 (constraining or hindering problem-solving): curiosity → puzzlement
→ bewilderment → frustration → anxiety → fear/despair → global structures of affect
(general self-concept structures, hate or rejection of mathematics and technology-aided
mathematics).
7. Now specify whether any of the aforementioned emotions were related to problem
visualization or representation and the exact part of the problem concerned.
The protocols and interviewee data were analyzed for their relationship to affect
as a representational system and the aspects described in section two.
4. Findings
The results shown here attempt to answer the concerns formulated in the
introduction. The affective pathways reported for each problem consistently showed: a)
TME, vol9, nos.1&2, p.71
the effect of subjects’ beliefs and goals on the preference and use of visual
thought/knowledge in computerized environments; b) that students proved to have a poor
command of the tools, especially the locus tool; c) that notwithstanding, beliefs on the
potential of GeoGebra helped them maintain productive affective pathways. As a
qualitative study, the aim here was to describe the findings in detail. Consequently, the
cases that best exemplified the results that were consistent across the entire group (30
students) and the nine volunteers were chosen and characterized by: gender,
mathematical achievement, visual style, beliefs about computer learning, computer
emotion, beliefs about visual thinking, feelings about visualization processes and global
affect.
4.1. Beliefs about visual reasoning and emotion typologies
The data showed that all students believed that visual thinking is essential to
solving mathematical problems. However, different emotions were associated with this
belief. Initially, these emotions toward the object were: like (77%), dislike (10%),
indifference (13%). The reasons given to justify these emotions were: a) pleasure in
knowing that expertise can be attained (30% of the students)6; b) pleasure when progress
is made in the schematization process and a smooth conceptual form is constructed
(35%); c) pleasure and enjoyment afforded by the generation of in-depth learning and the
control over that process (40%); d) pleasure and enjoyment associated with the
entertaining and intuitive aspects of mathematical knowledge (20%); e) indifference
about visualization (13%); f) dislike or displeasure when visualization is more
cognitively demanding (10%).
6
Some students cited several reasons.
Gómez-Chacón
A similar response was received when the beliefs explored related to the use of
dynamic geometry software as an aid to understanding and visualizing the geometric
locus idea. All the students claimed to find it useful and 80% expressed positive emotions
based on its reliability, speedy execution and potential to develop their intuition and
spatial vision. They added that the tool helped them surmount mental blocks and
enhanced their confidence and motivation. As future teachers they stressed that
GeoGebra could favour not only visual thinking, but help maintain a productive affective
pathway. They indicated that working with the tool induced positive beliefs towards
mathematics itself and their own capacity and willingness to engage in mathematics
learning (self-concept as a mathematical learner).
4.2. Cognitive and instrumental difficulties: student's geometric constructions
with GeoGebra
This section describes the solution typologies for the six problems.
Typology 1: static constructions (discrete treatment). In this typology, the
students used GeoGebra as a glorified blackboard (Pea, 1985), but none of its dynamic
features. They repeated the constructions for a number of points. To draw the geometric
locus, they used the “5-point conic” tool. This underuse of potential appeared in problems
1 and 4.
Typology 2: incorrect definition of the construction. The students solved the
problem (imprecisely), but with solutions that implied that the GeoGebra tools were
unusable. The “locus” tool can only be used if the defining points are correctly
determined (they may not be free points). Adopting this approach, at best the students
TME, vol9, nos.1&2, p.73
could build a partially valid construction, but since the GeoGebra tools couldn’t be used,
no algebraic answer was obtained.
This typology appeared in problems 2 and 4. In problem 2, the sheaf of lines had
to be defined by a point on an ancillary object such as a line, and not as a free point.
Otherwise, the approximate visual solution obtained was unusable with GeoGebra. The
students concerned were absolutely convinced that their solution was right and wholly
unaware of any flaw in the solution.
The difficulty in problem 4 was to define a point that was not the mid-point. The
locus tool could not be used for a free point on the ladder.
Typology 3: incorrect use of elements. For example, in problems 1, 2, 4 and 6,
some students used the “slider” tool to move the “mover point”. They realized that the
“mover point” had to be controlled, which is what the slider is for. In GeoGebra,
however, the slider is a scalar and can't be used with the locus tool.7
Problem 2 is a case in point. Some students defined the sheaf of lines as the lines
passing through the origin on a point in the circle, and this point in the circle was moved
with the slider. For example, student 9 said: “This problem is similar to the one before it.
I built the construction while reading the problem. The hardest step was to construct the
variable line. First, I thought I’d use a slider for the slope of the line passing through the
origin, but that way I never got a vertical line, so I used the slider as in the preceding
problem to build point C that revolves around the origin, and then to build the line
connecting C and O. After that, I just followed the instructions in the problem, and I was
very careful about the way I named the elements” (student 9, problem 2).
7
http://www.geogebra.org/help/docues/topics/746.html
Gómez-Chacón
Typology 4: failure to use the locus tool. Here, the construction was correct, but
the student did not use the locus tool. To use it, the point that projects the locus (tracer)
must be distinguished from the point that moves the construction (mover). The mover
must be a point on an object. Some students were apparently unable to make that
distinction, which prevented them from using the tool correctly.
This misunderstanding arose in problems 1, 2, 3 and 4. Student 8 exemplifies this
type of reasoning: “The first thing I had to do was find the center and radius of the circle
to draw, to complete the square in the equation: (x +2) ² + y ² = 4. Therefore, point C is in
a circle with a center at (-2, 0) and a radius of 2. (I didn’t actually need this because in
GeoGebra I could enter the equation directly and draw the circle). Now, to solve the
problem I had to know what a barycenter was. I took point C on the circle (creating an
angular slider so the point would run along the entire circumference of the circle) and
drew the triangle ABC. I calculated the triangle barycenter (I drew the medians as dashed
green lines to make it easier to see that G is the barycenter). Using animation to project
point G gave me the locus. Since the locus was a circle, I was able to solve the equation
by finding three points, G1, G2, G3, and activating the “circle through three points” tool.
Then I entered the data in GeoGebra: (x-0.66) 2 + y-1.34) 2 = 0.44" (student 8, problem
1).
4.3. Maintaining productive affective pathways
As noted in the preceding paragraph, the belief that visual thinking is essential to
problem-solving and that dynamic geometry systems constitute a visualization aid,
particularly in geometric locus studies, was widely extended across the study group. That
belief enabled students to maintain a positive self-concept as mathematics learners in a
TME, vol9, nos.1&2, p.75
technological context and to follow positive affective pathways with respect to each
problem, despite their negative feelings at certain stages along the way and their initial
lack of interest in and motivation for computer-aided mathematics.
A comparison of the affective pathways reported by the students revealed: a)
concurrence between the use of visualization typologies and associated emotion; b) that
the availability of and subsequent decision to use GeoGebra was often instrumental in
maintaining a productive affective pathway. This section addresses three examples, in
two of which the affective pathway remained productive and one in which it did not. It
discusses the determinants for positive global affect and positive self-concept as
mathematical learners. The key characteristics of the case studies are given in Table 4.
Table 4: Three case studies: characteristics
Case
Gender
Mathematical
achievement
Visual
style
Beliefs
about
computer
learning
Feelings
about
computers
Beliefs
about
visual
thinking
Feelings
about
visualization
Global
affect
Student
19
Male
High
Visualizing
student
Positive
Likes
Positive
Likes
Student
20
Female
Average
Positive
Dislikes
Positive
Dislikes
Student
6
Female
Low
Nonvisualizing
student
Style not
clear
Positive
Dislikes
Positive
Likes
Positive
selfconcept
Positive
selfconcept
Negative
selfconcept
Problem 4 (Table 1) was chosen for this analysis. The students’ affective
pathways for this problem are given in Table 5.
Student 19 is a visualizer. In the interview he said that the pleasure he derives
from visualization is closely associated with the mathematics view. He regards visual
reasoning as essential to problem-solving to monitor and generate in-depth learning, to
contribute to the intuitive dimension of knowledge and to form mental images.
Gómez-Chacón
When he was asked whether his feelings were related to visualization and
problem-solving and to specify the parts of the problem where they were, he replied:
“curiosity predominated in visualization. Since the problem was interesting and seemed
to be different from the usual conic problems, I was keen on finding the solution. I had a
major mental block when it came to representing the problem and later, as I sought a
strategy. I was unable to define a good strategy to find the answer. I was puzzled long
enough to leave the problem unsolved and try again later. When I visualized the problem
in a different way, I found a strategy: construct a circle with radius 5 to represent the
ladder and another smaller circle to represent the point in question. When I reached that
stage, I felt confident, happy and satisfied” (student 19).
Student 20 is a non-visualizing thinker with positive beliefs about the importance
of visual reasoning. However, she claimed that her preference for visualization depends
on the problem and that she normally found visualization difficult. It was easier for her to
visualize “real life” than more theoretical problems (the difference between problems 4
and 5, for instance).
Her motivation and emotional reactions to the use of computers were not positive,
although she claimed to have discovered the advantages of GeoGebra and found its
environment friendly. She also found that working with GeoGebra afforded greater
assurance than manual problem-solving because the solution is dynamically visible.
Convincing trainees such as student 20 that mathematical learning is important to
teaching their future high school students helps them keep a positive self-concept, even if
they don’t always feel confident in problem-solving situations (Table 5).
TME, vol9, nos.1&2, p.77
Student 6’s visual thinking style could not be clearly identified. She expressed a
belief in the importance of positive visual reasoning (“because visual reasoning helps
gain a better understanding of the problem and consequently the solution”). This
confirmed a liking for visualization and representation because it made it easier to
understand the problem and she found formalization helpful. She added, however, that
she felt insecure applying technological software to mathematics, although she believed
GeoGebra, specifically, to be useful. In her own words, “I don’t like it and never will. I
feel a little nervous and insecure, not because of GeoGebra but because computers
intimidate me because I don’t understand them completely. But when I managed to
represent the problem with GeoGebra, I felt more satisfied with the result than when I
solved it with paper and pencil”. Although student-6’s pathway was essentially negative
in problem 4, she persisted until she found the solution. In some cases students were
unaware of their mistakes and misunderstandings, however.
GeoGebra can be used to solve problem 4, although an average student cannot be
expected to build the entire construction from scratch. The visual and instrumental
challenge is to deploy the sliding segment, and that calls for an auxiliary circle (which
may be concealed to simulate the effect of the ladder). The point in the ladder must be
chosen with care to use the locus tool. Just any “point in segment” will not do; the
“middle point” tool or a more sophisticated construction must be used.
While none of the three students applied the “locus” command, student 19 used
the visual power of the technology to gain a better mathematical understanding of the
problem. That inspired a change in context which facilitated notion and property
applications. He used GeoGebra as a genuine mathematical modeling tool. He did not
Gómez-Chacón
solve the problem with the geometric locus command, however, even though he came up
with the right answer by modeling. A comparison of this student’s pathways in the six
problems revealed that the interaction between visual reasoning and negative feelings
arose around the identification of interactive representation strategies and the formulation
of certain representations in which the identification of parametric variations plays a role.
This student’s command of the use of concrete, kinesthetic and analogical images was
very helpful and contributed to his global affect and his positive overall self-concept
when engaging in computer-aided mathematics.
An analysis of the relationship between these three students’ affective pathways
(Table 5) and their cognitive visualization shows that visualization - negative feelings
interactions stem essentially from students’ lack of familiarity with the tools and want of
resources in their search for computer-transferable analogical images and their switch
from a paper and pencil to a computer environment in their interpretation of the
mathematical object.
Behavior such as exhibited by student 6 denotes a need to include construction
with locus tools in teacher training. Although no general methodology is in place, any
geometric problem that aims to determine locus must be carefully analyzed. This calls for
identifying three categories of geometric elements in such problems: fixed (position,
length, dimension); mobile (position, length, variable points); and constant (length,
dimension).
The data also revealed the relationship between beliefs, goals and emotional
pathways. The analysis of student 20’s responses showed that while she had no
inclination to use computers, the importance she attached to mathematics and IT in
TME, vol9, nos.1&2, p.79
specific objectives and the structuring of her overall objective kept her on a productive
affective pathway (McCulloch, 2011). Student 20’s solution to problem 5 (Table 1), for
instance, constitutes a good example of a productive pathway: despite negative feelings:,
she maintained a positive mathematical self-concept, which she reported when she
explained her global affect. (Her self-reported pathway in problem 5 was: curiosity
→confusion /frustration → desperation → puzzlement → satisfaction → a negative
mathematical self-concept in terms of technology for problem 5, but a positive global
affect regarding computer use in solving the six problems). Questions designed to elicit
the reasons for her positive mathematical self-concept in terms of technology showed that
objectives, purposes and beliefs were clearly interrelated. Her own words were: “I think
that computers, not only the GeoGebra program, are an excellent tool for anyone
studying mathematics. Nowadays, the two are closely linked: everyone who studies
mathematics needs a computer at some point… mathematics is linked to computers and
specifically to software like GeoGebra (if you want to teach high school mathematics, for
instance. I at least am trying to learn more to be a math teacher) (student 20)”.
Gómez-Chacón
Table 5: Affective pathways and visual cognitive processes reported for this problem by
three students
COGNITIVE-EMOTIONAL PROCESS
Problem 4
Student 19
Own pathway
Curiosity
Confusion
Reading and understanding problem
Drawing (patterns and lines/figure)
Analytical
Puzzlement.
Mental block
Confidence
Perseverance-motivation
Excitement and hope
(Search for mental image) (specific figure/illustration and dynamic image)
Confidence
Confidence, joy
Student 20
Own pathway
Technological manipulation with the computer
Representing circle radius (specific illustrations)
Interactive image generation,
slider (analogical)
Joy and happiness
Interactive image generation,
slider (analogical)
Perceived beauty
Specific illustration with interactivity (analogical)
Satisfaction
GLOBAL AFFECT
Analytical-visual
Memorized formulaic typology
Positive self-concept
Curiosity
Problem reading
Frustration
Global visualization of problem
Pictorial image
Search for mental image
Inability to visualize the ladder as the radius of a circle
Confusion
Puzzlement
GLOBAL AFFECT
Search for mental image
Dynamic and interactive image with GeoGebra
Technological manipulation with the computer
Pictorial representation with GeoGebra
Pictorial representation with “trace on” GeoGebra
Full construction from scratch
Come up with a final solution
Positive self-concept
Curiosity
Problem reading
Puzzlement
Bewildermen
Global visualization of problem
Pictorial image
Search doe an instrumental image with GeoGebra
Frustration
Computer handling skills
Anxiety
Inability to visualize the ladder as the radius of a circle and using “trace on”
Stimulus, motivation
Satisfaction
Student 6
Pathway-2
Search for mental image
Search for mental image
Physical manipulation - kinetics
Kinesthetic learning
Mental image Identification mathematical object
Fear/despair
Needing help to find the solution
GLOBAL AFFECT
Negative self-concept
TME, vol9, nos.1&2, p.81
Conclusion, limitations and further research
The results of this study suggest that various factors are present in conjunction
with visual thinking. The first appears to be the study group’s belief that visual thinking
and their goal to become teachers would be furthered by working with technology (Cobb,
1986). The data shows that all the student teachers believed that visual thinking is
essential to solving mathematical problems. That finding runs counter to other studies on
visualization and mathematical ability, which reported a reluctance to visualize (e.g.,
Eisenberg, 1994). However, different emotions were associated with this belief. The
belief about using computers and that software is a tool that contributes to overcoming
negative feelings has an impact on motivated behavior and enhances a positive selfconcept as a mathematical learner. Despite this advantage, however, student teachers may
still misunderstand or misinterpret and therefore misuse computer information,
unknowingly in some cases, and surrender all authority to the computer.
While prospective teachers resort to GeoGebra software to help maintain a
productive affective pathway and foster visual thinking, student 20’s experience with
problem 5 is significant, for it shows that the tool by itself is not enough. If the software
is unable to deliver the dynamic geometric capability that students want to use for the
concepts at hand, it is useless and may even have an adverse impact on their affective
pathway, possibly resulting in feelings of defeat such as reported by student 20. Her
experience provides further evidence of the importance and complexity of mathematics
teacher training, as documented by researchers studying the issue from an instrumental
approach (e.g., Artigue, 2002). The mere provision of tools cannot be expected to
necessarily raise the frequency of productive affective pathways. Rather, thought needs to
Gómez-Chacón
be given to how those tools are integrated into classrooms to support the development of
visualization skills. Some students (as in item 4.2) think of graphs as a photographic
image of a situation due to a primarily static understanding of functional dependence.
That might be attributed to the fact that the pointwise view of mathematical objects tends
to prevail in the classroom, where the dynamic view is underrepresented (institutional
dimension of visualization).
The results of this study bring to mind the progressive modelling in visual
thinking notion introduced by Rivera (Rivera, 2011: 270). Furthering visualization
processes in teaching involves more than just drawing “pretty pictures”: it requires
sequenced progression of the thought process. This in turn calls for awareness of the
transition in dimensional modelling phases from the iconic to the symbolic and the
change of mindset. For the problem proposed, “geometric locus”, each transition can be
associated with mathematical explanations and symbol notation and the proficient use of
the visual tool to reify the mathematical concept. Therefore, one question that would be
open for research is the definition of the components of an overarching theory of
visualization for problem-solving in technological environments where this progression is
explicit. While this study was conducted in a classroom context, it focuses on the
individual only, not on interaction among individuals. Future studies might profitably
explore the role of external affect and others’ (i.e., teachers’, community’s, institution’s)
external affective representations. Such interaction impacts meta-affect and may
potentially either help maintain or interrupt productive affective pathways.
Finally, as explained in the introduction, the teacher training model pursues the
development of students’ awareness and ability to apply their knowledge in complex
TME, vol9, nos.1&2, p.83
contexts, integrating knowledge with their own attitudes and values and therefore
developing their personal and professional behavior. From this standpoint, teacher
training programs should adopt a more holistic approach (cognitive, didactic, technical
and affective). The present paper aims to provide a preliminary framework to help
teacher educators or mathematical cognitive tool designers select and analyze interaction
techniques. A secondary aim is to encourage the design of more innovative interactive
mathematical tools.
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TME, vol10, nos.1&2, p.87
Young Children Investigating Advanced Mathematical Concepts With
Haptic Technologies: Future Design Perspectives
Stephen Hegedus1
Kaput Center for Research and Innovation in STEM Education
University of Massachusetts Dartmouth
1. Introduction
In this chapter, we focus on how new technologies can be used with young
children to investigate mathematical ideas and concepts that would normally be
introduced at a later age. In particular, we focus on haptic technologies that allow learners
to touch and feel objects through force feedback in addition to visual images on a screen.
The main purpose of this paper is to describe how these technologies can be used to
enable young learners to construct meaning about geometric shapes and surfaces as well
as attributes of particular mathematical constructions in multiple dimensions (particularly
2D and 3D for purposes of this chapter). Such learning environments enable various
forms of mediation both through the devices and software used as well as socially, as
students work together to develop meaning and create models of complex ideas.
We begin by describing how and why young learners in particular should be
working in such learning environments in order to provide a rationale for our work. In
Section 2, we provide some background on how these technologies have evolved and
their use in other disciplines and how we have built on prior research in the use of
dynamic geometry in mathematics education. Section 3 presents how relevant these new
learning environments can be with some specific examples from preliminary work at the
Kaput Center. The section also contains some theoretical reflections on how we can
1
[email protected]
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos.1&2, pp.87-108.
2013©The Author(s) & Dept. of Mathematical Sciences-The University of Montana
Hegedus
begin to analyze and understand how students work and construct meaning in such
environments. Section 4 then concludes by offering some design principles for future
research and development.
How Should Young Children be Doing Mathematical Problem-Solving in the
Future?
We believe that the answer to this question lies in three areas that focus on the
early introduction of mathematical ideas, the use of technology, and engagement.
Early introduction. Several researchers have promoted the idea of introducing
mathematical ideas earlier in the curriculum and even introducing the foundation of
advanced mathematical thinking in the early grades (Kaput, Carraher & Blanton, 2008;
Kaput, 1994). If not then, many children will never be exposed to important mathematics
and engage in fruitful and relevant investigations. This can have detrimental effects
throughout a child’s educational career, reducing their desire to want to learn
mathematics because of its lack of relevance or inaccessible representations.
Technology use. Technology is often not a major part of elementary school
classroom teaching due to a lack of resources and perception of its role and use. The
predominant form of technology use in most elementary school classrooms in the U.S. is
PowerPoint presentations. Some researchers (Carraher & Schliemann, 2000) believe that
the introduction of technology is not enough:
It is important to provide a social analysis in consonance with a cognitive
one. Because technology does not act directly on learners, but only exerts
an influence on the social activities and contexts in which it is employed,
introducing technology into the mathematics classroom ultimately entails
questions such as the following: What is the teacher’s role; what are the
students trying to achieve in the tasks … . (p. 174)
TME, vol10, nos.1&2, p .89
While we agree that these questions are important, new technologies can have a
more participatory and collaborative role rather than be a prosthetic device to prop up
existing pedagogical practices. New technologies can actually re-structure interaction in
the classroom and allow the introduction of advanced mathematical ideas through
radically new mathematical representation systems. The interactions of teachers, students
and technologies within a learning environment can modify and transform activity
structures (Jonassen, 2000).
Technological affordances can also be mathematical affordances providing a
symbiotic link between how mathematical activity can occur. Mathematizing
technological affordances is an important step and one we discuss in detail later.
Engagement. By integrating activity structures with the affordances of new
technologies, the learning environment should be simple enough to establish
engagement—to motivate curious young minds to explore, question, and be encouraged
to want to continue to learn. It should allow them to construct meaning in open-ended
tasks, which have been carefully designed to have mathematical purpose. It should allow
them to share, collaborate, and feel free to use non-scholastic language as they conduct
their mathematical investigation.
We take a very broad view of what is mathematical problem-solving viewing it as
an enterprise of collaborative investigation where multiple approaches are valid. It is not
just about solving a specific problem, which has a specific answer or application into the
real world, but rather it is an investigation that might have multiple approaches and where
students can make multiple observations. Also, most of our activities might best be
described as “tasks” rather than “problems.”—that is, they are goal directed activities.
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Students are seldom at a loss for ideas to pursue. They are not stuck; they are not
frustrated; and, their progress often does not fit the metaphor of moving along a single
path that is somehow temporarily blocked. Instead, our environments are carefully
engineered so that students can make parallel progress along a variety of interacting
paths. Our initial tasks involve exploring, categorizing attributes of geometric shapes or
objects, making sense of a set of objects and constructing broad and specific meaning.
These tasks, in a broad sense, could be described as modeling (Lesh, 2007). We will
continue to use the phrase mathematical problem-solving throughout this chapter but in
the spirit of the position described above.
We have referred to new technologies, but we focus on a particular type of
learning environment that utilizes haptic or multi-modal devices. Multiple modalities are
used in real-world applications. We make sense of problem conditions in the world by
using sight, touch, and hearing to name a few. Hence, in our research and development,
we have focused on new technologies that use multiple modes of input in early
mathematics classrooms. First, let us describe the evolution of such technologies in
contrast to the predominance of visualization software in mathematics education.
2. Background to New Technology
Haptic literally means “ability to touch” or “ability to lay hold of” (Revesz, 1950)
and has evolved to be an interface for users to virtually touch, push, or manipulate objects
created and/or displayed in a visual environment (McLaughlin, Hespanha, & Sukhatme,
2002). Recently, this has rapidly evolved to include multi-touch environments. In these
environments, learners literally lay their hands on objects via a screen interface,
TME, vol10, nos.1&2, p .91
mathematical objects can be manipulated and resultant actions be investigated. Let us
first examine the background of educational technology involved in dynamic visual
mathematics before extending to haptic technologies which is the focus of this chapter.
Traditionally, dynamic, interactive, mathematical, visual environments—
including Computer Algebra Systems such as Mathematica and Maple, as well as multidimensional Graphing software such as Avitzur’s Graphing Calculator—are used to aid
students to visualize complex surfaces in various coordinate systems and complete
computationally intensive tasks. The Geometer’s Sketchpad® is used in classrooms
ranging from elementary grades through to undergraduate programs to allow users to
construct, interact and explore geometric figures and shapes, and so engage in modeleliciting activities in various mathematical topics. But these environments are not
responsive to users’ physical interactions apart from mouse pointing.
The
experience
of
visual
mathematics,
particularly
three-dimensional
mathematics, is often very brief for U.S. mathematics and science students. Following a
school curriculum of Euclidean Geometry rarely expanding to non-Euclidean geometry
or solid geometry, there is a rapid progression in most university curriculum from threedimensional geometry, which is embedded in third or fourth semester Calculus courses,
to the abstract intangibles of higher dimensional mathematics. In fact, given the very
nature of multi-dimensional mathematics—that it can examine real life objects and
phenomena all around us—it is interesting that such a small proportion of a student’s
formal mathematical life is spent examining the subject. Such mathematics provides a
vocabulary for understanding fundamental modeling equations, for example, weather,
heat, planetary motion, waves, and later, multi-dimensional mathematics, finance,
Hegedus
epidemiology, quantum mechanics, bioinformatics and many more.
Yet, there is a
growing emergence of technologies in the scientific workplace that apply, manipulate,
and model three-dimensional representations
A wide range of technologies are used in the teaching and learning of multidimensional mathematics in various contexts, ranging from relatively expensive
Computer Algebra Systems (CAS) such as Mathematica™ and Maple™ (Meel, 1998;
Park & Travers, 1996), industrial design packages, (e.g., AutoCAD™), through to Java
Applets freely downloadable from the WWW. During reform periods, Mathematics and
Science departments have been encouraged to integrate CAS technology into their classes
as it can help students with visual and conceptual problems (Zorn, 1987; 1992). As
technology becomes more sophisticated, the opportunity cost of training time and money
spent on learning how to use a particular software and how to successfully integrate it
into school curriculum is sufficiently high to dissuade teachers from the investment.
Dynamic geometry environments offer point-and-click tools to construct
geometric objects that can be selected and dragged by mouse movements. All userdefined mathematical relationships are preserved, thus providing environments for
students to conjecture and generalize by clicking and dragging hotspots on the object.
These hotspots dynamically re-draw and update information on the screen as the user
drags the mouse, and in doing so, efficiently testing large iterations of the mathematical
construction (Moreno & Sriraman, 2005; Moreno & Hegedus, 2009; Moreno, Hegedus &
Kaput, 2008).
Such environments aim to develop spatial sense and geometric reasoning by
allowing geometric conjectures to be tested, offering “intelligent” constructivist tools that
TME, vol10, nos.1&2, p .93
constrain users to select, construct or manipulate objects that obey mathematical rules
(Mariotti, 2003)— that are largely used in secondary and not primary schools.
In summary, these dynamic mathematic environments are responsive to users’
interaction but are still more structured in their feedback and lack the expressive
capabilities of using physical interaction and force-feedback.
We believe that students naturally need more haptic, kinesthetic avenues through
such the combination of dynamic visual environments and haptic technologies to explore
the mathematics of change and variation in a more sensory environment to connect to the
symbolic formalisms of the mathematical ideas (Nemirovsky & Borba, 2003). Change
and variation occurs in multiple school subjects, in particular algebra, geometry and data
analysis. In allowing students the combined affordances of multi-touch interaction, visual
feedback and force feedback where possible, the technological environment can become
a semiotic mediator of mathematical thinking and investigation. Young learners can have
access to new forms of mathematical problem-solving or investigation through direct
manipulation of mathematical objects linked to varying attributes (e.g. area).
To this aim, we have focused on integrating two types of haptic technologies: (1)
Sensable’s PHANTOM Omni—a force-feedback device and (2) iPad with a dynamic
geometry application—a multi-touch/multi-input device.
Sensable’s
PHANTOM
Omni®
(http://www.sensable.com/haptic-phantom-
omni.htm)—hereon referred to as Omni—is a desktop haptic device with six degrees of
freedom for input (x, y, z, pitch, roll, yaw), and three degrees of output (x, y, z). The
Omni’s most typical operation is via a stylus-like attachment that includes two buttons
(see Figure 1a). The Omni has a very robust community and SDK behind it. The SDK
Hegeduus
(Open
nHaptics-Accademic Ediition) allowss for two levvels of progrrammatic coontrol: precisse
progrrammer creaated feedback
k—such as vibration—a
v
and pre-proggrammed feeedback—succh
as springs and dy
ynamic/staticc friction. Th
he Omni proovides up too 3 forces off feedback foor
x, y, and z. It is primarily used in reseaarch, with a significant ppresence in dentistry annd
medicine but grow
wing in math
hematics edu
ucation (Heggedus & Mooreno, 2011).
Figure 1b
b is an exaample of how
h
the Om
mni can bee used withh a graphicaal
envirronment. Users can clicck, push, or drag a “bugg” across a surface usinng the haptiic
devicce. Here, thee bug is bein
ng pushed around
a
a sadddle surface.. A saddle surface has a
maxim
mum and a minimum at the samee place (thee place wherre you wouuld sit on thhe
saddlle). A sample activity wo
ould be to assk the studennt to push thhe bug to thaat place on thhe
surface and desccribe how th
he surface changes
c
duriing its motion, both in terms of thhe
ortantly in terms
t
of itss feel. Mathhematically sspeaking this
visuaalization butt more impo
poten
ntially enablees an interprretation of differentiabil
d
lity of the suurface. This approach caan
offer new access to profound
dly importantt mathematiccal ideas forr younger chhildren.
Figurre 1a. Sen
nsable’s Om
mni Haptic Figure 1bb. Graphical interface proototype.
Devicce.
In additio
on to shapee, feedback
k from the Omni can be linked to particulaar
mathematical atttributes, parrticularly varrying quanttities. Figuree 2 highlighhts a trianglle
whosse shape can
n be dynamiccally transfo
ormed by mooving the veertices (A, B
B, or C) or bby
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E, vol10, noos.1&2, p .995
moving the paralllel lines apaart (via D). Here
H the feeddback can bee linked to thhe area of thhe
triang
gle and chiildren can investigate
i
what
w
properrties impactt the area before beinng
introd
duced to a fo
ormal equatiion.
Figure 2. Area of a triangle.
It should also be no
oted that many
m
mass-m
market whiteeboards, tabblet PCs annd
gestu
ure-based PD
DAs (e.g., iP
Pod Touch/iiPhone) andd larger form
ms such as tthe iPad now
w
featurre pressure-ssensitive sty
yli of some form;
f
this prressure-inputt becomes a limited form
m
of forrce-feedback
k in softwarre that “responds” differrentially to vvariable pressure. At this
stage, feedback is visual, an
nd does nott include thhe additionall sensory innformation oof
forces or kinesth
hetics that are directly linked to the propertiies of the oobjects beinng
manip
pulated on a screen. Bu
ut the grow
wing spread of these deevices and thheir use witth
educaational appliications creaates an entry
y point for the use of pperipheral ccomputationaal
devicces that exteend the use of mouse as a physicall pointing action, to an environmennt
where users can touch,
t
push or
o move objects more diirectly.
In multi-ttouch/multi-iinput platforrms such as the iPad, thee learner cann use multiplle
modees of input an
nd outputs—
— their naturral modes off seeing and feeling, to m
make sense oof
the taask. The iP
Pad offers a direct (alm
most zero-intterface) modde to touch and directlly
Hegedus
manipulate mathematical objects, and offer multiple inputs to one mathematical object
hitherto impossible on a single-input computer (mouse as pointer and selector).
With both hardware, the technological affordances are tightly coupled with
mathematical affordances in such that the technology offers mathematical meaningful
tools or avenues to investigate. Hence a new form of mathematical problem-solving
originates because of new mathematization routes. We will exemplify these affordances
in the next section. In terms of software deployment, key representational features
include high-resolution visualization of mathematical objects and constructions which
can be made transparent to see their interaction with other objects, and direct
manipulation of objects allowing users to rotate and navigate “around” objects and
flexible notation systems to allow users to observe outputs (e.g., changing area of a
shape) based upon their input.
We now describe how such environments are relevant to mathematics education
and offer examples of how they can advance mathematical investigations and inquiry.
3. Relevance: Future Mathematical Problem-Solving
Situations inside and outside of formal learning environments involve
visualization, multi-modal investigations, using and interpreting multiple representations,
connecting mathematical attributions and concepts to real world phenomenon, e.g., form,
shape of objects and models (visual surfaces), features and attributes. So what is
modeling in a problem-solving context for early learners? And is it relevant or necessary
for early learners to be introduced to such ideas? We think it is and it goes deep into what
a mathematical problem-solving environment is for a life-long learner. In addition,
TME, vol10, nos.1&2, p .97
making sense of the environment in a mathematical way is not just physical or visible
(i.e., tangible) but also occurs at the nano-level. Macro images, surfaces, and objects can
simulate phenomena, which cannot be seen or felt, e.g., cell structures. Census datasets
cannot be understood at a macro level without a deep understanding of the micro. What
constitutes a dataset? In a similar vein, what constitutes variation at all across
mathematical models? The heart of the research reported here is to establish conditions
by which early learners can advance their mathematical inquiry at stages that are hitherto
not required by standardized frameworks but are still complimentary. Our primary
research question is: Can we establish learning environments by which advanced
mathematical ideas can be more readily accessed, understood and used to solve
problems? Such understanding is mediated through the affordances of technological
devices and at the same time social interaction between peers. Young children can
construct meaning through collaboration (one form of mediation) but supported and
additionally mediated through the tools afforded to them through mathematicallyenhanced technologies.
Holland et al. (2004) outlines the role of a mediating device:
A typical mediating device is constructed by the assigning of meaning to
an object or a behavior. This symbolic object or behavior is then placed in
the environment so as to affect mental actions. (p.36)
In our design of a learning environment integrating new technologies in
mathematically relevant ways, we adhere to a socio-cultural perspective of learning and
analyze the interaction of the students in terms of mathematically-relevant discourse as
mediated by the various tools and supports available to them. The affordances of the
technological environment are cultural devices. Children can modify the environment to
make sense of the attributes of the geometric objects and configurations through
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investigation and interaction with each other. Vygotsky (1980) explains how activity
structures the social environment of interaction and the very behavioral routines of
members of that environment. We adhere to that position in our design and observe the
technological devices to not be the only mediating device in the learning environment but
the interaction between the children as meaning-making becomes a collaborative
enterprise. Both are forms of semiotic mediation and result from co-action (Moreno &
Hegedus, 2009) between the various participants. The children guide the discussion by
interacting with visuals on a screen, receiving visual and haptic feedback loops, which
are iteratively discussed and compared within the group and as such the technology
reciprocally guides the resulting investigation, decisions in how to further interact, and
conjectures or refutations from the resulting actions. Such embodied actions of pointing,
clicking, grabbing and dragging parts of the geometric construction also allows a
semiotic mediation (Falcade, Laborde & Mariotti, 2007; Kozulin, 1990; Mariotti, 2000;
Pea, 1993) between the object and the user who is trying to make sense of, or induce
some particular attribute of the diagram or prove some theorem.
Based upon this theoretical perspective, we present two different technologies
from our preliminary research and development at the Kaput Center. These have been
field tested in informal and formal learning settings. This preliminary work was
conducted with 4th graders in a high achieving elementary school in Massachusetts.
Omni Force-Feedback Device
In the Omni environment, we developed an exploration activity using solids and a
plane to explore how these objects interact—in particular, what different types of planar
intersections can be constructed. Our environment includes crisp visuals of these objects,
TME, vol10, nos.1&2, p .99
which can be navigated by dragging and moving the stylus on the Omni so that different
views of the objects could be explored. Through iterative design, we found that certain
colors and use of transparency helped the young learners focus their attention and
interpretation on the interaction and their reference to certain attributes. In addition, we
combined the haptic affordances of the Omni to add additional feedback to the
investigation. We found that magnetism was an important design principle to further aid
the learners to focus their attention and aid their discovery. In magnetizing the surfaces,
the children could lock onto the intersection of the two shapes and consider what they felt
in conjunction with what they saw. Two examples are shown in Figures 3a and 3b. The
first shows the planar intersection of a cube, which can result in a set of intersections
from a point (plane resting on a vertex), a line (plane on an edge), and 3-gon to 6-gons.
The second illustrates the planar intersection of a square based pyramid, which can result
in a similar set of intersections up to a 5-gon. Children, in groups of 4 with one device,
mainly explored a variety of triangles, quadrilaterals and pentagons. Such an activity is
challenging to undergraduates and the children had no prior experience with such an
investigation, but we discovered that their engagement in discovering various types of
intersection was immediate and endured for almost an hour. They did have prior
experience with 2D geometric shapes such as 3-gons to 5-gons but had only a basic
knowledge of the attributes of these shapes. For example, they did not classify 4-gons as
quadrilaterals but squares and rectangles. They did know how many sides each shape
should have which gave rise to interesting discussions as they explored what they saw
and how it contrasted with what they felt. In one investigation, the children thought they
saw a pentagon, but on tracing around the magnetized shapes they felt a 4-sided shape
Hegedus
(by counting edges) and concluded it was trapezoid through group discussion. This
illustrates a classic issue of cross-modality where our vision and touch can be in conflict.
The pseudo-3D representation on a flat screen is not sufficient, even with dynamic
interaction tools such as rotating and navigating the objects—more feedback is necessary
for young learners to make sense of certain specific mathematical attributes of the overall
geometric configuration. More work is needed in establishing activity structures that
help students make mathematical classifications of varying shapes. For example, can
force feedback help develop a sense of angle measure (acute, right, obtuse) in classifying
all types of triangles?
Figure 3a.
cube.
Planar intersection of a Figure 3b.
Planar intersection of a square-
based pyramid.
In collaboration with KCP Technologies, we developed a set of activities for use with
SketchExplorer for the iPad, a viewer application of the widely popular Geometer’s
Sketchpad® software. This application is available in the Apple Store. Activities were
constructed in Sketchpad and then transferred to the iPad through email or other forms of
file exchange. All activities are pre-configured for the children to use—as no construction
tools are presently available in this version for the iPad. Children directly interacted with
objects in the pre-configured activity including geometric objects (e.g., points), iterative
TME, vol10, nos.1&2, p .101
counters through flicking, or buttons that had been configured to perform a set of
operations (e.g., reflection of an image). Two examples are illustrated below. The first
(Figure 4a) allows students to make successive attempts at translating a pre-image onto
its pre-destined image (i.e., it has been fixed). They interact by moving the reflection line
and pressing the reflect button. This activity calls for two reflections to make one
translation. We found that all children in our preliminary field work in 4th grade
classrooms eventually discovered how to complete this activity through a variety of
methods, and develop an understanding of the relationship between reflections and
translations.
The second activity (Figure 4b) maximizes the affordance of multi-touch in a
mathematical way. Point 1 can be moved laterally and Point 2 vertically (they are
constrained to move along two perpendicular lines that have been hidden). The output of
these movements is a blob. This blob will simultaneously move in the directions of the
two input Points 1 and 2. The size of the blob can be changed by moving Point H along a
slider and the color can be changed by moving a point across the spectrum. In this
activity, we asked students to make the blob trace a circle. This was a rich mathematical
activity in that two inputs can make one output and many of the children in our
preliminary field work discovered this idea. More formally, the construction of a circle is
parameterized with two perpendicular actions. Again, this activity was extremely
engaging, especially when we added the time to establish a competition of who can make
the best circle in the least amount of time. Here, haptics is in the form of multi-touch and
can be done by one child (multiple fingers) or single-touch by multiple children. We
Hegeduus
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TME, vol10, nos.1&2, p .103
4. Future Design Principles
Technological affordances should become mathematical affordances and it is in
the mathematization of technological affordances that meaningful integration of new
multi-modal learning environments can be developed. We conclude with a set of design
principles that have evolved from our preliminary work, introduced in this paper, that
have the potential to profoundly affect teaching and student learning in the early grades.
Executable Representations
Mathematical objects and configurations should allow learners to dynamically
manipulate and execute operations on the representations in the learning environment.
Instead of dealing with static objects or computational outputs, representations that are
flexible allow young learners to adapt the configuration and test out their conjectures in
an iterative manner.
Co-action
The learner and learning environment should be collaborative. In dealing with
flexible and executable representations, the actions of the learner can guide the
environment (re-configure representations) and be guided by the resulting actions of the
learning environment.
Navigation
The integration of dynamic visuals with meaningful haptic feedback forms should
allow the learner to navigate the various attributes of the mathematical configuration and
construct meaning.
Hegedus
Manipulation and Interaction
Objects in such learning environments should be manipulable, and deformed into
a wide (if not infinite) set of similar objects, e.g., recall our triangle-area activity earlier,
in such a setup all triangles can be configured through direct manipulation.
Variance/Invariance
Understanding how quantities vary or not under certain interactions allows a large
wealth of mathematics to be explored. In addition to annotations such as measurement,
linking variation to force feedback allows meaningful feedback to help guide the learner
to make sense of important features, co-varying relationships or invariance.
Mathematically Meaningful Shape & Attributes
We naturally use touch to explore the composition of objects in nature as well as
varying attributes. In addition to shape, form and texture, haptic feedback can be linked
to attributes to aid the learner in their investigation.
Magnetism
A natural force is magnetism and this can be used to help learners focus on
particular features or relationships between geometric shapes and surfaces. Some objects,
or features of objects (where there is a particular mathematically-meaningful interest) can
be magnetized and all other attributes de-magnetized.
Pulse/Vibration
Pulse in the form of vibro-tactile feedback or oscillating devices (such as the
Omni) can similarly aid learners to focus their attention on certain parts of the activity, or
offer some form of numerical feedback. For example, the frequency and amplitude of the
pulse/vibration can be regulated to vary with some quantity.
TME, vol10, nos.1&2, p .105
Construction
Building on the affordances of dynamic geometry, allowing learners to use visual
and haptic tools to construct mathematical configurations can help learners to make sense
of what objects relate to each other (e.g., co-varying quantities) and communicate with
others their understanding or production of a mathematical model.
Aggregation
Learning environments often have the affordance of wireless connectivity.
Constructions, or evolving discoveries within the learning environment can be easily
shared across networks as part of larger models to be aggregated on another computer, or
to be contrasted with the work of other students working on the same project. Consider
transferring a haptic force with a visual across a network where others can “feel” what
you have felt.
We hope that these principles and our preliminary work provide ground-breaking
insights into effective generative activity design by future researchers and developers in
the future.
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Cognitive processes developed by students when solving mathematical
problems within technological environments
Fernando Barrera-Mora & Aarón Reyes-Rodríguez
Universidad Autónoma del Estado de Hidalgo
Abstract: In this paper we document and discuss how the use of digital technologies in
problem solving activities can help students to develop mathematical competences;
particularly, we analyze the characteristics of reasoning that students develop as a result
of using Cabri Geometry software in problem solving. We argue that the dynamical
nature of representations constructed with Cabri, and the availability of measure tools
integrated to it are important elements that enhance students’ ability to think
mathematically and foster the implementation of several heuristic strategies in problem
solving processes.
Keywords: Problem solving, digital technologies, mathematical thinking.
Introduction
Mathematical problem solving has been widely recognized as a framework to
analyze learning mathematical processes in which it plays dual relevant roles. On one
side, it guides performing research in mathematics education (Schoenfeld, 1985) and on
the other hand, it supports the development of curricular proposals (NCTM, 2000). In
learning approaches, based on problem solving, it is considered that students construct
mathematical knowledge by solving problems (Harel, 1994) in a community that fosters
development of an inquisitive attitude. Students' participation in a community of practice
has been recognized as a fundamental element of what constitutes mathematical thinking
(Schoenfeld, 1992; Santos-Trigo, 2010), since in this community they have opportunities
to reflect on their own thought processes through listening and reflecting upon ideas of
other members of it.
[In a community of inquiry] participants grow into and contribute to
continual reconstitution of the community through critical reflection;
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos.1&2, pp.109-136.
2013©The Author(s) & Dept. of Mathematical Sciences-The University of Montana
Barrera-Mora & Reyes-Rodríguez
inquiry is developed as one of the forms of practice within the community
and individual identity develops through reflective inquiry. (Jaworski,
2006, p. 202)
Problem solving is an activity involving conceptualization of the discipline “as a
set of dilemmas or problems that need to be explored and solved in terms of
mathematical resources and strategies” (Santos-Trigo, 2007, p. 523) and that promotes
students’ engagement in a variety of cognitive actions that can allow them to relate
diverse mathematical concepts, facts, procedures and forms of reasoning to construct
learning with understanding (Hiebert et al., 1997) through posing and pursuing relevant
questions.
In problem-solving learning approaches, students need to conceptualize the
construction of mathematical knowledge as an activity in which they have to actively
participate in order to identify and communicate ideas that emerge when they are
approaching mathematical situations (Moreno-Armella & Sriraman, 2005), as well as to
pose questions around problematic tasks that lead them to recognize relevant information
needed to give meaning to mathematical concepts. In this line of thinking, Santos-Trigo
(2010, p. 301) has stated that: “An overarching principle that permeates the entire
problem-solving process is that teachers and students should transform the problem
statement into a set of meaningful questions to be examined”.
Some classical approaches to problem solving have identified necessary steps for
solving problems (Polya, 1945), and central variables that influence students’ behaviors
and ways of reasoning. For instance, Schoenfeld (1985) considers four categories of
variables that are useful to characterize students’ mathematical performance: (i)
resources, (ii) heuristics, (iii) control and (iv) belief systems. However, since these
TME, vol10, nos.1&2, p .111
theoretical categories were developed based on experiences carried out in paper and
pencil environments, when using technological tools, those categories necessary have to
be reviewed since the use of technological tools offers students new opportunities to
discuss mathematical tasks from perspectives where visual and empirical approaches are
widely enhanced and by doing this, students can gain a deeper understanding of
mathematical concepts.
Technology based tools are now used on a daily basis in fields ranging
from the sciences to the arts and the humanities, as well as in professions
from agriculture to business and engineering […] And, these new
conceptual tools are more than simply new ways to carry out old
procedures; they are radically expanding the kind of problem solving and
decision-making situations that should be emphasized in instruction and
assessment. (Lesh & Doerr, 2003, p. 15)
Technological tools allow students experiment, observe mathematical relations,
formulate conjectures, construct proofs, and communicate results in ways that can
enhance and complement paper and pencil approaches, supporting mathematical learning
by offering opportunities to expand students’ capabilities to visualize, experiment, obtain
feedback, and consider the need to prove mathematical results (Arcavi & Hadas, 2000).
In order to examine the potential of using particular computational tools, in terms
of characteristics of reasoning developed by students when solving problems, and the
type of cognitive processes performed by learners as a result of the use of these tools, in
the international research agenda in mathematical problem solving it has been identified
some important questions that can shed light on our understanding about the effect of
using these tools in learning mathematics through problem solving, such as: To what
extent does the systematic use of technological tools help students to think
mathematically? Which aspects of mathematical thinking can be enhanced through the
Barrera-Mora & Reyes-Rodríguez
use of digital technologies in mathematical problem solving? What type of reasoning do
students develop as a result of using diverse computational tools in problem solving?
(Santos-Trigo, 2007). In this line of thinking, the aim of this paper is to identify and
analyze how the use of computational tools could help teachers, enrolled in a master
program in mathematics education, to propose problem solving strategies and give
arguments to justify and validate conjectures that emerge in the course of solving
optimization problems.
Digital technologies as cognitive reorganizers
According to Pea (1987), cognitive technologies are media that help us transcend
some limitations of mind such as capacity for storing and processing information based
only on biological memory. These cognitive technologies are characterized by
externalizing the intermediate thinking products, allowing us to operate, analyze and
reflect upon them. Furthermore, the representations that can be constructed with
technological artifacts are dynamical and manipulable. This dynamical character of
computational representations enable students to construct, for instance, families of
configurations, and to establish links among diverse representations, so that when a
representation is modified, the change is reflected immediately on the other
representations, allowing students to interact, operate or modify the representation and its
relations more directly than in a paper and pencil environment.
How does the systematic use of digital technologies impact cognitive structures?
Digital technologies can be considered as amplifiers or reorganizers of human cognition.
The term “amplify” means doing the same things that one could do without technology,
TME, vol10, nos.1&2, p .113
but performing it in a faster or a better way, without transforming qualitatively our
actions; for example, a calculator is an amplifier if it is used only to perform arithmetic
computations. On the other hand, “reorganize” means doing new things that one cannot
do without technology, or those that were not practical to do. A technological tool can be
considered as a reorganizer if it modifies cognitive processes and allows us to establish a
dialectical relationship among our actions, forms of thinking and tool’s functionalities,
which affect our modes of approaching the acquisition of knowledge.
The use of digital tools in learning activities, promotes that students pay attention
on the structural aspects of problem solving, by facilitating the performance of routine
procedures, opening the possibility of approaching problems which were difficult to
discuss within paper and pencil settings, and modifying the cognitive processes that they
develop to construct or to operate representations of mathematical objects. For instance,
to sketch the graph of a function within a paper and pencil environment, students could
proceed to explore and evaluate an algebraic expression, defining the function, for some
values of the variable, then those values need to be plotted in a coordinate system, and
finally students sketch the graph. However, graphing a function with a calculator or a
computational tool only requires introducing in the system the algebraic expression that
defines the function, and the software performs intermediate steps required to sketch the
graph. That is, computational tools simulate cognitive processes that formerly were
exclusive of human beings, attribute that Moreno-Armella and Sriraman (2005) have
called executability.
Although the use of computational tools offers students advantages to learn
mathematics, technological tools by themselves are not enough for constructing learning
Barrera-Mora & Reyes-Rodríguez
with understanding. Mathematical learning with understanding also requires developing
an appreciation to practice genuine mathematical inquiry, and disposition to construct
connections among diverse mathematical concepts, ideas and procedures. The ability to
construct connections is supported by the conceptual structure of the problem solver, a
term that we use to indicate how the problem solver’s resources are used to approach the
examination and solution of a problem; that is, the extent in what these resources can be
coordinated in order to articulate different concepts and results when students develop a
mathematical activity.
The use of technology to approach learning activities involves considering its
impact on the principles and concepts associated with the frame that guides research or
instructional processes. As Santos-Trigo and Barrera-Mora (2007, p. 84) have stated
“…any conceptual framework or perspective constantly needs to be examined, refined or
adjusted in terms of the development of the use of tools (particularly computational tools)
that influences directly the ways students learn the discipline”. Thus, it is important to
consider to what extent the systematic use of technology allows us to examine, test, refine
and expand some elements of mathematical thinking considered in problem solving
frameworks such as (i) students’ access to basic resources or knowledge, (ii)
implementation of problem solving strategies that involves ways to represent and analyze
the problems, (iii) the use of metacognitive strategies and, (iv) the construction of
justifications to validate conjectures and mathematical results.
[…] mathematical problem solving as a research and practice domain has
evolved along the development and availability of computational tools
and, as a result, research questions and instructional practices need to be
examined deeply in order to characterize principles and tenets that support
this domain. (Santos-Trigo, 2007, p. 524)
TME, vol10, nos.1&2, p .115
We argue that computational tools “incorporates a mathematical knowledge
accessible to the learner by its use” (Mariotti, 2000, p. 37), and by doing this, several
consequences arise. Among them, the use of technology allows that some resources
inherent to the tool could be incorporated to students’ resources when they solve
problems. For example, when students solve problems using computational tools, they
need a lesser amount of explicit mathematical resources to approach a task since students
can develop forms of reasoning based on visual and empirical approaches, enhanced by
the tools, and therefore their mathematical conceptual structure can be extended
incorporating to it some inherent tool’s characteristics.
Methodology
Six high school teachers (Jacob, Sophia, Daniel, Emily, Peter and Paul)
participated in three hours-weekly problem-solving and problem-posing sessions during
one semester. These teachers were enrolled in a master program in mathematics
education. They had some experience in using computational tools such as CabriGeometry and a hand-held calculator (Voyage 200). All teachers had completed a
Bachelor Science degree, majoring in mathematics or engineering, and they had teaching
experience ranging from one to five years.
During the semester there were twenty work sessions. The first two sessions were
employed to show teachers basic functionalities of Cabri-Geometry through the
construction of some common geometrical figures, and to illustrate the form to
implement several heuristic strategies such as: to consider that the problem has been
solved, relaxing problem conditions, add auxiliary elements to geometric configurations
Barrera-Mora & Reyes-Rodríguez
or to solve a simpler problem. The aim of these sessions was that teachers should
comprehend that a valid construction in Cabri geometry must be based in the properties
and relationships defining the geometrical figures, and that dynamic behavior of figures
is based on the hierarchy of construction procedure.
The core of the dynamics of a DGE figure, as it is realized by the dragging
function, consists of preserving its intrinsic logic, that is, the logic of its
construction. The elements of a figure are situated in a hierarchy of
properties; this hierarchy is defined by the construction procedure and
corresponds to a relationship of logical conditionality. (Drijvers, Kieran &
Mariotti, 2009, p. 119)
In the following three sessions, teachers discussed The Church View Task: A car
is driven on a straight roadway. Aside, there is an old church and the driver wants to stop
so that his friend (the passenger) can appreciate the facade of the church. At what
position of the roadway should the driver stop the car, so that his friend can have the best
view? In the process to solve this task, teachers used Cabri to construct a dynamic model
of the situation, and developed numerical and graphical strategies to quantify and
understand the relationship between the car’s position on the roadway, and the view of
the church’s facade. Besides, through exploration of relationships among elements of the
dynamic configuration, teachers transformed the original problem in an equivalent
geometrical problem: draw a tangent circle to line l (roadway) that passes through points
A and B (representing the church’s façade). They conjectured that tangency point of the
circle and line l is the place where the observer gets the best view of the church (SantosTrigo & Reyes-Rodríguez, 2011).
The analyzed tasks in this paper were developed within the sixth to eighth
sessions. During the sessions, the teachers were encouraged to use Cabri Geometry and a
TME, vol10, nos.1&2, p .117
hand held calculator to solve problems involving construction of dynamic configurations.
The teachers worked on solving problems that come from different contexts:
mathematical, hypothetical and real world (Barrera-Mora & Santos-Trigo, 2002). The
researchers documented how the tools helped teachers to propose strategies to solve the
problems and give arguments to justify and validate conjectures that emerged in the
course of the solution process.
The didactical approach employed during the sessions involved teachers working
in pairs and plenary discussions in which each pair of teachers communicated and
discussed their approaches and strategies employed to solve the problems. Two
researchers coordinated the sessions and participated as members of a community,
encouraging the development of an inquisitive approach to perform the tasks, and
promoting a collaborative work not only to solve the problems, but also to review and
reflect on mathematical content and ideas that emerged during problem-solving
processes.
The sessions were video recorded and recordings were transcribed. Each pair of
teachers handed in a report that included the software files. The transcripts and teachers’
reports constituted basic research data. The unit of analysis was the work shown during
the sessions by pairs of teachers, however sometimes attention was focused on the work
the entire community. The reduction of data was performed by identifying and selecting
some chunks of the transcripts or reports, which offered information about strategies
employed by teachers to solve the problems or forms of reasoning used to justify their
conjectures.
Barrera-Mora & Reyes-Rodríguez
The main tasks analyzed in this paper are three: (a) find the rectangle of
maximum area among all rectangles of given perimeter, (b) find the rectangle of
maximum perimeter among all rectangles of given area, (c) given a wire, split it into two
parts; and with one of the parts construct a square and with the other, construct a circle.
Where should you cut the wire so that the sum of the areas of the square and the circle
will be minimal?
First task
Peter and Paul constructed a dynamic configuration in Cabri to solve the first
problem. They drew a segment AB representing the given perimeter of the rectangle that
they wanted to construct. Then, they obtained midpoint M of segment AB, traced segment
MB and put a point C on segment MB. The teachers transferred measures MC and CB to
the horizontal and vertical axes, respectively, to construct a rectangle. After that, they
verified that the dynamic construction fulfilled the conditions of the problem (the
perimeter of the rectangle should be equal to the length of the segment AB) measuring the
length of the segments MC and CB, and comparing these lengths with the length of
rectangle’s sides (Figure 1). The aim of these actions was to verify that there were no
mistakes during the construction process, and to provide evidences that the dynamic
construction works properly.
Peter computed the rectangle’s area using Cabri tools, and dragged point C until
he obtained a numerical approximation of the maximum area, conjecturing that this one is
not attained when the rectangle is a square. His conjecture was based on numerical
results, since apparently the maximum area is reached when the measures of the
TME, vol10, nos.1&2, p .119
rectangle’s sides are 3.42 cm and 3.29 cm. In this phase of teachers’ activity, the tool
acted as a cognitive reorganizer since it enabled them to formulate conjectures based on
the relationship between visual and numerical representations mediated by dragging, as
well as to construct justifications supported and expressed via the software’s resources.
Figure 1. Dynamical model constructed in
Figure 2. Algebraic procedure developed
Cabri Geometry.
by Peter to obtain the problem solution.
Peter and Paul considered necessary to take an algebraic approach in order to
obtain the “exact” solution of the problem using calculus techniques. Peter and Paul
denoted by x and y the base and height of the rectangle, respectively. Then, they
represented algebraically the area A as a function of x, and differentiated this function to
obtain the critical points and the value that maximizes the area of the rectangle (Figure 2).
Based on this algebraic procedure, Peter and Paul were convinced that the maximum area
is attained when the rectangle is a square, and obtained evidence that their initial
conjectures was wrong. This conjecture was based on both, visual perception obtained by
manipulating the dynamic configuration, and prior problem solving experiences of Peter
with other optimization problems whose solution do not correspond to a square. For
instance, the following problem: find the rectangle of maximum area inscribed in a
semicircle (see bottom right corner from figure 2).
Barrera-Mora & Reyes-Rodríguez
Peter and Paul obtained additional certainty about the correctness of the solution
associating the geometric problem with a similar algebraic problem: maximize the
product of two numbers whose sum is given.
At first, I thought that the rectangle of maximum area would not be
square, since there is a classical problem of finding the maximum area of a
rectangle inscribed in a semicircle, and the square is not the figure with
maximum area. After we obtained the solution x = y algebraically, the
result of the problem became logical to me, because if the area is equal to
xy, you can prove that the product of two numbers, whose sum is given, is
maximum if both numbers are equal (Extracted from Peter and Paul’s
report corresponding to the sixth session).
Daniel and Emily drew the segment AB and its midpoint (C). After this, they
placed point E between points B and C, without considering that C should move on
segment BC. For this reason, point E can be dragged over the entire segment AB and not
only over BC. Teachers also drew point D, symmetric to point E respect to point C, but
this point was not used. Teachers transferred lengths EB and CE to the horizontal and
vertical axes, respectively, to draw a rectangle (Figure 3).
Daniel and Emily computed the rectangle’s area and transferred this value to the
vertical axis; then employed the “Locus” tool to construct a graphic representation of the
area function (Figure 3). The teachers were astonished to observe the graph (Figure 3),
since they expected that it was only a portion of a parabola. The graph behavior was due
to the way that Daniel and Emily developed the geometrical construction, since this
rectangle does not always meet the problem’s conditions. For some positions of point E,
rectangle’s perimeter is greater than the length of segment AB (Figure 3, right). It is
important to notice that Daniel and Emily, unlike Peter and Paul, did not verify that their
TME, vol10, nos.1&2, p .121
geometrical model satisfied the problem conditions, although the graph they visualized
on the screen did not correspond to what the teachers had anticipated.
Figure 3. Dynamical model elaborated by Daniel and Emily.
The problem solving behavior shown for this pair of teachers to approach the first
task is representative of the activity performed by them to approach all tasks. Daniel and
Emily had some difficulties to construct dynamic configurations that met the conditions
of problem statement. Additionally, in this task, they did not consider relevant to use the
resources offered by the software, such as measure tools to check the accuracy of their
dynamical construction. However, these teachers employed the graph of the function area
to conjecture that the maximum area is attained when the rectangle is a square, so the use
of the tool allowed teachers to formulate conjectures, which is an important element of
what constitutes mathematical thinking.
Other important feature of Emily and Daniel´s problem solving behavior was that
they showed difficulties to implement algebraic procedures to exploring solution routes,
although, plenary presentations allowed them consider the importance to develop this
type of strategies to prove or refute conjectures posed using the resources offered by
Cabri. In this context, the use of a dynamic software offered teachers opportunities to
Barrera-Mora & Reyes-Rodríguez
approach tasks with less amount of algebraic resources in relation to the requirements of
a paper and pencil setting.
Jacob and Sophia traced a segment AB, a point C on AB, and midpoint (M) of
segment AC. They transferred lengths of segments MC and CB to the vertical and
horizontal axes, respectively, and traced a rectangle based on these measures. Teachers
verified that rectangle’s sides had the same measures as segments CB and MC, however
they did not realize that rectangle’s perimeter is not the length of segment AB. The
mistakes made in the constructions process led them to formulate a wrong conjecture: the
base of the rectangle of maximum area must be twice its height (Figure 4).
Figure 4. Dynamic model elaborated by Sophia and Jacob.
Comments
The results of this task show that, in general, Cabri acted as a reorganizer, since it
allowed teachers to develop procedures to approach the task that could not be done in
paper and pencil environments, such as formulating conjectures based in the observation
of variation of numerical attributes of figures, as was the case of Peter and Paul's
approach; or the visualization of a relationship between two quantities obtained without
the previous formulation of an algebraic expression as in the approaches developed by
TME, vol10, nos.1&2, p .123
Daniel, Emily, Jacob and Sophia. That is, the teachers were able to access the resources
incorporated in the tool, specifically numerical and graphical resources available in
Cabri, to develop a particular form of thinking to approach the problem.
Concerning the justification process, Peter and Paul considered important to
verify empirically that the construction satisfied the conditions stated in the problem and
elaborated an algebraic proof of their conjecture. In this case, the use of measure tools
was a mean to establish the validity of their construction; and the algebraic proof was
employed to obtain an “exact” and not only an “approximated” solution. However, it can
be observed that not all teachers verified that geometric configurations were constructed
properly, neither all of them were aware of the importance to provide justifications using
the means offered by the tool or external to it. These results differ from other research
works that analyze the same problem. In those, it is concluded that the transition from a
geometric conjecture to an algebraic proof, emerges from a discrepancy between a
conjecture and the approximate results obtained with the tool, which suggested a different
result (Olive, 2000).
The plenary discussion supported Daniel, Emily, Jacob and Sophia to identify
pitfalls in their work and reflect about some important mathematical ideas such as the
domain of a function and the importance to provide justifications. Besides this, the
interaction among member of the learning community allowed Peter and Paul to
incorporate a visual approach to their repertoire of problem solving strategies.
Second task
Barrera-Mora & Reyes-Rodríguez
Peter and Paul selected a point A on the horizontal axis. The distance between
point A and the origin O of the coordinate system represents the length of a side of the
rectangle that teachers wanted to construct. Teachers used the “Numerical Edit” tool to
define a quantity that represents the area of the rectangle, and to calculate the length of
other side of it, they divided the length of segment AO by the area, and obtained the value
c. Then, they transferred c to the vertical axis and obtained point B. Then, teachers drew
rectangle OABC, calculated its area and dragged point A to verify that the area remain
constant. Next, teachers obtained the perimeter of rectangle OABC to construct a graph
relating a side of the rectangle and the corresponding rectangle´s area (Figure 5). In this
problem, Peter and Paul incorporated to their repertoire of strategies the graphical
approach discussed in the plenary session corresponding to the first task.
Figure 5. Perimeter of rectangle OABC, as a function of a length of side OA. (Graph,
elaborated by Peter and Paul)
Teachers conjectured that the graph of the perimeter, as a function of side OA,
consists of a branch of a hyperbola. Peter and Paul determined that although the locus
was split in two branches, it was enough to consider one of them. Teachers tried to test
their conjecture, first, by using the “Equation or Coordinates” tool, but the software did
TME, vol10, nos.1&2, p .125
not display the equation corresponding to the locus. Secondly, teachers selected five
points on one of the branches for tracing a conic that, visually overlapped the graph, and
by this mean they were convinced that the locus corresponded to a hyperbola. In the same
way as in the first task, the software acted as a reorganizer, since it allowed Peter and
Paul to develop graphical approaches to obtain evidence support their conjectures that are
difficult to implement in paper and pencil settings.
Peter and Paul also conjectured that minimum perimeter is reached when the
rectangle is a square. Teachers did not construct an algebraic proof of their conjectures;
they were convinced of their results based on the visual and numerical evidence provided
by the software. The problem solving behavior of these teachers differs from that shown
by them to solve the previous problem, in which they considered important to formulate
and solve the problem algebraically.
Sophia and Jacob approached this problem drawing a segment AB whose length
represents the rectangle´s area. Then, they put a point C on AB, and stated that the length
AC would represent one of rectangle’s sides. Sophia and Jacob transferred the measure of
AC to the horizontal axis to obtain a point X. Then, they obtained the length of the other
rectangle side computing the quotient AB/AC, and transferred this measure to the vertical
axis to obtain point Y, finally they drew the rectangle OXZY (Figure 6). Later, teachers
measured rectangle’s area to verify that this measure coincided with the length of
segment AB. In this action, it can be observed the effect of interaction in a learning
community, since in the previous problem; this pair of teachers does not considered the
use of measure tools to verify their construction was correct.
Barrera-Mora & Reyes-Rodríguez
Figure 6. Graph of the perimeter of rectangle OXZY, as a function of side OX, elaborated
by Sophia and Jacob.
Then, Sophia and Jacob constructed the graph that relates the perimeter of
rectangle OXZY and the length OX, and conjectured that the minimum perimeter is
reached when the rectangle is a square, based on dragging point C and the visualization
of perimeter function. They did not elaborate an algebraic justification of their conjecture.
Emily and Daniel had difficulties to build a rectangle of constant area in Cabri,
and they tried to solve a simpler problem with the aim of using this to solve the original
problem. They proposed constructing a triangle of constant area, and carried out the
construction fixing the triangle’s base, and putting the third triangle’s vertex on a parallel
line to the base of a triangle. Teachers verified, with the “Area” tool, that the triangle they
constructed satisfied the condition of having constant area, and conjectured that the
triangle of minimum perimeter is an isosceles triangle (Figure 7). In the process to
approaching this task it can be observed that Daniel and Emily incorporated the use of
measure tools to their repertoire of resources to verify accuracy of a dynamical
construction, strategy which was discussed during plenary discussion of the first task.
TME, vol10, nos.1&2, p .127
Figure 7. A family of triangles of constant area.
In the same line of thinking, Daniel and Emily considered relevant to provide
justifications. For instance, the teachers were able to justify that triangles they
constructed have constant area since the base is fixed and all triangles of the family have
the same height. Daniel and Emily also tried to use algebraic procedures to verify that the
triangle of minimum perimeter is an isosceles one, but they were unable to algebraically
formulate the problem, as can be observed in the figure 8. The analysis of the activity
developed by Daniel and Emily, allows us to obtain evidence that the use of Cabri
increases the number of problems that students, with a low ability to manage algebraic
procedures, can tackle.
Figure 8. Algebraic formulation for the minimum perimeter triangle problem. (Elaborated
by Daniel and Emily)
Barrera-Mora & Reyes-Rodríguez
Sophia and Jacob were interested in solving the previous problem, and they tried
to find, by algebraic means, the triangle of minimum perimeter given the conditions
stated by Emily and Daniel. The teachers formulated algebraically the problem (Figure 9,
left) and using calculus tools and a hand held calculator to perform algebraic operations,
they obtained the point that maximizes the perimeter of the triangle, and concluded that
the triangle of maximum perimeter is an isosceles triangle. (Figure 9, right).
Figure 9. Algebraic solution proposed by Sophia and Jacob.
It was observed that discussion developed into the community, influenced the
problem solving behavior of Sophia and Jacob, since these teachers incorporated the use
of algebraic procedures to their repertory of justifications. On the other hand, when
teachers solved this task, they used calculator Voyage 200 as an amplifier, since the tool
was only employed to perform computations such as the derivative of the perimeter
function
and to solve the equation
. However, the use of the calculator
allowed teachers to reflect about the results not encountered in paper and pencil settings.
When Sophia and Jacob solved the equation
, they obtained as a result
TME, vol10, nos.1&2, p .129
. Sophia commented that the expression
means that the
minimum perimeter is also attained if the base or height of the triangle is equal to zero,
but in this case the triangle dissapears.
Comments
This task allowed us to observe that Cabri transformed teachers’ forms of thinking
and reasoning. For instance, approaching tasks within a paper and pencil environment
leads to consider the meaning of a variable with restricted properties, basically based on
representing it with a symbol, say x. Meanwhile, using a dynamic software to approach
the task, allowed teachers to construct the idea of a variable, not only as a symbol, rather
as an amount that changes, as it can be observed when teachers dragged the point
representing the independent variable to approximate the value that produces the
minimum perimeter. That is, the use of Cabri, particularly the executability of
representations, gives rise to a different meaning of the concept of variable, since the tool
helps to perceive the idea of variation as the work of Peter, Paul, Jacob and Sophia has
shown. We argue that the exploration of ideas such as variation and co-variation, through
the use of a dynamic software, favors a reorganization of students’ cognitive processes,
since it helps them to give meaning to ideas and concepts involved in the solution of
optimization problems, such as the function concept. It is attained by means of
visualization and perceiving how one quantity changes when the other does.
Third task
Barrera-Mora & Reyes-Rodríguez
To approach this task Peter and Paul drew a segment AB that represents the length
of the wire. Then, they located a point P on AB. The lengths AP and PB were used to
construct the square and the circle, respectively (Figure 10). To construct the square Peter
and Paul divided the segment AB in four parts, the length of each of these parts is the
length of the square’s side. To construct the circumference, the teachers obtained the
radio using the calculator introducing the formula
, where perimeter is the
length of segment PB.
With the “Area” tool the teachers computed the area of each of the figures, added
them up and plotted the graph of area as a function of length AP. Based on visual
perception, teachers conjectured that the graph is a parabola and approximated visually
the value of segment AP that minimizes de sum of areas dragging point D.
Figure 10. Graph of sum of areas of a square and a circle as a function of a length AP.
To obtain the algebraic solution of the problem Peter drew on the board a segment
AB and a point P on the segment, in a similar way as he did in the software. He denoted
the length of segment AP as x, then he said that the length of segment PB is equal to
. Since AP is the perimeter of rectangle, then the area of this rectangle is equal to
TME, vol10, nos.1&2, p .131
. Moreover, the area of the circle can be computed as
(figure 11, left).
Then, Peter expressed the sum of areas as a function of x, and used the calculator to
obtain the derivative of the function and its critical points (Figure 11, right).
Figure 11. Algebraic formulation of the wire problem.
Peter expressed that in the dynamic configuration he approached the point which
minimizes the sum of the areas and compared it with the result obtained by substituting
the particular values into the algebraic solution.
The process employed by Daniel and Emily to solve the problem consisted in
drawing a segment AB to represent the wire, and put a point C on AB which is the point
where it is cut. Then, teachers constructed a square by considering as one of its sides the
segment AC, they traced midpoint (D) of segment CB, and drew a circle with center D
and radius DB. Daniel and Emily also computed the areas of the square and circle, and
computed their sum S. Finally, teachers constructed a graph relating length of segment
AC and the area S which is the sum of areas (Figure 12). The activity developed by the
Barrera-Mora & Reyes-Rodríguez
teachers showed that they did not understand the problem statement, since the perimeter
of the circle and square are not the lengths of segments AC and BC, respectively. Daniel
and Emily had difficulties to understand the problem, even after Peter and Paul showed
how they solved the problem; Daniel and Emily did not understand why the cable should
be divided into four equal parts to construct the square.
Figure 12. Dynamical configuration representing the wire problem, elaborated by Daniel
and Emily.
Comments
Approaching this task, using technology, required by the problem solvers to think
about the geometric objects in terms of actions, for instance, the actions to be consider to
construct a square given a segment, are different from those when paper and pencil
environment is used. The difference has to do with a “new quality” that the representation
of the objects have when using Cabri, the executability property.
In analogy with the previous tasks, the use of Cabri software allowed the problem
solvers to relate geometric and algebraic aspects of the problem as well as to coordinate
them into a wider conceptual network. For instance, it allowed them to assign the
variable, which represented the side of a square or the radius of a circle, a more concrete
TME, vol10, nos.1&2, p .133
meaning in terms of variation and not only its representations as a symbol. Besides, as in
the previous tasks, the idea of dependence between variables acquired a more robust
meaning in terms of the concept of a function. The plenary discussion allowed the
participants to reflect about the properties of a function concerning points where it
reaches maxima or minima as well as its domain.
Closing remarks
The use of digital tools allowed teachers not solely to remember facts or apply
algorithms, but most importantly, it helped them to formulate conjectures, and develop
visual schemas to provide justifications. Mainly, measuring attributes and dragging
elements in geometric constructions allowed teachers to formulate conjectures (Arzarello,
Olivero, Paola & Robutti, 2002) and observe relations among mathematical objects that
can be a departure point to develop a deeper mathematical understanding.
It was observed that the use of technology helped teachers to develop ways of
reasoning and forms of reflecting about the meaning and connections among
mathematical objects. For example, the dynamic software enabled teacher to search for
various forms of justifying a conjecture, in which the use and integration of visual,
empirical and deductive arguments were useful.
Based on the activities developed, we noticed that teachers founded their forms of
reasoning strongly on the visual representations, a result previously reported by Arcavi
(2003). The dynamism of representations helped teachers to think about variation of
particular instances and provided them with empirical basis to formulate conjectures. The
software provided feedback to the teachers (Arcavi & Hadas, 2000), but not all teachers
Barrera-Mora & Reyes-Rodríguez
were able to give meaning to this feedback, which was observed in the form that Sophia
and Jacob, and Emily and Daniel have solved the first task.
The analysis of the tasks has shown a way in which the conceptual structure of the
problem solver can be extended by incorporating the resources of the tool through the use
of it in the process of solving problems. This was explicit when teachers used the tool to
provide a visual representation of the information and by doing so, to approach a solution
of the problem, which used algebraic setting as well as visual ones. That is, the
capabilities of the tool as a cognitive reorganizer were based on the different possibilities
that the tool offers to establish connections and to act as an extension of the cognitive
structure of the teachers.
References
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TME, vol10, nos.1&2, p.137
Problem Solving and its elements in forming Proof
Joanna Mamona-Downs, Martin Downs
University of Patras, Greece
Abstract: The character of the mathematics education traditions on problem solving and
proof are compared, and aspects of problem solving that occur in the processes of
forming a proof, which are not well represented in the literature, are portrayed.
Keywords: heuristics; problem solving; proof
Introduction
Mathematics educators tend to compartmentalize the domains ‘problem solving’
and ‘proof and proving’. This detachment seems somehow artificial as both deal with
aspects of producing mathematical argumentation. However, problem solving tends to
emphasize the thought processes in furthering on-going work; in contrast the proof
tradition is concerned more in evaluating the soundness of the complete output. In this
paper, we shall respect the distinction made between problem solving and proof, but at
the same time we shall discuss issues that are common.
We use the words ‘culture’, ‘tradition’ and ‘agenda’ synonymously for general
views broadly adopted by the research community on any given educational perspective.
Both the problem-solving tradition and the proof tradition are diverse, so we restrict
ourselves to particular stances, mostly attending to the upper school and university level.
For problem solving, the subject is taken for it’s own sake; hence the full weight of selfconscious decision-making becomes the scope of investigation. For proof, we distinguish
the case where the practitioner possesses and implements the requisite mathematical tools
to fully articulate the proof from the case where he/she does not. The various types of
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos.1&2, pp.137-162
2013©The Author(s) & Dept of Mathematical Sciences-The University of Montana
Mamona-Downs & Downs
tools needed will be discussed, especially when the context lies in a mathematical theory
currently been taught: then tools are adapted and appropriated from techniques that the
theory avails. However, such tools have to be designed and then coordinated in the mind,
so within the processes in obtaining a proof it is evident that substantial elements of
problem solving must occur. The main focus of the paper is to give a preliminary
account of these elements.
In the next section, we shall present a short, rather personal, description of
problem solving. Largely supposing that the reader is familiar with the core principles
laid out by Polya, it discusses more practical issues like the role of the teacher,
implementation and assessment. The section that follows deals with the problem-solving
component in proof making. Here no attempt has been made to give a coherent picture of
the proof tradition; one reason is that proof and proving are, as an educational domain,
particularly prone to contrasting standpoints. Rather we limit our attention to those facets
of proof that differ from the problem-solving tradition but at the same time retain some
problem-solving elements. The choice of papers referred to is made with this in mind.
The discourse will not be unidirectional; some points made could be read as if the culture
of proof is supporting problem-solving activity. The extended example given in the
penultimate section illustrates this, as well as other matters. The epilogue, in part, raises
the question how well the problem-solving tradition (as it stands presently) is equipped to
cover the problem-solving elements in formulating proof.
On the problem-solving tradition and allied practical issues
The perspective of problem solving has a relatively compact core of ideas, mainly
centered on heuristics, meta-cognition including executive control, accessing and
TME, vol10, nos.1&2, p .139
applying the knowledge base, and identifying patterns of modes of thinking as students’
work progresses, following the pioneering work of Polya (e.g., Polya, 1973) and later by
Schoenfeld (e.g., Schoenfeld, 1985). However, problem solving, as a domain of
mathematical activity is very general; it concerns the student’s engagement on any
mathematical task that is not judged procedural or the student does not have an initial
overall idea how to proceed in solving the task. Many other perspectives taken by the
educational literature would embrace this same arena; for these the term problem solving
is likely to be invoked (after all it is a term that is quite natural to use generally), but it is
not a term around which the main focus revolves.
On the practical level, to deserve autonomous attention, problem solving must
have something to say about teaching and instruction. The function of problem solving
has been broadly characterized in three categories: teaching for problem solving, teaching
about problem solving, and teaching through problem solving (Schroeder and Lester,
1989). For the first category, tasks are chosen that force students to think more actively
about whatever mathematical topic that is being studied, the third is about building up
conceptualization via a program of deliberately sequenced tasks. For both, problem
solving is given a utilitarian role. On the other hand, for the second category, problem
solving is taken per-se as an integrated theme of discourse, and we will largely adopt this
perspective in this paper. The teaching must be directed to the student’s own awareness
of the influences that form his/her processes to reach a solution. The teacher then has to
teach not only mathematical content and method, but also general solving skills. Doing
this necessarily needs elements of intervention on the side of the teacher; he/she has to
induce habits of self-questioning and reflection that allow students to realize ideas critical
Mamona-Downs & Downs
in achieving a result. This means that there are aspects of teaching problem solving that
can be regarded authoritative (but not authoritarian), see e.g. B. Larvor (2010).
If a problem-solving approach is adopted in teaching, there are associated issues
about design and evaluation. What constitutes a ‘good’ problem? For this question, one
could simply say that any problem for which there is not immediately an obvious line of
attack is suitable. But do other factors come in? A sense of the attractiveness of the
solution is one, a sense of achievement the solver experiences is another. A measure of a
‘good’ problem is how far a solution, or an attempt to achieve a solution, would inspire
the solver to form related problems (Crespo & Sinclair, 2008). Another possible measure
might be the plurality of different directions that the problem can be treated so that
connections can be formed (Leikin & Levav-Waynberg, 2007), though problems that have
a particular ‘catch’ in the solution can also be useful because of the better control
afforded to the teacher/researcher. The evaluation of a student’s complete output, then, is
not straightforward; a model is given in Geiger & Galbraith (1998). Another factor is the
gap of experience between the setter and the solver; here lies the danger that either the
setter assumes that students have more experience than they really possess or the
experience of the setter leads him / her to expect an over-involved solution blinding a
more elementary path. Hence it is difficult to gauge how challenging a particular
problem is. Further if you credit an argument by its plausibility rather than its logical
security, you bring in a subjective factor. Such points of loss of control in terms of
evaluation makes problem solving, taken as an overall guiding principle in teaching, open
to criticism. For example, a recently published paper bears the rather provoking title:
“Teaching General Problem-Solving Skills Is Not a Substitute for, or a Viable Addition
TME, vol10, nos.1&2, p .141
to, Teaching Mathematics” (Sweller, Clark & Kirsher, 2010). The basis of the authors’
contention is well represented by their claim: “in over a half century, no systematic body
of evidence demonstrating the effectiveness of any general problem-solving strategies has
emerged”. The position taken by the paper might well seem to be extreme and partisan,
but it does reflect the difficulty in designing comprehensive, large-scale studies assessing
the success (or otherwise) of teaching about problem solving, not least on what exactly
should be measured.
The principles behind problem solving can be significant to the working of a
student of any age and of any ability, and there are numerous educational studies that
advance the cause of problem solving convincingly to populations ranging from preprimary school level to professional mathematicians. But making conscious decisions
about which heuristics to use as well as how other metacognitive dimensions should be
employed need mature deliberation. In danger of seeming elitist, we consider there are
two groups of students that are most able to cope with and profit from problem-solving
based instruction; the so-called mathematically ‘gifted’ student at school, and the
undergraduate student studying mathematics. (A third group might be teacher- students,
as they have to learn how to reflect on and attend to the difficulties of their future
students.) We are not saying that other students cannot gain from problem-solving
activities, but for them the gain could well be qualified. For instance, in Perrenet &
Taconis (2009), it is stated “(university mathematics) students show aspects of the
development of an individual problem-solving style. The students explain the shifts
mainly by the specific nature of the mathematics problems encountered at university
compared to secondary school mathematics problems”. Other papers offer models on
Mamona-Downs & Downs
how traits of thinking are different for the gifted and the expert over the ‘average’ solver
(e.g., Gorodetsky & Klavir, 2003).
What sources cater for these special groups? First, there is now a plethora of
‘problem solving’ texts on the market; these usually list many challenging problems with
exposition of some ‘model’ solutions. However, most are composed in the spirit ‘you
learn as you practice’, without much commentary on the educational level. Typically the
organization has some chapters based on general aspects of problem solving and others
on problem solving that is mathematically domain-specific. (It would be misleading to
identify such books as textbooks because the aim is not to cover a fixed curriculum of
mathematical content.) The style of presentation can be daunting, but some tomes are
particularly attractive and reader friendly. One, written by P. Zeitz (Zeitz, 2007, p. xi),
includes in its preface a list of principles guiding its writing that is surprisingly close to
the tenets held by educational research on problem solving:
Problem solving can be taught and can be learned.
Success at solving problems is crucially dependent on psychological
factors. Attributes like confidence, concentration and courage are
vitally important.
No-holds-bared investigation is at least as important as rigorous
argument.
The non-psychological aspects of problem solving are a mix of
strategic principles, more focused tactical approaches, and narrowly
defined technical tools.
Knowledge of folklore (for example, the pigeonhole principle or
Conway’s Checker problem) is as important as mastery of technical
tools.
Beyond problem-solving books, there is the collected ‘wisdom’ from the many
dedicated teachers involved in ‘training’ students for mathematics contests and special
examinations. There are now some regularly held conferences aimed not only to attract
such teachers but also researchers in education, such as one titled “Creativity in
TME, vol10, nos.1&2, p .143
Mathematics and the Education of Gifted Students”. The ensuing interaction between the
two interested communities, the one more theoretically inclined, the other more
practically minded, is valuable, and has enriched the educational literature published on
problem solving especially over the last decade or so. The facet of ‘training’ in particular
is interesting, for it does not at first seem to be quite consonant to the idea of flexible
thinking as espoused in the problem-solving tradition held by educators.
Another source is problem-solving courses offered in the curriculum of some
university mathematics departments. The content and ‘style’ of the delivery of a class
usually is a mixture of: introductory statements made by the instructor, students’
attempting to solve particular problems quite often conducted in small groups, students
criticizing peers work, a class discussion about how solutions were instigated and how
completed arguments functioned to realize the solution. The instructor perhaps in the end
winds up the session by relating the class activity to terminology found in the problemsolving culture. For such a free ranging course, an accompanying textbook is out of
place; rather a succession of class-plans by the individual teacher is followed ad lib. This
raises the issue of the demands put on a teacher when teaching a problem-solving course,
and whether they have to be trained to teach in a special way (see e.g. A. Karp, 2010).
Another awkward point concerning courses oriented towards problem solving is that the
level of interaction is high, so really are suitable only for classes of a relatively small size.
The yearly intake of students to a mathematics department can be in the hundreds, with
the result that a problem-solving course is usually selective and non-compulsory. Thus
the course effectively becomes a special interest class on a par to ‘standard’ courses that
present particular mathematical theory. Where then is the universal need for
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undergraduates to be instructed in problem solving? Indeed, it is reported in Yosof and
Tall, 1999, that students who took a problem-solving course mostly enjoyed the
experience but they found difficulties to apply what they learnt in other courses.
(At this point, we should clarify our position on the use of textbooks; as we
asserted above, textbooks perhaps do not have a place in teaching about problem solving,
but they certainly have their place in teaching for problem solving. The idea is that the
whole structure of the book consists of carefully sequenced problems leading up to major
theorems in a main field of mathematics. It replaces a ‘standard’ presentation of a topic
in the curriculum. An example is found in a book by Polya & Szego for Analysis first
published in 1924 (translated into English in 1978); more recently R. P. Burn has written
several textbooks (e.g., R. P. Burn, 1992) in the same kind of spirit. A natural question
arises: does a course based on such a textbook infuse general problem-solving
sensibilities?)
Much that we have discussed so far is addressed to practical matters; the focus for
the remaining part of the paper will be based on theoretical lines. There are many
expositions extant that have elaborated on the core ideas, i.e. heuristics, metacognition
and observing phase patterns during the solution process. Some have a local perspective,
some attempt to present overall models to refine the character of the whole field. For the
latter, (Carlson & Bloom, 2005) is a good example; the authors develop what they call a
‘Multidimensional Problem-Solving Framework’, which tabulates items along two axes:
activity phases against resources, heuristics, affect and monitoring. Within the
framework it is stressed that various aspects of cycling in types of activity occur in
problem-solving behavior. The paper clearly is in the fold of the problem-solving
TME, vol10, nos.1&2, p .145
tradition. But for many papers, this is not so clean-cut; in them there is substantial
material that seems to be in accord to the tenets of problem solving but the ostentatious
perspective lies elsewhere. In Mamona-Downs and Downs (2005), it was argued that if
we wished to form an ‘identity’ of problem solving we must examine how other
mathematics education topics impinge. One topic brought in was ‘proof’. The ‘terrain’
of proof obviously encompasses many reasoning processes that are common to problem
solving, so it is a natural candidate for comparison. In the next section we shall discuss
the confluences (and to some degree the disparities) between the domains; references are
made to papers that are ostensibly placed in the proof agenda but betray interesting
problem-solving traits.
The interface between problem solving and proving
Proof and proof production is associated with deductive reasoning. From the
perspective of problem solving, the notion of deductive reasoning can seem artificial;
employing deductive reasoning on its own cannot help students to build up the ideas
involved in the making of a strategy, it can only inform the student that any particular act
is ‘legal’ or the whole argument a-posteriori is logically sound. On the other hand
induction, i.e. obtaining evidence from considering cases that are not exhaustive, is useful
for explorative work but is insufficient to establish the desired result. Over the years
numerous models of reasoning have been put forward to fill the gap between inductive
and deductive argumentation. To mention a few, there is representational reasoning
(Simon, 1996), abductive reasoning due to Pierce (see Cifarelli, 1999, for a contemporary
summary), and plausible reasoning derived from Polya himself (Polya, 1954). Such
models differ in detail, but all deal in one way or another with a shift from a speculative
Mamona-Downs & Downs
mode of thinking to one that has an anticipatory character. Any type of mathematical
reasoning suggests an a-posteriori summation of the lines of thought taken before; the
question “what is your reasoning here” is a request to track back. Despite this, it still
concerns on-going working; an advance in reasoning depends on the problem-solving
decisions preceding it. In this respect some authors like to discriminate between
reasoning and argumentation; for example, Lithner (2001) takes argumentation as the
‘substantiation’ that convinces you that the “reasoning is appropriate”. Argumentation
then suggests a completion as well as a check that your reasoning is, in loose terms,
getting the job done; as such, argumentation should be closely related to proof, in
cognitive terms at least. It has always been contentious what a proof is; perhaps the
range of interpretation today is as wide as it has ever been. Here is not the place to give
even a skeleton sketch of the current views taken about the role and character of proof; a
comprehensive account from the mathematical education perspective is to be found in the
book Reid & Knipping (2010). In the last recommendation for ‘Directions for Research’
in this book, it is stated, “the relationship between argumentation and proof is far from
clear” (despite the numerous theoretical theses forwarded in this area). In explaining this
relationship it might be instructive to explore what different problem-solving skills we
would expect vis-à-vis argumentation and proof.
Is proof for every student and for every age? Leading up to answer this question
we regard ‘deductive’ or ‘formal’ proof as an ideal rarely adhered to. What, then, do
professional mathematicians tend to produce? We believe that this issue is nicely
expressed in a public lecture given by Hyman Bass (2009) where the image of a proof
providing certification is replaced by an image of a proof supporting a claim:
TME, vol10, nos.1&2, p .147
“ Proving a claim is, for a mathematician, an act of producing, for an
audience of peer experts, an argument to convince them that a proof of the
claim exists…the convinced listener feels empowered by the argument,
given sufficient time, incentive, and resources, to actually construct a
formal proof”.
p. 3
Hence a typical proof provided by a mathematician is an argument for which
there is a potentiality to convert it into formal proof (in principle at least). Bass then
considers the notion of generic proof (see also, e.g., Leron & Zavlasky, 2009) as a type of
proof that mathematicians often accept and adopt, and convincingly relates an incident
where a primary school child was able to express a generic proof (in joint work with D.
Ball). The child was able to explain the proposition that ‘the sum of two odds are even’
by mentally imagining two separate collections each with an odd number of objects,
pairing off objects that lie in the same collection as far as possible and pairing the two
objects ‘left over’ one from each collection. One might say the argumentation takes
place on the perceptional level and so cannot be regarded as a proof. On the other hand,
the reasoning is executed through properties that suggest both abstraction and
generalization are involved; from this criterion, perhaps it should qualify as a proof after
all. The obvious stance to take is to acknowledge different levels of proof. For example,
in the opening document for ICMI Study 19 (2009) on proof, the terms ‘developmental
proof’ and ‘disciplinary proof’ are introduced. A major factor in this distinction is that
students at school do not usually possess the requisite tools to allow them to articulate
disciplinary proof, whereas in the culture of advanced mathematical thinking (pertaining
mostly to tertiary level study) pains are taken to explicitly define the tools needed to
process proof in a particular field. For example, if the Intermediate Value Theorem of
Real Analysis is mentioned at school it is usually taken as a truism, but at university
either it has to be proved or explicitly recognized as a premise. Hence the learning of
Mamona-Downs & Downs
mathematics at the university level is ‘privileged’ in terms of proof making; in principle,
the tools are available, or the tools are at hand to ‘design’ further finely-honed tools for
your own specific purposes.
How does the above concern problem solving? As regard to ‘developmental
proof’, students have to rely on mental images and loosely grounded representations; the
processes in initializing, collating and monitoring the argumentation as it evolves are
very much in the field of problem solving. But because now mostly we are angling for an
‘informal’ justification of a general proposition, there is a tendency for properties to
determine objects rather than vice-versa. Here, the notion of characterization comes to
the fore; you ask which objects satisfy the conditions (rather than asking which properties
a given object satisfies). Even though there is no real difficulty in designing tasks asking
for a characterization in the problem-solving mold, this aspect is poorly represented as a
theme in the literature. What quite often occurs is that the solver identifies a class of
objects, C say, for which either all the objects that hold the given conditions are shown to
be in C or all the objects of C are shown to satisfy the conditions. Several rounds can be
made in restricting or expanding the class respectively, until analysis allows the removal
or inclusion of any remaining isolated exceptions. Such a program probably is best
illustrated in the literature by the framework of ‘example generation’, largely launched
through some work of Mason and his colleagues (e.g. Watson & Mason, 2005). Also
related is the Lakatosian notion of ‘heuristic refutation’, where counter examples are not
taken to disprove but in order to reformulate the premises (e.g. De Villiers, 2000).
For disciplinary proof, in principle the tools to prove are at hand. But an informal
discourse is usually kept to whilst engaged in the actual production of the proof, though
TME, vol10, nos.1&2, p .149
much technical terminology is retained. For the mathematician who holds well the topic
concerned, the technical terminology is not a barrier to understanding, to the contrary it is
empowering (Thurston, 1995). Whenever there is an interaction between an informal
language and one that is more documentarily directed, problem solving has its place;
strategy making is made in the informal language and ‘converted’ to the documentary
style. The problem-solving aspects so evoked would have restrictions compared to
general problem solving but they are nevertheless important. Below, particular angles of
this issue are mentioned. In the context of building up a mathematical theory, a metacognitive examination of a proof is required to understand what is important to retain in
the memory; the proof, the fact that the proof ascertains, both or none. For example, a
method can be extracted from a proof, which has a potential to be applied elsewhere; in
Hanna & Barbeau (2008), this phenomenon is discussed through various facets of
problem solving. Structuring mathematical work into semi-independent units acts not
only to produce a neat exposition but also represents essential ‘chunking’ of lines of
thought without which the mind could well be overstrained. The deliberate design of
modules in order to process the desideratum would seem to lie naturally in the realm of
executive control. (For an illustration, see the worked example appearing in the next
section.) For disciplinary proof the knowledge base typically is sophisticated and in flux;
one possible consequence of this is that the solver could be tempted to apply knowledge
that far exceeds the needs of the task/proof. An instance is given in Koichu (2010) from
an example-generating activity. Also, the detection of applicability of theoretical
knowledge is not often immediately apparent; in Mamona-Downs (2002) it is suggested
that part of the teaching of a mathematical theory should include what the author terms
Mamona-Downs & Downs
‘cues’, i.e., formats of knowledge that are not important theoretically but invite
realization of certain types of application. (An instance is to be found in the worked
example.) Changing track a little, we note that there is still a strand of informal discourse
in disciplinary proof, so one might expect that some students would like to exploit it more
than others. In this respect some research, mostly aimed at the tertiary level, has indeed
identified students that have a strong inclination to work consistently either semantically
or syntactically (e.g. Weber & Alcock, 2004). Such a marked preference must reflect the
character of the problem-solving tools with which a particular student feels most at ease.
Another but related issue is how to initiate a proof; Moore (1994) has noted that from
fairly elementary but formally defined systems many students cannot deduce the simplest
consequences, whereas Selden et al. (2000) talk about ‘tentative starts’. The first
suggests that students cannot interiorize the abstraction that confronts them, the second
suggests a more pro-active view that by ‘playing around’ with operations that they can
do, even ‘blindly’, students can ‘click’ to openings in the underlying (or accommodating)
structure yielded by the given situation. (Our worked task also features a tentative start.)
The notion of structure, even though being somehow vague, seems to be a natural
backdrop to combine the analytic tools given by formalization with problem-solving
input, see Mamona-Downs & Downs (2008).
We wind up this section by raising a few points for which the difference between
disciplinary and developmental proof is no longer relevant. First, whenever a shift of the
mode of thinking occurs, the character of the supportive components of problem solving
also changes. We have already discussed how an informal language reinforces
(disciplinary) proof. The making of a conjecture also marks a change in the mode of
TME, vol10, nos.1&2, p .151
thinking; one could compare the style of argumentation made before the conjecture to
that made after. If you adopt the more conventional propositional form of a proof
perhaps it would be more appropriate to compare pre-strategy work with work done in
effecting the strategy. Also, how does the reasoning allowing a first draft differ to the
reasoning leading to the final presentation of a proof? In particular, whilst obtaining a
proof sometimes recourse is made to diagrams and other kinds of representations that,
quite often, are not ultimately referred to in the presentation. Another differentiation in
mode of thought concerns what is processed in the mind against what is carried by
‘authorized’ symbolic usage. Even though these switches of thinking are documented in
the literature, they have not been thoroughly examined in the problem-solving tradition.
Second, we resume our discourse regarding the essential difference between proof
and problem solving. We center our problématique on discerning ‘problem solving’
tasks from ‘proof’ tasks. For a ‘problem solving’ task it is allowed to accept perceptual
indications, as long as they seem self-evident. Verification itself does not feature strongly
in the difference; rather a task requiring a proof differs in that the verification has to be
articulated in officially accepted mathematical language; one might say we require an
endorsement of the verification. Let’s have an example. The task is to identify the
different types of plane nets of a cube. It’s a problem that can be tackled by a bright
student of age eight, even though there are quite sophisticated things to do; first to
interpret from the task environment what is meant by a ‘type’, and after to validate the
answer by taking cases in an exhaustive manner. But the argument cannot be judged to
be a proof because each net is just recognized as being one; there is no explicit
mathematical warrant expressed that assures that when the squares are folded in the
Mamona-Downs & Downs
appropriate fashion they do indeed ‘form’ a cube. Furthermore there is no expectation
for the solver to undertake this ‘extra’ level of verification; thus what we have here is a
problem-solving task.
An interesting offshoot of the above concerns mathematical modeling. Suppose a
task is expressed in words that are not completely mathematical in form; it is modeled
into a self-contained mathematical system in which it can be treated as a proof.
However, the action of modeling certainly is subjective; so the task is better assigned as a
‘problem-solving’ task, but with a strong proof-making component. Also note that
informal argumentation carried out within the task environment sometimes can be
sustained right up to achieving a solution; if a more strict version is desired, modeling can
be made not only of the task but also of the line of the context-held reasoning. This can
be a valuable vehicle for students to realize the character of proof; proof in this regard is
a channel to provide the tools to fully articulate the output of a problem-solving activity,
see Mamona-Downs & Downs (2011).
An example
The example takes the form of an indicative solution path of a particular task; it
exemplifies some points made in the previous section.
The task is relatively complicated to solve, though no complex mathematics is
involved. A strategy is made without knowing which tools are needed to implement it.
Designing these tools requires anticipatory reflection but the form of them rests on simple
proofs, so we have a case where the proof culture contributes to problem solution
processes. The style of discourse indeed is made very much in a problem-solving vein,
but from it a presentation as a proof is readily extracted.
TME, vol10, nos.1&2, p .153
The task:
i
∞
Does the harmonic series
1
have a partial sum equal to an integer apart from
i=1
1?
Preliminary observations; sizing up the situation
From previous knowledge, we know the harmonic series ‘converges’ to , i.e. for
every nN, there is a partial sum that exceeds n. Hence there are ‘infinitely’ many
potential candidates for a partial sum to be an integer, so we cannot reduce the problem to
a finite number of cases.
The first partial sum equals 1; can we make a preliminary guess whether it is
likely for any other partial sum to be an integer? In response, take any nN\{1} and
consider the greatest rN for which
r
i=1
i
r
n-
1
must equal
i=1
1
< n. Then for n to be a partial sum necessarily:
i
1
.
r+1
This condition seems stringent because of the appearance of r in both terms, so
despite there being infinitely many candidates for n, it still would be a surprise to get a
solution apart from 1. But this does not really help us to get a start for a strategy that
would be conclusive.
Playing around; try out an action you can do
A partial sum is just a finite number of fractions added up; an action you can
perform is to render this into a single fraction, i.e.
Mamona-Downs & Downs
1i
r
i1
r! i
r
i1
r!
.
(1)
This move is made as a tentative start; it is speculative in the sense that there is a
hope (but not an expectation) that the new algebraic form might give us more handles to
attack the task. Notice that the new form is one natural number divided by another; the
question is reduced to analyzing whether the latter is a divisor of the prior.
Importing mathematical knowledge and its cue
The knowledge we import is the theorem that states any natural number greater
than one can be expressed (uniquely) by a product of prime powers. The cue relevant to
the task is if you think that n does not divide m, for m, n N, then there will be a prime p
such that the highest power of p dividing n is greater than the highest power of p dividing
m.
The strategy
Suppose that the solver believes that there is not a solution apart from 1. In
accordance he/she chooses the ‘simplest’ prime, i.e. 2, and tests whether the highest
power of 2 dividing r! is greater than the highest power of 2 dividing the nominator in
(1). (If this test fails, one might choose another prime to check, or change the method
from the experience gained by considering the case p=2).
Considerations how to implement the strategy
TME, vol10, nos.1&2, p .155
In order to analyze the highest power of 2 dividing
i
r
r!
i 1
you have to search for more elementary and general properties about the highest power of
2 dividing integers. For convenience, we introduce the rather eccentric (but standard)
symbolism: 2a || u where a, u N indicates that a is the highest power of 2 dividing u.
Designing the auxiliary tools
(i) Investigate: If 2a || u, 2b || v and 2c || (u+v), what is c in terms of a and b?
It is easy to demonstrate that
if ab, then c=min (a, b)
if a=b, then ca+b
but nothing further can be said in general for the second implication. Hence in the case
of ab we have more control than in the case of a=b.
(ii) In order to take advantage of the good control that occurs in the case of ab it
is useful to have this fact established:
If 2a || u and 2a || v (where u<v), then there is a natural number w such that w>u,
w<v and 2a+1 || w. An immediate corollary is:
Suppose that S:={1, 2, 3,...,r} and l : = Max{ l : kS s.t. 2 l || k}. Then there is
a unique element k of S for which 2l || k.
(2)
Implementation of the strategy
Here we show how the auxiliary tools facilitate the original situation:
Mamona-Downs & Downs
Let 2N || r! and 2 Ni || i whenever i ≤ r. Now the highest power of 2 dividing
r!
is equal to N-N i.
i
For k and l specified in (2), the highest power of 2 dividing
r!
is equal to N - l so
k
is less or equal to N - Ni; the equality holds only when i=k (second auxiliary tool).
Applying the first auxiliary tool (recursively) we know that
||
k
2
N-l
i=1
r!
.
i
Now Nk is simply an alternative symbol for l , so for i > k, N-Ni > N-Nk and we
can again apply the first auxiliary tool to obtain
r!
|| .
i=1 i
r
2
N-l
Now if r>1, l is a positive integer, so the highest power of 2 dividing the
nominator in (1) is less than that for the denominator. We are done.
Comments
1. The preliminary observations are made not only to ‘understand’ the
task, but include comments concerning a ‘feasibility test’ in order to
make an informed guess on what the most likely outcome would be.
Note that this guess is not made on experimental evidence.
2. The preliminary observations did not give a lead how to approach the
task. A blind algebraic operation is made, resulting in another
algebraic form and a new perspective. This action was not motivated,
so there is an element of luck here.
3. The new form is a quotient of natural numbers; the issue now is to try
to show that the denominator does not divide the nominator. (Guided
by our feeling that the solution space is empty). This issue then could
be resolved through an application of the fundamental theorem of
arithmetic. Would students notice this link? Suppose that this type of
application was taught, would they be more likely to catch the ‘cue’?
4. We now have a strategy; we take a particular prime p in the hope the
greatest power of p dividing the nominator is less than that of the
TME, vol10, nos.1&2, p .157
denominator. Things are still tentative; we don’t know as yet whether
the strategy is intractable or not, and we choose p=2 solely on the
grounds that 2 is probably the easiest prime to work with. Our
decisions here have as much to do with hopeful wishing as control.
However, somehow we judge the direction we have taken is
promising.
5. We have a strategy, but now we need a strategy to implement the
strategy. Modules of a more theoretical character are designed
deliberately directed to the implementation. These modules can stand
on their own merit outside of the context of the original task with
autonomous proofs. Designing such analytic tools vis-à-vis
envisaging how they would fit in the particular argumentation can be
challenging.
6. In the process of resolving the task, there are several places where the
solver is not completely controlling the effect of the decisions or
actions that have been made. Were we just lucky? No: luck comes in,
but from a certain stage there is an anticipation that things would work
out as envisaged. But this raises the question, how can we quantify the
grounds of this anticipation.
Epilogue
The main drive of this paper is to consider how elements of the problem-solving
tradition are evident in the formulation of a proof, and (to a lesser degree) vice-versa.
Also occasionally we have drawn out diverging tendencies between the two traditions.
One difference not as yet explicitly mentioned is that tasks in the domain of problem
solving are intended to be challenging, whereas for proof no such intention is imputed.
In this regard, it is now quite often to have a transition course aimed towards proof
practices in the curriculum of a mathematics department, especially in the United States;
the ‘content’ of the examples presented tend to be relatively elementary in order to
concentrate on the exposition of logically based argumentation. Such courses have a
completely different ‘feel’ to problem-solving courses. We have stated that there is some
doubt that taking a problem-solving course really will help the student when more
theoretical courses are studied, but a ‘proof’ course could also be problematic in other
Mamona-Downs & Downs
ways. For example, in Alcock (2010) it is reported that some mathematicians felt that
those students who can pass a proof course would be the ones that did not really need to
take the course anyway. Given this, a mix of the two kinds of courses might be the most
profitable. It would bring up, for example, situations for which a student can achieve a
result informally, and then can be challenged to articulate it with consummate reasoning
and grounds; also it would bring up situations for which the student cannot proceed
without building up constructions of a formal character. Proof should not be shown just
as an imposition, but as a channel that enhances our range of mathematical thought and
potentialities. The two kinds of situations mentioned above surely pertain to problem
solving as much as to proof, but bring out a tempered outlook towards the current
tradition of problem solving. To the mathematician, answering a typical problem-solving
task often is an enjoying and rewarding pursuit, but can seem frivolous if aspects resting
on perception are not tied down mathematically.
In the paper by Alcock referred to above, the author identifies four modes of
thinking whilst forming a proof (drawing on the comments of a small population of
mathematicians); instantiation, structural, creative and critical. These bear a striking
semblance to the set of phases in problem solving famously put forward by Polya, i.e.,
getting acquainted, working for better understanding, hunting for the helpful idea,
carrying out the plan, looking back (Polya, 1973, p.p. 33-36). The main discrepancy
between the two taxonomies is that ‘structural thinking’ is characterized by introducing
appropriate definitions and by working according to the rules of logic. This is consonant
to the notion of definitional tautness introduced in Mamona-Downs and Downs (2011)
that is not currently stressed in the problem-solving tradition. But the meta-cognitive
TME, vol10, nos.1&2, p .159
thought involved in forming definitions whilst partially envisaging how to control their
consequences to particular ends has problem-solving elements that should not be ignored;
then, perhaps, problem solving might be just as relevant in proving as it is in more
relaxed forms of argumentation. In this paper we have examined elements of problem
solving in the context of proof construction, admittedly in a rather eclectic way. We
suggest that further research in this direction should be undertaken in the future,
involving both researchers primarily ‘affiliated’ to proof and those primarily concerned
with problem solving.
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TME, vol10, nos.1&2, p.163
Becoming Aware Of Mathematical Gaps In New Curricular Materials: A
Resource-Based Analysis Of Teaching Practice
José Guzman1
Dept. of Mathematics Education, CINVESTAV-IPN, Mexico City
Carolyn Kieran2
Département de mathématiques, Université du Québec à Montréal
Abstract: The study featured in this article, with its central focus on resources-in-use,
draws upon salient aspects of the documentational approach of didactics. It includes an a
priori analysis of the curricular resources being used by a teacher for the first time,
followed by detailed in situ observations of the unfolding of her teaching practice
involving these resources. The central mathematical problem of the lesson being analyzed
deals with families of polynomial functions. The analysis highlights the teacher’s
growing awareness of the mathematical gaps in the resources she is using, which we
conjecture to be a first step for her in the evolutionary transformation of resource to
document, as well as an essential constituent of her ongoing professional development.
Keywords: documentational approach of didactics, documentational genesis, curricular
resources in mathematics, families of polynomial functions, mathematical gaps in
resources, ongoing professional development, resources-in-use, research on teaching
practice with new curricular resources.
Introduction
Mathematical problems suitable for use in high school classrooms can be obtained
from a variety of resources, including the internet, newspapers and books, colleagues, and
of course textbooks. There is general consensus that most mathematics teachers rely on
textbooks for their day-to-day fare of problem-solving items for students (Schmidt,
2011). Over time, these problems and the ways in which they are presented to students
get tinkered with and gradually become refined (Gueudet & Trouche, 2010, 2011).
However, we are only now beginning to learn a little about the ways in which teachers
1
[email protected]
2
[email protected]
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos.1&2, pp.163-190
2013©The Author(s) & Dept of Mathematical Sciences-The University of Montana
Guzman & Kieran
interact with the mathematical resources available to them (Gueudet, Pepin, & Trouche,
2011). Chevallard and Cirade (2010) have raised an additional issue, that of the lack of
adequate mathematical resources for teachers when the school program is changed and
new problems and problem-solving approaches are introduced. Moreover, as pointed out
by Artigue and Houdemont (2007), many teachers who teach mathematics – even at the
level of secondary school – are not mathematics specialists and “are quite often not
proficient in mathematics, and that the mathematics and didactic formation they receive
during their training does not compensate for these limitations” (p. 376). Although a
focus on the mathematical resources available to teachers, their supportive role, and their
adaptation and adoption is not one that, up to now, has been central to the research
agenda of the problem-solving research community, its importance can be argued for, at
the very least, on pragmatic grounds: The ways in which resources support (or do not
support) teachers in their problem-solving efforts in class clearly impact upon the
problem-solving experience of students.
According to Remillard (2005) who conducted a seminal review of teachers’ use
of curricular materials, the process by which mathematics teachers appropriate and
transform such resources, as well as the support that these resources offer, is rather
unexplored terrain. In 2000, Adler similarly proposed that “mathematics teacher
education needs to focus more attention on resources, on what they are and how they
work as an extension of the teacher in school mathematics practice” (Adler, 2000, p.
205). In one such study of teachers using reform-based curricular materials, Manouchehri
and Goodman (1998) reported what they viewed as shortcomings in the guidance for
teachers provided by the curricula, saying that the curricula “did not provide the teachers
TME, vol10, nos.1&2, p .165
with detailed methods of how to address the content development” (p. 36). Teaching with
new resources can thus lead to situations where teachers are not suitably prepared, but
which can provide the impetus for new awarenesses of both a mathematical and
didactical nature. In this regard, Gilbert (1994) has said: “reflection-in-action occurs
when new situations arise in which a practitioner’s existing stock of knowledge – their
knowledge-in-action – is not appropriate for the situation” (p. 516). This reflection-inaction, which involves critical examination and reformulation of one’s existing knowings,
is intimately connected to, and synergistic with, one’s evolving appropriation and
transformation of resources, according to the documentational approach of didactics
(Gueudet & Trouche, 2009, 2011).
The Documentational Approach of Didactics
Gueudet and Trouche (2009, 2011) have developed a theoretical research
framework based on the premise that documentation work is at the core of teachers’
professional activity and professional change. Documentation work includes selecting
resources, combining them, using them, and revising them. Even outside a particular
reform or professional development program context, such work is deemed central to
teaching activity. Gueudet and Trouche employ the term “resource” to describe the
variety of artifacts that they consider – such as a textbook, a piece of software, a student’s
work sheet, a discussion with a colleague. Like Adler (2000), a key aspect of Gueudet
and Trouche’s (2011) approach is resource-in-use (in-class and out-of-class).
One of the pivotal constructs of their theory is that ‘resources’ become
transformed into ‘documents’ via a process of documentational genesis – a construct
Guzman & Kieran
inspired by and adapted from the parallel process in the instrumental approach whereby
artifacts become transformed into instruments via instrumental genesis (Rabardel, 1995).
The instrumental approach distinguishes between an artifact, available for a given user,
and an instrument, which is developed by the user – starting from this artifact – in the
course of his/her situated action. Similarly, a document is developed by a teacher, starting
from a resource, in the course of his/her situated action. Gueudet and Trouche represent
this process of documentational genesis with the following simplified equation, where the
‘scheme of utilisation’ refers to the various personal adaptations that are made with
respect to using the resource in accordance with a teacher’s evolving knowledge and
beliefs (Gueudet & Trouche, 2009, p. 209): “Document = Resources + Scheme of
utilization”. Documentational genesis is therefore considered to be a dialectical process
involving both the teacher’s shaping of the resource and her practice being shaped by it.
In their description of this theoretical approach and its accompanying
methodological principles, Gueudet and Trouche (2011) emphasize the professional
growth that is intertwined with documentational genesis. They argue that:
Teachers “learn” when choosing, transforming resources, implementing
them, revising them etc. The documentational approach proposes a
specific conceptualisation of this learning, in terms of genesis. A
documentational genesis induces evolutions of the teacher’s schemes,
which means both evolutions of the rules of action (belonging to her
practice) and of her operational invariants (belonging to knowledge and
beliefs). Documentation being present in all aspects of the teacher’s work,
it yields a perspective on teachers’ professional growth as a complex set of
documentational geneses. (Gueudet & Trouche, 2011, p. 26)
The study featured in this article, with its central focus on resources-in-use within
actual teaching practice, draws upon salient aspects of the documentational approach of
didactics. More specifically, our research question centered on uncovering key moments
TME, vol10, nos.1&2, p .167
of teacher awareness, particularly those of a mathematical nature, in the process of using
new curricular resources in class. We begin with an a priori analysis of the curricular
resources being used by a teacher for the first time, followed by detailed in situ
observations of the unfolding of her teaching practice involving these resources. The
analysis highlights the teacher’s growing awareness of the mathematical gaps in the
resources she is using – conjectured to be a first step for her in the evolutionary
transformation of resource to document, as well as an essential constituent of her ongoing
professional development.
Methodological Aspects of the Study
The present study is situated within a multi-phase program of research whose
current phase is the study of teaching practice in mathematics classes involving the use of
digital technology in the teaching of algebra, in particular, the use of Computer Algebra
System (CAS) technology. Previous phases of the research integrated tasks that had been
designed by members of the research team (see, e.g., Kieran, Tanguay, & Solares, 2011).
This phase examines teaching practice in technology-supported classroom environments
where commercially-developed curricular resources, such as textbooks, are in use.
Participants in this phase of the study included three teachers from three different
public high schools. They responded positively to our request for volunteers who were
using technology in their regular teaching of high school algebra and who would be
willing to be observed and interviewed for our research study. We observed and
videotaped each teacher’s practice for five consecutive days in all of their regular
mathematics classes. We intended to capture, as much as would be possible under the
videotaping circumstances, their natural teaching practice involving whatever resources
Guzman & Kieran
they had chosen to make use of. We also interviewed each teacher twice – once at the
beginning of the week and once at the end. The analysis presented in this article focuses
on the practice of one of the three teachers, Mae (a pseudonym), during one of her
lessons of the week.
Mae taught all three of the senior year (17-year-old students) mathematics classes
in her school. She was one of the pioneers of her school on the use of technology in the
teaching of mathematics. In her own classes, she regularly used a whiteboard hooked up
to her computer and all students had CAS calculators available to them. She was
technically very savvy and could respond easily to all students’ questions regarding the
use of technology. Her academic background included a doctorate in education with a
thesis on the use of graphing calculator technology. Her mathematical knowledge
seemed, however, less developed than her technological skill. She made a regular
practice of asking students to read ahead in the text because – as she mentioned during an
interview – they would soon be graduating and had to learn to be autonomous adults who
were responsible for their own learning. However, this practice also led students to pose
questions of a mathematical nature that went beyond what they had been able to extract
from their textbook. Such questions were not, in general, handled with the same expertise
and knowledge base with which Mae handled their technological questions.
The analysis of Mae’s teaching practice that we present in this article does not
focus on her integration of technology into the teaching of mathematics, but rather on the
mathematical content at stake in her lesson within the framework of the documentational
approach of didactics (Gueudet & Trouche, 2009), a key construct of which is the
evolutionary nature of documentational genesis whereby resources gradually become
TME, vol10, nos.1&2, p .169
transformed into documents. The resources that Mae was using during the period in
which our classroom observations occurred were new to her that year. The provincial
curriculum guidelines had changed the year before and new textbooks were developed
that would adhere more closely to the new guidelines. Mae tended to rely on both the
student textbook and accompanying teacher guide to plan the mathematical content of her
lessons. We were interested in following the process of her integration of these resources
into her teaching practice, the way in which she was adapting and transforming them, and
the way in which they might be co-transforming her practice and her knowledge – that is,
in capturing the reciprocal nature of the documentational genesis that was occurring.
Although the analysis we present in this article is focused on a very small part of
Mae’s teaching practice, on one lesson in fact, the approach to our analysis is broader in
scope. We begin with an analysis of the two text-based resources she used for her lesson
on families of polynomial functions, tracing back in these resources to some of the earlier
notions that served as foundation for the development of the lesson’s mathematical
content. Then we analyze the videotape of the unfolding of the classroom lesson. This
latter analysis attempts to draw out the dynamics and forces that came into play as the
prepared mathematical content was elaborated in the classroom setting, examining in
particular those moments that seemed critical to the further development of her teaching
practice and to the evolutionary process whereby a resource becomes a document. The
videotapes of the interviews with the teacher also serve to illuminate some of the
underlying aspects of her teaching practice.
Analysis of the Resources Used by the Teacher in Preparing her Lesson
Guzm
man & Kieraan
Herein wee present pertinent extraacts from thee resources uused by the tteacher, Mae,
in prreparing herr lesson, ass well as some of ourr own mathhematical annd didacticaal
comm
mentary relaated to thesse extracts. The two reesources shee used weree the studennt
textbook Advancced Function
ns 12 (Erdman, Lenjosekk, Meisel, & Speijer, 20008a) and thhe
accom
mpanying teacher guide Advanced Functions
F
122, Teacher’s Resource (E
Erdman et all.,
2008b
b). The lesson was on Families
F
of Polynomial
P
Functions (S
Section 2.4 of Erdman eet
al., 2008a).
2
Our analysis of the resourcees used by tthe teacher focuses prim
marily on thhe
issue of the math
hematical lin
nks between
n factors wriitten in the fform (x – b//a) versus thhe
form (ax – b) for given families of polyno
omial functiions.
The Student
S
Tex
xtbook
Backgrou
und mathem
matical mateerial from Section 2.22. In Sectioon 2.2 of thhe
textbook (Erdmaan et al., 2008a), studen
nts are preseented with tthe Factor T
Theorem (seee
Figurre 1). This section
s
of th
he textbook provides
p
som
me of the suupport for thhe theoreticaal
affirm
mations regaarding the deesired form for factors oof a polynom
mial that aree made in thhe
later section on Families
F
of Polynomial
P
Functions
F
wiith respect too rational rooots.
m (drawn fro
om p. 95 of E
Erdman et all., 2008a)
Figurre 1. The Facctor Theorem
This theo
orem allows for determiining whethher a certainn binomial iis or is not a
factorr of a given polynomial on the basiss of numericcal evaluatioon. The textbbook does noot
provee this theoreem; it merely
y provides th
he followingg affirmationn, which alloows for som
me
misin
nterpretation
n: “With th
he factor th
heorem, youu can deterrmine the factors of a
TME, vol10, nos.1&2, p .171
polynomial without having to divide” (p. 95). No explanation is provided as to why the
numerical evaluation Pb / a , when it yields zero, should in fact be sufficient for
determining a factor of the polynomial. However, the central issue for our analysis is the
following: if Pb / a = 0, why write the factor in the form (ax – b) and not in the form of
(x - b/a)? It clearly makes for an easier long-division calculation when written in the form
of (ax – b). But what happens, mathematically speaking, when one expresses (x – b/a) as
(ax – b)? Are the two forms equivalent? What mathematics is hidden in expressing the
former form as the latter? How does one convert one form to the other and maintain
equivalence?
Subsequent pages of the student textbook expand on the Factor Theorem by
means of two additional theorems, the Integral Zero Theorem (p. 97) and the Rational
Zero Theorem (p. 100), illustrated in Figure 2. However, once again, no further
explanation is provided for the case of the polynomial P(x) having a rational zero a/b,
either as to why a should be a factor of the leading coefficient of P(x) or the issue
regarding the form to be used for the factor of P(x) corresponding to the rational zero.
Figure 2. The Integral Zero and Rational Zero Theorems
(drawn from Erdman et al., 2008a, pp. 97 & 100)
The textbook provides several examples that show the advantages of using these
two latter theorems when the task is to find the factors of a polynomial. However, the
relevance of writing the factor in the form (ax – b) when x = b/a is a root of the
Guzman & Kieran
polynomial P(x) is never discussed. This can have repercussions, didactically speaking, at
the moment when the teacher introduces the theory underlying families of polynomial
functions, coming up in Section 2.4. The intervening section 2.3, on the solving of
polynomial equations, adds no further theory related to the Factor Theorem.
Families of Polynomial Functions. Before giving a general definition of families
of polynomial functions, the textbook offers several examples that illustrate that one
obtains different members of the same family of polynomial functions for different values
of the parameter k (see Figure 3 for one such example).
Figure 3. Algebraic representation of a family of polynomial functions (drawn from
Erdman et al., 2008a, p. 115)
As is illustrated in Figure 3, the family of polynomial functions that has as zeros 2
and –3 can be represented algebraically as y k ( x 2)( x 3) . But, if we look at part (b)
of the solution of this example, the information that is given suggests that different values
of k yield different members of the same family of polynomial functions. This can lead
those who are using this textbook as a resource to a false mathematical conception if they
TME, vol10, nos.1&2, p .173
do not distinguish the crucial role being played by the root of the polynomial in terms of
whether it is a whole number or a rational. In other words, if the zeros of the polynomial
are not whole numbers, but rather rational numbers, then the value of k can vary
according to the form of the factor, without changing the member of the polynomial
family. For example, if the zeros of a family of polynomial functions are 3 and –1/2, then
the family has as its function P( x) k ( x 3)( x 1 / 2) . And so, a member of this family
is: P( x) 2( x 3)( x 1 / 2) , if k = 2. But if we write the factor ( x 1 / 2) as (2x + 1), the
value of k changes from 2 to 1 for the same member of the polynomial family, that is,
P ( x) 1( x 3)(2 x 1) . The algebraic transformation involved in changing the form of
the factor ( x 1 / 2) to (2x + 1) is as follows: ( x 1 / 2) = 2/2 ( x 1 / 2) = 1 / 2 (2x + 1).
Thus, the conversion of ( x 1 / 2) to (2x + 1) involves also multiplying the rest of the
expression by 1 / 2 , thereby yielding the new k-value of 1 (from multiplying the previous
k-value of 2 by 1 / 2 ). This example shows that, if we have a family of polynomial
functions expressed algebraically as P( x) k ( x a1 )( x a2 )...( x an ), k , k 0, we
cannot say that different values of the parameter k necessarily imply different members of
a given family of polynomial functions, unless of course all the zeros are whole numbers.
The examples provided in the textbook are then followed by the general definition
of families of polynomial functions (see Figure 4).
Guzm
man & Kieraan
Figurre 4. Definitiion of familiies of polyno
omial functioons
(draw
wn from Erdm
man et al., 2008a,
2
p. 118
8)
With thiss general deefinition, wh
here there aare no condditions on thhe zeros of a
nomial functtion, the earllier suggestiion to write the factor as ( ax b) w
when the zerro
polyn
is rational receivees no consisstent theoretiical support.. In fact, the general deffinition woulld
m to suggest that the faactored form
m for a giveen zero a, bbe it a rational or wholle
seem
numb
ber, is of the form ( x a) .
The Teacher
T
Gu
uide
We now examine th
he nature off the supporrt offered inn the teacheer guide witth
respeect to preparring lessons on families of polynom
mial functions (Erdman eet al., 2008bb).
This resource presents
p
only
y a few teeaching sugggestions, m
most of whiich could bbe
consiidered, at thee very least, quite incom
mplete from a didactical ppoint of view
w (see Figurre
5).
on found in the teacher guide from
m the sectionn dealing witth
Figurre 5. A typiccal suggestio
famillies of polyn
nomial functiions (drawn from Erdmaan et al., 20008b, p. 52)
TME, vol10, nos.1&2, p .175
In fact, the guidance noted in Figure 5 where students are to be encouraged to use
fractions and not decimals is contradicted in another suggestion within the same resource
a few lines later (see Figure 6).
Figure 6. An explicit suggestion in the teacher guide (Erdman et al., 2008b, p. 52) to
write all factors involving rational zeros in the form (ax b)
The advice displayed in Figure 6 is not accompanied by any justification for the
use of the form (ax – b), nor is there any discussion as to how a teacher might respond to
potential students’ questions regarding the issue as to why they are to use the form
(ax b) and not (x – b/a). In fact, the teacher is not even alerted to the possibility that
such a question might arise. Additionally, no explanation is provided as to why “all
equations should be expanded and simplified.” Question 10, to which the suggestion
given in Figure 6 refers, reads as follows: Determine an equation for the family of quartic
functions with zeros –5/2, –1, 7/2, and 3. In accordance with the directive given in Figure
6, the equation for the given family of quartic functions ought to be written as
P( x) k (2 x 5)( x 1)(2 x 7)( x 3). But an obvious question is why one might not
instead write the function in the following form: P( x) k x 5 / 2( x 1) x 7 / 2( x 3).
Analysis of the Unfolding of Mae’s Lesson on Families of Polynomial Functions
The mathematical problem on which Mae had decided to focus in her lesson on
families of polynomial functions was one that involved a rational root. It was a variation
of an example that was worked out in the student textbook (see Figure 7).
Guzm
man & Kieraan
Figurre 7. The mathematicaal problem that was ussed as a baasis for Maee’s lesson oon
famillies of polyn
nomial functiions (drawn from Erdmaan et al., 20008a, pp. 117--118)
Mae had taught the same
s
lesson in two otheer Grade 122 classes earrlier the sam
me
day. We note th
hat during eaach class, in
ncluding thee third, she had emphaasized that aall
memb
bers of a faamily of po
olynomial fu
unctions shaare the sam
me x-interceppts. The onlly
featurres that diffe
fered for each member of
o a given faamily, she saaid, were thee value of thhe
leadin
ng coefficien
nt k and thee accompany
ying stretchinng or comprressing of thhe graph. Shhe
had also
a drawn students’
s
atteention to thee fact that thhere would bbe only one correct valuue
of thee parameter k for any given
g
membeer of a familly of polynoomial functions. We now
w
TME
E, vol10, nos.1&2, p .1777
recou
unt the unfollding of the lesson as it occurred in the third claass of the daay. The readeer
will notice
n
how closely
c
Mae follows thee textbook prresentation oof the probleem she chosse
to usee in her math
hematics lesson.
The lesso
on on Familiies of Polyn
nomial Functtions began with Mae ppresenting thhe
follow
wing three functions – all of them
m being mem
mbers of onne family oof polynomiaal
functtions:
f1 ( x) 2 x 3 4 x 2 10 x 12
f 2 ( x) x 3 2 x 2 5 x 6
f3 ( x) 2.5 x3 5 x 2 12.5 x 15 .
Mae displlayed the deffinitions of the
t functionss on the whiiteboard at thhe back of thhe
room
m [see Figuree 8] and ask
ked students to copy theem down annd then to grraph them oon
their CAS calculaators: “Open
n up a graphs page on yoour calculatoor. Given f1 , f 2 , and f 3 ,
what do you noticce about all three functio
ons?”
Figurre 8. The opeening of the lesson, with
h its accompaanying CAS
S
and whiteboard
w
teechnology
Guzman & Kieran
After students had spent some time trying to find appropriate graphing windows,
Mae asked them what common characteristics the functions shared. One student
mentioned that they were all of degree three and another that they had the same
x-intercepts. Following up on the latter idea, Mae asked if they were able to tell from
looking at the given expanded forms that the three functions had the same x-intercepts.
“So how could you make it more obvious?”, she asked. When one student suggested
“factoring them”, Mae responded: “Yes, when you factor them, you have your function in
a form where you can easily see that the x-intercepts are similar.” She then asked students
to split their graphs page in two so that they could insert a calculator page for the
factoring of the three functions. It is noted that a certain amount of time was devoted to
taking care of technical aspects of the CAS, such as splitting a page in two and then
copying the three functions to that page.
The factored form of the three functions was as follows:
f1 ( x) 2( x 2)( x 1)( x 3)
f 2 ( x) ( x 2)( x 1)( x 3)
f 3 ( x) 2.5( x 2)( x 1)( x 3)
Mae then continued with her lesson, as illustrated by the following extract of
classroom dialogue. It was soon to lead to the problem associated with a factor that
corresponds to a given rational zero.
Teacher: So, in factored form, right away you can see that they all share –2, 1, and
3 as x-intercepts. So, if you are looking at all of these three graphs and they all share the
x-intercepts, why do they look so different on your graphs page?
Student1: Coefficients and translations.
Student2: Leading coefficients.
TME, vol10, nos.1&2, p .179
Teacher: So you can express it in different ways: leading coefficients, stretches,
compressions. OK, so if you look at the leading coefficients, there’s a two in one of them,
negative one in the other one, and negative two point 5. Alright.
So, this section (2.4) is titled, Families of Polynomial Functions. And by
definition if you have polynomial functions, all with the same x-intercepts, they’re within
the same family. Is everyone OK with that?
So another way I can ask you questions would be something like this. So here
[referring to the whiteboard where the general form for families of polynomial functions
was displayed: f ( x) k ( x a1 )( x a2 )( x a3 )...( x an ) , where k , k 0 ] is the
basic definition of the functions you were dealing with before, where if you have all the
zeros, all the x-intercepts being the same, and the only thing that differs is your value –
and here they label it k – in front, basically you can say that this family of polynomials,
they share the same characteristics, they’re in the same family.
Then I can ask you something like question #3 [which was then displayed on the
whiteboard]:
A function has x-intercepts –3, –(1/2), 1, and 2, with point (–1, –6) on the
function. Determine the equation of the polynomial function.
What #3 is asking you to do, you’re given specific x-intercepts, they want you to
find the equation of the polynomial function. But along with the four x-intercepts, they
also give you a point. What do you think the point is going to help you determine?
Student1: the k.
Teacher: Right, the k. Thank you very much. So try to give me the equation of the
polynomial function. And remember there are two ways to present the equation of a
Guzm
man & Kieraan
polyn
nomial functtion, or two forms. It’s up to you w
which form yyou want too present. Buut
obvio
ously in facto
ored form, you
y can get th
he k easily.
After worrking on thee problem, various
v
studdents stated the values tthat they haad
obtain
ned for k, not all of them
m arriving at
a the same vvalue. So, M
Mae asked a few studentts
with different ansswers to go to
t the board to show theeir work, butt first insistedd that they aall
use th
he basic form
m, which shee wrote at th
he board as fo
follows:
f ( x) k ( x a1 )( x a2 )( x a3 )( x a4 ) . She also asked that everyoone show thhe
factorrs they weree using and how they were
w
substituuting-in the coordinatess of the giveen
pointt.
One student
s
begaan writing at the board th
he followingg equation (see Figure 9)):
f ( x) k ( x 3)(2 x 1)( x 1)( x 2) , clearly using the fa
factored form
m (ax b) tto
represent the ratio
onal zero –1/2.
Figurre 9. At the blackboard,
b
one
o student writing his vversion of thhe requested equation
TME, vol10, nos.1&2, p .181
The other students who were working at the board used a similar form for the
second factor. This was clearly a reflection of the work they had done earlier in the week
on the Factor Theorem. Despite the fact that Mae had just a few minutes earlier
mentioned that they all should employ the general form, whose factors were of the form,
x a , she did not remark on the students’ use of the form ax b . It conformed, after all,
to the form suggested in the teacher’s guide. The student, after substituting-in the
coordinates of the point for the x’s and f(x), obtained the result of 1 / 2 for k. So too did all
of the others who were showing their work at the board. The various erroneous values
that they had earlier obtained for k were self-corrected.
Teacher: Well, so, we all got a half. So you all determined your polynomial
function equation all in the same way. Did anyone happen to write their function
differently?
Student1: Well, you could expand your function first and then plug it in.
Teacher: Actually, that’s correct. So, it actually turns out to be the same thing. But
did anyone write this part differently [pointing to the four factors of the expression]? [No
one said anything]. So, everyone was able to write their factors as either x plus or minus
b, or ax plus or minus b. Is everyone OK with that?
Student3: Why can’t you use ( x 0.5) for the x-intercept of 1 / 2 ? Like for
(2 x 1) .
The teacher seemed unsure as to what Student3 was proposing. So, she asked him
to come forward to write it at the board, which he did: [he wrote x 0.5 ].
Guzman & Kieran
Teacher: Ooh! Very good question. So. Let’s all try this. Instead of using (2 x 1)
, use ( x 1 / 2) . Tell me what happens when you use ( x 1 / 2) instead of (2 x 1) .
Student 4: You get 1.
Teacher: OK, you get 1. So you get something completely different. Right. So
why do you get something completely different?
Student 5: Divide that part by 2 and then write in the rest of it [clearly referring to
the 2 x 1 , but his technique was not clearly and completely stated].
Teacher: OK, good [without expanding on the student’s partial suggestion], so
your entire expression is actually completely different.
Here in lies the crux of the mathematical difficulty. The teacher appears to see the
function with its different value of k as another member of the family of polynomial
functions, and not as the same member: that is, that f ( x) 1 / 2( x 3)(2 x 1)( x 1)( x 2)
and f ( x) 1( x 3)( x 1 / 2)( x 1)( x 2) are two different members of the same family.
We reiterate that neither of the resources she was using had led her to think otherwise.
She attempted to explain this phenomenon to the class in the following manner, focusing
on the fact that the zeros were the same, but the k’s were different:
Teacher: So your x-value here [in 2x + 1] is –1, so when you go 2 times –1 plus 1,
you get –1. But when you put –1 in here [in x 1 / 2 ] plus 1 / 2 , you get 1 / 2 . Right, so
you get two totally different values, so your k will be different.
Student1: Isn’t that also right though?
TME, vol10, nos.1&2, p .183
Teacher: Is this [pointing to 2x + 1] a different intercept from this [pointing to
x 1 / 2 ]? We have 2x + 1 and x 1 / 2 [she writes on the board 2 x 1 0 and
x 1 / 2 0 ]. So, what does x equal in the two cases? So, they’re the same answer, right
[i.e., the same zero or x-intercept]. But we’re getting different values [for each] because,
in 2 x 1 0 , you double something and then you add, and in this [ x 1 / 2 ] you just add
something. So, according to the order of operations, you get different values of k here.
Right.
1/ 2 ?
Student6: So how do you know it’s not (4 x 2) , because the x-intercept is still
Teacher: That’s very good, but you actually don’t know that. You don’t know if
that would be (4 x 2) . Although again what you’re trying to do is figure out what kind
of leading coefficient you have there. OK.
Mae’s ‘explanation’ of the phenomenon at hand showed her to be oblivious at that
moment to any consideration that the two algebraic forms might be equivalent. If she had
realized that the factoring of (2 x 1) as 2( x 1 / 2) , followed by the multiplication of the
2 with the k-value of 1 / 2 , would yield an equivalent second form of the given
expression, the problem might have been resolved. Furthermore, Student6’s question
regarding the possibility of using (4 x 2) for the (2 x 1) factor (or any of an infinite
number of other possibilities for the factor representing the x-intercept of 1 / 2 ) might
have been discussed in terms of there being no difference whether one uses one form of
the factor or another, because the resulting different value of k would maintain the
equivalence.
The
following
are
all
equivalent:
1( x 3)( x 1 / 2)( x 1)( x 2) ;
Guzman & Kieran
1 / 2( x 3)(2 x 1)( x 1)( x 2) ; 1 / 4( x 3)(4 x 2)( x 1)( x 2) ; and so on. They are all
the same member of a certain family of polynomial functions, despite their having
different k’s. Mae’s distinction between different members of the same family, based on
the criterion of having different k’s, had failed to take into account the role played by
different possible forms of a factor that represent the same x-intercept, or zero, when it is
a rational number. The textbook resources she had just begun to use had not alerted her to
this phenomenon.
As if to prove her point about the two functions being distinct members of the
same family, Mae then suggested to the class that they expand the two – but was
somewhat taken aback by the result. When the expanded results came out to be the same,
the teacher wondered aloud if she had not mistakenly entered the same expression twice
into her computer, which was hooked up to the whiteboard. The following classroom
discussion ensued.
Student1: Even though the k is different, it is still the same thing. Whatever you
do to the factor, you are also doing to k [not quite correct, but on the right track]
Teacher: I am not sure that you are all following this. For the second one, we got
a different value of k. And what do you find when you do it [that is, expand the
expression: 1( x 3)( x 1 / 2)( x 1)( x 2) ]?
Several students at once: The same thing!
Student1: Witchcraft!
Teacher: [recovering somewhat from her surprise, but still at a loss for words]
Does it make a difference? [Looking around the class] Do you understand why that, even
TME, vol10, nos.1&2, p .185
though, because of how you are phrasing the question, or your factors, you are going to
get your different values of k. Remember some people were saying that when you expand
it, you should still get the same thing anyway [what had actually been suggested earlier
by one of the students was related to expanding just one expression that was in factored
form and not expanding two seemingly different expressions]. Well, when you expand it
[the two seemingly different factored forms], you can see that the functions are still the
same. Generally, we do use the ax b form, but obviously you can see that we are
dealing with the same function. Right. So thank you very much for your question,
Student3.
At this moment, the teacher quickly brought her lesson on families of polynomial
functions to an end. The mathematical issues that had arisen clearly required further
reflection on her part.
Discussion
The issues we wish to discuss here are threefold: the mathematical gaps of
textbook resources, the process of becoming aware of and overcoming these gaps which
constitutes a form of ongoing professional development for a teacher, and the
evolutionary nature of documentational genesis whereby resources gradually become
transformed into documents.
The new textbook and teacher guide that Mae had used as resources for her lesson
had not provided the level of mathematical support that she needed. They had not alerted
her to the issues surrounding the two forms of a factor representing a given rational zero
of a function, and the accompanying impact on the value of the parameter k. The
Guzman & Kieran
resources had been silent about both the technique for converting from one factored form
to another and the equivalent nature of the two. Chevallard and Cirade (2010) have
discussed the question of missing mathematical resources and have identified this as a
major praxeological problem for the profession.
It was in the act of teaching her three classes on a given day that Mae became
aware of the mathematical deficiencies of the textual resources with which she had
prepared her lesson on families of polynomial functions. She had not been equipped to
handle the questions put to her by her students and had to react on the fly in an ad hoc
and inadequate fashion. Nevertheless, she seemed to learn from the experience.
Zaslavsky and Leikin (2004) have pointed out that, by listening to students and observing
their work, and by reflecting on this work, teachers learn through their teaching. Mason
(1998) has emphasized that it is one’s developing awareness in actual teaching practice
that constitutes change in one’s knowledge of mathematics and mathematics teaching and
learning.
By taking seriously her students’ questions regarding the relationship between
two seemingly different factored forms of a function, Mae became sensitized to
mathematical aspects of the given area of study that she had not heretofore considered.
Her knowledge of families of polynomial functions was in the process of being
transformed by what transpired in her class, especially by the thought-provoking queries
of her students. According to Zaslavsky and Leikin (2004), such in-practice activity can
be an effective vehicle for teachers’ own professional growth. Although Mae’s primary
preoccupation was the teaching of the material on families of functions, she was at the
same time engaging in the problem that she was putting to the students. She, with the
TME, vol10, nos.1&2, p .187
collaboration of her students, was developing her knowledge of the mathematics of this
area.
In their theoretical paper on documentation systems for mathematics teachers,
Gueudet and Trouche (2009) introduce a general perspective for the study of teachers'
professional evolution, where the researcher's attention is focused on the resources, their
appropriation and transformation by the teacher or by a group of teachers working
together. Their approach aligns with Adler’s (2000), who claims that, “in mathematics
teacher education, resources in practice need to become a focus of attention” (p. 221) and
with Remillard’s (2005) whereby the evolution of the curriculum material actually used
and a teacher's professional development are viewed as two intertwining processes.
With respect to this intertwining process, Gueudet and Trouche (2009) point out
that:
A teacher draws on resource sets for her documentation work. A process
of genesis takes place, producing what we call a document. … A given
teacher gathers resources: textbooks, her own course, a previously given
sheet of exercises... She chooses among these resources to constitute a list
of exercises, which is given to a class. It can then be modified, according
to what happens with the students, before using it with another class
during the same year, or the next year, or even later. The document
develops throughout this variety of contexts. (p. 205)
We suggest that the awarenesses acquired by Mae in her teaching of families of
polynomial functions with new resources will be instrumental in enabling her to modify
these resources, thereby leading to the gradual transformation of a resource into a
document for her. However, Gueudet and Trouche (2009) emphasize that
“documentational genesis must not be considered as a transformation with a set of
resources as input, and a document as output. It is an ongoing process … that continues in
usage. We consider here accordingly that a document developed from a set of resources
Guzman & Kieran
provides new resources, which can be involved in a new set of resources, which will lead
to a new document etc. Because of this process, we speak of the dialectical relationship
between resources and documents” (p. 206).
We close our discussion by turning to a relevant comment made by Adler (2000)
that puts the focus not on producing more (or better) resources, but rather on better
understanding how teachers use the resources they have, change them, and in the process
engage in a form of ongoing, personal, professional development: “Our attention shifts
away from unproblematised calls for more [resources] and onto the inter-relationship
between teacher and resources and how, in diverse, complex contexts and practices,
mathematics teachers use the resources they have, how this changes over time, and how
and with what consequences new resources are integrated into school mathematics
practice” (p. 221). In this article, we have attempted to illustrate the complex
interrelationship within actual teaching practice between a teacher and a new set of
resources, by describing the nature of the classroom experiences whereby a teacher
becomes aware of the mathematical gaps of new resources and thus better positioned to
make changes to them over time. Such an approach both situates resources and their
adaptive use within a documentational framework and re-centers professional
development within the actual practice of teaching.
Acknowledgments
The research presented in this article was made possible by grants from the Social
Sciences and Humanities Research Council of Canada (Grant # 410-2007-1485), from the
Québec Ministère des Relations Internationales, and from CONACYT of Mexico (Grant
TME, vol10, nos.1&2, p .189
# 80454). We express our appreciation to the teacher and her class of students who
participated in this research.
References
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Mathematics Teacher Education, 3, 205-224.
Artigue, M., & Houdement, C. (2007). Problem solving in France: Didactic and
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et genèses documentaires. In G. Gueudet & L. Trouche (Eds.), Ressources vives. Le
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From text to ‘lived’ resources: Mathematics curriculum materials and teacher
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adaptation within classroom teaching practice: Shaping both the implicit and the
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Rabardel, P. (1995). Les hommes et les technologies: Approche cognitive des instruments
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Remillard, J.T. (2005). Examining key concepts in research on teachers’ use of
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Schmidt, W.H. (2011). Measuring content through textbooks. In G. Gueudet, B. Pepin, &
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TME, vol10, nos.1&2, p.191
Mathematical Problem Solving in Training Elementary Teachers from a
Semiotic Logical Approach
Martín Socas
Josefa Hernández
University of La Laguna, Spain
Abstract: The aim of this article is to consider the professional knowledge and
competences of mathematics teachers in compulsory education, and to propose basic
tasks and activities in an initial training programme in the framework of a global proposal
for “Immersion” in the curriculum of the educational phase which the trainee teacher
would go on to work in. Problem-solving, in this context, is considered as being an
inherent part of mathematics and this is described in terms of problem-solving,
establishing connections between concepts, operations and implicit processes in the
mathematical activity (conceptual field) and their relationships problem-solving; and it is
assumed that the learning of problem-solving is an integrated part of learning in
mathematics.
Keywords: Problem Solving, Teacher Training, Didactical Analysis, Semiotic Logical
Approach (SLA).
Introduction
The analysis of the results obtained, in recent years, in different national (in
Spain) and international assessments shows that the knowledge of mathematics (Problem
Solving) of students in compulsory education (K-9 Grades) is insufficient in terms of the
desired curriculum. What needs to be done to improve the learning and teaching of
mathematics and, in particular problem solving in this educational stage? This question is
addressed here by reflecting on the role played by teachers in primary and secondary
education in the pursuit of an effective learning of mathematics and problem solving. At
present, the initial training of teachers in primary and secondary education takes place
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos.1&2, pp.191-218
2013©The Author(s) & Dept of Mathematical Sciences-The University of Montana
Socas & Hernández
within the European Higher Education Area, where primary school teachers need four
years training and secondary school teachers are required to have completed a mandatory
Professional Masters degree. This initial teacher training has a great opportunity for
improvement.
Problem solving in mathematics education
Problem solving has always been regarded as a basic component in the
construction of mathematical culture. However, when mathematical knowledge is
presented in its final state, what prevails is the conceptual organization of the objects of
such knowledge in which problem solving appears again as a core of relevant
mathematical knowledge. In the early eighties, in view of the primacy of the concepts and
their properties as well as their algorithmic use, problem solving was vindicated as a key
activity in the learning of mathematics, which has led to the development of an emerging
theoretical and practical body of research in mathematical education, and a notable
increase of its presence in the curriculum, either as a further block of contents or as cross
content but specific to mathematics at the corresponding level (Santos-Trigo, 2007,
Castro, 2008). The follow-up research on problem solving clearly shows that, despite all
amount of effort, there are no significant data on the improvement in this on the part of
the students and different questions arise ranging from the need to establish relationships
and existing connections between the development of the understanding of mathematical
contents and problem solving skills, to the need of having theoretical bases to guide
problem solving (Lester and Kehle, 2003).
TME, vol10, nos.1&2, p .193
Some authors such as Lesh and Zawojewski (2007), suggest that the rise of
research in problem solving was very important between 1980 and 1990, and that some
trends are presently aimed at putting an emphasis on critical thinking, technology and
mathematical problem solving, and analysis of how mathematics is used in other sciences
and professions that does not match the way mathematics is taught in school, or the
development of problem solving in other settings or contexts such as situated cognition,
communities of practice or representational fluency. These directions and perspectives in
solving mathematical problems are, at the present, promising lines of research.
The knowledge and professional skills of a mathematics teacher
The concern, from the point of view of mathematics education, regarding
teacher’s knowledge and professional skills has been and is, a constant research topic,
and is based on the following conjecture: The knowledge and professional skills of the
mathematics teacher must be acquired through different scientific domains: mathematics,
mathematical didactics and educational sciences. The initial teacher training should
enable the trainee teachers to increase their knowledge about mathematics and
mathematical didactics as a specific field of professional competence (mathematics
education) and a field of research, along with other issues arising from educational
sciences.
Shulman pointed out in 1986, for the first time, the importance of the specific
subject to teach in teacher training. This author identified three categories of professional
knowledge of teachers: Knowledge of the specific subject, pedagogical content
knowledge (PCK) or in the context here: the didactical content knowledge (DCK) and
curricular knowledge. Subsequently, Bromme (1988, 1994) described the qualitative
Socas & Hernández
characteristics of the major areas of professional knowledge: Knowledge of mathematics
as a discipline, knowledge of mathematics as school subject, philosophy of mathematics
schools, pedagogical knowledge and specific pedagogical knowledge of mathematics.
The author proposed that the teachers' professional knowledge is not simply a
conglomeration of these domains of knowledge, "but an integration of the same", which
occurs during teaching practice or during professional teaching experience.
Semiotic Logical Approach (SLA). Assumptions
Semiotic Logical Approach (SLA) (Socas 2001a and b, 2007), when understood
as a theoretical-practical proposal (formal-experimental), aims to provide tools for the
analysis, description and management of problematic situations or phenomena of a
mathematical didactical nature from a perspective based on semiotics, logic and
competence models (semiosis), and takes one of the great problems of mathematics
education, the study of difficulties and errors of students in learning mathematics as a
reference (Freudhental, 1981). Logical and Semiotic Aspects of SLA uses Peirce's
Phenomenology (1987) as a reference. Peirce, starting from the logic conceived of as a
science of language, describes the development of a science of signs and meanings called
semiotics which can be used to analyse, within the semiotic constructs, different
phenomena of logic, mathematics, physics and even psychology, which is why this
phenomenology is used here. Semiotics is a theory of reality and knowledge that one can
have of phenomena through signs which are the only means available. Semiotic inference
emerges in sign analysis where what is analysed are the trademarks or observable and
overt expressions of inference, which Pierce organized as a logical theory (semiotics) that
has three references closely linked to one another. Therefore, if the aim is to study any
TME, vol10, nos.1&2, p .195
phenomenon (problem situation), which is the starting point in SLA, this will always be
analysed from a given context and by means of three references organized as first, second
and third, which is defined as primary or basic semiotic function which is determined by
the sign, object and meaning references (Socas, 2001b). This can be used to determine
the notion of representation as the semiosis determining such references. Therefore, the
representation is a sign that:
(1) has certain characteristics that are proper (context)
(2) sets a dyadic relationship with the meaning
(3) establishes a triadic relationship with the meaning via the object, this triadic
relationship being such that it determines the sign of a dyadic relationship with
the object and the object to a dyadic relationship with the meaning (Hernández,
Noda, Palarea and Socas, 2004).
As far as the Educational System is concerned, SLA uses the Begle’s diagram of
school mathematics as a reference, which shows the mutual relationships between the
different components in the training process and defends the need to set multiple
perspectives and procedures in the field of the teaching / learning of school mathematics
(cited in Romberg, 1992). To do this, two different parts must be distinguished: the
"educational macro system", where both disciplinary knowledge and the institutions or
persons involved intervene in the education system, and, the "educational micro system",
which is made up of three references or basic elements: mathematical knowledge
(mathematics), students and teachers, and their relationships in a context determined by
the following components: social, cultural and institutional, which is shown in the figure
below:
Socas & Hernández
ult
ur
tio
tit u
l
na
So
cio
-C
Ins
al
Teacher
Mathema
tics
Student
Educational micro system
Figure 1: Elements of educational micro system
The three essential relationships are:
Relationship1: Between the mathematical knowledge and the student, which is
called "school mathematics learning as a conceptual change".
Relationship 2: Between the mathematical knowledge and the teacher, called:
"adapting the curricular mathematics content to be taught".
Relationship 3: Between the mathematical knowledge and the teacher via the
student which is called: "interactions".
Thus, the three elements and the three essential relationships contextualized in the
three components of the context determine the teaching / learning process in the regulated
systems, thereby characterizing the six core contents that are a part of the mathematics
teacher's professional knowledge, in addition to those derived from the three previous
mentioned relationships: mathematical knowledge in a disciplinary sense, the curricular
mathematics knowledge and the mathematics curriculum of an educational stage. It is in
this framework that the difficulties, obstacles and errors that students have or make in the
construction of mathematical knowledge are examined. SLA organizes three models of
competence: Formal Mathematical Competence (FMC), Cognitive Competence (CC) and
Teaching Competence (TC), which constitute the references that define the General
TME, vol10, nos.1&2, p .197
Semiosis which plans and manages research in the educational micro system (Socas,
2010a, 2001b, 2007 and 2010).
This General Semiosis can be used to plan and manage both the problems of
teaching and learning in the educational micro system, and the didactical mathematical
problems to be studied.
The Formal Mathematics Competence Model (FMC) can be used to describe the
conceptual field of the mathematical object in the thematic level in which both their
functions and phenomenology are being considered.
The Cognitive Competence Model (CC) is the second reference and takes into
account the above mentioned Formal Mathematic Competence Model, it refers to the
specific cognitive functions of students when they use the mathematical objects in
question and structural aspects of learning.
The Teaching Competence Model (TC) is the third reference, and it also considers
the above mentioned aspects (formal mathematical competence and cognitive
competence) and describes the actions of the subjects involved, the communication
processes, the mediators, the situations, the contexts, which occur in education.
Three basic assumptions of SLA are now proposed here: Mathematical Content
Analysis, Didactical Analysis and the Curricular Organization.
The didactical analysis and the curricular organization are the concepts that SLA
uses to characterize the knowledge of mathematical content from the professional point
of view. The didactical analysis allows the comprehension of the professional problem,
while the organization curricular plans his development.
Mathematical Content Analysis
Socas & Hernández
The Formal Mathematics Competence Model (FMC) sets out the conception of
mathematical literacy and the different relationships between the elements characterizing
it. The FMC is organized by means of the semiosis that characterizes and relates the
conceptual, phenomenological and functional aspects of mathematical content involved
in the problematic situation to be addressed in the educational micro system, and would
appear as below:
MODELLING
Processes
FORMAL SUBSTITUTION
TECHNIQUES
GENERALIZATION
STRUCTURE
Structures
Operations
OPERATIONS
ALGORITHMS/
RULES
CONCEPTS/
DEFINITIONS
PROPERTIES
Figure 2: Domains of mathematical activity
The different domains of mathematical activity are expressed within this model in
relation to the conceptual field from a formal perspective and its different relationships,
i.e., it describes the duality of mathematical objects in relation to conceptual/procedural
mathematical knowledge of the field in question. Any activity is described in relation to
the three components: operations, structures and processes, and relationships, which we
explain later. Each component, in turn, is determined by three others that describe a new
semiosis: 1) The operations component for the semiosis: operations, algorithms (rules)
and techniques, 2) the structures component for: concepts (definitions), properties and
TME, vol10, nos.1&2, p .199
structure, and 3) the processes component for: formal substitutions, generalization and
modelling. This organization of the conceptual fields is contextualized in the problematic
situations that are addressed in the language (representations) and in the arguments
(reasoning) that are used in developing it.
The three context components are similarly determined by the respective
semiosis. In the case of problematic situations: identification, approach and resolution; in
representations (language): recognition, transformation (conversion) and elaboration
(production), and in arguments: description, justification and reasoning. This organization
of mathematics by the FMC can be used to consider problem solving as an inherent part
of mathematics and to describe it in terms of problem solving. Hence, the following
aspects characterize mathematical culture in SLA:
1. Mathematics is a multifaceted discipline
2. Mathematical culture emerges and develops as a human activity of problem
solving
3. The problems have one common feature: the search for regularities
(identification, approach and resolution). Modelling is the mathematical process
par excellence
4. Mathematical culture creates a system of signs able to express regular
behaviour
5. The set of regularities is organized into conceptual fields
6. The conceptual elements of these fields are mathematical objects
The Formal Mathematical Competence Model can also be used to establish the
connections between concepts, operations and processes involved in mathematical
Socas & Hernández
activity and their relationship to problem solving, which is, generally speaking, relevant
for mathematics education, and particularly for problem solving.
Didactical Analysis in the Semiotic Logical Approach (SLA)
Semiosis can be used to identify and understand the didactical mathematical
problem, whose reference framework is comprised of the curriculum organizers (Rico,
1997) and the initial notion that Freudhental (1983) put forward for didactic analysis as
follows: "the analysis of the curricular content of mathematics is performed to serve the
organization of its teaching in educational systems".
Didactic analysis is organized according to the following triad: formally described
curricular mathematics, semiotic representations and difficulties, obstacles and errors, it
also facilitates the identification and understanding of the didactic problem to be
addressed.
Didactic analysis implies, in relation to the curricular component, a review of the
curricular contents from the formal perspective: operational, structural and processual
(using processes), but also implies a necessary relationship with the students linked to
their interests and motivation.
The semiotic representations component involves a review of the curricular
content in relation to different forms of representation of the objects in question, as well
as the presentation of information to students. The following states of the historical
development of the mathematical object are considered in this section: semiotic,
structural and autonomous, that also implies a necessary relationship with the students
linked to the coordination between the forms of expression and representation and the
interests and motivation of the students. The component difficulties, obstacles and errors,
TME, vol10, nos.1&2, p .201
require a review of the curricular content in relation to these three aspects, with a dual
aim of prevention and remedy, making it possible, for example, from the perspective of
prevention, to set the levels or cognitive skills required of students in relation to the
mathematical object in question. The identification of the errors generated by the students
needs analytical tools which can get into the complexity of learning difficulties in
mathematics. One way to address this would be, as reported by Socas (1997), to take the
three directions of analysis into consideration, like three coordinated axes which would
more accurately identify the origins of the error and would enable the teachers to devise
more effective procedures and remedies. These three axes would be determined by their
origin: i) in an obstacle, ii) in the absence of meaning; iii) in affective and emotional
attitudes.
Curricular Organization
Not only do mathematics teachers need knowledge about the discipline of
mathematics and the curriculum, but they also require didactical mathematical knowledge
(DMK) in order to organize the mathematical content for teaching.
This is professional knowledge that includes the appropriate elements of analysis
to understand, plan and do a professional job. The teacher needs to expand and connect
different perspectives on the curricular mathematics content, in such a way that its
consideration is not only from the internal logic of the discipline, which may emerge as
being too restrictive, formal and technical, but from the curricular dimension, a more
open perspective and one which integrates the teaching of mathematical knowledge more,
and this is not possible to put into practice from only the theoretical consideration of
knowledge about the discipline of mathematics and the curriculum, to convert this into
Socas & Hernández
the mathematical knowledge to be taught. This professional knowledge develops in the
subject Didactic of the Mathematics for teachers, structured according to the didactical
analysis and the curricular organization.
The curricular organization emerges from the organizers of the curriculum (Rico,
1997), and must be understood as those teaching skills that can be used to plan
mathematical content for teaching, i.e., planning and evaluating mathematics classroom
schedules, which is determined by the following triad: context, teaching/learning and
assessment.
In so far as the context reference is determined by the semiosis described by the
problem situation, which refers to the environments in which the activities take place, the
contextualization, which is determined by the specific goals, specific skills and teaching
content involved in the activity, and the levels, referring to the complexity of
mathematical tasks: reproduction, connection and reflection, skills demanded by the
same, taken from the PISA Project (Rico and Lupiáñez, 2008), or to stages of
development: semiotics, structural and autonomous, taken from SLA (Socas, 1997).
Proposal for training mathematics teachers
The different areas of knowledge (mathematical and didactical mathematical) that
can be used to support the training proposal have been described here in general terms.
But before going on, it is necessary to make a few comments about the trainee teachers
who this training proposal is aimed at. Several studies conducted at the University of La
Laguna (Spain), in which students from several other Spanish universities have also
participated, show that the students who start teacher training courses in primary
TME, vol10, nos.1&2, p .203
education teaching have huge gaps in basic mathematical knowledge. As regards problem
solving, the situation is that 18 year old students, with more than 12 years of studying
mathematics in the educational system and learning to solve problems, still tend to
concentrate on the data of the problem as their general cognitive strategy, without
demonstrating a clear understanding of the problem and without identifying operational,
structural (conceptual) or processual relationships, given in the data, often providing
solutions that cannot be valid for the conditions of the problem, which furthermore
clearly shows a lack of cognitive strategies (heuristic methods) and a lack of critical
thinking (Palarea, Hernández and Socas, 2001; Hernández, Noda, Palarea and Socas,
2002 and 2003). Subsequent studies show no improvement on the previous results,
finding that students show a predominance of operational rather than structural and
processual thinking, and it is this thinking that is mostly behind the solution to any
mathematical task, which many times is unsuccessful, even when the applied operational
knowledge is correct. This suggests that the emphasis that the teaching of mathematics
puts on operational knowledge may be creating difficulties and obstacles for the student
to apply, for example, heuristics and strategies to solving problems that are more
associated with structural and even processual thinking, which creates difficulties in
achieving mathematical competence (Socas et al., 2009).
As regards trainee teachers of mathematics in secondary education, the starting
assumptions were that the design of the plan should take two essential aspects into
account, on the one hand the mathematical training of future teachers (graduates in
mathematics) and, on the other hand, the lack of a specific didactic training for
professional work (teacher), except for that formed by existing knowledge, implicit
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theories, values and beliefs that had come from their experiences as students of
mathematics during their schooling, and which are, in many cases, an obstacle to properly
channelling many aspects of professional thinking. The analysis of educational reforms
leads one to believe that such reforms require the teacher to be able to take on the new
curriculum changes which actually means confronting new tasks. The latter necessarily
implies significant changes in training mathematics teachers which can be summarized in
the following points:
- Scientific and educational training tailored to this new curricular change.
- Training to work with students who have a high degree of heterogeneity in basic
skills, interests and needs.
- A change in attitudes among teachers so that they can develop the educational
aspects of teaching, adopt flexible approaches and delve into a more interdisciplinary
vision of culture.
- A conception of the curriculum as a research tool that can be used to develop
concrete methods and strategies of consolidation and adaptation.
- Assessment and exercising of teamwork as well as the development of a strong
professional autonomy (Camacho, Hernández and Socas, 1998).
Fundamentals of the Proposal
The analysis of the knowledge and skills that a maths teacher must have in
compulsory education, shows that two essential questions need to be answered: What are
the basic tasks and activities in an initial training plan for maths teachers in compulsory
TME, vol10, nos.1&2, p .205
education? And whether the theory and practice dichotomy is enough to provide a
response to the tasks and basic activities of teacher training?
Llinares (2004, 2009 and 2011) proposes the articulation of three systems of
activities or tasks to develop the knowledge and skills of a mathematics teacher:
"Organise the mathematical content to teach it", "Analyze and interpret the production of
the students" and "Manage the mathematical content in the classroom". A reflection and
analysis of the two questions leads one to consider that the three aforementioned
activities systems are at least necessary. These are the activities that also emerge as
necessary and essential in all three relationships in the Semiotic Logical Approach
(Socas, 2001a and 2007). As for the second question, one can see the need to make
progress in the dichotomy between theory and practice with knowledge to develop the
professional skills to design and manage teaching practice in mathematics. The general
aspects of the basic proposal take the following as a reference: the analysis of
mathematical content, the didactic analysis of curricular content and organization. This is
a comprehensive proposal for the training of mathematics teachers, which aims to
facilitate a reconciliation between disciplinary mathematical knowledge (DMK) to
curricular mathematical knowledge (CMK), to pedagogical mathematical knowledge
(PMK) and knowledge of educational practice (KEP). This can be achieved by means of
a proposal that ranges from the general comprehensiveness of the curriculum and of the
disciplinary mathematical knowledge, to the organized totality of curricular content as
content to be taught. The situation is depicted in the graph below, which expresses the
cyclical nature of the proposal.
Socas & Hernández
Practices of Learning and Teaching
Curricular Organization
Didactic Analysis
The Mathematical Curriculum
in the Compulsory Education
Mathematics
fot teachers
Figure 3: Proposal for the training of mathematics teachers
The analysis of the mathematical content plays a role, in this proposal, in the reteachers’ conceptualization of mathematics, and together with the didactic analysis of
curricular content and organization, in the development of the school subjects of the
didactics of mathematics and teaching practice of mathematics, where the three
previously mentioned professional activities have a place.
Professional activity will be considered first, "organizing the mathematical
content to teach it". This deal with solving a professional problem that requires analysis,
understanding and planning, and can be represented by the following semiosis: curricular
mathematical content, disciplinary mathematical content, and mathematical content for
teaching.
First, the teacher needs to organize the curricular mathematical content (CMC),
the desired mathematical content that is definable in the domain of the disciplinary
mathematical content, although it is not organized under that logic. This CMC is
TME, vol10, nos.1&2, p .207
extracted via precise and precise mechanisms and organizations from the disciplinary
content and is inserted in the curriculum. Once these actions have been performed by
different elements of the educational system, curricular mathematical content knowledge
is intrinsically different to the disciplinary knowledge, at least in its epistemological
aspect, and supports interpretations from different perspectives, for example functional,
as part of a common basic culture (Rico and Lupiáñez, 2008), the second derives from
the discipline itself, scholarly mathematical knowledge, which we call disciplinary
mathematical content (DMK) or formal mathematical knowledge (Socas, 2010a) and the
third is the mathematical content for teaching (MCT), which includes both the taught and
the mathematical content assessed (Hernández et al., 2010). The three components are
interrelated in a process called transposition or adaptation of mathematical content, but
have their own independent organization. The organization of curricular mathematical
content comes from a pedagogical order implicit in the curriculum designers, and is
associated with basic mathematical competence as part of a common culture. The
organization of the mathematical content for teaching is compiled using the didactic order
as a starting point, and is associated with the subjects’ competence in didactical
mathematical knowledge (DMK) and determines the sequence and level of the
mathematical content in the teaching proposal with regard to basic mathematical skills
and the other basic skills.
The professional task of organizing the mathematical content for teaching
involves competence in the three areas of mathematical content. The question is now
what happens to our students and how does one involve them in professional tasks that
enable them to be competent professionals who can identify, analyze, understand and
Socas & Hernández
plan for these three areas of mathematical content? As has been shown, students who
begin teacher training for primary school have huge gaps in basic math skills which is
why they need a revision of the discipline in terms of some "mathematics" to train them
professionally, to improve not only their knowledge but their beliefs about the ends of
this knowledge in compulsory education (Socas et al., 2009).
Mathematics for teachers in compulsory education
Teacher training programs have generally been designed to include in subjects,
like mathematics, mathematical content as disciplinary knowledge, which is developed
by explicating the different conceptual fields, and by considering mathematics as a
fundamentally instructive tool that is organized primarily from the point of view of its
internal logic, which means characterizing mathematical knowledge by using an
organization based on its key concepts and on an introduction using a logical sequencing,
i.e. the material is organized in the way a mathematicians would. On the other hand, the
mathematical content of the curriculum that the teacher must impart has been determined
by various agents of the educational macro system via a process that is generally
unknown to the future teacher. The curriculum is organized by a list of contents that are
related to the skills and competencies to be developed, the same happens with the
evaluation process, and is immersed in a particular conception of understanding teaching
and learning. Therefore, the curricular organization of the mathematical content, the
object of education in a stage of education, needs to be seen as a systematic organization,
which considers mathematical content as a fundamentally cultural and basic element,
which is organized from an epistemological and phenomenological perspective capable
of developing basic mathematical skills, and is introduced by means of an educational
TME, vol10, nos.1&2, p .209
organization as well as criteria for assessing the acquired knowledge and skills. The
subject: mathematics for teachers in compulsory education would deal with revising
different aspects of curricular mathematical content relevant to the stage of education in
which the teachers have to exercise their profession from the disciplinary mathematical
perspective, facilitating the teachers with a re-conceptualization of curricular
mathematical content. This is a process of immersing the trainee teacher in curricular
mathematical content which they will have to organize for teaching afterwards. This is,
ultimately, a proposal for basic training in a closed curricular structure, which is
approached from formal mathematical competence and basic mathematical competence,
i.e. the analysis and understanding of curricular mathematical content in disciplinary
terms with epistemological, phenomenological and applicability references, in which
students complete their basic training related to such issues at the level of conceptual
systems involved: operations, structures and processes in problem-solving situations,
using the reasoning and the appropriate language for the thematic level in question by
means of tasks and activities of differing natures but necessary for linking the school
tasks and activities.
The didactic of mathematics for teachers in compulsory education
The next item to be considered is the second group of activities and tasks to be
developed by the trainee teacher: "Analyzing and interpreting students’ production"
which refers to the knowledge and ability to mobilize different resources: analogical and
digital mathematical representations, difficulties, obstacles and errors associated with the
object of teaching mathematical content. Take, for example, the role of the difficulties,
obstacles and errors of students in this analysis and interpretation. It is known that
Socas & Hernández
learning mathematics creates many difficulties for the students and that these differ in
nature. Some difficulties originate in the educational macro system, but generally
speaking, they originate in the educational micro system: student, subject, teacher and
educational institution. These difficulties are connected and reinforced in complex
networks that, in practice, materialize in the form of obstacles and are manifested by the
students in the form of errors. The error will have different roots, but will always be
considered as the existence of an inadequate cognitive schema in the student, and not
only because of a specific lack of knowledge or an oversight. The difficulties may be
grouped into five major categories associated to 1) the complexity of the objects of
mathematics, 2) mathematical thinking processes, 3) the teaching processes developed for
the learning of mathematics, 4) cognitive development processes of students and 5)
affective and emotional attitudes toward mathematics (Socas, 1997). In addition to the
curricular and disciplinary mathematical knowledge, the trainee teacher of mathematics
requires didactical mathematical knowledge (DMK) to be able to organize the
mathematical content for teaching. This is specific professional knowledge that has to be
provided by the subjects belonging to the didactics of mathematics, which includes the
elements of analysis for adequately understanding, planning and conducting professional
work. This knowledge is developed under the two constructs discussed above, didactic
analysis and curricular organization.
Best Practices
The proposed teacher training should focus on the organization and development
of best practices for the attainment of the skills required, these have to be developed
within the framework of problem solving of a professional nature and associated with the
TME, vol10, nos.1&2, p .211
knowledge and resources that the teacher must mobilize to obtain the solution to the
problem.
Thus, mathematical problems emerge from the situations developed in the
curriculum and are addressed from the FMC in the subject of mathematics for the
teachers in terms of the language and reasoning involved in the conceptual field in
question. This immersion of the student continues because the problem solving must be
organized for teaching, usually in the context of a classroom program, which must be
considered from the training analysis. It involves incorporating the consideration of the
difficulties, obstacles and errors of students in the different domains of mathematical
activity. The trainee teachers of mathematics perform different activities and tasks of
application, related to the various mathematical fields, and conclude in all situations with
the elaboration of a map of the mathematical knowledge being dealt with, organized in
terms of the six disciplinary mathematical content areas according to the FMC model,
i.e., operations, structures, processes, representations, problems and reasoning. Certain
tasks developed by the trainee teachers of mathematics in the course in a report format,
all of which are from a questionnaire, are presented below:
Task 1: Report on numeration systems and decimal system
(For example, the first questionnaire has questions about the relations between the
different numerical systems, the description of the numerical systems from the decimal
representation and the representation in the number line of the different numbers).
- Analysis of the errors made and of the blank responses, as well as determination
of their cause or origin.
Socas & Hernández
- To characterize the D numeration system (Decimals) as is clear from the
answers to the questions.
- Analysis of the representational procedures on the number line of the numbers
proposed in the questionnaire.
- Decimal numbers in the curriculum of compulsory primary or secondary
education.
- To elaborate a map of the numbers in the primary or secondary education.
- To elaborate a map of the procedures for representing numbers in primary or
secondary education.
Task 2: Report on operational, structural and processual knowledge in
mathematics
- Analysis and evaluation of the mathematical discipline according to SLA.
- Analysis of unanswered questions and the mistakes made in the questionnaire,
determining the source of errors.
- Analysis of operational, structural and processual knowledge used in the
questionnaire responses, both correct and incorrect.
- Self-evaluation of the type of knowledge used in the answers.
- Analysis of the mathematics curriculum in primary or secondary education.
Choosing a course and a content block about numbers, algebra and functions, and
analysing them from an operational, structural and processual perspective,
identifying the systems they use for representing mathematical objects, the
problems they give rise to and the reasoning they propose, with special emphasis
on identifying the heuristic content.
TME, vol10, nos.1&2, p .213
- Analysis of a mathematics textbook in compulsory secondary education.
Choosing two consecutive themes on numbers, algebra and functions, and then
analyzing them from the aforementioned perspective.
Task 3: Report on mathematical problem solving
- Solving the problems correctly in various sessions.
- Analysis of the difficulties and errors made in the different sessions in solving
the problems of the questionnaires.
a) To identify the following phases in each problem: acceptance, blockage and
exploration
b) To determine the source of the difficulties and errors.
- To identify the different reasoning (and different heuristics) used in the given
questionnaire responses.
- To analyze the map of the contents involved in solving the mathematical
problems proposed in the questionnaire, paying special attention to the
mathematical tools and reasoning (heuristics) used.
- To develop a new map of knowledge involved in the correct resolution of the
proposed problems.
Final considerations
A proposal is suggested here, for training student teachers in primary and
secondary mathematics to improve the learning and teaching of mathematics in these
education stages because as Sowder said (2007), many of the difficulties that
mathematics students have are to do with the teaching they receive, but what does
preparing a trainee maths teacher competently really involve? This proposal opts to
Socas & Hernández
develop three systems of basic activity that can determine the knowledge and skills of the
teacher, presented as professional tasks from a global perspective in the context of
problem solving in the case of their profession.
The three systems of professional activities categorize teachers according to
different skills, for example, in the case of the activity: organizing the mathematical
content to teach it, puts students in these skills areas: knowledge of the contents of
mathematics from a global perspective in which the resolution of problems is an inherent
part of the mathematical culture that should be taught and the ability to translate this into
learning expectations, and the design and planning of learning sequences. In the case of
the activity of analyzing and interpreting the students’ mathematical production places
students in the skills area regarding understanding and working based on the students’
representations including their idiosyncrasies, and knowing and working with the
difficulties, obstacles and errors of the students.
As regards the activity of knowing how to manage mathematical content in the
classroom, this places students in the skills area of designing and controlling problematic
situations appropriate to the different levels and possibilities of the students, and
observing and assessing students in learning situations. The case of training maths
teachers leads one to consider the basic situations of meaningful and effective work and
how these should be dealt with by a professional comprehensive approach. The
comprehensive approach is set in the context of trainee teachers, and it articulates and
connects different subjects in a global proposal which seeks to ensure a comprehensive
and inclusive vision of mathematics and of teaching and learning mathematics,
TME, vol10, nos.1&2, p .215
encouraging the active participation of students, which shows how we get closer to
understanding reality through mathematical culture and how it is perceived by them.
Research has shown that when mathematics programs, from the disciplinary
approach, are used with trainee teachers as a finished product, they are insufficient.
Providing trainee teachers with an epistemological and phenomenological analysis of
mathematical objects of teaching involves not only knowing the conceptual systems
involved, their languages and problems, but also the usefulness of mathematical objects
and their use, which could be successfully used to deal with the interpretation of the aims
of the mathematics curriculum in this educational stage and confidently take on the
didactical mathematical knowledge. The organization of mathematical knowledge using
the phenomenology / epistemology pairing involves paying special attention to the use,
management and function that this knowledge can have at a given time, without losing
sight of its internal logic. Finally, it is important to emphasize that this global proposal
for training mathematics teachers by "immersion" in the curriculum of the educational
stage where they will work in the future, will allow them to develop, in this environment,
the knowledge and skills needed in their professional work.
This research was partially supported by the National Plan of Research of the Ministry of Science
and Innovation: Mathematical Competence, problem solving and technology in Mathematical
Education (EDU2008-05254).
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Greenwich, CT: NCTM.
TME, vol10, nos.1&2, p.219
Developing problem solving experiences in practical action projects
François Pluvinage
Departamento de Matemática Educativa, Cinvestav - I.P.N., México
Abstract: Problem solving doubtless is an essential element of mathematical learning, so
that mathematics educators often are satisfied when finding situations that lead their
students to such activity. But in many cases, the chosen situations and the ways to guide
students' works are not sufficiently analyzed from a didactic point of view. Our goal in
the present analysis is to underline the possible ways for managing the situations, and to
exhibit the parameters that educators have at their disposition within their role as
mediator between students and mathematical knowledge and know-how.
Keywords: Problem solving, problem for investigating, project based learning, a priori
analysis, mediation.
1. Classical problem solving
Problem solving at its broad extent is clearly exposed by Alan Schoenfeld
(Schoenfeld, 2006, p.41): “[Problem solving] includes a child’s actions in interacting
with its parents, a student working on a mathematics problem in class or in the
laboratory, and a teacher’s decision-making while teaching a mathematics (or other)
lesson.” In this paper, we will only consider problem solving in mathematics instruction.
Nevertheless, what we mostly found in mathematics educational literature and in learning
material designed for students is a restricted use, with sometimes a reduction to drill for
improving skills. In our opinion, this approach is too limited. Thus we first analyze some
typical situations used in mathematics education with a problem solving purpose.
Our goal in the following analysis is to underline the possible ways for managing
the situations, and to exhibit the parameters that educators have at their disposition within
their role as mediator between students and mathematical knowledge and know-how.
1.1. Too easy… and how to do it better
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos.1&2, pp.219-244
2013©The Author(s) & Dept. of Mathematical Sciences-The University of Montana
Pluvinage
The following Math Problem Solving example comes from a worksheet of Rhl
School. We may consider it as a representative of certain trend in exploiting problem
solving in mathematics education.
“Ryan’s Class
There are exactly twelve children in Ryan’s class. Only four of the children are
boys. The following questions refer to a time when all the children are present in the
class. There are no visitors in the class. There might be more than one correct answer to a
question.
1. Which of the lettered statements must be true?
2. Which of the lettered statements cannot be true?
3. Which of the lettered statements could be true or not true?
a. There are twice as many girls as boys in Ryan’s class.
b. There are eight more girls than boys in Ryan’s class.
c. There are four more girls than boys in Ryan’s class.
d. If Ryan is sitting at a table with all the girls, there are exactly nine
children at that table.
e. If only three of the boys are standing on their heads, one of the boys is
not standing on his head.”
Math Problem Solving, Vol. 8, No. 1, May 5, 2003. Retrieved from
http://www.rhlschool.com/math8n1.htm
Comments about the task: Giving an answer to the lettered statements a, b, c and e
only supposes to understand them. The only true problem here comes from the fact that
we don’t know Ryan’s gender (while mostly used for boys, Ryan has been used for girls
in the United States since the 1970s). And for the lettered statement d, if Ryan is a boy,
the statement is true, and if Ryan is a girl, the statement is false; then the correct answer
for us is the answer 3, but the correct answer would be different for somebody who
TME, vol10, nos.1&2, p .221
knows Ryan’s gender. So we note that there is a subtle semantic distinction in that
statement; nevertheless, in terms of mathematics knowledge, we will say that the
proposed task is too easy. That means that the mathematical activity for answering is
poor, even though understanding the statements may have a relatively high intellectual
cost for the students. Compare with the situation that would result from the following
questions, without changing the data (i. e. a class of 12 students, four of them being
boys):
We try to form groups, all having the same number of students (for
instance two groups of six students each), or all having the same numbers
of girls and boys (for instance two groups, each one composed of four
girls and two boys). What numbers of groups can we form in each case?
For such reasons, researchers later followed in France by official curricular
commissions, introduced precisions about problem solving, considering open problems
and problem for investigating. What follows is translated from a French text by Roland
Charnay (Charnay, 1993).
“The team of the IREM of Lyon offers the following definition.
An open problem is a problem that has the following characteristics:
The statement is short.
The statement does not induce either method, no solution (no
questions intermediate or questions like "show that"). This solution should
never be reduced to the use or the immediate application of the latest
results presented in class.
The problem is in a conceptual domain to which students have
enough familiarity. Thus, they can easily take "possession" of the situation
and engage in testing, conjecture, draft resolution, counterexamples.
Example 1 (extracted from "Rencontres Pédagogiques", n°12, INRP)
I have 32 coins in my piggy bank.
I only have coins of 2 F and 5 F.
The total amount of my 32 coins is 97 F.
How many coins of each value are in my piggy bank?
Example 2 (extracted from "Situations problèmes", APMEP, Elemmath IX).
Pluvinage
What is the biggest product of two natural numbers that we can obtain
using each digit from 1 to 9 once for writing the numbers?”
The reader can find more precisions about Problem for Investigating in
(Houdement, 2009), in particular a list of 24 representative statements, heuristic elements
and perspective for research.
1.2. OK but limited except for students at an advanced level…
In this section, we present a situation well appropriate for sequences of
investigation by 4 or 5 grade students. Thus it generates a good students’ training in
numerical treatments. But its institutionalization in Brousseau’s meaning will be difficult
until an advanced level, for instance undergraduate, because of the difficulty for
enunciating and proving the general result. So in this case, the teacher can exploit the
situation for younger students with an objective of arithmetical training or acquisition of
methods, but not of learning new mathematics knowledge.
“A statement for students of all levels:
The number 23 can be written in many ways as a sum of natural numbers.
For example:
23 = 11 + 5 + 7. Among these sums, find the one whose product of the
terms is the biggest (in our example, the product was 11×5×7 = 385).
Other statement is: Among the additive decompositions of a natural
number, find the one, whose product of its terms is the biggest.”
Retrieved from http://educmath.inrp.fr/applet/exprime/plgrprge.pdf and from
http://gilles.aldon.free.fr/ensemble/Ensemble/syplgrprtel.pdf
In this situation, we can first discover the answer, either in particular cases (such
as 23, that produces 4374 = 3×3×3×3×3×3×3×2 as highest product) or in the general case
in a descriptive way. At more advanced levels, a new possibility is to deepen the
TME, vol10, nos.1&2, p .223
problem, and express and prove the general solution in an algebraic way. So does Gilles
Aldon when he writes what follows.
The Sloane Encyclopedia (the OEIS Foundation) gives some properties of
this biggest product at http://oeis.org/A000792. It quotes a dozen of
different definitions of the number a(n) equal to the biggest product
obtained with partitions of n in sum. If we take a(0) = 1, then we obtain
the induction formula: a(n) = max{(n-i)a(i), i<n}. This is the definition
given in the Encyclopedia.
In such a situation, a teacher does not have possibility to adapt the situation with
some changes in the statement. He/she has only to know that the situation is convenient
for arithmetic training of young students, and for improving the use of induction by older
students.
1.3. Too difficult because insufficiently explored by the interested teachers…
Analyzing a problem of a mathematical contest: the “Math Rally of Alsace”
We translate here the statement of one of the problems of a mathematical contest,
the “Math Rally”, followed by the solution and comments given by the organizing team,
retrieved from http://irem.ustrasbg.fr/php/index.php?frame=.%2Fcompet%2Fcompet.php&m0=ral&categ=rallye.
Exercise 3 (grade 11)
In the Valley of Bruche River, a clearing has a circular shape. A treasure is
buried near the clearing. An old parchment shows the location of the
treasure: “From the great fir located on the circle, go to the poplar in the
clearing. Then turn right at a right angle and walk to the edge of the
clearing. Still turn right at a right angle and walk as many steps as from fir
to poplar. There is buried treasure.”
The clearing has a radius of 20 meters, and the only tree on the clearing is
a poplar located at 4 meters from the centre. Unfortunately, on the edge of
the clearing, firs disappeared long time ago. Can you find the distance
between the centre of the clearing and the treasure?
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Solution
As is often the case in Math Rally, the statement tells a story, placed here in
Alsace. The story is about a treasure, a fir tree, a poplar. Why not denote them T, S, P and
label O the centre of the clearing, i.e. the circle.
Figure 1: Geometric representation of the Math Rally problem
In the Cartesian coordinate system (O, I, J) with x- axis parallel to the line (SP),
let A and B the points obtained by orthogonal projection of P on the coordinate axes. We
know that OP = 4, OS = OD = 20. As SPDT is a rectangle, SP = DT and PD = ST. Let a
= OA, b = OB, c = PS and d = PD. Applying the Pythagorean Theorem to the triangles
OBS, OAD and OAP, we obtain the relationships:
OS2 = OB2 + BS2 = b2 + (a + c)2 = 202
(1)
OD2 = OA2 + AD2 = a2 + (b + d)2 = 202
(2)
OP2 = OA2 + OB2 = a2 + b2 = 42 = 1 (3)
Computing (1) + (2) – (3), we obtain:
(a + c)2 + (b + d)2 = 784.
TME, vol10, nos.1&2, p .225
Since (a + c)2 + (b + d)2 = OT2, then OT = 28.
Thus the distance that we sought is 28 meters.
Comments by the organizing team
This is a geometrical exercise and it was the less successful for students. Yet
its resolution only involves the Pythagorean Theorem, applied to several right
triangles of course.
Candidates who worked on this exercise drew a figure that includes letters
often undefined. Didn’t they have the habit of labeling a figure in order to make it
understandable? The resolution led them to the correct answer (28 meters), but most of
them relied on a particular case – for example points S, O and P collinear – and they
do not consider the general case. Some say that they were only considering a particular
case, but many students seem not to realize it. (…)
Our analysis
Thus in its report, the organizing staff itself recognizes that this problem was
unsuccessful. But we think that this poor result originates in a lack of mathematical
analysis by the organizing team. The members of this team are very experienced teachers
and competent mathematicians, but they were more preoccupied in this case by the
design of an amazing story than by didactical considerations. As a matter of evidence, we
refer to the solution given in the report, which is for us not satisfactory, because it is not
convincing. It is like the rabbit that comes out of the hat of a magician: we measured
lengths, and it appears that the final result does not depend on the variable elements. A
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“good” solution of a problem produces a change in our mind with respect to the situation:
from mysterious the result becomes evident for the reader. As an example, let us consider
a classical result: The three altitudes of a triangle ABC intersect in a single point, called
the orthocenter of the triangle. There are several proofs of this theorem, for instance by
considering geocenter and circumcenter of the given triangle ABC, or studying angles
ˆ tan B,
ˆ tan Cˆ is the searched
and finding that the point of barycentric coordinates tanA,
orthocenter of the triangle. These proofs are of interest, but their cognitive routes are
relatively complex. In contrast, the proof illustrated by Figure 2 easily could produce the
change of mind that we want to emphasize: We trace respectively by A, B and C the lines
parallel to the opposite side of the triangle, that create a new triangle A’B’C’; the
altitudes of ABC are the perpendicular bisectors of the sides of A’B’C’. Thus it is
obvious that they intersect at the point O equidistant of the three vertices A’, B’ and C’.
Figure 2: Altitudes of a triangle as perpendicular bisectors of a bigger triangle
For the Math Rally problem, we present below a solution approach that intends to
promote the reader’s mathematical reflection and analysis. Then we will consider what
changes could produce a better result in implementing this problem in the classroom.
To conclude that the distance between the center O of the circle and the point T
does not depend on the location of S on the circle is the same as to affirm that the locus of
TME, vol10, nos.1&2, p .227
T when S moves along the circle is a new circle with O as its center. This assertion does
not seem easier than the given statement of the problem (Figure 3 left), but…
Figure 3. Left: incomplete figure, right: completed figure
If we observe that the situation is mathematically incomplete, our vision of the
problem completely changes. And completing the figure is a natural idea when working
with software such as Cabri or Geogebra. This involves drawing a complete line as a
perpendicular to a given line or segment. After completing (figure above, right), a new
statement is: “Given a circle and a point P inside the disk, we consider two perpendicular
lines passing through P and respectively cutting the circle at D and D’, and at S and S’.
Then the rectangle obtained by tracing by these points parallels to DD’ and SS’ has its
center at O.” And this is a quite evident result, because the perpendicular bisectors of SS’
and DD’ are the axes of symmetry of the rectangle and pass through the center O of the
circle.
The given statement of the problem did not lead the candidates, of the
mathematical contest, a way to focus on a geometric construction. But we see that a
teacher in his (her) class can present the same mathematical situation through another
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statement and in a different environment that changes the didactical situation and
facilitates the students’ access to it. In this case, we pretend that a modified presentation
could change the vision that students have of that geometric situation.
As a conclusion at this point, we will assert that the kind of students’ work
generated by dealing with a problem strongly depends on the way showed by the
statement of the situation. But the form of working, particularly when using the
technology, has an influence too. This enforces the role of mediator devoted to the
teacher, not only when preparing the lesson but also during the class.
2. Project based learning (PBL or PjBL)
In the expression “Problem Solving” appears the word “Problem”, which is also
present in the pedagogical method called “Problem Based Learning” and often designated
by the initials PBL. This method is in use in various disciplines such as biology or
medicine or engineering, but problems for the associated topics do not refer to the same
kind of mathematical problems that we studied in §1. For our part, in empiric
experiments, we studied the potential impact of projects of practical action on the
teaching-learning process. We were applying a teaching strategy, presented in Cuevas &
Pluvinage (2003), close to the method denominated Project Based Learning, which
sometimes is also designated by the initials PBL. Here, like other authors, we use the
letters PjBL in order to avoid confusions with Problem Based Learning.
Important features of our teaching strategy lie in the systematic use of registers of
representation according to Duval’s terminology (Duval, 1995): formation rules,
treatments within a register, and conversions from one register to other. For this reason, a
TME, vol10, nos.1&2, p .229
first step in the study of a project of practical action is a descriptive phase, the objective
of which being to introduce the formation in one register or various registers.
2.1. A paradigmatic example: concrete and virtual Tangram
Mexican educative institution presents Tangrams as a medium to organize
learning activities at school. The Instituto Latinoamericano de la Comunicación
Educativa (ILCE) has a web page on this subject at
http://redescolar.ilce.edu.mx/educontinua/mate/imagina/mate3z.htm, with a link to
instructions for cutting the puzzle. We retrieved the figure below from this web site. In
English, for instance Wolfram’s project gives a version at
http://demonstrations.wolfram.com/Tangram/.
Figure 4: The seven pieces of a Tangram and shapes to be built
The web page of ILCE presents both possibilities for using Tangrams: concrete
and virtual (on line). Wolfram’s proposal is only a virtual one. We actually observed
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some use of a concrete Tangram in class and related problems, but it remains to
experiment the more extended use that we present in this section.
There are many problems that the students are able to pose by themselves. For
example: Can we build a triangle with the seven pieces? With the seven pieces, can we
construct a triangle different from the preceding? Can we build with the seven pieces a
rhombus different from a square? How many distinct rectangles can we build with the
seven pieces? How many distinct rectangles can we build with some of the pieces, but not
necessary all pieces? Etc. In that situation, teacher’s role can be to encourage the students
to invent problems and not only to solve given problems.
An interesting activity in a class might be to classify these distinct problems.
Organizing principles for such a classification arise from considerations of shape
resulting from values of lengths, angles and areas. For example, the shape of each piece is
a polygon with angles 45º, 90º or 135º, and this leads us to a strength restriction for the
possible shape of a triangle made with the seven pieces: a triangle necessarily is a right
isosceles triangle. With exact lengths, we enter in the symbolic world, because the
lengths of the sides of a piece in two distinct directions are incommensurable magnitudes.
In the table below, we present the exact lengths of the sides of pieces with the choice of
the length of the shortest side as unit.
TME, vol10, nos.1&2, p .231
Table 1. Names of the Tangram pieces and exact lengths of their sides
Name
Side 1
Side 2
Side 3
Side 4
Square
1
1
1
1
Parallelogram
1
2
1
2
Triangle 1
1
1
2
Triangle 2
1
1
2
Triangle 3
2
2
2
Triangle 4
2
2
2 2
Triangle 5
2
2
2 2
Figure 5: A shape impossible to construct with Tangram
Then, for instance when studying the situation of a rhombus different from a
square (Figure 5), we can first observe that its angles might be 45º and 135º and then use
a c
the algebraic register and employ inferences like a b 2 c d 2
.
b d
Problem to be solved by the reader: Prove the impossibility of constructing a
rhombus different from a square with the seven pieces of Tangram.
We suggest that all the problems we have seen can be included into a project of
practical action, and in order to do this, we will replace the concrete material by a virtual
one. Then the first step is to construct the material. We used Cabri geometer for this
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purpose, but the design would be approximately the same if we were using Geogebra.
Thus it is necessary to solve a first problem in our project: How to move a piece in the
plane?
There is a natural relationship between movements and geometric
transformations. Nowadays, it is controversial if geometric transformations have a place
in the curricula at elementary level. In our opinion, moreover with the reference to a
genetic point of view, the geometric transformations are a topic of high interest in
mathematics education. We suggest using the introduction of transformations in the
learning of geometry at secondary level in order to facilitate the transition from Geometry
I (natural geometry) to Geometry II (natural axiomatic geometry) (Houdement &
Kuzniak, 1999).
1
2
3
4
Figure 6. Modeling the displacement of a quadrilateral in the plane: 1- Choosing a center
(here the midpoint of two midpoints), 2- Translating the piece, 3- Rotating the translated
piece around its center, 4- Showing only the final figure with the two director points
(black and red)
TME, vol10, nos.1&2, p .233
Figure 6 illustrates a solution obtained by a Cabri program. For each piece, we
choose a center point that is represented by a black dot, and a vertex that is represented
by a red dot. When moving the black dot, we translate the piece, and when moving the
red dot, we rotate the piece around the black dot. In order to do that, first we build each
piece and mark the black and red dots. Then we define the translation and the rotation
and apply them to the piece. Finally we hide the initial piece and only show its last
image.
The reader can imagine alternative possibilities, and also complete the universe of
allowed transformations by adding the reflection of the parallelogram that we do not
introduce in our program (with a concrete jigsaw puzzle, we can turn over the
parallelogram). And one can find many interactive programs on line in Spanish and in
English, but almost all such web pages are only game oriented. For example, we don’t
see there any place for verbal descriptions or for problems whose solutions are
“impossible” to achieve. So, in this case, it seems to us preferable to promote students’
work in classroom or at home. The teacher can construct a virtual Tangram and give
his/her students a web page like that illustrated in the figure below, or ask them for
constructing the virtual Tangram.
Pluvinage
Figure 7: Virtual Tangram designed for students
Assigning the task of constructing a virtual Tangram to students by using
geometric software is a good example of what Martin Wagenschein (1977) presents as
Exemplarisches Lehren. The translation “exemplary teaching” does not exactly reflect the
meaning of this expression, which refers to a specific kind of case analysis. The
specificity of this method is that the studied cases are chosen as representative of the
situations to be mastered by the student at the end of the learning process. Here, we
pretend that the achievement of the construction of Tangram is significant for
acquisitions of certain level of competency in mastering geometric transformations.
2.2. Some didactic observations resulting from empirical studies using projects
Most projects of practical action include spatial situations. So their modeling
supposes to connect spatial geometry (3D-geometry) and 2D-geometry. This can be made
TME, vol10, nos.1&2, p .235
by perspective representations or plane sections and also by nets, a net being an
arrangement of edge-joined polygons in the plane which can be folded (along edges) to
become the faces of the polyhedron (see
http://en.wikipedia.org/wiki/Net_%28polyhedron%29). A great family of projects of
practical action is the construction of solids subject to certain constraints.
The study of nets associated to polyhedra is a particular case of these spatial
projects, which can be applied at high school or more advanced level. When we were
experimenting in undergraduate classes, we realized that the majority of students do not
master the easiest spatial situations, because of the lack of both knowledge and
experience. For instance it was a great surprise for them to know that all convex
polyhedron has a net, and that the “tower” made by the superposition of two cubes, the
edges of the higher being a third of the edges of the lower, is a polyhedron whose
construction with thin cardboard requires two disconnect nets.
Figure 8. “Tower”: Non convex polyhedron that does not have a net
The preceding example illustrates one of the natural problems that arise from
studying the nets associated to polyhedra. Other problems are those of uniqueness: Can
we obtain two different polyhedra with the same plane net?
Pluvinage
This image cannot currently be displayed.
Figure 9: Same net for two different polyhedra?
Problem to be solved by the reader: The net above allows constructing a
regular octahedron by folding paper. Try to find another (non convex)
octahedron that the same net allows constructing.
Modeling spatial situations goes further than 3D-geometry. There is a lot of
possible projects about containers (glasses, bottles, cans, recipients, etc.). A task for our
students of the Master degree in mathematics education is to design a didactical project,
and some of them choose this kind of subject. We present below the statement of a
problem, which is a part of such a project that a student presented in a Web site, followed
by its English translation and the representation of the solution that we made with
Geogebra.
Optimización
Escrit o por Paulo Angel Garcia Regalado
Dom ingo, 05 de Diciem bre de 2010 01: 32
1. Se pretende fabricar una lata de refresco de 335 mililitros de capacidad. ¿Cuáles
deben ser sus dimensiones para que se utilice el mínimo posible de metal?
Retrieved from http://grupolycaon.com/matedu/
Translation of the statement to English:
TME, vol10, nos.1&2, p .237
We want to make a soda can of 335 ml capacity. What should be its size in order to use
the minimum amount of metal?
Figure 10. Problem to be solved by the reader: In the solution
represented above for the problem of constructing a can with minimal
quantity of metal, it seems that the height of the can is equal to the
diameter of both disks at the bottom and the top. Is that rigorously exact?
In this last example, we saw calculus beside geometry. That is only a sample of
the variety of mathematical domains and theories that the study of projects led us to
encounter. Particularly, the project of modeling how recipients are filled by a regular
flow of water has been a wealth field.
For instance 15-year students in Germany and France were asking by Stölting
(2008) to represent in this situation the height of water in the recipient as a function of the
time. We reproduce below the answers given to Stölting by a student. The last figure on
the right is very interesting, because it shows a trend to the discretization of the
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phenomenon. Previously we emphasized the importance of verbal descriptions of a
phenomenon to be studied. Here we observe another important step in an investigation
process for a complex phenomenon: a qualitative approach with the help of a
representation. Both verbal description and figural representation were actually present in
Stölting’s research. Hence, students were interviewed to explain ways they drew the
figures.
Figure 11: Answers of a (high achiever) French student
Interviews with high school students in order to deepen these observations with
learning objectives are reported by Pluvinage & Marmolejo (2012) (see an illustration on
next page), who describe the application in a class of the collaborative method
ACODESA (Hitt, 2007), characterized by distinct phases of work: individual, group and
collective, for exploiting the complete situation of filling recipients. We observed that a
first step is important, namely to consider various cylindrical recipients. Indeed, many
TME, vol10, nos.1&2, p .239
students think that, in the case of a cylindrical recipient, diameter and height play the
same role for the volume. So it is important to pay attention to this case studying the
volumes of various cylindrical recipients, for instance as pouring water from one
recipient to another that has double diameter and half height. Many students are surprised
by the fact that the second recipient is not full. Moreover the same behavior as that
related before was observed with our students. With access to the convenient
mathematical tools, this would lead us to use an ordinary differential equation for solving
the problem.
Nevertheless there is another easier way for modeling the phenomenon, changing
the theoretical frame of reference. It is the consideration of reciprocal function. Indeed,
the volume of a truncated cone of height h, radio of lower disk a, and radio of upper disk
b, is given by Vtotal
h
3
a
2
ab b2 . Thus we can represent the volume as a function
of the height, and then by symmetry we can conversely obtain the height as a function of
the height. We used Geogebra in the illustration. But, and we do not know the precise
reason, students do not spontaneously go on that way. This needs a help to the students
from the teacher.
Pluvinage
Figure 12. Answer given by an interviewed high school Mexican student: “The same
amount a of water produces a rise that always diminishes when the height increases”
Figure 13: Solution resulting from consideration of the reciprocal function of volume as
function of height
TME, vol10, nos.1&2, p .241
How could a teacher exploit these reported observations and considerations, for
designing a possible instructional route? The first stages to do this are common to many
practical action projects:
(a) A diagnostic, here with a sheet of questions as we have seen about
comparing volume of different cylindrical recipients and representing
how various recipients are filled by a regular flow;
(b) Some concrete experiment, for instance here to pour water or sand
from a recipient to another and observe if the second is fulfilled or
not. In our environment it is easy to get recipients that have the
required characteristics: shape approximately cylindrical and
respective diameters and heights approximately having simple ratios
(for instance a box of coarse salt and a glass, or tin cans of varying
size).
At the end of each stage, a discussion and a synthesis (e.g. volume of cylinder)
will be useful. For the following stages, with the use of computation, the knowledge of a
truncated cone and its volume formula are necessary. The teacher has the choice between
giving the students these elements, and asking the students to obtain them by searching
on Internet. Then the stages are:
a) Figural representation of the transversal section of a truncated cone of
varying height and diameters of upper disc and lower disc (see figure
on the preceding page), and graphical representation of its volume V as
a function of the height h of liquid in the recipient;
b) In order to find now the inverse of the obtained function, i.e. the height
h of liquid as a function of the volume V, first observe the difficulty to
work only with a formula and, as a consequence of this difficulty,
introduce the students into the work with geometric register,
particularly with the use of reflection. Construction depends on
software in use, for instance Cabri could reflect the graph of the
function V(h) and Geogebra not, so with Geogebra we had to locate a
point on the graph and then to reflect this point and eventually obtain
its locus.
The time devoted to this project is a good investment with the objectives of
improvement of the functional thinking and acquisition of the difficult procept (amalgam
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of process, mathematical object and symbol introduced by Eddie Gray and David Tall) of
an inverse function.
3. Concluding
Teaching ways that we found rich of learning perspectives for the students consist
in managing practical action projects at school, particularly in collaborative environment,
for instance with the so called ACODESA method (Hitt, 2007). An important feature of
these teaching ways is that the teacher is not the only one who is posing problems or
ideas of problems. Nevertheless, his/her experience is essential for redacting in a correct
mathematical language the statements suggested by students. And this step of writing
mathematics also is extremely important for the students in terms of improvement of their
mathematical experience.
The author is grateful to the reviewers, whose pertinent suggestions allowed to improve
the first version of this paper.
References
Charnay, R. (1993) Problème ouvert, problème pour chercher, Grand N, nº51, retrieved
from http://www.crdp.ac-grenoble.fr/imel/nx/n51_6.htm
Cuevas, C. A. & Pluvinage, F. (2003). Les projets d’action pratique, éléments d’une
ingénierie d’enseignement des mathématiques, Annales de didactique et sciences
cognitives, Vol. 8, 273–292.
Duval, R. (1995), Sémiosis et pensée humaine : registre sémiotique et apprentissages
intellectuels, Peter Lang, Berne.
Hitt, F. (2007). “Utilisation de calculatrices symboliques dans le cadre d’une méthode
d’apprentissage collaboratif”, de débat scientifique et d’autoréflexion. In M. Baron,
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D. Guin et L. Trouche (Éds.), Environnements informatisés pour l’éducation et la
formation scientifique et technique : modèles, dispositifs et pratiques. Paris: Hermes.
Houdement, C. (2009), Une place pour les problèmes pour chercher, Annales de
Didactique et de Sciences Cognitives, Volume 14, 31-59
Houdement, C. & Kuzniak, A. (1999). Un exemple de cadre conceptuel pour l’étude de
l'enseignement de la géométrie en formation des maîtres, Educational Studies in
Mathematics, Volume 40, Number 3, 283-312
Pluvinage, F. & Marmolejo Rivas, E. (2012) Une recherche didactique recourant à la
modélisation et au travail collaboratif : un cas d’étude de paramètres, in Formation à
la Recherche en Didactique des Mathématiques, Loze-Dion éditeurs, Québec,
Canada, pp. 11-24.
Schoenfeld, A. (2006). Problem Solving from Cradle to Grave. Annales de Didactique et
de Sciences Cognitives, Volume 11, 41- 73
Stölting, Pascal, (2008), La pensée fonctionnelle des élèves de 10 á 16 ans. Une étude
comparative de son enseignement en France et en Allemagne, Thèse de doctorat,
Université de Paris Diderot
Wagenschein, M. (1977), Verstehen lernen, genetisch-sokratisch-exemplarisch, Beltz
Verlag: Basel; Weinheim.
Pluvinage
TME, vol10, nos.1&2, p.245
Thoughts About Research On Mathematical Problem- Solving
Instruction
Frank K. Lester, Jr.
Indiana University, Bloomington, USA
Abstract: In this article, the author, who has written extensively about mathematical
problem solving over the past 40 years, discusses some of his current thinking about the
nature of problem-solving and its relation to other forms of mathematical activity. He
also suggests several proficiencies teachers should acquire in order for them to be
successful in helping students become better problem solvers and presents a framework
for research on problem-solving instruction. He closes the article with a list of principles
about problem-solving instruction that have emerged since the early 1970s.
Keywords: mathematical activity, problem solving, problem-solving instruction,
proficiencies for teaching, craft knowledge, research design, teaching as a craft, teacher
planning, metacognition..
Introduction
My interest in problem solving as an area of study within mathematics education
began more than 40 years ago as I was beginning to think seriously about a topic for my
doctoral dissertation. Since that time, my interest in and enthusiasm for problem solving,
in particular problem-solving instruction, has not waned but some of my thinking about it
has changed considerably. In this article I share some of my current thinking about a
variety of ideas associated with this complex and elusive area of study, giving special
attention to problem-solving instruction. To be sure, in this article I will not provide
much elaboration on these ideas and careful readers may be put off by such a cursory
discussion. My hope is that some readers will be stimulated by my ideas to think a bit
differently about how mathematical problem solving, and in particular problem-solving
instruction, might be studied.
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos.1&2, pp.245-278
2013©The Author(s) & Dept of Mathematical Sciences-The University of Montana
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Setting the stage
Most mathematics educators agree that the development of students’ problemsolving abilities is a primary objective of instruction and how this goal is to be reached
involves consideration by the teacher of a wide range of factors and decisions. For
example, teachers must decide on the problems and problem-solving experiences to use,
when to give problem solving particular attention, how much guidance to give students,
and how to assess students’ progress. Furthermore, there is the issue of whether problem
solving is intended as the end result of instruction or the means through which
mathematical concepts, processes, and procedures are learned. Or, to put it another way,
should teachers adopt “teaching for problem solving, ”—an ends approach—or “teaching
1
via problem solving” —a means approach? (I say more about means and ends later in
this article.) In my view, the answer to this question is that both approaches have merit;
problem solving should be both an end result of learning mathematics and the means
through which mathematics is learned (DiMatteo & Lester, 2010; Stein, Boaler, & Silver,
2003). Whichever approach is adopted, or if some combination of approaches is used,
research is needed that focuses on the factors that influence student learning.
Unfortunately, as far as I know, no prolonged, in-depth, programmatic research of this
sort has been undertaken and, as a result, the accumulation of knowledge has been very
slow. Moreover, the present intense interest in research on teachers’ knowledge and
1
It has become more common to refer to the “means” approach to teaching as teaching through
problem solving. In Schroeder and Lester (1989) we discuss three approaches to problem-solving
instruction: teaching about, for, and via problem solving. Teaching “via” problem solving is
essentially the same as teaching through problem solving. Today, teaching about problem
solving is not generally regarded as a legitimate instructional method, although I suspect that
some (many?) teachers and curriculum writers subscribe to this approach.
TME, vol10, nos.1&2, p .247
proficiencies demands that future problem-solving research pay close attention to the
mathematical and pedagogical knowledge and proficiencies a teacher should possess (cf.,
Ball, Thames, & Phelps, 2008; Hill, Sleep, Lewis, & Ball, 2007; Moreira & David, 2008;
Zazkis & Leikin, 2010).
But before discussing problem-solving instruction, let me first say a few things
about mathematical problem solving. This short discussion will highlight how my
thinking has changed about the nature of problem solving and other forms of
mathematical activity.
Some claims about Problem Solving
Among the many issues and questions associated with problem-solving
instruction I have worried about during my career, several have endured over time. In
this section I make five claims related to these enduring issues and offer brief
discussions of my current thinking about them.
Claim 1. We need to rethink what we mean by “Problem” and “Problem Solving”
Although there have been at least four distinct problem-solving research traditions
within (namely, Gestalt/Cognitive, Learning/S-R, Computer/Information Processing, and
Psychometric/Component Analysis), they all agree that a problem is a task for which an
individual does not know (immediately) what to do to get an answer (cf., Frensch &
Funke, 1995; Holth, 2008). Some representative definitions illustrate this fundamental
agreement:
A problem arises when a living creature has a goal but does not know how this
goal is to be reached. (Duncker, 1945, p. 1)
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A question for which there is at the moment no answer is a problem. (Skinner,
1966, p. 225)
A person is confronted with a problem when he wants something and does not
know immediately what series of actions he can perform to get it. (Newell &
Simon, 1972, p. 72)
Whenever you have a goal which is blocked for whatever reason . . . you have a
problem. (Kahney, 1993, p. 15)
These definitions have two common ingredients: there is a goal and the
individual (i.e., the problem solver) is not immediately able to attain the goal. Moreover,
researchers irrespective of tradition, view problem solving simply as what one does to
achieve the goal. Unfortunately, these definitions and descriptions, like most of those
that have been given of mathematical problem solving – including those I and other
mathematics educators have proposed – are unhelpful for thinking about how to teach
students to solve problems or to identify the proficiencies needed to teach for or via
problem solving. A useful description should acknowledge that problem solving is an
activity requiring an individual (or group) to engage in a variety of cognitive actions,
each of which requires some knowledge and skill, and some of which are not routine.
Furthermore, these cognitive actions are influenced by a number of non-cognitive factors.
And, although it is difficult to define problem solving, the following statement – which
Paul Kehle and I devised a few years ago– comes much closer to capturing what it
involves than most of those that have appeared in the literature.
Successful problem solving involves coordinating previous experiences,
knowledge, familiar representations and patterns of inference, and intuition in an
TME, vol10, nos.1&2, p .249
effort to generate new representations and related patterns of inference that
resolve some tension or ambiguity (i.e., lack of meaningful representations and
supporting inferential moves) that prompted the original problem-solving activity.
(Lester & Kehle, 2003, p. 510)
The advantage of this description of problem solving over the others lies in its
identification of several key ingredients of success: coordination of experience,
knowledge, familiar representations, patterns of inference, and intuition. So, to be a
successful problem solver, an individual must have ample relevant experience in learning
how to solve problems, strong content knowledge, proficiency in using a variety of
2
representations and a solid grasp of how to recognize and construct patterns of inference.
Moreover, it recognizes the importance of intuition in successful problem solving3. With
the possible exception of intuition, each of these ingredients should be attended to any
program aimed at equipping prospective teachers with the proficiencies needed to teach
mathematics either for or via problem solving. I say more about the implications of this
description for the education of mathematics teachers later in this article. But first, let me
continue with a few more observations about the nature of problem solving.
Claim 2. We know very little about how to improve students’ metacognitive abilities.
So much has been written about metacognition and its place both in the teaching
and learning of mathematics that a few comments about this elusive construct seem
warranted. I remain convinced that metacognition is one of the driving forces behind
2
The research perspective on the role of representation in doing mathematics provided by Goldin
(2003) is particularly relevant to this discussion.
3
A reviewer pointed out to me that intuition is itself a very subtle notion and as such the
definition we propose is unhelpful. To be sure, intuition is a subtle idea, but I think it essential to
include it in any description of what problem solving entails because it serves to point out just
how subtle the act of problem solving can be and, consequently, how difficult it has been to make
progress in learning how to teach students to be better problem solvers.
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successful problem solving (Garofalo & Lester, 1984), but we really know almost
nothing about what teachers should do to develop students’ metacognitive abilities. To
be sure, it is essential that successful problem solvers be able to monitor and regulate
their cognitive behaviors. But, almost no research has been done that demonstrates that
students’ can be taught these behaviors. Within the mathematics education community
both Schoenfeld (1992) and I (Lester, Garofalo, & Kroll, 1989), among others, have
conducted research aimed at enhancing students’ metacognitive abilities, but neither of us
has identified the proficiencies teachers need to do this. Instead, we have offered
suggestions, with too little evidence to support them. So, any program designed to
enhance mathematics teachers’ proficiencies that pays heed to metacognition should do
so only after acknowledging that there is no conclusive research evidence to support any
particular method of metacognition instruction over another.
Claim 3. Mathematics teachers needn’t be expert problem solvers; they must be serious
students of problem solving.
It is natural to suggest that teachers must themselves be expert problem solvers
before they are to be considered proficient mathematics teachers. But, I think this is
asking too much of them! George Polya was an expert (and, hence, proficient) problem
solver as well as an expert teacher of mathematics, but to expect all teachers to be experts
is both unreasonable and unnecessary. After all, expert basketball coaches needn’t have
been expert basketball players and expert violin teachers needn’t have been
concertmasters. Of course, teachers should be experienced problem solvers and should
have a firm grasp of what successful problem solving involves, but care should be taken
TME, vol10, nos.1&2, p .251
not to confuse proficiency in teaching students to solve problems with expertise as
problem solvers.
Claim 4. Problem solving isn’t always a high-level cognitive activity.
A fourth observation is that the description of problem solving I have given above
blurs the distinction between problem solving and other types of mathematical activity—I
have more to say about this blurring later. The distinctions that led historically to the
isolation of mathematical problem solving as a research focus from other areas of study
and the subsequent distinctions that resulted from this isolation are due in part to strong
traditions of disciplinary boundaries (Lester & Kehle, 2003).4 This isolation led to
subsuming mathematical understanding under problem solving. But, the inverse makes
more sense; that is, to subsume problem solving (and problem posing) under
mathematical understanding and, hence, under mathematical activity. By so doing,
emphasis is placed on several other constructs that are important in being able to do
mathematics — e.g., model building, generation of representations, constructing patterns
of inference — that too often are not considered when problem solving is isolated from
other forms of mathematical activity.
Claim 5. Research tells us something about problem-solving instruction, but not nearly
enough.
Although research on mathematical problem solving has provided some valuable
information about problem-solving instruction, we haven’t learned nearly enough (but
see also the very last section of this article). In a paper I co-authored about 20 years ago,
4
Indeed, some years ago, one mathematics education researcher asked me why I (and most other
problem-solving researchers) studied problem solving in isolation from learning specific
mathematics concepts and processes. I had no good answer; she was correct and I couldn’t
provide a compelling reason why we did so. Today, I think the reason stems from our reliance on
cognitive science for guidance in developing our research agendas and methods.
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my co-author and I identified four reasons for this unfortunate state of affairs: (1)
relatively little attention to the role of the teacher in instruction; (2) too little concern for
what happens in real classrooms; (3) a focus on individuals rather than small groups or
whole classes; and (4) the largely atheoretical in nature of problem-solving research
(Lester & Charles, 1992). I have discussed the fourth reason elsewhere (Lester, 2005), so
will not discuss it here. Instead, let me comment on the other three reasons. (Interested
readers may wish to read the provocative analysis of the state of mathematical problemsolving research written by Lesh and Zawojewski (2007). In their analysis they take
issue with the nature and direction of nearly all the research over the past 50 years.)
The role of the teacher. More than twenty-five years ago, Silver (1985) pointed
out that the typical research report might have described in a general way the
instructional method employed, but rarely was any mention made of the teacher's specific
role. Some progress has been made since then (see, e.g., the edited volumes by Lester
and Charles (2003) and Schoen and Charles (2003) and the review by Schoenfeld
(1992)). But, as useful as these efforts have been, they fall short of what is needed.
Instead of simply considering teachers as agents to effect certain student outcomes, their
role should be viewed as one dimension of a dynamic interaction among several
dimensions of a system involving: the role of the teacher, the nature of classroom tasks,
the social culture of the classroom, the use of mathematical tools as learning supports,
and issues of equity and accessibility. Changing any of the dimensions of this system
requires parallel changes in each of the other dimensions (Hiebert et al., 1997).
Observations of real classrooms. Several years ago, my colleague, Randy
Charles, and I conducted a large-scale study of the effectiveness of an approach to
TME, vol10, nos.1&2, p .253
problem-solving instruction based on ten specific teaching actions (Charles & Lester,
1984). The research involved several hundred fifth and seventh grade students in more
than 40 classrooms over the period of one full school year. The results were
encouraging: students receiving the instruction benefited tremendously with respect to
several key components of the problem-solving process. However, despite the promise
of our instructional approach, the conditions under which the study was conducted did
not allow us to make extensive, systematic observations of classrooms. Ours is not an
isolated instance. In particular, there has been a lack of descriptions of teachers'
behaviors, teacher-student and student-student interactions, and the type of classroom
atmosphere that exists. It is vital that such descriptions be compiled if there is to be any
hope of deriving sound prescriptions for teaching problem solving. In the final section of
this article I present a framework for research that, if used, might provide the sorts of
rich, detailed descriptions I think we need.
Focus on individuals rather than groups or whole classes. Throughout most of
the history research in mathematical problem solving (dating back about 50 years) the
focus has been on the thinking processes used by individuals as they solve problems or as
they reflect back on their work solving problems. When the goal of research is to
characterize the thinking involved in a process like problem solving, a microanalysis of
individual performance seems appropriate. However, when our concerns are with
classroom instruction, we should give attention to groups and whole classes. To be sure,
small groups can serve as an appropriate environment for research on teaching problem
solving, but the research on problem-solving instruction cannot be limited to the study of
small groups. In order for the field to move forward, research on teaching problem
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solving needs to examine teaching and learning processes for individuals, small groups,
and whole classes.
5
A Model of Complex Mathematical Activity
In addition to the lack of attention to the role of the teacher in real classrooms and
the focus on individuals rather than whole classes, the relative ineffectiveness of
instruction to improve students' ability to solve problems can be attributed to the fact that
problem solving has often been conceptualized in a simplistic way. This naive
perspective has two levels or "worlds": the everyday world of things, problems, and
applications of mathematics and the idealized, abstract world of mathematical symbols,
concepts, and operations. In this naive perspective, the problem-solving process typically
has three steps. Beginning with a problem posed in terms of physical reality, the problem
solver first translates the problem into abstract mathematical terms, and then operates on
this mathematical representation in order to come to a mathematical solution of the
problem. This solution is then translated into the terms of the original problem.
According to this view, mathematics may be, and often is, learned separately from its
applications and (too often) with no attempt to connect new mathematics concepts to old
ones. Teachers who adhere to this perspective are very concerned about developing
skillfulness in translating (so-called) real-world problems into mathematical
representations and vice versa. However, these teachers tend to deal with problems and
applications of mathematics only after the mathematical concepts and skills have been
introduced, developed, and practiced. Many of the “problems” found in textbooks often
5
The discussion in this section is excerpted with only minor revision from Lester and Kehle
(2003).
TME, vol10, nos.1&2, p .255
can be solved exactly as this naive perspective indicates. But for more challenging,
substantive problems, the problem solver cannot simply apply a previously learned
procedure to solve the problem. In addition to translation and interpretation, these
problems also demand more complex processes such as planning, selecting strategies,
identifying sub-goals, choosing or creating appropriate representations, conjecturing, and
verifying that a solution has been found. For non-routine tasks, a different type of
perspective is required, one that emphasizes the making of new meanings through
construction of new representations and inferential moves (refer back to the description
of problem solving Kehle and I (2003) have proposed).
The new perspective, like the previous one, also contains two levels representing
the everyday world of problems and the abstract world of mathematical concepts,
symbols and operations. In this perspective, however, the mathematical processes in the
upper level are "under construction" (i.e., being learned, as opposed to already learned;
coming to be understood, as opposed to being understood) and the most important
features are the relationships between steps in the mathematical process (in the
mathematics world) and actions on particular elements in the problem (in the everyday
world). It is in the forging of these relationships that results in the meaning making that
is central to mathematical activity of all kinds. At times the problem solver is learning to
make abstract written records of the actions that are understood in a concrete setting.
This involves the processes of abstraction and generalization. And, at other times the
problem solver attempts to connect a mathematical process to the real-world actions that
the mathematical process represents. Also, a problem solver who had forgotten the
details of a mathematical procedure would attempt to reconstruct that procedure by
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imagining the corresponding concrete steps in the world in which the problem was posed.
As a result, typically the problem solver moves back and forth between the two worlds—
the everyday problem world and the mathematical world—as the need arises.
But, although this perspective is an improvement over the original, it too falls
short of what is needed because it does not account for many of the most important
actions (both cognitive and non-cognitive) involved during real problem solving. Even
the modified perspective regards problem solving as somehow being different from other
sorts of mathematical activity. In my view, what is needed is to subsume problem
solving within a much broader category, "mathematical activity,” and to give a prominent
6
role to the metacognitive activity engaged in by the individual or group.
Figure 1 below depicts mathematical behavior as a complex, involved,
multiphase process that begins when an individual, working in a complex context (Box
A), poses (or is given) a specific task to solve (the solid arrow between A and B). To
start solving the task, the individual simplifies the complex setting by identifying those
concepts and processes that seem to bear most directly on the problem. This simplifying
and problem posing phase involves making decisions about what should be attended to
and what can be ignored, developing a sense about how the essential concepts are
connected, and results in a realistic representation of the original situation. This realistic
representation is a model of the original context from which the problem was drawn
because it is easier to examine, manipulate, and understand than the original situation.
Next comes the abstraction phase (solid arrow from B to C), which introduces
mathematical concepts and notations (albeit perhaps idiosyncratic). This abstraction
6
I should point out that this depiction is a representation of ideal, rather than typical, performance
during an individual's work on some mathematical task. It is ideal in the sense that it denotes key
actions in which the individual should engage in order to obtain acceptable results.
TME, vol10, nos.1&2, p .257
phase involves the selection of mathematical concepts to represent the essential features
of the realistic model. Often the abstraction phase is guided by a sense of what a given
representation makes possible in the subsequent computation phase. The explicit
representation of the original setting and problem in mathematical symbolism constitutes
a mathematical representation of both the setting and the task/problem.
Once a problem solver has generated a mathematical representation of the original
situation, the realistic problem now becomes a specific mathematical problem related to
the representation. This mathematical problem acquires a meaning all its own, becoming
an isolated, well-defined mathematical problem (Box C).
The third phase of the process (from C to D) involves manipulating the
mathematical representation and deducing some mathematical conclusions—depicted in
the figure by the “computing” arrow. During this phase, the person draws upon her or his
store of mathematical facts, skills, mathematical reasoning abilities, and so forth. For
example, the problem might call for a solution of a system of equations and solving this
system of equations does not depend on the original context of the initial problem. The
final phase (from D to A, D to B, and D to C), then, should involve the individual in
comparing the conclusions/results obtained with the original context and problem, as well
as with the mathematical representation (refer to the dashed arrows between boxes). But,
the act of comparing does not occur only after conclusions are drawn and a solution is
obtained. Rather, it might take place at any time and at any point during the entire
process. Indeed, this regular and continual monitoring—metacognitive activity—of one's
work is a key feature of success on complex mathematical tasks. In general, the act of
comparing the current state of one's work, thinking, and decisions denotes how complex
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mathematical activity can be. The degree to which the individual chooses to compare her
or his current state with earlier states can be considered a determinant of task complexity
and, in fact, is the primary way to distinguish “routine” from “non-routine” tasks (i.e.,
problems). For example, performing routine calculations using whole numbers typically
requires little comparing, whereas work on more complex tasks might necessitate quite a
lot of comparing throughout ones work on it. In brief, then, the degree to which a task
can be considered problematic can be determined by the amount of “comparing”
involved.
Figure 1. A model of complex mathematical activity
To sum up, what Kehle and I have proposed is a blurring of the distinction
between problem solving and other mathematical activity emerging from research on
mathematical problem solving and constructivist thinking about learning. Furthermore,
we have proposed a blurring of task, person, mathematical activity, non-mathematical
TME, vol10, nos.1&2, p .259
activity, learning, applying what has been learned, and other features of mathematical
problem solving. A consequence of this blurring is that it necessitates some rethinking
about the proficiencies mathematics teachers need. In the next section I discuss these
proficiencies in light of the preceding discussions.
Proficiencies for Teaching Mathematics
The debate over the merits of direct (explicit) instruction versus constructivist
instruction has been raging for at least 50 years and any consideration of the
mathematical proficiencies needed for teaching mathematics must be made in view of
this debate. More specifically, the identification of such proficiencies must take into
account the assumptions that are being made about the nature of mathematics learning
and instruction, as well as about instructional goals. For example, a proponent of direct
instruction (e.g., Kirschner, Sweller & Clark, 2006) might view learning as simply a
matter of making a change in students’ long-term memories. But for a constructivist
teacher, in addition to making a change in students’ long-term memories, learning
involves much more. Constructivist teachers are concerned with (among other things)
how to help students select and use good procedures for solving problems (Gresalfi &
Lester, 2009). Clearly, these quite different perspectives on what mathematics learning
involves will have a tremendous influence on what teachers must be able to do in their
classrooms (i.e., what proficiencies they need). Furthermore, there is the matter of the
teacher’s goals. If problem solving is intended as the end result of instruction, one set of
proficiencies for teaching is needed, but if problem solving is the means through which
mathematical concepts, processes, and procedures are learned, then a different set of
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proficiencies may be called for. For example, the teacher for whom problem solving is a
means, would likely need to be very proficient at listening to and observing students as
they work on mathematical tasks (Davis, 1997; Yackel, 2003). And, quite naturally,
listening to students would play a much less important role for a teacher who mostly
lectures. Put more directly, consideration of how to include problem solving in a
mathematical-proficiencies-for-teaching framework should be done in view of the
assumptions the teacher makes about the nature of mathematics learning and the goals of
instruction.
But, what of the proficiencies needed to help students learn how to solve
problems? One consequence of subsuming problem solving under the broader heading
“mathematical activity,” is that it becomes more difficult to specify a precise set of
proficiencies teachers need. To illustrate, consider the task “Which is more 2/3 or 2/5?”
Does this task involve any problem solving on the part of the student? Maybe, maybe
not! Of course, one can “cross multiply” to determine that 2/3 is more (or use some other
previously learned procedure), or one may have had sufficient experience with fractions
to simply “know” that 2/3 is more. In these instances, one could argue that no problem
7
solving is going on. But suppose you are a 3rd grader who does not know of any
procedures to decide which is more. Without a prescribed method of attack, this task
might be used to help you better understand the meanings of numerator and denominator
and also help you see how useful one half can be as a fraction benchmark (Van de Walle,
2003). This is at the heart of what it means to teach via problem solving. But, what
7
One needn’t be a 3rd grader to find this task problematic. Over the past 4 years I have been
tutoring unemployed adults who hope one day to pass the US high school equivalency exam
(GED). Almost to a person, they do not know how to solve this task when they begin to study
with me.
TME, vol10, nos.1&2, p .261
proficiencies must teachers have who subscribe to a teaching via problem solving
approach? Of course, they must be adept at selecting good problems, at listening and
observing, at asking the right questions, at knowing when to prod and when to withhold
8
comment, as well as a host of other actions . These actions make up what Moore (1995)
has called the “craft of teaching.” Moore’s “image of a [proficient] teacher is that of a
skilled craftworker, a master machinist say, who knows exactly what she must do, brings
the tools she needs, does the work with straightforward competence, and takes pleasure in
a job well done. She does her work right every day, and every day's work fits the larger
plan of her project” (p. 5). For Moore a craft is a “collection of learned skills
accompanied by experienced judgment” (p. 5). So, the question is “How does one
become a craftsman?”
Thirty years ago, Randy Charles and I wrote a book in which we laid out an
instructional plan for teachers to follow in order to be effective in teaching students how
to solve mathematics problems (Charles & Lester, 1982). The plan focused on three
phases of instruction—Before, During, and After—and was organized around 10
“teaching actions.” Since then, the three phases have appeared in different guises in
various American elementary and middle school textbook series (e.g., the middle grades
Connected Mathematics series organizes activities around Launch, Explore, and
Summarize (Pearson Education Inc. 2011)). The features of our plan that most clearly
distinguish it from more “traditional” instructional plans have to do with the teacher’s
role and the nature of the classroom environment. However, this is far from sufficient;
knowing about the teaching actions is simply not enough! In addition to knowing what to
8
I, and various collaborators over the years, have used the word (teaching) “actions” to refer to
what the teacher does during the act of teaching.
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do, the teacher must also know when to do it and what the implications might be of the
action taken. In particular, teachers must be adept at: (1) designing and selecting tasks
and activities, (2) listening to and observing students as they engage with problemsolving activities, (3) making sure that instructional activities remain problematic for
students, (4) focusing on the methods students use to solve problems and being familiar
with problem-solving methods (e.g., heuristics, strategies) that are accessible to students,
and (5) being able to tell the right thing at the right time (cf., Cai, 2010; DiMatteo &
Lester, 2010; Hiebert, 2003). Moreover, teachers and students share responsibility for
creating and maintaining a classroom atmosphere that is conducive to exploring and
sharing ideas, cooperating with each other, and risk taking (Stephan & Whitenack, 2003).
Thus, for me, in addition to myriad other knowledge and skills, a proficient mathematics
teacher must be skillful at —
Designing and selecting appropriate tasks for instruction
Making sense of and taking appropriate actions after listening to and
observing students as they work on a task
Keeping tasks appropriately problematic for students
Paying attention to and being familiar with the methods students use to solve
problems
Being able to take the appropriate action (or say the right thing) at the right
time
Creating a classroom atmosphere that is conducive to exploring and sharing.
To be sure, teachers who have command of these and related teaching actions and
who also have considerable mathematics content knowledge appropriate for the level at
TME, vol10, nos.1&2, p .263
which they are teaching might be considered craftsmen. But, I think what separates a
craftsman from others has to do with the amount of planning and reflection that he or she
has done prior to and after instruction. Unfortunately, even though it seems clear to me
that the type and amount of planning a teacher does have tremendous impact on what
happens during instruction, teacher planning has been largely ignored as a factor of
importance in research on problem-solving instruction. Indeed, in most studies teacher
planning has not even been considered because the teachers in these studies have simply
implemented a plan that had been predetermined by the researchers, not the teachers.
Furthermore, it is no longer warranted to assume that the planning decisions teachers
make are driven totally by the content and organization of the textbooks used and,
therefore, need not be considered as an object of research. The challenge, then, is to
determine ways to provide these teachers with opportunities to acquire the proficiencies
needed to become craftsmen; opportunities that in my view are best provided through
apprenticeship experiences in their real-world context and situation (Collins, Brown, &
Newmann, 1990). To date, too little attention has been paid to studying the design and
implementation of apprenticeship programs for teacher education. This lack of attention
is unfortunate because I think apprenticeship training is the approach most likely to result
in highly proficient teachers—that is, teacher craftsmen skilled at teaching mathematics
via problem solving.
A Framework for Research on Problem-solving Instruction
Twenty years ago, Randy Charles and I developed a framework for research on
problem –solving instruction that was a synthesis of previous conceptualizations of
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teaching in general and mathematics teaching in particular (Lester & Charles, 1992).
Unfortunately, to my knowledge, other researchers have not adopted this framework. I
still think it could serve us well in designing research on problem-solving instruction and
I bring this article to a close by offering a slightly-modified sketch of what it consists of.
The framework is comprised of four broad categories of factors that we consider
essential in the conceptualization and design of research studies: (1) Non-classroom
factors, (2) Teacher planning, (3) Classroom processes, and (3) Instructional outcomes.
Of course, the categories overlap and the factors within each interact both within and
across categories.
Category 1: Non-classroom Factors
What goes on in a classroom is influenced by many things that exist or take place
apart from actual classroom instruction. For example, the teacher's and students'
knowledge, beliefs, attitudes, emotions, and dispositions all play a part in determining
what happens during instruction. Furthermore, the nature of the tasks used as well as the
contextual conditions present outside the classroom also affect instruction (e.g., course
schedules, school structures). There are six types of factors: teacher presage
characteristics, student presage characteristics, teacher knowledge and affects, student
knowledge and affects, tasks features, and contextual (situational) conditions.
Teacher and student presage characteristics. These are characteristics of the
teacher and students that are not amenable to change but which may be examined for
their effects on classroom instruction. In addition, presage characteristics serve to
describe the individuals involved. Typically, in experimental research these
characteristics have potential for control by the researcher. But, awareness of these
TME, vol10, nos.1&2, p .265
characteristics can useful in non-experimental research as well by helping researchers
make sense of what they are observing. Among the more prominent presage
characteristics are age, sex, and previous experience (e.g., teaching experience, previous
experience with the topic of instruction). Factors such as previous experience may
indeed be of great importance as we learn more about the ways knowledge teachers glean
from experience influences practice.
Teacher and student knowledge and affects. The teacher's and students'
knowledge (both cognitive and metacognitive) and affects (including beliefs) can
strongly influence both the nature and effectiveness of instruction. As a category, these
teacher and student traits are similar to, but quite different from, presage characteristics.
The similarity lies in the potential for providing clear descriptions of the teacher and
students. The difference between the two is that affects and knowledge may change, in
particular as a result of instruction, whereas presage characteristics cannot.
Task features. Task features are the characteristics of the tasks used for
instructional or assessment purposes. Historically, at least five types of features serve to
describe tasks: syntax, content, context, structure, and process (see Goldin & McClintock,
1984). Syntax features refer to the arrangement of and relationships among words and
symbols in a task. Content features deal with the mathematical meanings in the problem.
Two important categories of content features are the mathematical content area (e.g.,
geometry, probability) and linguistic content features (e.g., terms having special
mathematical meanings such as "less than," "function," "squared"). Context features are
the non-mathematical meanings in the task statement. Furthermore, context features
describe the problem embodiment (representation), verbal setting, and the format of the
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information given in the problem statement. Structure features can be described as the
logical-mathematical properties of a task. Structure features are determined by the
particular representation that is chosen for a problem. For example, one student may
choose to represent a task in terms of a system of equations, while another student may
represent the same problem in terms of some sort of guessing process. Finally, process
features represent something of an interaction between task and student. That is,
although problem-solving processes (e.g., heuristic reasoning) typically are considered
characteristics of the student, it is reasonable to suggest that a problem may lend itself to
solution via particular processes. A consideration of task process features can be very
informative to the researcher in selecting tasks for both instruction and assessment.
Contextual conditions. These factors concern the conditions external to the
teacher and students that may affect the nature of instruction. For example, class size is a
condition that may directly influence the instructional process and with which both
teacher and students must contend. Other obvious contextual conditions include
textbooks used, community ethnicity, type of administrative support, economic and
political forces, and assessment programs. Also, since instructional method provides a
context within which teacher and student behaviors and interactions take place, it too can
at times be considered a factor within this category. I should add that that these six areas
of consideration do not necessarily cover all possible influences; it is likely that there are
other influences that may be at least as important as the ones I have mentioned. Rather,
my intent is to point out the importance of paying attention to the wide range of factors
that can have an impact on what takes place during instruction.
Category 2: Teacher Planning
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Teacher planning is not clearly distinct from the other categories, in fact, it
overlaps each of them in various ways. Of particular interest for research are the various
decisions made before, during and as a result of instruction about student presage
characteristics, instructional materials, teaching methods, classroom management
procedures, evaluation of student performance, and amount of time to devote to particular
activities and topics. Unfortunately, teacher planning has been given too little attention
as a factor of importance in problem-solving instruction research. Indeed, in most studies
teacher planning has not even been considered because the teachers in these studies have
simply implemented a plan that had been predetermined by the researchers, not the
teachers. Furthermore, it is no longer warranted to assume that the planning decisions
teachers make are driven totally by the content and organization of the textbooks used
and, therefore, need not be considered as an object of research. A teacher's behavior
while teaching either for or via problem solving is certainly influenced by the teacher's
knowledge and affects. However, some of this behavior is likely to be determined by the
kinds of decisions the teacher makes prior to entering the classroom. For example, a
teacher may have planned to follow a specific sequence of teaching actions for delivering
a particular problem-solving lesson knowing that the exact ways in which these teaching
actions are implemented evolve situationally during the lesson. Or, if the knowledge
teachers use to plan instruction is knowledge gleaned from previous instructional
episodes, then we would search for those cases that significantly shape the craft
knowledge teachers use as a basis for planning and action. Future research should
consider how teachers go about planning for problem-solving instruction and how the
decisions made during planning influence actions during instruction.
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Category 3: Classroom Processes
Classroom processes include the host of teacher and student actions and
interactions that take place during instruction. Four dimensions of classroom processes
are apparent: teacher knowledge and affects; teacher behaviors; student knowledge and
affects; and student behaviors.
Both the teacher's and the students' thinking processes and behaviors during
instruction are almost always directed toward achieving a number of different goals,
sometimes simultaneously. For example, during a lesson the teacher may be assessing
the appropriateness of the small-group arrangement that was established prior to the
lesson, while at the same time trying to guide the students' thinking toward the solution to
a problem. Similarly, a student may be thinking about what her classmates will think if
she never contributes to discussions and at the same time be trying to understand what the
task confronting her is all about. In our framework, we have restricted consideration to
what the teacher thinks about and does to facilitate the students’ thinking and what the
student thinks about and does to solve a problem. We have not attempted to include a
complete menu of objects or goals a teacher might think about during instruction.
Teacher knowledge and affects. These processes include those attitudes, beliefs,
emotions, cognitions and metacognitions that influence, and are influenced by, the
multitude of teacher and student behaviors that occur in the classroom during instruction.
In particular, this dimension is concerned with the teacher's thinking and affects while
facilitating students’ attempts to understand a task, develop a plan for solving it, carry
out the plan to obtain an answer, and look back over the solution effort.
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Teacher behaviors. A teacher's knowledge and affects that operate during
instruction give rise to the teacher's behaviors, the overt actions taken by the teacher
during problem-solving instruction. Specific teacher behaviors can be studied with
regard to use (or non-use) as well as quality. The quality of a teacher behavior can
include, among other things, the correctness of the behavior (e.g., correct mathematically
or correct given the conditions of the problem), the clarity of the action (e.g., a clear
question or hint), and the manner in which the behavior was delivered (e.g., the verbal
and nonverbal communication style of the teacher).
Student knowledge and affects. Similar to the teacher, this subcategory refers to
the knowledge and affects that interact with teacher and student behaviors. The concern
here is with how students interpret the behavior of the teacher and how the students'
thinking about a problem, their affects, and their work on the problem affects their own
behavior. Also of concern here is how instructional influences such as task features or
contextual conditions directly affect a student's knowledge, affects, and behaviors.
Student behaviors. These behaviors include the overt actions of the student
during a classroom problem-solving episode. By restricting our attention to the problemsolving phases mentioned earlier, we can identify several behaviors students might
exhibit as they work on a task.
Category 4: Instructional Outcomes
The fourth category of factors consists of three types of outcomes of instruction:
student outcomes, teacher outcomes, and incidental outcomes. Most instruction-related
research has been concerned with short-term effects only. Furthermore, transfer effects,
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effects on attitudes, beliefs, and emotions, and changes in teacher behavior have been
considered only rarely.
Student outcomes. Both immediate and long-term effects on student learning are
included in this category, as are transfer effects (both near and far transfer). Illustrative
of a student outcome, either immediate or long-term, is a change in a student's skill in
implementing a particular problem-solving strategy (e.g., guess and check, working
backwards). An example of a transfer effect is a change in students' performance in
solving non-mathematics problems as a result of solving only mathematics problems.
Also, of special importance is the consideration of changes in students' beliefs and
attitudes about problem solving or about themselves as problem solvers and the effect of
problem-solving instruction on mathematical skill and concept learning; for example,
how is computational skill affected by increased emphasis on the thinking processes
involved in solving problems?
Teacher outcomes. Teachers, of course, also change as a result of their
instructional efforts. In particular, their attitudes and beliefs, the nature and extent of
their planning, as well as their classroom behavior during subsequent instruction are all
subject to change. Each problem-solving episode a teacher participates in changes her or
his craft knowledge. Thus, it is reasonable to expect that experience affects the teacher's
planning, thinking, affects, and actions in future situations.
Incidental outcomes. Increased performance in science (or some other subject
area) and heightened parental interest in their children's school work are two examples of
possible incidental outcomes. Although it is not possible to predetermine the relevant
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incidental effects of instruction, it is important to be mindful of the potential for
unexpected “side effects.”
Research on teaching in general points to the important role a teacher's knowledge
and affects play in instruction. Questions such as the following need to be investigated :
What knowledge (in particular, content, pedagogical, and curriculum knowledge) do
teachers need to be effective as teachers of problem solving? How is that knowledge best
structured to be useful to teachers? How do teachers' beliefs about themselves, their
students, teaching mathematics, and problem solving influence the decisions they make
prior to and during instruction?
The forgoing analysis of factors to be considered for research on problem-solving
instruction is intended as a general framework for designing investigations of what
actually happens in the classroom during instruction. As I mentioned earlier, there may
be other important factors to be included in this framework and that certain of the factors
may prove to be relatively unimportant. Notwithstanding these possible shortcomings,
this framework could serve as a step in the direction of making research in the area more
fruitful and relevant.
A Final, More Positive Note
I do not intend for my remarks to give the impression that I think mathematical
problem solving research has not amounted to much during the past 40 years or that
current research efforts are misguided. Indeed, quite the opposite is the case! Several
important principles have slowly emerged from the research since the early 1970s. I end
this article by listing these principles without comment: each principle could serve as the
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basis for an article or monograph. My hope is that this list, like much of the rest of my
article, will stimulate discussion among those who are interested in pursuing a research
agenda that includes problem solving at its core.
1. The prolonged engagement principle.
In order for students to improve their
ability to solve mathematics problems, they must engage in work on
problematic tasks on a regular basis, over a prolonged period of time.
2. The task variety principle. Students will improve as problem solvers only if
they are given opportunities to solve a variety of types of problematic tasks (in
my view, principles 1 and 2 are the most important of the seven).
3. The complexity principle. There is a dynamic interaction between
mathematical concepts and the processes (including metacognitive ones) used
to solve problems involving those concepts. That is, heuristics, skills, control
processes, and awareness of one’s own thinking develop concurrently with the
development of an understanding of mathematical concepts. (This principle
tells us that problem- solving ability is best developed when it takes place in
the context of learning important mathematics concepts.)
4. The systematic organization principle. Problem-solving instruction,
metacognition instruction in particular, is likely to be most effective when it is
provided in a systematically organized manner under the direction of the
teacher.
5. The multiple roles for the teacher principle. Problem-solving instruction that
emphasizes the development of metacognitive skills should involve the
teacher in three different, but related, roles: (a) as an external monitor, (b) as a
TME, vol10, nos.1&2, p .273
facilitator of students' metacognitive awareness, and (c) as a model of a
metacognitively-adept problem solver.
6. The group interaction principle. The standard arrangement for classroom
instructional activities is for students to work in small groups (usually groups
of three or four). Small group work is especially appropriate for activities
involving new content (e.g., new mathematics topics, new problem-solving
strategies) or when the focus of the activity is on the process of solving
problems (e.g., planning, decision making, assessing progress) or exploring
mathematical ideas.
7. The assessment principle. The teacher's instructional plan should include
attention to how students' performance is to be assessed. In order for students
to become convinced of the importance of the sort of behaviors that a good
problem-solving program promotes, it is necessary to use assessment
techniques that reward such behaviors.
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Framing the use of computational technology in problem solving
approaches
Manuel Santos-Trigo
Cinvestav-IPN, Mexico
Matías Camacho Machín
University of la Laguna, Spain
Abstract: Mathematical tasks are key ingredient to foster teachers and students’
development and construction of mathematical thinking. The use of distinct
computational tools offers teachers a variety of ways to represent and explore
mathematical tasks which often extends problem solving approaches based on the use of
paper and pencil. We sketch a framework to characterize ways of reasoning that emerge
as result of using computational technology to solve a task that involves dealing with
variation phenomena.
Keywords: problem solving, framework, the use of computational tools.
Introduction
It is widely recognized that the use of computational technology offers teachers
and students different ways to represent and explore mathematical problems or concepts.
There is also evidence that different tools might offer learners different opportunities to
think of problems in order to represent, explore, and solve those problems. What tools
and how should teachers integrate them in their teaching environments? What
instructional goals should teachers aim with the use of technology? In accordance to
Hegedus & Moreno-Armella (2009) “technology is here to transform thinking, and not to
serve as some prosthetic device to prop up old styles of pedagogy or curriculum
standards” (p. 398). Thus, it becomes important for teachers to discuss approaches to use
technology in order to guide their students to develop ways of thinking that favour their
comprehension of mathematical concepts and problem solving experiences. In particular,
teachers should discuss the extent to which the use of the tools helps them represent and
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos.1&2, pp.279-302
2013©The Author(s) & Dept of Mathematical Sciences-The University of Montana
Santos-Trigo & Camacho-Machín
explore mathematical tasks in ways that enhance and complement problem solving
processes that rely on the use of paper and pencil environment. The use of computational
tools in learning scenarios implies that teachers need to pay attention to and reflect upon
aspects that involve:
(a) The process shown by the subject to transform the artefact (material object)
into an instrument to represent, to comprehend mathematical ideas, and to solve
problems;
(b) The type of tasks used to foster students’ mathematical thinking;
(c) The ways of reasoning exhibited by the subjects during problem solving
activities;
(d) The role of teachers during problem solving sessions; and in general,
(e) The structure and dynamics of scenarios that promote the use of different tools
to learn mathematics and solve problems.
We introduce a pragmatic framework for teachers to organize learning activities
that promote the systematic use of technology. The framework provides teachers with the
opportunity to discuss aspects related to the presentation and exploration of mathematical
tasks through the use of a dynamic software in problem solving environments. The aim is
to identify and reflect on possible routes that teachers or researchers can follow to
structure and organize problem-solving activities that enhance the use of technology with
the purpose of furthering mathematics learning. We highlight a set of questions that
teachers can think of as a way to delve into the problem through the use of technology.
TME, vol10, nos.1&2, p .281
To this end, we chose a generic1 task that involves a variation phenomenon to illustrate
how the use of the tool fosters an inquiring approach to make sense of the posed
statement and to promote different ways of reasoning to explore and solve the task
(NCTM, 2009). Thus, focusing on ways to represent a variation phenomenon through the
tool demands that teachers identify, express, and explore mathematical relationships in
terms of visual, numeric, graphic, and algebraic approaches. “Conceptualization of
invariant structures amidst changing phenomena is often regarded as a key sign of
knowledge acquisition” (Leung, 2008, p. 137). Thus, teachers need to work on tasks
where the use of the tools provides them a set of affordances to identify and perceive
what parameters vary and what are maintained invariant within the problem structure.
Background and Rationale
Lester (2010) quotes the online Encarta World English Dictionary to define a
framework: “a set of ideas, principles, agreements, or rules that provides the basis or the
outline for something that is more fully developed at a later stage” (p. 60). Our notion of
framework includes initial arguments that describe patterns associated with the use of a
dynamic software in mathematical problem solving. “ A framework tells you what to
look at and what its impact might be” (Schoenfeld, 2011, p. 4). It is a pragmatic
framework that consists of episodes that could help practitioners re-examine and contrast
those frameworks that explain learners competences exhibited in paper and pencil
environments. It becomes a scaffolding tool to reflect on issues related to the use of tools
in learning scenarios.
1
Generic in the sense that the task represents a family of tasks where it is possible to
explore or examine optimization behaviours of the parameters involved in the task.
Santos-Trigo & Camacho-Machín
Schoenfeld (1985) proposed a framework to explain students’ problem solving
behaviours in terms of what he calls basic resources, cognitive and metacognitve
strategies, and students’ beliefs. Schoenfeld’s framework came from analyzing and
categorizing experts and students’ problem solving approaches that involve mainly the
use of paper and pencil tools. What happens when subjects use systematically
computational tools to make sense of problem statement, represent, explore and solve
problems? We argue that the use of technology introduces new information to
characterize the problem solver’s proficiency. For instance, one of the tasks used by
Schoenfeld involves asking the students to draw with straightedge and compass a circle
that is tangent to two intersecting lines where one point of tangency is a given P on one
line. Schoenfeld reports that students formulated several conjectures about the position of
the centre of such a tangent circle: (a) The centre of the tangent circle C is the midpoint
of the line segment between P and the point Q, where P and Q are equidistant from the
point of intersection V (Figure 1a); (b) The centre of the circle is the midpoint of segment
of the circular arc from P to Q that has centre V and radius |PV| (Figure 1b), etc.
(Schoenfeld, 2011, p. 31).
P
P
V
C
Q
Figure 1a: A student conjecture
V
C
Q
Figure 1b: Another student conjecture
Schoenfeld stated that the students picked up the straightedge and compass, tried
out their conjecture, and either accepted or rejected it on the basis of how good their
TME, vol10, nos.1&2, p .283
drawing looked. With the use of a dynamic software “good drawing” doesn’t depend on
subject’s skills to manage the straightedge and compass; rather, the tool provides the
affordances (precision of drawings, parameter movement, quantification of parameters,
loci, etc.) to deal or explore conjectures. That is, the use of a dynamic software provides
teachers ways to initially visualize and test empirically conjectures and, they often access
or develop relevant knowledge needed to verify and prove those conjectures (MorenoArmella & Sriraman, 2005; Santos-Trigo, 2010). For example, in Figures 2a and 2b, the
dotted circle drawn with the software provides elements to reject the corresponding
conjectures. Thus, the use of the tool offers relevant information to characterize and
foster the students’ problem solving competences. For example, students can explore
visually that the centre of the tangent circle lies on the perpendicular line to line PV at P
(Figure 2c) and use that information to construct a formal approach based on properties
embedded in that visual approach.
P
P
R
V
V
C
C
R'
Q
Q
Figure 2b: Another student conjecture
Figure 2a: A student conjecture
P
V
C
Q
Santos-Trigo & Camacho-Machín
Figure 2c: The centre of the tangent circle lies on the perpendicular to PV that passes
through P.
We argue that practitioners interested in using computational tools in their
learning activities can find in the problem solving episodes described in the next section a
quick reference to the type of mathematical discussions that might emerge during the
problem solving sessions. In addition, the episodes might provide directions to structure
a lesson plan where empirical, visual, graphic, and formal approaches can be considered
to organize a didactic route. We contend that the episodes can provide relevant
information that relates to what Jackiw and Sinclair (2009) call first and second order
effects of the use of the software (referring to The Geometer’s Sketchpad) in learning.
“First-order effects are a direct consequences of the affordance of the environment;
second-order effects are then a consequence of these consequences, and usually relate to
changes in the way learners think, instead of changes in what learners do” (p. 414). That
is, teachers could use the affordances associated with the software to encourage their
students to think of novel ways to represent dynamically problem situations. Software’
affordances (dragging, finding loci, quantifying parameters, etc.) provide ways to observe
changes or invariance of involved parameters. As a consequence, the use of the tool
allows the problem solver to develop ways of reasoning to examine parameters
behaviours that emerge as a result of moving mathematical objects within the task
representation or configuration. Heid & Blume (2008) stated “[t]he nature of a
mathematical activity depends not only on the mathematical demands of the task but also
on the process of the task as constructed by the doer” (p. 425). Thus, teachers with the
TME, vol10, nos.1&2, p .285
use of the tool might guide their students to think about the problem in different ways and
to discuss concepts and processes that appear during the exploration of the task.
A problem-solving episodes to deal with phenomena of variation
An example is used to illustrate, in terms of episodes, a route to think of the use of
technology to represent and explore the area variation of an inscribed parallelogram. The
first episode emphasizes the relevance for the problem solvers to comprehend the
statement in order to construct a dynamic representation that can help them visualize
parameter behaviours.
The task
Given any triangle ABC, inscribe a parallelogram by selecting a point P on one of
the sides of the given triangle. Then from point P draw a parallel line to one of the sides
of the triangle. This line intersects one side of the given triangle at point Q. From Q draw
a parallel line to side AB of the triangle. This line intersects side AC at R. Draw the
parallelogram PQRA (Figure 3). How does the area of inscribed parallelogram APQR
behave when point P is moved along side AB? Is there a position for point P where the
area of APQR reaches a maximum value? (Justify).
C
R
A
Q
P
B
Santos-Trigo & Camacho-Machín
Figure 3: Drawing a parallelogram inscribed into a given triangle.
Comprehension Episode
Polya (1945) identifies the process of understanding the statement of a problem as
a crucial step to think of possible ways for solving it. Understanding means being able to
make sense of the given information, to identify relevant concepts, and to think of
possible representations to explore the problem mathematically. The use of technology
could help teachers focus on the construction of a dynamic model as a means to pose and
explore questions that lead them to comprehend and make sense of tasks.
The comprehension stage involves questioning the statement and thinking of the
use of the tool to make sense and represent the task. For instance, what does “for any
given triangle” mean and how this can be expressed through the software?, what
information does one need to draw any triangle?, are there different ways to inscribe a
parallelogram into a given triangle?, and how can one draw a dynamic model of the
problem? are examples of questions where the problem solver could rely on the tool to
explore and discuss the problem. Thus, a route to answer these questions might involve
using Cabri-Geometry or The Geometer’s Sketchpad to draw triangle ABC (Figure 4) and
from P on AB draw a parallel line to CB (instead of AC). This line intersects side AC and
from that point of intersection, one can draw a parallel line to AB that intersects BC, thus,
the two intersection points and point P and B form an inscribed parallelogram, the
problem solver can ask: how is the former parallelogram related to the one that appears in
Figure 3? Do they have the same area for the same position of P? How can we recognize
that for different positions of point P the area of the parallelogram changes? This problem
TME, vol10, nos.1&2, p .287
comprehension phase is important not only to think of the task in terms of using the
software commands, but also to identify and later examine possible variations of the task.
For example, how does the area of a family of inscribed parallelograms, generated when
P is moved along AB, change (Figure 4)?
C
Q
A
R
P
S
B
Figure 4: Another way to inscribe a parallelogram in a given triangle.
Comment
Making sense of the problem statement is a crucial step in any problem solving
approach. The use of a dynamic software plays an important role in initially
conceptualizing the statement as an opportunity to pose and explore a set of questions.
That is, the use of the tool demands that the problem solver thinks of the statement in
terms of mathematical properties to use the proper software commands to represent and
explore the problem (Santos-Trigo & Espinosa-Pérez, 2010). In this case, teachers can
work on the task in order to identify task’s sketches that can help their students focus
their attention to particular concepts or explorations. Of course, the posed questions don’t
include all possible routes to examine the statement; rather they illustrate an inquiry
method to guide the problem solver’s reflection.
Santos-Trigo & Camacho-Machín
A Problem Exploration Episode
Teachers can use the software to draw a triangle by selecting three non- collinear
points and discuss conditions needed to draw it when for example three segments (instead
of three points) are given (the triangle inequality). The use of the software allows moving
any vertex to generate a family of triangles. This process broadens the cases for which the
problem can be analyzed. Then, they can select a point P on side AB to draw the
corresponding parallels to inscribe the parallelogram. With the help of the software it is
possible to calculate the area of the parallelogram and observe area values change when
point P is moved along side AB. Thus, it makes sense to ask whether there is a position of
P in which the area of the inscribed parallelogram reaches either its maximum or
minimum value. By setting a Cartesian system (an important heuristic) as a reference and
without using algebra, it is possible to construct a function that associates the length of
segment AP with the area value of the corresponding parallelogram. Figure 5 shows the
graphic representation of that function. The domain of the function is the set of values
that represents the lengths of AP when point P is moved along side AB. The range of that
function is the corresponding area values of the parallelogram associated with the length
AP. This graphic representation can be obtained through the software by asking: What is
the locus of point S (the coordinates of point S are length AP and area of APQR) when
point P moves along the side AB? It is important to observe that the graphic
representation is obtained without defining explicitly the algebraic model of the area
change of the parallelogram.
TME, vol10, nos.1&2, p .289
Figure 5: Representation and visual exploration of the problem.
This graphic approach to solve the problem provides an empirical solution. Both
visually and numerically it is possible to observe that in the given triangle the maximum
area of the inscribed parallelogram is obtained when P is situated at 2.30 cm from point
A. At this point, the area value of the parallelogram is 8.56 cm2. Based on this
information a conjecture emerges: When P is the midpoint of segment AB, then the
corresponding inscribed parallelogram will reach the maximum area value. Graphically
the behaviour of tangent line to the curve behaves at different points can be observed
(Figure 5). It can be seen that when the slope of the tangent line to the area graph is
positive the function increases, but when the slope is negative the function area
decreases.
Are there other ways to inscribe a parallelogram in triangle ABC? Figure 6 shows
three ways to draw an inscribed parallelogram and all of them have the same area for
different positions of point P. Also, Figure 7 shows that when point P is the midpoint of
side AB then triangle ABC can be divided into four triangles with the same areas.
Santos-Trigo & Camacho-Machín
Figure 6: Inscribing three parallelograms in triangle ABC.
Figure 7: When point P is situated at the midpoint of segment AB, then triangles APR,
PQR, PBQ, and RQC have the same area.
From Figures 6 and 7 two conjectures emerge: (i) the three inscribed
parallelograms always have the same area for different positions of point P, and (ii) when
point P is the midpoint of segment AB, the four triangles always have the same area and
the maximum area of the inscribed parallelogram is half the area of the original or given
triangle. Thus, the use of the tool provides an opportunity for the problem solver to
simultaneously examine properties of figures that within the configuration. These
conjectures are proved further down.
Comment
TME, vol10, nos.1&2, p .291
The dynamic representation becomes a source that generates mathematical
conjectures as a result of moving objects within the configuration. Exploring different
ways to inscribe the parallelogram leads to formulate two related conjectures. In addition,
the use of the tool allows graphing the area’s variation without defining explicitly an
algebraic model. Thus, it is possible to think of a functional approach, without defining
the function algebraically, that associates the position of point P (for example, the
distance between AB, BP or AC) with the corresponding area value. Figure 5 provides a
visual and numerical approach to describe the parallelogram’s area behaviour.
The Searching for Multiple Approaches Episode
We argue that if students are to develop a conceptual understanding of
mathematical ideas and problem solving proficiency, they need to think of different ways
to solve a problem or to examine a mathematical concept. In this context, the visual and
empirical approaches used previously to explore the problem provide a basis to introduce
other approaches. We argue that each approach to the problem demands that the problem
solver not only think of the problem in different ways; but also to use different concepts
and resources to solve it.
Analytical approach
In this approach, the use of the Cartesian system becomes important to represent
the objects algebraically. The problem can be thought in general terms as shown below.
Santos-Trigo & Camacho-Machín
y
C
y = m1x
Q
R
y = m1(x - x1)
y = m3(x - x2)
1
x
1
A
P(x1, 0)
B(x2, 0)
Figure 8: Using a Cartesian system to construct an algebraic model of the problem
General case
Without loosing generality, we can always situate the Cartesian system in such a way
that one side of the given triangle can be on the x-axis and the other side on line
y m1 x (Figure 8). Point P will be located on side AB and its coordinates will be
P(x1 ,0). Point B(x 2 ,0)is vertex B of the given triangle (Figure 8). The general goal
is to represent the area of parallelogram APQR in terms of known parameters. This
process leads to represent the area in terms of one variable ( AP x1) as:
A(x1 )
m1 m 3 x12 x 2 x1
m1 m 3
. The roots of A(x1) (a quadratic function) are 0 and x2.
m1 m 3
0
m
m
3
. We are
Also, this function has a maximum value if and only if 1
assuming that m1 0. The assumption on the triangle location guarantees that
m 3 and m1 m 3 have opposite signs. By a symmetric argument, A(x1 ) reaches its
x
x1 2
0,
x
2 , that is, at
2.
maximum at the midpoint of the interval
TME, vol10, nos.1&2, p .293
Another way to determine the maximum value of this expression is by using calculus
A'(x1 )
concepts:
we have that
x1
m1 m 3 (2x1 x 2 )
m1 m 3
, the critical points are obtained when A'(x1 ) 0,
x2
2 which is the solution of the equation, then the function A(x1 )
will reach its maximum value at
x1
x2
2 . This is because A''(x1 ) m1m3 0 . Thus,
m1 m3
this result supports the conjecture formulated previously in the graphic approach.
General case
It is possible to use a hand-held calculator to find the maximum area for the case
shown in Figure 9. In this case, we have that m1 72 / 85; m 3 10.33; and
x 2 6.6cm .
Figure 9: Finding the equations of lines with the use of the tool.
Figure 10 shows the algebraic operation carried out to get the point where the
function reaches its maximum value and Figure 11 shows its graphic representation.
Santos-Trigo & Camacho-Machín
Figure 10: Using the derivative to find the maximum of the function area.
Figure 11: Graphic representation of the function area.
A Geometric approach
The goal is to use geometric properties embedded in the problem’s representation
to construct an algebraic model. In Figure 12, it can be seen that:
Triangle ABC is similar to triangle PBQ , this is because angle PQB is
congruent to angle ACB (they are corresponding angles) and angle ABC is the same as
angle PBQ. Based on this information,
h1
h( a x )
a
and the area of APQR can then be
h(a x)
A(x) x
a . This latter expression can be written as
expressed as A xh1 , that is,
A( x) xh
hx 2
A'(x) h 2hx
a . This expression represents a parabola.
a , now if
TME, vol10, nos.1&2, p .295
A'(x) h 2hx 0
a
, then x a /2. Now, we observe that A' ' 0 for any point on the
domain defined for A(x), therefore, there is a maximum relative for that value.
C
A
x
R
h
AP = x
M
Q
h1
P
a-x
N
B
AB = a
Figure 12: Relying on geometric properties to construct an algebraic model.
During the comprehension and exploration episodes two conjectures emerged, the
first one (area of parallelogram APQR is the same as area of parallelogram PBQ’R’) can
be proved by considering parallelogram APTR’ (Figure 7). It is observed that triangles
APR’ and TR’P are congruent and triangles RR’T’ and TQQ’ are also congruent (SSS).
Then, we have that quadrilaterals APT’R and T’QQ’R’ have equal areas, also, the area of
triangle PQT’ is the same as the area of triangle PQB. Based on this information, we have
that the area of APQR is the same as area of PBQ’R’.
The second conjecture that involves showing that the four triangles have the same
area can be proved by observing that the triangles are part of three parallelograms
(APQR, PBQR and PQCR) that overlap each other (Figure 7). Then the overlapping
triangle PQR has the same area as the others because they share a diagonal as a side of
each corresponding parallelogram. Therefore, the maximum value of the inscribed
parallelogram is half the area of the given triangle (∆ABC).
Comment
Santos-Trigo & Camacho-Machín
An important feature of the frame is that teachers should always look for different
ways to solve and examine the tasks. The common goal in the task is to represent and
explore the area model, however the approaches used to achieve this goal offer tearcher
the opportunity to focus on diverse concepts and resources as a way to construct the
model. For example, the algebraic model relies on representing and operating
mathematical objects analytically while the geometric approach is based on using
triangles’ properties to define the area model. It is also observed that the general model
can be tested by assigning particular coordinates to the original triangle vertices. Thus,
problem solvers have the opportunity to test their initial conjectures obtained visually and
empirically by using now the general result (Figure 10 and 11). The use of a hand-held
calculator, in general, makes easy to operate the algebraic expressions and as a
consequence learners could focus their attention to discuss the meaning of the results.
Each approach relies on using different concepts and ways to deal with the involved
relations. As a consequence, the problem solver can contrast strengths and limitations
associated to each approach.
An extension
In figure 13, we draw a line passing through points PR (vertices of parallelogram
APQR). With the use of the software, we ask for the locus of line PR (envelope) when
point P is moved along side AB. Visually, the locus (tangent points) seems to be a conic
section, the goal is to show that it holds properties that define that figure.
TME, vol10, nos.1&2, p .297
Figure 13: What is the locus of line PR when point P is moved along side AB?
Again, with the use of the tool it is shown that the locus is a parabola whose focus
and directrix are identified in Figure 14. It is also shown that when point M is moved
along the locus the distance from that point to the directrix (L) and to point F (focus of
the parabola) is the same (this property defines a parabola).
Figure 14: The locus of line PR when point P is moved along segment AB is a parabola.
Comment
Some serendipitous results or relations might appear as a result of introducing
other objets within the configuration. In this case, adding a line PR to the configuration
led to identify a conic section. Thus, the use of the tool offers a means to think of
Santos-Trigo & Camacho-Machín
mathematical connections that are not easy to identify with the only use of paper and
pencil approaches.
The Integration Episode and Reflections
It is important and convenient to reflect on the processes involved in the distinct
phases that characterize an approach to solve mathematical problems that fosters the use
of computational technology. Initially, the comprehension of the problem’s statements or
concepts involves the use of an inquiry approach to make sense of relevant information
embedded in those concepts or statements. This enquiry process provides the basis to
relate the use of the tools and ways to represent dynamically the problem or situation. A
dynamic model becomes a source from which to explore visually and numerically the
behaviour of parameters, as a result of displacing some elements within the problem
representation. In particular, it might be possible to construct a functional relationship
between a variable, for example the variation of the side AP of the parallelogram and its
corresponding area.
Two distinct ways to construct an algebraic model of the area variation
were pursued; one involves the use of the Cartesian system to identify the equations
associated with some elements of the model. The second way relies on identifying similar
triangles in the inscribed parallelogram whose properties led to the construction of the
area model. Both approaches, the analytic and geometric, converge in the search for the
algebraic model. The algebraic model represents the general case and it can be
“validated” by considering the information of the triangle used to generate the visual
model. In addition, it can be used to explore some of the relations that were detected
TME, vol10, nos.1&2, p .299
during the visual approach. For example, to identify the intersection points of line y k
A(x1 )
m1 m 3 x12 x 2 x1
and the area model
k
m1 m 3
m1 m 3 x12 x 2 x1
m1 m 3
(Figure 5) we solve the equation
for x1. Thus, the discriminant of this quadratic equation
provides useful information to interpret the
relationship between line y k and the graph of the area model
A(x1 )
m1 m 3 x12 x 2 x1
m1 m 3
. When the discriminant is zero the line intersects the graph at
the maximum point, when it is greater than zero, there are two intersection points and
when the discriminant is less than zero, then the line does not intersect the area’s graph.
Concluding, the systematic use of computational tools in problem solving
approaches led us to identify a pragmatic framework to structure and guide learning
activities in such a way that teachers can help the students develop mathematical
thinking. A distinguishing feature of the problem solving episodes is that constructing a
dynamic model of the phenomena provides interesting ways to deal with them from
visual and empirical approaches. Later, analytical and formal methods are used to support
conjectures and particular cases that appear in those initial approaches. The NCTM
(2009) recognizes that reasoning and sense making activities require for students to
gradually develop levels of understanding to progress from less formal reasoning to more
formal approaches.
The use of computational tools provides a basis not only to introduce and connect
empirical and formal approaches, but also to use powerful heuristics as dragging objects
Santos-Trigo & Camacho-Machín
and finding loci of particular objects within the dynamic problem representation. As
Jackiw & Sinclair (2009) pointed out “Dynamic Geometry is revealed as a technological
capability to produce seemingly limitless series of continuously-related examples, and in
so doing, to represent visually the entire phase-space or configuration potential of an
underlying mathematical construction” (p. 414). Throughout the problem solving
episodes we show that it is important for teachers to conceive of a task or problem as an
opportunity for their students to represent, explore and examine the task from diverse
perspectives in order to formulate conjectures and to look for ways to support them. The
diversity of approaches allows them to contrast and relate different concepts and ways to
reason about their meaning and applications. In this context, the use of the tools opens up
new windows to frame and encourage teachers and students’ mathematical discussions
Remarks
Is there any way to characterize forms or ways of mathematical reasoning that
emerge as a result of using computational tools in problem solving approaches? In which
ways does this reasoning complement problem solving approaches that rely on the use of
paper and pencil? Thinking of the task in terms of the affordances provided by the tools
demands that problem solvers focus their attention on ways to take advantage of the
opportunities offered by the tool to represent and explore the problem. For example, the
use of the tool to construct a dynamic model of a task not only becomes relevant to
identify and formulate series of conjectures or mathematical relations but also to reason
about the task in terms of graphic and visual approaches without relying, at this stage, on
an analytic model. In addition, with the use of the software becomes natural and easy to
TME, vol10, nos.1&2, p .301
extend the analysis of a case to a family of cases. For example, by moving any vertex of
triangle ABC, it is possible to verify that all the relations found during the analysis of the
task are also true for a family of triangles that result when moving one the vertices. With
the use of the tool it is often possible to generate loci of points or lines within the model
or to identify parameter behaviours without defining the corresponding algebraic model.
In addition, the empirical and visual approaches often provide important information to
present formal arguments to support conjectures. In this context, it is clear that the
software approach could play an important role to complement and construct formal or
analytic approaches.
Acknowledgement: We thank the support received from projects EDU2011-29328, EDU2008-05254, and
Conacyt-168543 during the development of this article.
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TME, vol10, nos.1&2, p.303
Proof and Problem Solving at University Level
Annie Selden and John Selden
New Mexico State University, USA
Abstract: This paper will be concerned with undergraduate and graduate students’
problem solving as they encounter it in attempting to prove theorems, mainly to satisfy
their professors in their courses, but also as they conduct original research for theses and
dissertations. We take Schoenfeld’s (1985) view of problem, namely, a mathematical task
is a problem for an individual if that person does not already know a method of solution
for that task. Thus, a given task may be a problem for one individual, who does not
already know a solution method for that task, or it may be an exercise for an individual
who already knows a procedure or an algorithm for solving that task.
Keywords: Hungary, mathematics education, mathematics competition, Olympiads,
international comparative mathematics education, problem solving, creativity,
mathematically talented students.
A Continuum of Tasks from Very Routine to Very Non-routine
While what is a problem depends on what a solver knows, it is possible for most
mathematics teachers to judge what is difficult for most students in a given class. Thus,
we see mathematical tasks for a given class, such as a calculus class, on a continuum
from those that are very routine to those that are genuinely difficult problems (Selden,
Selden, Hauk, & Mason, 2000). At one end, there are very routine problems which mimic
sample worked problems found in textbooks or lectures, except for minor changes in
wording, notation, coefficients, constants, or functions that are incidental to the way the
problems are solved. Such problems are often referred to as exercises (and might not be
considered to be problems at all in the problem-solving literature).
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos.1&2, pp.303-334
2013©The Author(s) &Dept. of Mathematical Sciences-The University of Montana
Selden &Selden
The vast majority of exercises in calculus textbooks are of this nature. Lithner
(2004) distinguished three possible solution strategies for typical calculus textbook
exercises: identification of similarities (IS), local plausible reasoning (LPR), and global
plausible reasoning (GPR). In IS, one identifies surface features of the exercise and looks
for a similar textbook situation -- an example, a rule, a definition, a theorem. Without
consideration of intrinsic mathematical properties, one simply copies the procedure of
that situation. In LPR, one identifies a slightly similar textbook situation, but one in
which a few local parts may differ. The solution strategy is to copy as much as possible
from that similar situation, modifying local steps as needed. In GPR, the strategy is
mainly based on analyzing and considering intrinsic mathematical properties of the
exercise, and using these, a solution is constructed and supported by plausible reasoning.
Lithner selected a textbook used in Sweden [Adams' Calculus: A Complete Course (5th
ed.), Addison-Wesley], and worked through and classified solution strategies for 598
single-variable calculus exercises. He found 85% IS, 8% LPR, and 7% GPR.
Furthermore, he concluded that "it is possible in about 70% of the exercises to base the
solution not only on searching for similar situations, but on searching only the solved
examples."
Moving toward the middle of the continuum, there are moderately routine
problems which, although not exactly like sample worked problems, can be solved by
well-practiced methods, for example, ordinary related rates or change of variable
integration problems in a calculus course.1 Moving further along the continuum, there are
moderately non-routine problems, which are not very similar to problems that students
1
Sandra Marshall (1995) has studied how students can develop schema (well-practiced
routines) to reliably guide the solution of arithmetic word problems.
TME, vol10, nos.1&2, p .305
have seen before and require known facts or skills to be combined in a slightly novel
way, but are "straightforward" in not requiring, for example, the consideration of multiple
sub-problems or novel insights. This is the type of problem we used on the non-routine
test in our three studies of undergraduate students’ calculus problem solving. One of
those problems was: Find values of a and b so that the line 2x+3y=a is tangent to the
graph of f ( x ) bx 2 at the point where x=3. (Selden, Selden, Mason, & Hauk, 2000, p.
133).
Finally, at the opposite end of the continuum from routine problems, there are
very non-routine problems which, while dependent on resources in one’s knowledge
base, may involve considerable insight, the consideration of several sub-problems or
constructions, and use of Schoenfeld's (1985) behavioral problem-solving characteristics
(heuristics, control, beliefs). For such problems a large supply of tentative solution starts
(Selden, Selden, Mason, & Hauk, 2000, p. 145), built up from experience, might not be
adequate to bring to mind the resources needed for a solution, while for moderately novel
problems it probably would. Often the Putnam Examinations include such very nonroutine problems.2
2
The following problem was on the 59th Annual William Lowell Putnam Mathematical
Competition given on December 5, 1998: Given a point (a, b) with 0 < b < a, determine
the minimum perimeter of a triangle with one vertex at (a, b), one on the x-axis, and one
on the line y = x. You may assume that a triangle of minimum perimeter exists.
This appears to be a calculus problem, but it only requires clever use of geometry.
An elegant solution (posted by Iliya Bluskov to the sci.math newsgroup) involves
extending the construction "outward" by reflecting across both the lines y = x and the xaxis and noticing that the perimeter of the triangle equals the distance along the path from
(b, a) to (a, -b). Probably only very experienced geometry problem solvers could have
previously constructed images of problem situations containing a tentative solution start
that would easily bring this method to mind.
Selden &Selden
Most U.S. university mathematics teachers would probably like undergraduate
students who pass their lower-level courses, such as calculus, to be able to work a wide
selection of routine, or even moderately routine, problems. In addition, we believe that
many such teachers would also expect their better students to be able to work moderately
non-routine problems, and would think of the ability to do so as functionally equivalent
to having a good conceptual grasp of the course. In other words, we conjecture that the
ability to work moderately non-routine problems based on the material in a university
mathematics course, such as calculus, is often considered part of the implicit curriculum
and taken as equivalent to good conceptual grasp. However, no research has yet been
done to substantiate this conjecture.
Tentative Solution Starts
An individual who has reflected on a number of problems is likely to have seen
(perhaps tacitly) similarities between some of them. He or she might recognize (not
necessarily explicitly or consciously) several overlapping problem situations, each arising
from problems with similar features. For example, after much exposure, many lowerlevel university students would probably recognize a problem as one involving, for
example, factoring, several linear equations, or integration by parts.3 Such problem
situations can act much like concepts (perhaps without signs or labels). While they may
lack names, for a given individual they are likely to be associated with mental images,
3
Although the kinds of features noticed by students in mathematical problem situations
do not seem to have been well studied, the features focused on in physics problem
situations have been observed to correspond to an individual’s degree of expertise.
Novices tend to favor surface characteristics (e.g., pulleys), whereas experts tended to
focus on underlying principles of physics (e.g., conservation of energy) (Chi, Feltovich,
& Glaser, 1981).
TME, vol10, nos.1&2, p .307
that is, strategies, examples, non-examples, theorems, judgments of difficulty, and the
like. Following Tall and Vinner's (1981) idea of concept image, we have called this kind
of mental structure a problem situation image and have suggested that some such images
may, and others may not, contain what we have called tentative solution starts (Selden,
Selden, Hauk, & Mason, 2000, p. 145). These are tentative general ideas for beginning
the process of finding a solution. The linking of problem situations with one or more
tentative solution starts is a kind of (perhaps tacit) knowledge. For instance, the image of
a problem situation asking for the solution to an equation might include "try getting a
zero on one side and then factoring the other." It might also include "try writing the
equation as f(x) = 0 and looking for where the graph of f(x) crosses the x-axis," or even
"perhaps the maximum of f is negative so f(x) = 0 has no solution." We suggest that an
individual’s problem-solving processes are likely to include the recognition of a problem
as belonging to one or more problem situations, and hence, bring to mind one or more
tentative solution starts contained in that individual's problem situation image. This, in
turn, may mentally prime the recall of resources from that individual's knowledge base.
Thus, a tentative solution start may link recognition of a problem situation with the recall
of appropriate resources. We have suggested that problem situations, their images, and
the associated tentative solution starts all vary from individual to individual and that the
process of mentally linking recognition (of a problem situation) to recall (of requisite
resources) through problem situation images might occur several times in solving a single
problem, especially when an impasse occurs (Selden, Selden, Mason, & Hauk, 2000, pp.
145-147).
Selden &Selden
The Genre of Proof
We consider proofs, those that occur in advanced university mathematics
textbooks and research journals, as being written in a special genre. It is clear that not
every mathematical argument can be considered a proof, and much has been written in
the mathematics education research literature about the distinction between
argumentation and proof. (See, for example, Duval, as reported in Dreyfus, 1999, and
Douek, 1999.) In this paper, we are considering proofs of the sort that advanced
undergraduate students and beginning graduate mathematics students are expected to
produce for their professors. We are aware that many upper-level U.S. mathematics
majors just beginning their study of proof-based courses such as abstract algebra and real
analysis often have great difficulty producing such proofs, despite the fact that many of
them have previously taken a transition-to-proof course (Moore, 1994), usually in their
second year of university. Students in such transition-to-proof courses often have trouble
knowing what to write, especially when asked to prove simple set theory theorems,
perhaps because the results are “too obvious” or are verifiable using examples or Venn
diagrams. Thus, learning the genre of proof is important. Indeed, to help students learn
the genre of proof, we have considered two aspects (or parts) of a final written proof: the
formal-rhetorical part and the problem-centered part. The formal-rhetorical part of a
proof (also sometimes referred to as a proof framework) is the part of a proof that
depends only on unpacking and using the logical structure of the statement of the
theorem, associated definitions, and earlier results. In general, this part does not depend
on a deep understanding of, or intuition about, the concepts involved or on genuine
problem solving in the sense of Schoenfeld (1985, p. 74). We call the remaining part of a
TME, vol10, nos.1&2, p .309
proof the problem-centered part. It is the part that does depend on genuine mathematical
problem solving, intuition, and a deeper understanding of the concepts involved. (See
Selden & Selden, 2009).
A sample proof framework is given below for a proof of the following theorem: If
f and g are real valued functions of a real variable continuous at a, then f + g is
continuous at a.
Proof. Let f and g be functions and suppose they are continuous at a. Suppose
is a number > 0. Because f is continuous, there is a f >0 so that for all x, if | x – a| < f,
then | f(x) – f(a) | < ____. Also because g is continuous, there is a g >0 so that for all x, if
| x – a | <g, then
| g(x) – g(a) | < ____ . Let = ____. Note that > 0. Let x be a number.
Suppose that | x – a | < . Then | f(x) + g(x) – ( f(a) + g(a) ) | = … < . Thus, | f(x) +
g(x) – ( f(a) + g(a) ) | < . Therefore f + g is continuous at a.
The problem-centered part of the proof consists of cleverly filling in the blanks
using, for example
2
, minimum, and the triangle inequality. This is not to say that
filling in the blanks is easy. Indeed it can be very difficult for an individual with little or
no experience with such real analysis proofs.
Being able to write a proof framework can be very helpful for students because it
not only improves their proof writing, bringing it in line with accepted community norms,
but also because it can reveal the nature of the problem(s) to be solved. Having once
learned to write a number of proof frameworks, students can then concentrate their
creative energies on solving the actual mathematical problems involved. In addition, for
students, writing the formal-rhetorical part of a proof, and whatever else they can
Selden &Selden
regarding the actual problem(s) to be solved, can enable their university mathematics
teachers to give more helpful and targeted criticism and advice.
The Close Relationship Between Problem Solving and Proving
A number of authors have remarked on the close relationship between problem
solving and proving (e.g., Furinghetti & Morselli, 2009; Mamona-Downs & Downs,
2009; Moore, 1994), and our division of proofs into their formal-rhetorical and problemcentered parts (described above) can make this explicit for students. However, having
good ideas for how to solve the problem-centered part of a proof is not equivalent to
having a proof. Mamona-Downs and Downs (2009) have given university students
informal arguments suggesting a way to solve tasks and asked them to convert those
arguments into acceptable mathematical form. They concluded that “proof production
[from an intuitively developed argument] can involve significant problem solving
aspects. … A particularly frustrating circumstance for a student is when he/she can ‘see’
a reason why a mathematical proposition is true, but lacks the means to express it as an
explicit [mathematical] argument. ” Thus, there are actually two distinct kinds of problem
solving that can occur during proof construction, namely, solving the actual mathematical
problem(s) that enable one to get from the given hypotheses to the given conclusion, and
converting one’s (informal) solution into acceptable mathematical form. Neither of these
problem-solving tasks is easy and students may require instruction and practice with
each. How informal arguments are converted into acceptable mathematical form has been
very little researched.
TME
E, vol10, nos.1&2, p .3111
Sometimees one’s info
ormal argum
mentation (deeveloped durring one’s initial problem
m
solvin
ng) can be converted ratther seamlessly into a wrritten proof tthat is accepptable to the
mathematical com
mmunity. Bo
oero, Douek, Morselli, aand Pedemonnte (2010, p.. 183)
suggeested that “in
n some casess this [probleem-solving] argumentattion can be eexploited by
the sttudent in thee construction
n of a prooff by organizinng some of tthe previoussly producedd
argum
ments into a logical chain
n,” and in su
uch situationns, writing prroofs is easier for
studeents. They reefer to this siituation as on
ne of structuural cognitivee unity.
We have experienced
d such structu
ural cognitivve unity withh beginning ggraduate
mathematics stud
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f
the follo
owing theoreem comparat
atively easy tto prove: If
c
at a and f is a function coontinuous at g (a) , then f g is
g is a function continuous
continuous at a. We
W conjectu
ure that, if th
hey think of aan “appropriiate represenntation,” (seee
Figurre 1), they caan sketch an epsilon neig
ghborhood oof f(g(a)), annd invoking the
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n
d of g(a). Thhen they cann use that
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w epsilon in applying thee continuity of g to arrivve at the needded final
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Selden &Selden
Figure 1. A “picture” of g f with the epsilon and delta neighborhoods indicated.
However, the theorem whose proof framework was illustrated above, namely, If f
and g are real valued functions of a real variable continuous at a, then f + g is
continuous at a, and whose proof involves using minimum and the triangle inequality
cannot be easily obtained from informal intuitive argumentation about adding together
the ordinates of the Cartesian graphs of f and g.
The Importance of Problem Reformulation and Selection of Appropriate
Representations
A number of researchers (e.g., Boero, 2001; Gholamazad, Liljedahl & Zazkis,
2003; Zazkis & Liljedahl, 2004) have noted that reformulating a problem by making an
appropriate choice of representation is often useful, sometimes even necessary, to make
progress. Furinghetti and Morselli (2009) reported the unsuccessful problem-solving
behavior of two fourth-year Italian university mathematics education students during
attempts to prove that The sum of two numbers that are prime to one another is prime to
each of the addends. One student, with the pseudonym Flower, after some initial panic
and working with examples, succeeded in producing a potentially helpful representation
(using the Prime Factorization Theorem), but could not exploit it. The other student, with
the pseudonym Booh, first chose the representation of Least Common Multiple that
“synthesizes [captures the essence of] the property, but doesn’t allow algebraic
manipulation … being non-transparent” and realized that it was “without future.” So he
considered another representation (also using the Prime Factorization Theorem), but also
could not exploit it. In a separate earlier paper, Furinghetti & Moriselli (2007) reported
TME, vol10, nos.1&2, p .313
the unconventional, metaphorical thinking of another student who chose to think of, and
draw, a frog jumping from stop to stop (i.e., from integer to integer on the number line)
and successfully proved the same theorem. They noted that “The choice of the
representation … may foster or hinder transformational and anticipatory thinking, which
are two key issues in the proving process” (Furinghetti & Morselli, 2009, p. 74).
Concepts can have several (easily manipulated) symbolic representations or none
at all. For example, prime numbers have no such representation; they are sometimes
defined as those positive integers having exactly two factors or being divisible only by 1
and themselves. It has been argued that the lack of an (easily manipulated) symbolic
representation makes understanding prime numbers especially difficult, in particular, for
preservice teachers (Zazkis & Liljedahl, 2004).
Symbolic representations can make certain features transparent and others
opaque.4 For example, if one wants to prove a multiplicative property of complex
numbers, it is often better to use the representation rei , rather than x iy , and if one
wants to prove certain results in linear algebra, it may be better to use linear
transformations, T, rather than matrices. Students often lack the experience to know when
a given representation is likely to be useful. More research is needed on the effect of
one’s choice of representation(s) on successful problem-solving behavior.
How Mathematicians Solve Problems
4
For example, representing 784 as 282 makes the property of being a perfect square
transparent and the property of being divisible by 98 opaque. For more details, see Zazkis
and Liljedahl (2004, pp. 165-166).
Selden &Selden
It would be very informative to have research on how advanced university
mathematics students or mathematicians actually construct proofs in real time, but such a
study has not yet been conducted. However, there is research on how mathematicians
solve mathematical problems of various kinds: Carlson and Bloom (2005) investigated
how mathematicians manage their well-connected conceptual knowledge and make
decisions during problem solving; DeFranco (1996) replicated Schoenfeld’s work on the
use of resources, heuristics, control, and beliefs with mathematicians; and Stylianou
(2002) investigated how mathematicians use diagrams in problem solving.
However, the problems given to the mathematicians in these studies were not the
sort encountered by advanced undergraduate or graduate mathematics students when
constructing proofs for their courses or by professors when conducting research. For
example, one problem whose solution Carlson and Bloom (2005, p. 55) discussed at
length was: A square piece of paper ABDC is white on the front side and black on the
back side and has an area of 3 square inches. Corner A is folded over to point A’ which
lies on the diagonal AC such that the total visible area is ½ white and ½ black. How far
is A’ from the fold line? One problem used by DeFranco (p. 212) was: In how many ways
can you change one-half dollar?
Still some of the results are interesting, so we briefly recall them here. Based on
interviews with 12 mathematicians, Carlson and Bloom (2005) developed a “problem
solving framework” that has four phases (orientation, planning, executing, and checking).
As part of the planning phase, there was a conjecture-imagine-evaluate sub-cycle, in
which the mathematicians typically imagined a hypothetical solution approach, followed
by “playing out” and evaluating whether that approach was viable. If it was not viable,
TME, vol10, nos.1&2, p .315
the conjecture-imagine-evaluate sub-cycle was repeated until a viable solution path was
identified. Carlson and Bloom (2005, p. 45) stated, “The effectiveness of the
mathematicians in making intelligent decisions that led down productive paths appeared
to stem from their ability to draw on a large reservoir of well-connected knowledge,
heuristics, and facts, as well as their ability to manage their emotional responses.”
DeFranco (1996) studied the problem-solving behaviors of eight research
mathematicians who had achieved national or international recognition in the
mathematics community (e.g., had altogether 12 honorary degrees and had been awarded
prizes such as the National Medal of Science) and eight who had not achieved such
recognition, but had published from three to 52 articles. He concluded that the former
were problem-solving experts, as well as content experts, and had superior metacognitve
skills, whereas the latter were content experts with only modest problem-solving skills.
Stylianou (2002) was interested in the interplay between visualization and
analytical thinking and asked mathematicians the following problem: Given a right
circular cylinder cut at an angle (shown in her accompanying diagram), describe the
resulting truncated cylinder’s net, that is, the “unrolled” truncated cylinder. She
observed that the “mathematicians consistently attempted to infer additional
consequences from their visual action. Each time a mathematician either constructed a
new diagram or modified a previously constructed one, he took a few seconds to ‘extract’
any additional information . . . and to understand any possible implications.”
In addition, there have been studies (e.g., Burton, 1999) that have included the
reflections of mathematicians upon their own ways of working; however, these are often
too general to be useful for an in-depth understanding of problem solving or for obtaining
Selden &Selden
suggestions for teaching. For example, Burton (1999) found some of the mathematicians
likened problem solving and research to working on jigsaw puzzles or to climbing
mountains.
The Role of Affect in Proving and Problem Solving
While strong affect can play both a positive and a negative role during proving
and problem solving, more research is needed on the role of various kinds of affect from
beliefs and attitudes to emotions and feelings. Furinghetti and Morselli (2009, p. 82)
considered how negative affective factors influenced the problem-solving behavior of
their two unsuccessful students. They noted that Flower panicked immediately after
reading the statement of the theorem writing, “Help! I’m not familiar with prime
numbers!” Later, after constructing some examples, Flower wrote, “Help! I cannot do it, I
do not see anything. Deepest darkness.” Then when she came up with the prime
factorizations, Flower apparently expected to “conclude the proving process in an almost
automatic way … [without] the possibility of dead ends and failures,” illustrating that
beliefs and expectations are also important factors influencing problem-solving
outcomes.
In their study of mathematicians’ problem solving, Carlson and Bloom (2005)
concluded, “The effectiveness of the mathematicians … appeared to stem from their
ability to draw on a large reservoir of well-connected knowledge, heuristics, and facts, as
well as their ability to manage their emotional responses [italics ours].” In a study of
non-routine problem solving, McLeod, Metzger, and Craviotto (1989) found that both
experts (research mathematicians) and novices (undergraduates enrolled in college-level
TME, vol10, nos.1&2, p .317
mathematics courses), when given different experience appropriate problems, reported
having similar intense emotional reactions such as frustration, aggravation, and
disappointment, but the experts were better able to control them.
DeBellis and Goldin (1997, 2006) have considered affect (i.e., values, beliefs,
attitudes, and emotions) as an internal representational system that is not merely auxiliary
to cognition, but as “a highly structured system that encodes information, interacting
fundamentally – and reciprocally – with cognition.” They have introduced the construct
of meta-affect, by which they mean not only affect about affect, but also cognition that
acts to monitor and direct one’s emotional feelings. They also suggested that one might
characterize individuals’ affective competencies, such as the ability to act on curiosity or
to see frustrations as a signal to modify strategy, but did not suggest how to do so.
In addition, we see nonemotional cognitive feelings of appropriateness and of
rightness or wrongness as giving direction to one’s problem-solving efforts. As Mangan
(2001, Section 6, Paragraph 3) said, “In trying to solve, say, a demanding math problem,
[a feeling of] rightness/wrongness gives us a sense of more or less promising directions
long before we have the actual solution in hand.” Below we give the example of Mary, a
returning graduate student, who did not get a feeling of appropriateness with regard to
using fixed, but arbitrary elements in her real analysis proofs for at least half a semester.
Also there have been working groups on affect and mathematical thinking at
several recent CERME conferences,5 but the discussions there seem to have been mainly
concerned with methodological issues and such topics as changing teachers’ and
students’ motivation and attitudes towards mathematics. Still we feel that the interplay
5
CERME is the Congress of the European Society for Research in Mathematics
Education, many of whose Proceedings are available online.
Selden &Selden
between cognition and affect during problem solving and proving needs further
investigation.
How University Mathematics Students Prove Theorems
Much of the research on university students’ proving has been concerned with
difficulties they encounter or competencies needed. These include the use of logic,
especially quantifiers (Dubinsky & Yiparaki, 2000; Epp, 2009; Selden & Selden, 1995);
the necessity to employ formal definitions (Edwards & Ward, 2004); the need for a
repertoire of examples, counterexample, and nonexamples (Dahlberg & Housman, 1997);
the requirement for a deep understanding of the concepts and theorems involved (Weber,
2001); the need for strategic knowledge of which theorems are important (Weber, 2001),
the selection of appropriate representations (Kaput, 1991); and the importance of being to
be able to validate (i.e., read and check) one’s own and others’ proofs for correctness
(Selden & Selden, 2003).
Teaching Proving to University Mathematics Students
For several years, we have been developing methods for teaching proof
construction to advanced undergraduate and beginning graduate mathematics students.
We have developed an inquiry-based Modified Moore Method course (Mahavier, 1999;
Coppin, Mahavier, May, & Parker, 2009) for advanced undergraduate and beginning
graduate mathematics students who need help with proving (hereafter referred to as the
“proofs course” and described in Selden, McKee, & Selden, 2010) and a voluntary
TME, vol10, nos.1&2, p .319
proving supplement for undergraduate real analysis (hereafter referred to as the
“supplement” and described in McKee, Savic, Selden, & Selden, 2010).
In the proofs course, the students are given self-contained notes consisting of
statements of theorems, definitions, and requests for examples, but no proofs. The
students construct their proofs at home and present them in class. The proofs are then
critiqued, sometimes extensively, and additionally suggestions for improvements in the
notation used and the style of writing are given. There are no formal lectures, and all
comments and conversations are based solely on students’ work. The specific topics
covered are of less importance than giving students opportunities to experience as many
different kinds of proofs as possible so we select theorems from sets, functions, real
analysis, semigroups, and topology.
We have developed some theoretical underpinnings for the two courses. One such
theoretical underpinning involves having students develop a proof framework first in
order to reveal the mathematical problem(s) to be solved. (See the above description in
“The Genre of Proof” section.) While students with little experience in proof writing, at
first can find constructing a proof framework to be a problem of moderate difficulty,
eventually through practice, writing a proof framework can become routine or very
routine.
In addition, our proofs course notes are constructed to give students opportunities
to prove theorems that are successively more non-routine. But non-routineness is not
unidimensional; it is not simply a matter of whether the students have seen the concepts
before or have the necessary factual knowledge but cannot bring it to mind (as was the
case for the students in our calculus studies). In our proofs course notes we have “built
Selden &Selden
in” non-routine theorems, which we refer to as theorems of Types 1, 2, and 3. Type 1
theorems have proofs that can depend on a previous result in the notes. These theorems
are included to encourage students to look for helpful previous results, as we have found
that students often attempt to prove theorems directly from the definitions without
recourse to previous results. Type 2 theorems require formulating and proving a lemma
not in the notes, but one that is relatively easy to notice, formulate, and prove, whereas
Type 3 theorems require formulating and proving a lemma not in the notes, but one that is
hard to notice, formulate, and prove. An example of a Type 3 theorem is: A communtative
semigroup S with no proper ideals is a group, given after a brief introduction to the ideas
of semigroup and ideals thereof. What is needed for a proof of this theorem is the
observation that aS is an ideal and hence aS=S. (This is the first lemma needed.) This is
followed by the nontrivial observation that aS=S implies that equations of the form ax=b
are solvable for any b in S. Using some clever instantiations of this equation, one can
obtain an identity and inverses, and hence, conclude S is a group. To date only two
students have been able to produce a proof without help or hints, and several mathematics
faculty (whose speciality is not semigroups) have found that proving this theorem takes
time and a certain amount of reformulation. This convinces us that this theorem can be
considered at least moderately non-routine. More research is needed on what makes a
problem non-routine (for an individual or a class), that is, what are the various
dimensions or characteristics contributing to non-routineness.
The Co-construction of Proofs in the Supplement
We have implemented this method three times to date in small (at most 10
students) supplementary voluntary proving classes for real analysis. The supplement is
TME, vol10, nos.1&2, p .321
intended for students who feel unsure of how to proceed in constructing real analysis
proofs. At the beginning of a supplement class period, the statement of a theorem entirely
new to them, but similar to a theorem assigned for homework, but not a template
theorem, is written on the board. The students themselves, or one of us if need be, offer
suggestions about what to do, beginning with the construction of a proof framework. For
each suggested action, such as writing up the hypotheses or an appropriate definition,
drawing a sketch, or introducing cases, one student is asked to carry out the action at the
blackboard. The intention is that all students reflect on the actions and later perform
similar actions autonomously on their assigned homework (McKee, Savic, Selden, &
Selden, 2010).
For example, if the students are to prove a sequence {an }n1 converges to A , they
would typically begin by writing the hypotheses, leave a space, and write the conclusion.
After unpacking the conclusion, they would write “Let 0 ” immediately after the
hypotheses, leave a space for the determination of N , write “Let n N ”, leave another
space, and finally write “Then | an A | ” prior to the conclusion at the bottom of their
nascent proof. This would conclude the construction of a proof framework and bring
them to the problem-centered part of the proof (Selden & Selden, 2009), where some
“exploration” or “brainstorming” on the side board would ensue. The entire coconstruction process, and accompanying discussions, is a slow one – so slow that only
one theorem can be proved and discussed in detail in a 75-minute class period. More
research is needed on how to foster such mathematical “exploration” and
“brainstorming.”
Selden &Selden
Theoretical Underpinnings: Actions and Behavioral Schemas
Actions in the Proving Process
We see proving as an activity, that is, as a sequence of actions, that are either
physical (such as writing or drawing) or mental (such as attempting to recall a definition
or theorem). Each action is paired with, and is a response to, a situation in a partly
completed proof. By a situation we mean a reasoner’s inner, or interpreted, situation as
opposed to an outer situation that may be visible to an observer. Although we are
referring to a person’s inner situation, we have found in teaching that we can often gauge
approximately what the inner situation is from the outer, observable, situation and the
ensuing action. For example, below we will interpret Sofia’s staring blankly at the
blackboard during tutoring sessions as the situation of not knowing what to do next.
If a person engages in proving several theorems, then he or she is likely to
experience a number of similar situations yielding similar actions. The first such
situation-action pair is likely to have a conscious warrant based on, say, heuristics, logic,
strategy, or known mathematics. However with time and (sometimes considerable)
repetition, the need for a conscious warrant may disappear. The situation may then
become linked, in an automated way, to a tendency to carry out the corresponding action;
and the individual will not be conscious of anything happening between the situation and
the action. We see such automated situation-action pairs as persistent mental structures
and have called the smallest of them behavioral schemas (Selden, McKee, & Selden,
2010; Selden & Selden, 2011). By a small situation-action pair, we mean one that is not
equivalent to any sequence of smaller such pairs. While the word “schema” has been
used in several ways in the literature, we only mean such a persistent mental structure.
TME, vol10, nos.1&2, p .323
Behavioral schemas
The formation of behavioral schemas, whether beneficial or detrimental, requires
the development of a way of recognizing particular kinds of situations, and in response,
enacting particular kinds of actions. It is possible that neither the kind of situation nor the
kind of action for a potential behavioral schema exists as a concept in the surrounding
culture. In that case, constructing a behavioral schema entails noticing, either explicitly or
implicitly, similarities among situations and among the corresponding actions, and
eventually reifying these into what amounts to conceptions (usually without any need for
formal designations).
Properties of Behavioral Schemas
(1) Within very broad contextual considerations, behavioral schemas are
immediately available. They do not normally have to be recalled, that is, searched
for and brought to mind.
(2) Behavioral schemas operate outside of consciousness. A person is not aware
of doing anything immediately prior to the resulting action – he/she just does it.
Furthermore, the enactment of a behavioral schema that leads to an error is not
under conscious control, and one should not expect that merely understanding the
origin of the error would prevent its future occurrences.
(3) Behavioral schemas tend to produce immediate action, which may lead to
subsequent action. One becomes conscious of the action resulting from a
behavioral schema as it occurs or immediately after it occurs.
(4) A behavioral schema that would produce a particular action cannot pass that
information, outside of consciousness, to be acted on by another behavioral
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schema. The first action must actually take place and become conscious in order
to become information acted on by the second behavioral schema. That is, one
cannot “chain together” behavioral schemas in a way that functions entirely
outside of consciousness and produces consciousness of only the final action. For
example, if the solution to a linear equation would normally require several steps,
one cannot give the final answer without being conscious of some of the
intermediate steps.
(5) An action due to a behavioral schema depends on conscious input, at least in
large part. In general, a stimulus need not become conscious to influence a
person’s actions, but such influence is normally not precise enough for doing
mathematics. Also, non-conscious stimuli that lead to action usually originate
outside of the mind, not within it (as often happens in proof construction).
(6) Behavioral schemas are acquired (learned) through (possibly tacit) practice.
That is, to acquire a beneficial schema a person should actually carry out the
appropriate action correctly a number of times – not just understand its
appropriateness. Changing detrimental behavioral schemas, many of which have
been tacitly acquired, requires similar, perhaps longer, practice (Selden, McKee,
& Selden, 2010; Selden & Selden, 2011).
Implicitly acquired detrimental behavioral schemas can be enacted automatically
in problem-solving situations. For example, some experienced teachers may have noticed
that giving a counterexample to a student who consistently makes an errorful calculation,
such as (3a b) / 3c (a b) / c or
a 2 b 2 a b , is often not very effective. This can
be so even when the student seems to understand the counterexample. Our view of
TME, vol10, nos.1&2, p .325
behavioral schemas suggests an explanation. If an incorrect algebraic simplification is
caused by the enactment of a behavioral schema, then the resulting action (the incorrect
simplification) would follow directly from the situation, that is, would not be under
conscious control. To change the student’s behavior, one might try to change the
detrimental behavioral schema not only by providing a counterexample, but also by
suggesting a number of relevant problems and some monitoring.
Using our Theoretical Underpinnings to Teach Proving
Having students write a proof framework first, enables them to “get started” on
writing a proof and reveals the mathematical problem(s) to be solved. What happens next
depends on a student’s ability to solve various mathematical problems. Informally, one of
our graduate students has reported that writing a proof framework helped her organize
her thoughts on her high stakes mathematics PhD comprehensive examinations. Also,
looking for students’ detrimental behavioral schemas and trying to help them replace
them with beneficial schemas has enabled us to help students with proof construction.
Sometimes acquiring a beneficial schema can take a long time.
Mary’s Reaction to Considering Fixed, but Arbitrary Elements
There are theorems, particularly in real analysis, that involve several quantifiers.
For example, proving a function f is continuous at a involves proving that for all
there is a
0
so that for all x, if |x-a|<
0
then |f(x)-f(a)|< . For such proofs, one
needs to consider a fixed, but arbitrary . Students are often reluctant to do this. We
conjecture this is because they do not feel it right or appropriate to do so.
Selden &Selden
Mary, an advanced mathematics graduate student, was interviewed about events
that took place two years earlier when she was taking both a pilot version of our proofs
course and Dr. K’s graduate real analysis course. In the homework for Dr. K’s course,
Mary needed to prove many statements that included phrases like ‘For all real numbers
0 ,’ where
represented a variable (the situation). In her proofs, Mary needed to
write something like ‘Let
0 ,’ where represented an arbitrary, but fixed number (the
action).
When Mary was interviewed about this situation-action pair she said the
following:
Mary:
At that point [early in Dr. K’s real analysis course] my biggest idea
was, well he said to “do it”, so I’m going to do it because I want to get
full credit. And so I didn’t have a sense of why it worked.
Interviewer:
Did you have any feeling … if it was positive or negative, or extra …
Mary:
Well, I guess I had a feeling of discomfort …
Interviewer:
Did this particular feature [having to fix ] keep coming up in proofs?
Mary:
… it comes up a lot and what happened, and I don’t remember
[exactly] when, is that instead of being rote and kind of
uncomfortable, it started to just make sense … By the end of the
semester this was very comfortable for me.
Mary told us that, after completing each such proof, she attempted to convince
herself that considering a fixed, but arbitrary element resulted in a correct proof.
However, only after repeatedly executing this situation-action pair, and convincing
TME, vol10, nos.1&2, p .327
herself that her individual proofs were correct, did she develop a feeling of
appropriateness.
Willy’s Focusing Too Soon on the Hypotheses
We have observed that after writing little more than the hypotheses, some
students turn immediately to focusing on using the hypotheses, rather than unpacking the
conclusion to see what is to be proved, after which they often cannot complete a proof.
For example, late in our proofs course, Willy was asked to prove the theorem: Let X and
Y be topological spaces and f : X Y be a homeomorphism of X onto Y . If X is a
Hausdorff space, then so is Y . Because only ten minutes of class time remained and
Willy had indicated that he had not yet proved the theorem, we asked him to “do the setup”, that is, construct a proof framework (Selden, McKee, & Selden, 2010; Selden &
Selden, 2011).
On the left side of the blackboard, Willy wrote:
Proof. Let X and Y be topological spaces.
Let f : X Y be a homeomorphism of X onto Y .
Suppose X is a Hausdorff space.
...
Then Y is a Hausdorff space.
Then, on the right side of the board which was for scratch work, he listed one
after the other: “homeomorphism, one-to-one, onto, continuous (f is open mapping)”. He
then looked perplexedly back at the left side of the board. Even after two hints to look at
the final line of his proof, Willy said, “And, I was just trying to just think,
Selden &Selden
homeomorphism means one-to-one, onto, …” After some discussion about the meaning
of homeomorphism, the first author said, “There is no harm in analysing what stuff you
might want to use, but there is more to do before you can use any of that stuff”, meaning
that the conclusion should be examined and unpacked first.
We inferred that Willy was enacting a behavioral schema in which the situation
was having written little more than the hypotheses, and the action was focusing on the
meaning and potential uses of those hypotheses before examining the conclusion. We
conjectured that Willy and other students, who are reluctant to look at, and unpack, the
conclusion feel uncomfortable about this, or perhaps feel it more appropriate to begin
with the hypotheses and work forward.
Sofia’s Reaction to Not Having an Idea
Sofia was a diligent first-year graduate student; however, as our proofs course
progressed, an unfortunate pattern in her proving attempts emerged. When she did not
have an idea for how to proceed, she often produced what one might call an “unreflective
guess” only loosely related to the context at hand, after which she could not make further
progress. Although we could sometimes speculate on the origins of Sofia’s guesses, we
could not see how they could reasonably have been helpful in making a proof, nor did she
seem to reflect on, or evaluate, them herself. We inferred that Sofia was enacting a
behavioral schema: she was recognizing a situation, that is, that she had written as much
of a proof as she could, and had a feeling of not knowing what to do next. This situation
was linked in an automated way to the action of just guessing any approach that usually
was only loosely related to the problem at hand without much reflection on its usefulness.
TME, vol10, nos.1&2, p .329
Using our idea of behavioral schemas, we devised an intervention that was used in
tutoring sessions with Sofia. We attempted to deflect implementation of her “unreflective
guess” schema, by suggesting that she write the first and last lines of a proof, unpack the
conclusion, and then do something else, such as draw a diagram, review her class notes,
or reflect on everything done so far. These suggestions and guidance helped Sofia
construct a beneficial behavioral schema. As the course ended, this intervention of
directing Sofia to do something else was beginning to show promise. For example, on the
in-class final examination Sofia proved that if f, g, and h are functions from a set to
itself, f is one-to-one, and f g f h , then g h . Also on the take-home final,
except for a small omission, she proved that the set of points on which two continuous
functions between Hausdorff spaces agree is closed. This shows Sofia was able to
complete the problem-centered parts of at least a few proofs by the end of the course, and
suggests her “unreflective guess” behavioral schema was weakened (Selden, McKee, &
Selden, 2010; Selden & Selden, 2011).
Future Research on Proof and Problem Solving
The above discussion has not only synthesized some of the literature on proof and
problem solving, it has highlighted several areas that could use more research. These are:
how informal arguments are converted into acceptable mathematical form; how
representation choice influences an individual’s problem-solving and proving behaviour
and success; how students’ and mathematicians’ prove theorems in real time (especially
when working alone); how various kinds of affect, including beliefs, attitudes, emotions,
and feelings, are interwoven with cognition during problem solving; which characteristics
Selden &Selden
make a problem non-routine (for an individual or a class), that is, what are the various
dimensions contributing to non-routineness; and how one might foster mathematical
“exploration” and “brainstorming” as an aid to problem solving.
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TME, vol10, nos.1&2, p.335
Cognition and Affect in Mathematics Problem Solving with
Prospective Teachers1
Lorenzo J. Blanco
Eloisa Guerrero Barona
Ana Caballero Carrasco
Didáctica de la Matemática, Universidad de Extremadura, Spain
Abstract: Recent studies relating the affective domain with the teaching and
learning of mathematics, and more specifically with mathematics problem solving,
have focused on teacher education. The authors of these studies have been ever more
insistently pointing to the need to design educational programs that take an integrated
cognitive and affective approach to mathematics education. Given this context, we
have designed and implemented a program of intervention on mathematics problem
solving for prospective primary teachers. We here describe some results of that
program.
Keywords: Mathematics Teaching, Problem Solving, and the Affective Domain
Problem solving (PS) has always been regarded as a focal point of mathematics, and in
the last 30 years its presence in curricula has increased notably (Castro, 2008; Santos,
2007). It is regarded as the methodological backbone to approach mathematics content
since it both requires and helps develop skills in analysis, comprehension, reasoning, and
1This
work forms part of a study carried out under the Research Project "Elaboración de
instrumentos reflexivos de intervención en la formación, inicial y permanente, del profesorado de
Matemáticas en Primaria y Secundaria" [Development of reflexive tools of intervention in the initial
and continuing education of mathematics teachers in primary and secondary education] approved in the
III Regional Plan for Technological R&D (2008–11), awarded by the Junta de Extremadura regional
government (Reference: PRI08B034).
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos.1&2, pp.335-364
2013©The Author(s) & Dept. of Mathematical Sciences-The University of Montana
Blanco, Guerrero & Caballero
application. At the same time, it is now being proposed as an item of curricular content in
its own right as a core competence that students need to acquire. Castro (2008) and
Santos (2008) recognize that attempts to teach students general PS strategies have been
unsuccessful. Also, it seems important to emphasize the lack of attention in textbooks to
learning heuristic problem solving strategies (Schoenfeld, 2007; Pino & Blanco, 2008).
The results of the Programme for International students Assessment (PISA) of
2003, 2006, and 2009 have highlighted the importance of mathematics problem solving
(MPS) in compulsory education. One of the aspects tacitly accepted in the curricula at
this educational level is the influence of affect on the teaching and learning of
mathematics in general, and of MPS in particular. Already in the 1980s, Charles & Lester
(1982) were observing that: "The problem solver must have sufficient motivation and lack
of stress and/or anxiety to allow progress towards a solution" (p.10). In their work, they
recognized that factors involving cognition, experience, and affect influence the MPS
process. Among the affective factors that they explicitly noted were interest, motivation,
pressure, anxiety, stress, perseverance, and resistance to premature closure. It is currently
accepted that the cognitive processes involved in PS are susceptible to the influence of
the affective domain in its three fundamental areas: beliefs, attitudes, and emotions
(Sriraman, 2003).
TME, vol10, nos.1&2, p .337
Initial Primary Teacher Education, the Affective Domain (Beliefs, Attitudes, and
Emotions), and Problem Solving
Research on the affective domain has also expanded to the field of initial teacher
education and the professional development of in-service teachers. It is considered that,
in their actions in the classroom, teachers cannot dissociate affect from content when
faced with a specific activity for pupils at a specific level.
Influence of beliefs
"When prospective primary teachers enter an Initial Education Centre
they bring with them the educational baggage of many years in school.
They thus naturally have conceptions and beliefs concerning Mathematics
and the teaching/learning of Mathematics that derive from their own
learning experience" (Blanco, 2004, p.40). Furthermore: "Few apparent
changes in their beliefs were affected as a result of traditional
mathematics method courses" (Chapman, 2000, p.188).
It is important to distinguish the beliefs of prospective primary teachers (PPTs)
about mathematics as an object – about its teaching and learning, beliefs which depend
on affect – and their beliefs about themselves as learners – beliefs related to their selfconcept, self-confidence, expectations of control, etc.
Beliefs about mathematics and problem solving
According to Llinares and Sánchez (1996), prospective teachers acquire a technical
school culture that conditions their approach to mathematics tasks and learning as future
teachers. For them mathematics teaching is the transmission of specific information and
mathematics learning is done through repetition. The teacher's role consists of presenting
the content in a way that is clear and concise, and the learner's role consists of listening
and repeating.
Blanco, Guerrero & Caballero
According to Szydlik, Szydlik & Benson (2003), research has shown that
prospective teachers tend to "see mathematics as an authoritarian discipline, and that
they believe that doing mathematics means applying memorized formulas and procedures
to do textbook exercises" (Szydlik, Szydlik & Benson, 2003, p.254).
Prospective primary teachers have a very traditional idea of mathematics
problems that are quite different from the suggestions of current curricular proposals
(Blanco, 2004; Johnson, 2008). This leads to "a contradiction between their personal
experience, which they judge as having been negative and monotonous, and their
conception of mathematics as linked to reasoning and rigour" (Blanco, 2004, p.42).
Furthermore, these beliefs constitute a kind of lens or filter through which they interpret
their own personal learning processes and orient their teaching experiences and
behaviours (Chapman, 2000), thus limiting their possibilities for action and
understanding (Barrantes & Blanco, 2006). Moreover, "these beliefs are [internally]
consistent" (Blanco, 2004, p.41).
For Schoenfeld (1992), beliefs form a particular view of the world of
mathematics, setting the perspective from which each person approaches that world, and
they can determine how a problem will be tackled, the procedures that will be used or
avoided, and the time and intensity of the effort that will be put into the task.
Consequently, these beliefs need to be taken into account in teacher education, which, if
necessary, will have to try to promote their change and the generation of new beliefs
(Blanco, 2004).
Prospective primary teachers regard MPS as a rote mechanical process, have few
resources with which to represent and analyse problems, never look for alternative
TME, vol10, nos.1&2, p .339
strategies or methods for their solution, and make no use of the different guidelines and
hints they might be given to help them towards a solution (Blanco, 2004; Córcoles &
Valls, 2006; NCTM, 2003), thereby generating a vision of themselves as incompetent
problem solvers (NCTM, 2003).
The beliefs that most influence motivation and achievement in mathematics are
students' perceptions about themselves in relation to mathematics (Kloosterman, 2002;
Skaalvik & Skaalvil, 2011). Hernández, Palarea & Socas (2001) and Blanco et al. (2010)
note that PPTs generally do not see themselves as capable or skilled as problem solvers,
with most of them experiencing feelings of uncertainty, despair, and anxiety which block
their approach to the task – in sum, most of them consider themselves to be incompetent
at PS. A major difference between successful and unsuccessful problem solvers lies in
their beliefs about MPS, about themselves as solvers, and about how to approach the task
(NCTM, 2003).
Influence of attitudes
What students think about mathematics influences the feelings that surface
towards the subject and their predisposition to act in consequence. That is, if students
have negative beliefs about mathematics or its teaching, they will tend to show adverse
feelings towards related tasks, in particular presenting avoidance behaviour or simple
rejection of those tasks. This predisposition towards certain personal intentions and
behaviours is what one calls attitudes.
One can distinguish between mathematics attitudes themselves and attitudes
towards mathematics as a subject. Mathematics attitudes have a marked cognitive
component, and relate to general cognitive skills that are important in mathematics tasks.
Blanco, Guerrero & Caballero
Studies in Spain have shown that PPTs have few mathematics attitudes in this sense on
aspects related to PS (Blanco, 2004; Corcoles & Valls, 2006).
In attitudes towards mathematics, the affective component predominates. It is
manifest in interest, satisfaction, and curiosity, or, on the contrary, in rejection, denial,
frustration, and avoidance of mathematics tasks. Positive interest and attitudes towards
mathematics seem to decline with age, especially during secondary education (Hidalgo,
Maroto, Ortega & Palacios, 2008).
Influence of emotions
The emotions aroused in students by a mathematics task are affective responses
characterized by high intensity and physiological arousal reflecting the charge of positive
or negative meaning that a task has for them Studies of emotion have focused on the role
of anxiety and frustration and their impact on achievement in mathematics, noting that
one of the difficulties of mathematics education is seeing its teaching as essentially
cognitive, and detached from the field of emotions (De Bellis & Goldin, 2006).
Emotions appear in response to an internal or external event which has a charge
of positive or negative meaning for the person. Thus, in facing a mathematical task a
pupil may encounter difficulties which lead to the frustration of their personal
expectations, causing the appearance of essentially negative valuations of the subject.
Various authors agree that anxiety interacts negatively with cognitive and motivational
processes, and therefore with the pupil's overall performance (De Bellis & Golding,
2006; Zakaria & Nordim, 2008). In this regard, Hidalgo, Maroto, Ortega & Palacios
TME, vol10, nos.1&2, p .341
(2008) found a strong negative correlation between pupils' levels of anxiety towards
mathematics and their final marks at the end of the course. This correlation is also
present when comparing the levels of anxiety and positive attitudes towards mathematics.
The relationship between anxiety and mathematics education has also been transferred to
the case of prospective teachers, for which there is already a substantial literature (Peker,
2009).
Recent work has established relationships between anxiety and self-confidence.
Thus, pupils with more anxiety towards mathematics have less confidence in their
mathematical abilities and as learners of mathematics (Gil, Blanco & Guerrero, 2006;
Isiksal, Curran, Koc & Askun 2009). "Many of the negative emotional attitudes towards
mathematics are associated with anxiety and fear. Anxiety to be able to complete a task,
fear of failure, of making mistakes, etc., generate emotional blockages of affective origin
that have a repercussion on the students' mathematics activity" (Socas, 1997, p.135).
Zevenbergen, (2004) notes that PPTs show "low levels of mathematics knowledge as well
as considerable anxiety towards the subject" (Zevenbergen, 2004, p.5).
With respect to mathematics teaching and learning, there are various moments at
which the relationship between emotions and cognitive processes becomes visible: when,
following the proposal of a mathematics task, the structure of the activity is understood or
relevant information is retrieved; when problem-solving strategies are being designed,
including the recall of formulas or mechanical procedures; and when the PPTs are
involved in the process of the control and regulation of their own learning coupled with a
clear methodological approach to teaching the mathematics which they had come to
reject.
Blanco, Guerrero & Caballero
It therefore seems appropriate to consider studying the beliefs, attitudes, and emotions of
prospective teachers when they are dealing with PS. The lack of reflection on these issues
is one reason for the persistence of PPTs' inappropriate conceptions and attitudes. In their
passage through initial teacher education, they have not been led to re-conceptualize their
role as primary teachers. Authors such as Mellado, Blanco & Ruiz (1998), Chapman
(2000), Uusimaki & Nason (2004), and Malinsky, Ross, Pannells & McJunkin (2006)
suggest that the origin of the negative beliefs of prospective teachers in their initial
teacher education could be attributed to the influence of their own experiences as learners
of mathematics, i.e., to their experiences when they themselves were being taught
mathematics in school and to their teachers at that time, and to the mathematics courses
in their teacher education programs.
A Research Project with Prospective Primary Teachers on Cognition and Affect in
Problem Solving
The above references clearly show that the cognitive and the affective are closely related,
that beliefs, attitudes, and emotions influence knowledge, and that knowledge in turn
affects those same three aspects.
The study of this relationship between affect and cognition has also been explored
with teachers. Teachers' concepts and values determine the image of mathematics in the
classroom, and condition the type of teacher-pupil relationship. Conceptions influence
attitudes, and both of them influence the teacher's behaviour and the pupils' learning
(Ernest, 2000). In order to foster change in our prospective teachers' views of teaching,
TME, vol10, nos.1&2, p .343
we shall have to incorporate conceptions and attitudes as part of a process of discussion
and reflection in our initial teacher education programs (Mellado, Blanco & Ruiz, 1998;
Stacey, Brownlee, Thorpe & Reeves, 2005; Johnson, 2008). There thus seems to be a
clear interest in studying the affective and emotional factors involved in the mathematics
education of PPTs since, as future teachers, their beliefs and emotions towards
mathematics will influence both the level of achievement and the beliefs and attitudes
towards the subject of their pupils.
De Bellis & Goldin (2006) and Furinghetti & Morselli (2009) note that studies of
students' performance and problem solving have traditionally concentrated primarily on
cognition, less on affect, and still less on cognitive-affective interactions. However, a
growing number of studies recognize the importance of integrating the affective and
cognitive dimensions into the teaching and learning of mathematics (Amato, 2004; Zan,
Bronw, Evans & Hannula, 2006; Furinghetti & Morselli, 2009; Blanco et al., 2010).
Some authors, such as Furinghetti & Morselli (2009), specifically note the need to
simultaneously develop cognitive and affective factors in teacher education programs. In
this regard: "The role of teacher education is to develop beginning teachers into confident
and competent consumers and users of mathematics in order that they are better able to
teach mathematics" (Zevenbergen, 2004, p.4).
In this context, we considered that there was a need to undertake a research
project on MPS in initial primary teacher education with the consideration of its cognitive
and affective aspects. Initial teacher education is conceived of as being just one part of a
continuous and permanent process in a teacher's professional life in which emotional
education is an indispensable complement to cognitive development. Indeed, cognitive
Blanco, Guerrero & Caballero
and affective aspects are essential elements in the development of teaching as a
profession.
We believe that gaining the capacity to solve mathematics problems should be an
achievable goal in an educational environment in which students are allowed to generate
their own PS strategies and compare them with other alternatives. In particular, we
believe that the way in which PS is approached is highly personal. Each student will have
to be helped to discover their own particular style – their own capabilities and limitations.
We must avoid conveying to our students only heuristic rules or methods, but instead be
sure to help them develop positive attitudes and emotions towards MPS based on their
own past and present experiences.
Objective of the research project
In our research study, we set ourselves the following general objective: to design,
develop, and evaluate an intervention program to enhance the performance of PPTs in
MPS, and to lay the foundations for them to learn to teach MPS at the primary school
level, integrating in a single model cognitive aspects of PS and emotional education
(Annex 1).
Additionally, we set different specific objectives relating to the study population,
two of which were:
To describe the participating prospective teachers' conceptions about MPS.
To describe and analyse their attitudes, beliefs, and emotions related to
MPS, and in particular their expectations of control.
TME, vol10, nos.1&2, p .345
In addition, two specific objectives relating to the teacher education program were
pursued:
To evaluate the development of the program with respect to the PPTs'
levels of anxiety.
To describe the aspects of the program which they found to be most
significant.
During the 2007-08 academic year, we conducted a pilot study that served to finetune the program. We performed the actual field-work during 2008-09.
Data collection and analysis
The nature of the research problem and the data collection led us to use a combination of
qualitative and quantitative methods to relate, compare, and contrast the different types of
evidence. The implementation of the program followed an action research approach since
the ultimate goal is to help the participants develop their thinking, modify their attitudes,
and seek ways to overcome their difficulties in MPS.
Annex I presents the plan of the 13 sessions comprising the program, specifying
the objectives of each session, the instruments used to obtain information (open and
closed questionnaires, diaries, and forums), the nature of the data, and the corresponding
type of analysis. The participants in the program were a core group of 55 PPTs in the
Education Faculty of the University of Extremadura (Spain) in the third year of their
course.
All the program sessions lasted two hours, and were audio and video recorded,
accompanied by field notes. The Moodle Virtual Platform was used as support for the
program's documents, information, and forums, as also is indicated in Annex I.
Blanco, Guerrero & Caballero
Apart from the open and closed questionnaires specifically indicated in Annex I,
the following research tools were employed:
Observation of the behaviour in the classroom of both teacher and
students, video recorded with two cameras, with subsequent transcription and analysis.
The Moodle platform (Universidad de Extremadura) is a useful tool for the
presentation of information and communication. It allows information to be stored for
later analysis (both qualitative and quantitative), with the date and time and the subject
contributing the information being reliably logged. It allows one to evaluate the
participation, and to see whether the students have attained specific learning objectives,
providing feedback as well as motivation to the students. A reference to the use of this
platform in the present research can be found in Caballero, Blanco & Guerrero (2010).
platform.
Diaries (Nichols, Tippins & Wieseman, 1997), kept on the Moodle virtual
These
allow
the
collection
of
observations,
sensations,
reactions,
interpretations, anecdotes, introspective remarks about feelings, attitudes, motives,
conclusions, etc.
A forum, also via the Moodle virtual platform, on some of the specific
content or situations arising in class.
For the data analysis, we used the program packages SPSS 15.00 for the quantitative
analysis of the questionnaires that we are given in Annex I. For the qualitative analysis,
we followed the recommendations of Goetz & LeCompte (1984) and Wittrock (1986),
establishing a process similar to that described in Barrantes & Blanco (2006) based on
units of analysis (Goetz & LeCompte, 1984) and the categories noted in the instruments.
TME, vol10, nos.1&2, p .347
Analysis of Results and Discussion
The breadth of the research study and the characteristics of this present communication
only allow us to present some partial results. In particular, we shall refer to some of the
results on the PPTs' conceptions about PS, on certain aspects related to the affective
domain, particularly those concerning the students' expectations of control, and on some
general aspects of the program's evaluation.
What do the prospective teachers understand by a mathematics problem?
Our analysis of the questionnaires showed the prospective teachers to hold very
traditional conceptions about mathematics problems. Thus, they referred to them as
closed statements which explicitly or implicitly indicate the procedure to follow for their
solution. The responses to the items of the questionnaire on "What do I understand by a
mathematics problem?" (Annex II) reflect the classifications noted by some authors in the
literature. In this sense, their formulation of a problem is in the form of a text which gives
all the information to be resolved, which Borasi (1986, p.135) calls a "word problem";
and the method of solution explicitly or implicitly suggested in the text involves a
translation of the words of the problem to a mathematical expression, which Charles &
Lester (1982, p.6) call a "simple or complex translation problem"; and the solution of this
expression involves using a known algorithm, which Butts (1980, p.24) refers to as an
"application problem".
The basic referents of their problems are arithmetic operations, algebraic
algorithms, or, to a lesser extent, calculations of areas. It was interesting to note that the
contexts they describe are those that have been traditional in mathematics textbook
problems since the late nineteenth century. Thus, in both years of the study, there are
Blanco, Guerrero & Caballero
references to problems of taps, the number of heads and legs of farm animals, trains and
distances, and the comparison of ages. It stood out that in no case was there any reference
to specific situations of their or their potential pupils' immediate environment, or to such
everyday resources as mobile phones or personal hobbies. This result, which we did not
find in the literature we reviewed, seems especially important because it is necessary to
link problem solving with the pupils' interests and relate the problems to their immediate
environment.
Of a total of 178 problems, 126 (70.8%) were arithmetic with a structure
involving addition or multiplication, representing elementary shopping or business
situations2. Another 31 (17.4%) were questions of arithmetic proportionality3. There were
7 problems (3.9%) involving equations in which the situations were related to ages, taps,
speeds of trains or cars, and farm animals4. Geometry problems accounted for 5.7%, and
were very basic, referring to the calculation of areas5.
In analysing this PS situation in class with the students (4-XI-2008), we thought
that it was convenient to focus on the following question:
Do you think there are other types of problems? If so, write down two
examples.
2
There are 47 apples in an apple tree. Mary has picked 37 apples. How many apples are left in the tree?
In a fruit shop, 1 kg of apples costs 1.75 euros. If Laura buys two kilos, how much money has she spent
altogether?
3 We know that Juan has eaten 2/3 of a cake, and his brother Sergio 5/6 of the rest. How much of the
cake
is
left?
Three friends have 40 euros to spend. The first spent 2/5 of the total, and the second 2/3 of what the first
spent. How much did the third spend?
4 On a farm there are horses and chickens. In total there are 74 feet and 12 beaks. How many horses and
how many chickens are there on the farm?
5 Calculate the area of a square whose side is 2 cm.
TME, vol10, nos.1&2, p .349
Observation of the recordings and the analysis of this last question brought out the
difficulty they were having in establishing mathematics activities that were different from
those they had proposed, and which had been analysed previously in that same program
session.
Thus, 14 participants answered directly that there are no different types of
problem. Two examples of these responses are the following:
"I think not, because throughout my school life I always had problems of
the same type."
"The truth is that I have no idea. The maths problems that I know are those
of always."
Another 34.5% again insisted on the same kinds of problem noted above, but
involving situations concerned with other mathematics content such as statistics or
probability that had not specifically appeared previously.
The question prompted some students to guess that there really must exist other
types of mathematics problems, but they found themselves unable to give any examples:
"After what we have seen today, there must be other types, but right now I
can't think of any."
"Clearly there must be, but I am unable to find any examples."
This conception of MPS contrasts with what is imagined in today's curricula
which consider a much broader view of problems, different perspectives (in terms of
content, application, and methods), and in the usual classifications such as those we
Blanco, Guerrero & Caballero
presented in the program which show a variety of different possibilities. Consequently,
initial teacher education programs should intensify the attention given to these issues.
The PPTs' expectations as problem solvers themselves
For the 5th session, we adapted the Battery of Scales of Generalized Expectations of
Control, BEEGC-20 (Palenzuela et al., 1997), to the context of PS. This adaptation
consisted of a closed questionnaire, with responses on a scale of 1 to 10, targeted at
determining the students' expectations of control when faced with MPS. We wanted to
examine whether they believed their success or failure in PS would be a true reflection of
their actions, or rather be simply at the mercy of luck or chance. We also wanted to
determine their expectations of self-efficacy, i.e., to what extent they felt themselves
capable of solving mathematics problems. This was the second specific objective that we
indicated above in Sec. 3, and whose partial results we shall consider in the following
paragraphs.
The results showed the participating students had a high expectation of
contingency on their actions (perseverance, effort, commitment, ability), and a low
external locus-of-control reflecting luck or chance.
Thus, their responses to Item 1 ("My success in solving mathematics problems
will have much to do with the effort I put into it"), with a mean score of 6.69, showed that
they see effort as being crucial for success in MPS. The result was similar for Item 15 ("If
I try hard and work, I will be able to solve successfully the mathematics problems that I
am set") which was directly related to the dependence of success in problem solving on
effort and application. Additionally, 54.9% indicated on Item 11 ("In general, success or
TME, vol10, nos.1&2, p .351
failure in solving a mathematics problem will depend on my actions") that success would
depend on their own actions.
Rinaudo, Chiecher & Donolo (2003) and Martínez (2009) also refer to high levels
of control, and the subjects studied by Orozco-Moret & Diaz (2009) and Yara (2009)
attributed success in MPS to ability and effort. However, many prospective teachers
become blocked when faced with these mathematics tasks, and in many cases end up by
abandoning the effort, as was shown in the observations of their behaviours in the fourth
and eighth sessions. This reflected a certain contradiction between what they expressly
stated and their actions in class in dealing with these mathematics tasks. These
observations also revealed their lack of knowledge of procedures and heuristics with
which to tackle mathematics problems.
With respect to their expectations of self-efficacy, these prospective teachers
showed little confidence in themselves and their abilities when solving quite normal
problems of mathematics. Thus, 70.58% said they had "thoughts of insecurity when doing
MPS" (Item 14) and 64.7% "had doubts about their ability to solve mathematics
problems" (Item 2). In this regard, Caballero, Guerrero & Blanco (2008) and Hernández,
Palarea & Socas (2001) also note that PPTs in initial teacher education do not see
themselves as capable or skilled in mathematics.
That the PPTs mainly attributed their success or failure in solving mathematics
problems to their own actions and not to helplessness or luck means that they are largely
attributing success to internal, unstable, and controllable factors. This is beneficial for
their future learning situations. On the contrary, their low expectations of self-efficacy,
Blanco, Guerrero & Caballero
i.e., their lack of confidence in their capacity to solve the mathematics problems they will
be set, would seem to be prejudicial for the future learning.
Their high expectations of contingency together with low expectations of selfefficacy foster the development of negative attitudes towards solving mathematics
problems, leading the PPTs to consider that failure in this activity is due to a lack of
ability rather than to any lack of effort. As suggested by Martínez (2009), the result is to
severely lower their expectations of success, and to encourage them to abandon any
persistence in trying to learn how to solve mathematics problems. Similarly, their low
expectations of self-efficacy and their not very high expectations of achievement are
suggestive of an algorithmic approach to learning.
Some results of the program of MPS and emotional education
To evaluate the program, one of the instruments we used was the State/Trait Anxiety
Index (STAI) self-assessment questionnaire adapted from Spielberger (1982). We
presented this questionnaire on three occasions – at the beginning of the program, on its
completion, and four months after its completion. In the present communication, we shall
compare the results of four of its items:
When I am solving mathematics problems I feel calm. (calm)
When I am solving mathematics problems I feel secure. (security)
I feel comfortable when I am solving a mathematics problem. (comfort)
I feel nervous when I am faced with a mathematics problem. (nervousness)
The results indicated a positive trend in the period covered by the program, with a
decrease in anxiety about MPS continuing four months after the program. Even though
TME, vol10, nos.1&2, p .353
there was a slight regression relative to the actual moment of completion of the program,
the data were better than those obtained at the beginning. This reflects a major advance
with respect to the control of anxiety following participation in the intervention program.
The participants also declared a change in attitude: "The program has changed
our attitude to MPS, even though the content we have acquired throughout our lives is
impossible to change in just 13 sessions" (10 FS 1). Other evidence shows their desire to
integrate cognitive with affective aspects: "This program has also been useful in that,
when we are teaching, we will know to take into account not only what our pupils know,
but also their attitudes and emotions, which, by my own experience, I know have a great
influence both positive and negative" (7 DP 4).
The debate that took place in the evaluation session and in the forums showed
important reflections and concerns which we interpreted as an attempt to approach PS in
a systematic and orderly fashion, as a result of the procedures and heuristics worked on in
the general model during the program. Thus, in the evaluation session (Session 13), one
student notes that: "The execution of the steps [in a problem solving strategy] helps me to
concentrate, to analyse the problem, and to understand it better" (18 MV 2). Another
student says: "In my case, it helped me to be more orderly in presenting problems" (10 FS
2), and expresses a desire to apply it in her future work as a teacher: "We dealt with
aspects that we shall subsequently transmit to our own pupils, and with methods that we
will use such as the steps to follow for problem solving and relaxation methods" (18 MV
3).
Blanco, Guerrero & Caballero
Conclusions
The present work has confirmed the importance of considering in an integrated form the
cognitive and affective aspects of mathematics teaching and learning at different levels,
especially in initial teacher education. This can help foster the change in our prospective
teachers' beliefs and attitudes along the line laid out in current curricular proposals.
As one of the students stated, it is difficult in just 13 class sessions for our PPTs to
learn both MPS for themselves and how to teach it to their future primary pupils. But it is
gratifying to note that a change in attitude was initiated, and that they themselves valued
positively their first-person experience in the program, and that the content of the
program fell within their expectations as future teachers.
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Science & Technology Education, 4(1), 27-30
Zan, R., Bronw, L., Evans, J. & Hannula, M.S. (2006). Affect in Mathematics education:
an introduction. Educational Studies in Mathematics, 60, 113–121.
Zevenbergen, R. (2004). Study groups as tool for enhancing preservice students’ content
knowledge. Mathematics Teacher Education and Development vol.6, 4-22.
Zurilla, T.J. & Goldfried, M.R. (1971). Problem solving and behavior modification.
Problem solving and behavior modification. Journal of Abnormal Psychology, Vol
78(1), Aug 1971, 107-126.
TME, vol10, nos.1&2, p .361
Blanco, Guerrero & Caballero
Annex I. Intervention Program
The program ran from October to December 2008 except for the evaluation session
which took place in April 2009, four months after completion of the program.
Session 1. (27 and 28 October) Presentation of the program to students. 55 PPTs.
27-X. The students were provided with extensive information about the
workshop, working methods, and objectives.
27-X. Initial questionnaire - Commitment to the workshop.
Objective: To determine the participants' self-perception as problem solvers, and
their
degree
of
commitment
to
the
workshop.
(Open questionnaire; qualitative analysis.)
28-X. Conceptions and knowledge of MPS. What do I understand by a
problem? Objective: To analyse students' conceptions and knowledge about
mathematics
problems.
(Open questionnaire; qualitative analysis.)
28-X.
Affective
domain
in
MPS.
Objective: To examine the students' affective factors (beliefs, emotions, and
attitudes) that influence their development in MPS. Adaptation to MPS of the
questionnaires of Gil, Blanco & Guerrero (2006), and Caballero, Guerrero &
Blanco
(2008)
on
the
affective
domain
in
mathematics.
(Closed questionnaire; quantitative analysis.)
Session 2. (4 November) Conceptions and affective aspects of MPS. 53 PPTs.
Presentation and discussion of the results of the previous questionnaires. We
analyse the PPTs' conceptions of MPS, comparing them with the perspectives
outlined in the curriculum (as specific content and as method) and with those
described by different authors. Likewise, the results of the questionnaire on
affect are discussed, expanding them with other previous results (Gil, Blanco &
Guerrero, 2006; Caballero, Guerrero & Blanco, 2008).
Session 3. (7 November) Problems vs exercises. 55 PPTs.
Differentiation of exercises and problems, and presentation of other types of
problems based on different classifications (Borassi, 1986; Butts, 1980; Charles
& Lester, 1982; etc.).
Forum on the Moodle platform concerning the content of Sessions 2 and 3.
Session 4. (11 November) How do I approach MPS? Before, during, and after. 55
PPTs.
Pre-test.
Objective: To evaluate the participants' own impressions that arise at different
moments of MPS. Two problems proposed for solution which will allow us to
observe their knowledge and affects at different stages of the PS process at this
early stage of the program. They will be asked to respond to the same open
questionnaire on three occasions – before seeing the problem, while they are
solving it, and after having had to deal with the activity.
(Open questionnaire; qualitative analysis.)
TME, vol10, nos.1&2, p .363
Adaptation of the STAI (state / anxiety) to MPS – pretest.
Objective: To determine the level of anxiety that MPS provokes in the students.
Adaptation
of
the
STAI
(Spielberger,
1982)
to
MPS.
(Closed questionnaire; quantitative analysis.)
Session 5. (14 November) Personal involvement. Causal attributions, and
behaviour and stress. 54 PPTs.
BEEGC-20
adapted
to
MPS.
Objective: To examine the causal attributions relating to MPS (expectations of
success and of the locus-of-control, of helplessness, and of self-efficacy).
Adaptation of the BEEGC-20 Questionnaire of Palenzuela et al. (1997) to MPS.
(Closed questionnaire; quantitative analysis.)
Initiation of a discussion in class on the content of the questionnaire, which
will be followed by a specific forum on the Moodle virtual platform.
Session 6. (18 November) Emotional coping: relaxation, breathing, and selfinstruction. 55 PPTs.
Presentation of results of the previous questionnaire, and analysis of the
interventions in the forum.
Information and explanation of different aspects of emotional education and
its relationship with PS. Mainly the topics covered in the questionnaire.
Session 7. (21 November) Overview of the Integrated Model6 of MPS I. 55 PPTs
Sessions 8–10. (25 November, 2 December, 5 December) Application of the
Integrated Model, with specific problems.
Session 11. (9 December) Specific activities of the Integrated Model for primary
pupils. 55 PPTs.
In this session, we present specific activities adapted to the primary level that
allow the PPTs to work with pupils aged 6 to 12 at different stages of the general
model – basically, the comprehension and analysis of problems, and the design
of strategies.
Session 12. (12 December) General model of MPS. 55 PPTs.
The PPTs work specifically on problems in a complete and continuous
application of the Integrated Model.
STAI
adapted
to
MPS
(state
/
anxiety).
Post-test
I.
Objective: To determine the level of anxiety that MPS provokes in the students
6
Blanco et al. (2010) proposed a theoretical model based on general models of PS (Bransford &
Stein, 1987; Polya, 1957; Santos, 2007), on the cognitive-behavioural models of Zurilla & Goldfried
(1971) and on the systemic model of De Shazer and the Milwaukee group (De Shazer, 1985). It
consists of a process of experimentation and reflection based on the general model, and structured into
five steps: (i) accommodation / analysis / understanding / familiarization with the situation; (ii) search
for and design of one or more PS strategies; (iii) execution of the strategy or strategies; (iv) analysis of
the process and the solution; and (v) How do I feel? What have I learnt?
Blanco, Guerrero & Caballero
after the workshop. (One problem.) Adaptation of the STAI of Spielberger
(1982)
to
MPS.
(Closed questionnaire; quantitative analysis.)
Session 13. (16 December) Evaluation of the PPTs and the workshop. 55 PPTs.
Evaluation: Proposal of a problem for the PPTs to solve by following the
Integrated Model, in order to evaluate the knowledge they have acquired about
the General Model.
Workshop evaluation. How have I managed my resources?
Objective: To determine the strengths and weaknesses of the workshop, and the
progress
the
participants
have
made
in
MPS.
(Open questionnaire; qualitative analysis.)
Classroom discussion about how the workshop has functioned in relation to
its proposed objectives, and about the participants' individual goals and
commitments. Audio and video recordings and field notes, and opening a forum
on the Moodle virtual platform.
Session to evaluate the research. (April 2009) 34 PPTs.
STAI
(state
/
anxiety).
Post-test
II.
Objective: To determine the level of anxiety that MPS provokes in the students
four months after completion of the program.
Annex II
"What do I understand by a mathematics problem?"
a. Full name __________________________________________
1. Formulate the statement of three mathematics problems.
2. Indicate why mathematics problem solving is important in compulsory
education.
3. Write down some personal reflections about your own experience in
mathematics problem solving in primary and secondary school.
4. What consideration does mathematics problem solving merit on your part?
5. Add something that you find significant, and have not written.
TME, vol10, nos.1&2, p.365
Developing the art of seeing the easy when solving problems
Alfinio Flores & Jaclyn Braker
University of Delaware
Introduction
For Leonardo da Vinci “saper vedere”, that is, knowing how to see, or having the
art to see, was the key to unlocking the secrets of the visible world. Saper vedere
included a precise sensory intuitive faculty as well as artistic imagination (Heydenreich,
1954) which were at the root of Leonardo’s inventiveness and creativity. According to
Leonardo, to understand, you only have to see things properly (Bramly 1994, p. 264).
Knowing how to see is also important in mathematics. The Italian mathematician Bruno
de Finetti (1967) stresses this importance in his book on “Saper vedere” in mathematics.
He highlights several aspects of knowing how to see in mathematics, such as knowing
how to see the easy, how to see the concrete things, and how to see the economical
aspects. He also discusses in what ways knowing how to see also helps us to better
recognize the meaning of general and systematic methods of mathematics represented in
formulas. His book starts by highlighting the importance of reflection for learning the art
of seeing.
Reflection also plays a central role in Polya’s Looking back stage in problem
solving. Polya’s heuristics also provide a language to help problem solvers think back
about their problem solving experiences. As Lesh and Zawojewski (2007) point out, “by
describing their own processes, students can use their reflections to develop flexible
prototypes of experiences that can be drawn on in future problem solving” (p. 770).
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos.1&2, pp. 365-378
2013©The Author(s) & Dept of Mathematical Sciences-The University of Montana
Flores & Braker
Reading Polya’s heuristics and looking at the examples he gives, we can concur with
Lesh and Zawojewski that Polya’s heuristics are intended to help students go beyond
current ways of thinking about a problem, rather than being intended only as strategies to
help students function better within their current was of thinking.
Lesh and Zawojewski (2007, p. 769) point out that when solving problems in
complex problematic situations the abilities related to “seeing” are as important as
abilities related to “doing”. Schoenfeld (1985) found that individuals select solution
methods to problems based on what they “see” in problem statements. Schoenfeld’s data
indicate that mathematical experts decide what problems are related to each other based
upon the deep structure of the problems, whereas novices tend to classify problems by
their surface structure (p. 243). Krutetskii (1976) found in his research that one trait of
mathematically able students was to strive for a clear, simple, short, and thus “elegant”
solution to a problem (p. 283). He also mentions that “a striving for simplicity and
elegance of methods characterizes the mathematical thought of all prominent
mathematicians” (p. 283-284). Krutetskii also describes how all the capable students,
after finding the solution to a problem, continued to search for a better variant, even
though they were not required to do so (p. 285). In contrast, average students paid no
particular attention in his experiments to the quality of their solutions if there were no
special instructions from the experimenter in that respect. Krutetskii observed that
capable students “were usually not satisfied with the first solution they found. They did
not stop working on a problem, but ascertained whether it was possible to improve the
solution or to do the problem more simply” (p. 285-286).
TME, vol10, nos.1&2, p .367
In this article we will focus on learning the art of seeing the easy, by using an
example of a
problem posed to future secondary mathematics teachers. De Finetti
indicates that it is often difficult to see the easy things, that is, to be able to distinguish, in
the complexity of circumstances present in a problem, those that are enough to formulate
the problem or that allow one to do the formulation as several successive steps that can be
carried out easily.
The problem presented below was posed as part of a modeling course. Lesh and
Doerr (2003) point out that from a modeling perspective, traditional problem solving is
viewed as a special case of model-eliciting activities. Lesh and Doerr emphasize that “for
model-eliciting activities that involve a series of modeling cycles, the heuristics and
strategies that are most useful tend to be aimed at helping students find productive ways
to adapt, modify, and refine ideas that they do have.” (p. 22). According to Lesh and
Doerr, we need to put “students in situations where they are able to reveal, test, and
revise/refine/reject alternative ways of thinking” (p. 26).
We will first present the strategy used by a group of future teachers, and then an
approach gained by looking back at the problem and trying to see it at a glance. We finish
with a brief discussion of why it would be worthwhile for prospective teachers to look
back at the this and other problems..
The problem
During a course for prospective high school teachers, one of the assignments was
to present a problem to their fellow students that could be modeled or solved with high
school mathematics. The second author posed the following problem to her classmates.
Flores & Braker
You are attempting to bathe a cat in your kitchen. Unfortunately, the cat is not as
open to the bath as you were hoping, and as a result you spill 3 gallons of water in
your kitchen. Which brand of paper towel should you use to clean up the spill?
Brand A
Brand B
Paper towel is 1/32 inches thick
Paper towel is 1/64 inches thick
Total diameter of roll is 5 inches
Total diameter of roll is 6 inches
Diameter of hollow inside is 2 inches
Diameter of hollow inside is 2 inches
One sheet absorbs 1.5 fluid ounce
One sheet absorbs 1 fluid ounce
Each sheet is 10 inches long
Each sheet is 10 inches long
The assumption is that the price for the roll is the same for both brands. Notice
that it is not necessary to know the width of the sheets, because we know how much each
sheet absorbs for each brand. Remember that 1 gallon = 128 fluid ounces.
Figure 1a. Cross section of Brand A roll
Figure 1b. Cross section of Brand B roll
The future teachers used an approximation by modeling the spiral cross section of
the role of paper as a series of concentric circles. Each successive layer was a little longer
because the thickness of each sheet increased the diameter. The approach used by all the
future secondary teachers to solve the problem was to find how many rolls of each brand
TME, vol10, nos.1&2, p .369
were needed to clean up the spill. To find this number they decided to compute how
much water can be absorbed by one roll of each brand, finding first how many sheets are
in each roll.
Figure 2. Concentric layers
Thus for Brand A the first layer has a length of
C1 2 1
inches.
For the second and third layers the length in inches is
1
C2 2 1
32
2
C3 2 1
32
and in general, the length in inches of the k-th layer is
k 1
Ck 2 1
32
.
The number of layers, n, is given by dividing the thickness of the roll, 1.5 inches,
n
by the thickness of each sheet, 1/32 inches, so
1.5
48
1
32
.
Flores & Braker
The total length is thus
1 47 48
2 48
32
2
96
⋯
⋯
⋯
141
523 inches.
2
Thus the total length of a roll of brand A is 523 inches. The length of each sheet is
10 inches, so there are about 52 sheets per roll. These sheets together can absorb
52 1.5 78 ounces of water. Thus each roll absorbs 78 fl. oz. of water. To clean 3
384 fl. oz
4.9
fl. oz
78
roll
gallons = 3 128 fl. oz. = 384 fl. oz. we need
rolls. That is, we need
almost 5 rolls of Brand A to clean the spilled water.
k 1
For Brand B the length of each layer is Ck 2 1
inches and the
64
number of layers is n
2
128 . The total length is
1
64
⋯
⋯
1 127 128
2 128
1602
64
2
inches. The total length is thus about 1602 inches.
Because each sheet is 10 inches long, that is about 160 sheets. Each sheet absorbs one
TME, vol10, nos.1&2, p .371
fluid ounce, so one roll absorbs 160 fl. oz. To clean 3 gallons we need
384 fl. oz
2.4
fl. oz
160
roll
rolls. Brand B is clearly the better choice for this problem.
Looking back
Polya points out that when we have obtained a long and involved solution we
naturally want to see whether there is a more direct and clear way to solve the problem.
He advises one to ask the questions: Can you derive the result differently? Can you see it
at a glance? (Polya 1973, p. 61). He also points out that even when we have found a
satisfactory solution we may still benefit from finding a different solution, which may
give us further understanding or allow us to look at the problem from a different
perspective. Polya encourages us to study a result and try to understand it better, to see a
new aspect of it (p. 64). In the same way that we might get a better perception of an
object by using two senses, we might get a better understanding of a problem by finding a
solution in two ways. Future teachers need to learn to guide their students on how to find
in a result itself indications of a simpler solution.
The approach used by the future teachers described above has several advantages.
One is that it highlights the use of an arithmetic sequence and how the average of its
terms can be used to obtain their sum. Because 1 + 2 + … + n is an arithmetic series, the
average of all the terms will be the average of the first and last terms,
read the formula for 1 + 2 + … + n =
n 1
. One way to
2
n(n 1)
is that to obtain it we multiply the average
2
Flores & Braker
of thee terms,
n 1
, by the nu
umber of terrms n. Anothher advantagge is that wee can actuallly
2
find how
h many ro
olls of paperr we need.
In terms of the orig
ginal problem
m posed w
we may wannt to look bback and assk
ourseelves what are
a the essen
ntial differen
nces betweenn the two tyypes of papeer rolls in this
probllem. In the situation
s
desscribed, we really want to comparee the efficienncy to absorrb
waterr of the rolls relative to each
e
other in
n order to dettermine whiich brand to use. Once w
we
determ
mine what brand
b
to use then we can
n compute hoow many rollls of that braand we needd.
Use of
o proportio
onal reasoniing to comp
pare paper ttowels
A key inssight for solving this problem in a ddifferent waay is to realiize that wheen
comp
paring the ro
olls, we need
d to comparee their ratioss with respeect to the diffferent factorrs
that affect
a
their number
n
of sh
heets and abssorption capaacity.
In the solu
ution above,, the averagee of the lenggths of the laayers playedd an importannt
role. Here we wiill see how we can use the averagee in a differrent way. Thhe number oof
w be proporrtional to thee area of thee circular ring cross sectiion . The areea
sheetts in a roll will
of thiis ring can be
b obtained by
b multiplyin
ng the circum
mference off the averagee circle by thhe
width
h of the ring (Figure 3). If
r1
is the radius
r
of thee hollow circcular center,
r2
the radiuus
of the paper rolll, and d1 an
nd d2 their respective
r
ddiameters, thhen the areaa of the crosss
sectio
on is given by
b
(1)
TME
E, vol10, nos.1&2, p .3773
Exerccise 1. Deriv
ve formula (1) for the area
a
of the rring using thhe differencce of areas oof
conceentric circless.
Exerccise 2. Discu
uss in what ways
w
formula (1) is analoogous to thee formula forr the area off a
trapezzoid.
Figure 3. The averagee circle
ood way to compute
c
how
w many sheeets are arounnd the roll iss by using thhe
Thus, a go
circum
mference off the averagee circle, in this
t case thee mid circle between thee hollow corre
and th
he outer layeer. For Bran
nd A this aveerage circle has a diameter of
for Brand
B
B a diameter
d
of
25
3.5 inches,
2
26
4 inches. Thhe number oof sheets inn each roll is
2
propo
ortional to the
t diameterrs of its mid circles, annd proportioonal to the useful crossssectio
onal width of
o the rolls. The
T correspo
onding ratioss comparing Brand B to Brand A wiill
Flores & Braker
be thus
4
2
for the diameters of the average circles, and
for the useful cross3.5
1.5
sectional widths of the rolls. The number of sheets will be inversely proportional to the
thickness of each sheet, so the ratio between Brand B and Brand A is
=
64 2
. The
32 1
two brands have the same lengths of sheets, so to get the ratio of the number of sheets of
Brand B to the number of sheets of Brand A, we just need to multiply these three ratios,
which yields
4 2 2
. Because the ratio of the absorption efficacy per sheet of
3.5 1 1.5
Brand B to Brand A is
1
, the ratio of number of ounces of water absorbed by a roll of
1.5
Brand B to the number of ounces of water absorbed by Brand A is
1 128
4 2 2
. So Brand B is about twice as good as Brand A for this task.
3.5 1 1.5 1.5 63
This agrees with our previous result that the ratio of rolls needed is
4.9
. With this
2.4
alternative approach of multiplying ratios it would be easy to make adjustments in case
the length of the sheets or the price was not the same for both brands. All we would have
to do is to multiply the previous product of ratios by the ratios of the prices, and by the
ratio of the length of the sheets. In these cases, as with the thickness of the sheets, we
would be dealing with inverse ratios.
To find how many rolls of Brand B we would actually need, we can find the
number of sheets in a roll, using the average circumference (4π), multiplying it by the
number of layers that fit in the usable cross-sectional width ( 2
1
), and dividing by the
64
TME, vol10, nos.1&2, p .375
length of the sheets (10). So the number of sheets is 4 128 10 161 . (Notice that
this result is very close to the result obtained with the other method.) Because each sheet
of Brand B absorbs one ounce of water, this is also the number of fluid ounces that each
roll can absorb. The total number of rolls required is
384
2.4 .
161
Exercise 3. Derive formula (1) as the limit of polygonal rings formed by trapezoids (see
Figure 4).
Exercise 4. Discuss in what ways is formula (1) analogous to the formula for the volume
of a torus obtained by rotating a circle around an axis outside the circle. The volume of
the torus is equal to the product of the area of the circle times the circumference traced by
its center.
Figure 4. A ring formed by trapezoids
Concluding remarks
When teachers pose a mathematical problem to their students, they often do so
because the problem can be solved with a mathematical approach that the teachers want
to illustrate. Problem solving can be used as a powerful means to learn mathematical
concepts and procedures (Lester & Charles, 2003; Schoen & Charles, 2003). In the above
problem, the intent of the preservice teacher who posed it was that students have an
Flores & Braker
opportunity to use an arithmetic progression and the formula for its sum. Problems can be
excellent ways to foster the development and understanding of particular mathematical
concepts and procedures. However, students might use an alternative solution process
that does not require the concept or process that the teacher wanted to emphasize.
Teachers thus need to be aware that students might find alternative solutions that do not
involve those concepts or procedures. In that case, the teachers needs to decide at what
point, and to what extent, they should discuss those alternative approaches. It is important
that teachers look at problems they pose from multiple perspectives, and try to foresee
alternative solutions. That way teachers can better plan how and when to use those
alternatives so that it becomes an enriching experience for all the students, rather than
becoming a situation where some students have the opportunity to develop their thinking
with respect to specific mathematical concepts and methods and others do not. Of course,
sometimes students may surprise us and find an approach we did not foresee.
Learning to see the easy is one of the possible benefits of looking back at a
problem and reflecting on its solution. Finding a simpler solution does not mean that our
original approach was less valuable. The first method that occurred to us very likely gave
us some insight into mathematical relations of a certain kind in the given situation, and
perhaps used mathematical ideas that were freshest in our minds. Furthermore, often we
find a simpler path only after we are able to solve the problem in another way. By taking
time to consider alternatives once they have found a solution, students may find an easier
solution. Students may realize it is not always necessary to apply the most complicated
mathematical concepts that they know in order to solve even what appear to be difficult
problems.
TME, vol10, nos.1&2, p .377
However, even when we find a simple solution first, it is worthwhile to take a
second look at a problem and look for a different solution. The second solution may give
us a different kind of insight. As Polya points out, there are also other benefits of looking
back, such as establishing connections. A few connections were hinted at above, but a
full treatment would go beyond the main focus of this paper.
There are other authors who emphasize the importance of reflection when solving
problems. Shulman states that “the more complex and higher-order the learning, the more
it depends on reflection—looking back—and collaboration—working with others.”
(Shulman 2004, p. 319). The importance of reflection is not restricted to mathematics
learning. Shulman also describes how studies of expertise in the solving of physics
problems indicate that the most able problem solvers do not learn by just doing, that they
do not learn from simply practicing the solving of physics problems. Rather they learn
from looking back at the problems they have solved and learn by reflecting on what they
have done to solve them. Able problem solvers learn, not just by doing, but also by
thinking about what they have done. (Shulman, 2004, p. 319).
Good teachers understand and convey to their students the benefits of looking
back at a problem. Learning to see the easy is one of the benefits.
Flores & Braker
References
Bramly, S. (1994). Leonardo: The artist and the man. New York: Penguin Books.
De Finetti, B. (1967). Il “saper vedere” in matematica. Torino: Loescher Editore.
Heydenreich, L. H. (1954). Leonardo da Vinci. New York: Macmillan.
Krutetskii, V. A. (1976). The psychology of mathematical abilities in schoolchildren.
Chicago, University of Chicago Press.
Lesh, R. & Doerr, H. M. (2003). Foundations of a models and modeling perspective on
mathematics teaching, learning, and problem solving. In R. Lesh & H. M. Doerr
(Eds.), Beyond constructivism: models and modeling perspectives on mathematics
problem solving, learning, and teaching (pp. 3 – 33).
Mahwah, NJ: Lawrence
Erlbaum and Associates.
Lesh, E. & Zawojewski, J. S. (2007). Problem solving and modeling. In F. Lester (Ed.),
Second Handbook of Research on Mathematics Teaching and Learning (pp. 763804). Charlotte, NC: Information Age Publishing.
Lester, F. K., Jr., & Charles, R. I. (Eds.). (2003). Teaching mathematics through problem
solving: Pre-K–grade 6. Reston, VA: National Council of Teachers of Mathematics.
Polya, G. (1973). How to solve it: A new aspect of mathematical method. (2nd edition).
Princeton, NJ: Princeton University Press.
Schoen, H. L., & Charles, R. I. (Eds.). (2003). Teaching mathematics through problem
solving: Grades 6–12. Reston, VA: National Council of Teachers of Mathematics.
Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando FL: Academic Press.
Shulman, L. S. (2004). Teaching alone, learning together: Needed agendas for the new
reform. In L. S. Shulman, The wisdom of practice: Essays on teaching, learning, and
learning to teach (pp. 309-333). San Francisco, CA: Jossey-Bass.
TME, vol10, nos.1&2, p.379
Two-step arithmetic word problems
Enrique Castro-Martínez1
University of Granada, Spain
Antonio Frías-Zorilla2
University of Almería, Spain
Abstract: This study uses the perspective of schemes to analyze characteristics of
arithmetic word problems that can influence the process of translation from the verbal
statement to an arithmetical representation. One characteristic that we have detected in
the two-step word problems is the presence of one or two connections (nodes) in schemes
that represent them, and this paper explores whether the number of nodes affects the
activation of the associated schemas. With students from the 5th and 6th grades of
elementary school (11 and 12 years of age), we analyze the written productions and
would stress that the number of connections influences the activation of the right schema.
Results show that the double connection implicate a greater difficulty for obtaining a
correct arithmetical representation. Likewise, the presence of a simple or double
connection between the two relationships means that the students commit specific errors
that we associate with this characteristic.
Keywords: Two-step word problems, arithmetic, schemes, double node, errors.
Introduction
Research on problem solving on mathematics education is a wide and varied field,
and it is not limited to a single study focus; nor is it performed within a single theoretical
framework (Castro, 2008; Santos, 2008). A good number of studies have centered on the
use of arithmetic operations to solve word problems. Verschaffel, Greer, & Torbeyns,
(2006) distinguish four focuses in the study of arithmetic problems: (a) conceptual
structures (schemes) for representing and solving word problems; (b) word problems
viewed from a problems-solving perspective; (c) a sociocultural analysis of performance
on arithmetic word problems; and (d) the modeling approach.
1
2
Author’s address: Facultad de Ciencias de la Educación, 18071, Granada, Spain. E-mail: [email protected]
Author’s address: Facultad de Ciencias de la Educación. Almería, Spain. E-mail: [email protected]
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos.1&2, pp.379-406
2013©The Author(s) & Dept. of Mathematical Sciences-The University of Montana
Castro & Frías-Zorilla
Since the 1990s, there have been a tendency in Mathematics Education to
undervalue the educational value of word problems and stress situated and socially
mediated approaches to solving authentic, complex problems. Despite this focus,
Jonassen (2003) indicates that “story problems remain the most common form of problem
solving in K-12 schools and universities” (p. 267). This paper treats arithmetic word
problems whose statement contains two relationships between the data and that therefore
require more than one operation to solve them (two-step arithmetic word problems). We
perform our analysis from the perspective of schemes (Hershkovitz, & Nesher, 2003) and
focus on characterizing the double node in two-step arithmetic word problems and the
schemes to which they give rise, and on studying the influence of the double node on the
activation of the schemes and the errors this causes. Enright, Morley, & Sheehan (2002)
indicate that problem features such as those described previously can be related
theoretically to individual differences in cognition (p. 51).
The scheme approach
From the semantic perspective on one-step arithmetic problems, once the
concepts and relationships involved are understood, the child has only to choose the
correct operation and apply it (Quintero, 1983, p. 102). In problems with several steps, it
is also necessary to understand the concepts and relationships, but additional issues are
involved as well. Quintero (1983) indicates that the child must plan and organize the
order in which to apply the operations and identify the pairs of numbers to which to apply
each operation. Shalin and Bee (1985) analysis of two-step problems leading to specific
structures is based on the possible logical combinations of one-step problems. They
TME, vol10, nos.1&2, p .381
represent the corresponding scheme of a simple arithmetic word problem by means of a
diagram (Figure 1) of three connected components in terms of part-whole relationships.
Figure 1. Notation of the triad of components present in the part-whole relationship
If the diagram in Figure 1 represents a mathematical object, we can construct
more complex mathematical relationships from it using more than one diagram and
connecting them to each other, forming networks. Following this idea, Shalin & Bee
(1985) obtain the structure of a two-step problem by combining two triads based on local
relationships. Each of the different ways of combining two triads like that in Figure 1
constitutes a different global problem structure. These combinations (Figure 2) define
three structures of two-step problems: hierarchy, sharing the whole and sharing a part.
Figure 2. Hierarchical scheme, sharing the whole and sharing a part
Nesher & Herskovitz (1994, 2003) research the influence that the three schemes
(Figura 2) have on the index of difficulty of composite problems. With a sample of
students from third, fourth, fifth, and sixth grades in Israel, they find that the variable
type of scheme has a significant effect on the index of difficulty of these problems. The
“hierarchical” scheme is the easiest, followed by the “shared whole” and finally the
“shared part” scheme. The study by Shalin & Bee (1985) also showed that children in the
Castro & Frías-Zorilla
3rd, 4th
and 5th grades (elementary school) had a higher rate of success with the
hierarchical scheme. In the following section, we will see that the results can be altered
by other cognitive variables.
Decrease-increase relationship
In the research performed by the Numerical Thinking Group of the University of
Granada, with 4th, 5th, and 6th grade elementary school children from Granada (Spain), the
results obtained by comparing the indices of difficulty of the different combinations of
the relationships of increase or decrease show that the combinations of increase and
decrease affect the difficulty of the two-step problems (Castro, Rico, Castro, & Gutiérrez,
1994; Castro, et al., 1996); Rico, Castro, González, & Castro, 1994; Rico, et al., 1997).
The four classes of problem are determined by whether the relationship is one of
increase or decrease.
Type (I, I). Two relationships of increase. The whole of the first initial
relationship is a part of the second relationship (hierarchical scheme).
Type (D, D). Two relationships of decrease. One part of the first
relationship is the whole of the second relationship (hierarchical scheme).
Type (I, D). First relationship of increase and second of decrease. The two
relationships share the whole (sharing the whole scheme).
Type (D, I) The first relationship is one of decrease and the second one of
increase. The two relationships share a part (share a part scheme).
Presented in order of increasing difficulty, they are:
(I, I), (I, D), (D, I) and (D, D)
TME, vol10, nos.1&2, p .383
where the type (D, D) is the most difficult. These results contradict the argument that the
hierarchical scheme is generally less difficult than the other two schemes. Shalin & Bee
(1985) and Nesher & Hershkovitz (1994) find that the problems associated with the
hierarchical scheme are less difficult than the others. However, in Castro, et al., (1996)
study with additive problems, the problems corresponding to the two extreme
combinations—the easiest, increase-increase (I, I) and the most difficult decreasedecrease (D,D)—correspond to the same scheme: the hierarchical scheme. The difficulty
of the hierarchical scheme is consequently affected by the relationships of increase or
decrease used to state the problem. Other cognitive variables also appear in two-step
problems, however, such as the number of connections between the components of the
basic structure, as we will see in the next section.
Problems with two nodes
One of the key issues in understanding the structure of two-step word problems is
understanding the nature of the two elements that compose the basic triad of the partwhole scheme and the way of connecting these elements between two triads. To
determine how this is done, Nesher & Hershkovitz (1994) perform a textual analysis of
the problems, breaking them into components. They distinguish three components in a
one-step problem. Two of these provide numerical information explicitly (complete
components) and the other, the question, is missing numerical information (incomplete
component).
In the composite schemes for two-step problems (Nesher & Hershkovitz, 1994),
the connection between the two one-step problems is created by a new component, which
Castro & Frías-Zorilla
they call the latent component of the problem (see Figure 3) and which is common to the
two simple structures.
Figure 3. Latent component
From a representational point of view, we say in this situation that there is a nexus
or node between the two simple structures that produce the corresponding composite
scheme. Thus, the two simple structures share a component within a two-step problem.
For example, in Problem 1:
Problem 1. I bought 5 books. Each book cost 8 euros. If I pay 50 euros,
how much money will I get back?
In the first structure, the latent component is the question of the first problem:
I bought 5 books
Each book cost 8 euros
How much do all of the books cost?
In the second structure, the latent component becomes a complete component:
All of the books cost 40 euros.
I pay 50 euros.
How much money will I get back?
In this problem, the latent component (the price of all the books) is shared by the
first and second arithmetic structures. This latent component, which is not stated
TME, vol10, nos.1&2, p .385
explicitly in the wording of the problem, connects the two structures. The price of the
books is obtained in the first structure, where it has the function of incomplete
component. This price is then used in the second structure as a complete component. This
function of connection between the two structures is what leads us to call it a node or
nexus of union between the two.
In the schemes of two-step problems defined by Shalin & Bee (1985) and
subsequently used by Hershkovitz & Nesher (1996) and Nesher & Hershkovitz (1994,
1996, 2003), the latent quantity is the only nexus of union between the two simple
structures. However, the condition of a node does not imply being a latent quantity, nor
does it mean that this is the only quantity with this condition. The node can also be a
piece of information given explicitly in the statement and that is shared by more than one
simple structure within a two-step problem. It is possible to find two-step problems that
have two simple structures connected by two nodes, as occurs in the following problem:
Problem 2. John has 5 balls. His grandfather gives him triple the number
he had. How many balls does John have now?
This problem 2 combines two simple schemes: one multiplicative scheme and one
additive. Both schemes have two quantities, “John’s 5 balls” and “the balls that John’s
grandfather gives him,” which are shared. In Figure 4, we see the representation of the
two simple schemes and how both contain the shared quantities. This kind of two-step
problem has only two pieces of information or, from another perspective, three pieces,
but one of these is repeated or has a double function. Therefore, two components are
shared by the two simple structures, one of these the latent component (balls that the
grandfather gives) and the other the repeated piece of information (John’s 5 balls) in the
problem.
Castro & Frías-Zorilla
Figure 4. Simple schemes of a two-node problem
The quantities that are shared by various simple structures within a composite
problem have, therefore, the condition of node, independently of whether these quantities
are given pieces of information or intermediate unknowns (latent quantities) in the
problem.
Types of two-node schemes
The problem we have used as an example of a double node is hierarchical in kind
(see Figure 5), and the 15 balls constitute the latent variable, which is the intermediate
unknown quantity.
Figure 5. Hierarchical scheme
We can see that the quantity of 5 balls appears in the two simple structures. If we
merge both boxes into a single one, as shown in Figure 5, we get a sub-scheme of the
hierarchical scheme, but one in which two quantities are shared by the two simple
schemes.
TME, vol10, nos.1&2, p .387
Figure 6. Composite scheme HP
When a part and the whole of one simple scheme matches with the part and part
of the other simple scheme (P, W = P, P), we call this composite scheme HP, since it can
be obtained from the hierarchical scheme H, since one part of each simple scheme
coincides with the other (see Figure 6).
In the other two structures of two-step problems, sharing part and sharing whole,
new substructures also emerge with this condition of considering the double node. In the
case of the structure “sharing part” (SP), we can generate a substructure by making it
agree with the other part of the two simple schemes (see Figure 7). We call this
substructure SPP.
Figure 7. Composite scheme SPP
An example of a problem corresponding to the structure SPP is
Castro & Frías-Zorilllla
Problem 3.
3 John has 7 pieces of clothing,
c
3 sshirts and thhe rest pants. How
many wayys can he com
mbine shirtss and pants tto get dresseed?
A new su
ubscheme em
merges wheen one part and the whhole of a sim
mple schem
me
hes with onee part and th
he whole of the other (P
P, W = P, W
W). In this caase, we obtaiin
match
the composite sccheme that we
w labelled SWP (Shariing Whole aand Part) (ssee Figure 88).
This subscheme can
c be obtain
ned both fro
om the compposite schem
me “sharing w
whole” SW, if
m
a partt of each siimple schem
me coincide . It can alsso be obtainned from thhe
we make
comp
posite schem
me “sharing part”
p
SP, if we
w match thee whole of eaach simple sscheme.
Figurre 8. Composite scheme SWP
The follow
wing problem
m is an exam
mple of the sscheme SWP
P:
Problem 4. Peter hass 15 marbless. Peter has 10 marbless more than John.
How many
ny times moree marbles do
oes Peter haave than Johnn?
In summaary, if we co
onsider two nodes to coonnect the tw
wo simple sstructures thaat
form a two-step problem,
p
we get the follo
owing compposite subschhemes (see F
Figure 9):
Figurre 9. Composite subscheemes with tw
wo nodes
TME, vol10, nos.1&2, p .389
The double node as a characteristic of some two-step problems can be related
theoretically from the cognitive point of view to individual differences or different
success rates (Frías, & Castro, 2007). This is due, for example, to the limited capacity of
the work memory or, as Embretson (1983) suggests, to the fact that “the characteristics of
the stimuli of the items in the tests determine the components that are involved in finding
the solution” (p. 181). From the foregoing considerations, we find it important to study
whether the two-node problems have different cognitive effects on the subjects.
For the specific case of two-step problems, the variable node takes more than one
value. We have described two-step problems with one node and two-step problems with
two nodes. We now ask whether the number of nodes in a scheme is a cognitive variable
that can influence the problem-solving process for two-step problems in students who are
finishing their elementary education.
Our conjecture is that the number of nodes in a composite two-step problem
affects the way in which the advanced elementary school students represent two-step
problems internally. This difference should become visible in issues such as the success
rate and the emergence of errors specifically involving the number of nodes.
Method
Participants
We performed a study to compare the competence of students from the fifth and
sixth grades of elementary education (ages ranging from ten to twelve) in two-step
arithmetic problems and to determine whether the number of nodes in the problem
influences the process of solving it. 172 students from public elementary schools in the
Castro & Frías-Zorilla
city of Almería (Spain) participated in the study, 86 students from 5th grade and 86 from
6th.
Variables
Given the wide variety of two-step problems, we limited the study to using a
carefully-defined set of problems. The first condition we imposed on the two-step
problems used in the study was that the semantic category corresponding to the first
simple structure of the problem be comparison (additive or multiplicative) and the
semantic category corresponding to the second simple structure of the problem be
combination, whether additive or a cartesian product. We imposed this restriction to
control for the possible effect that the kind of semantic category in each of the simple
schemes could have on the overall solving of the two-step problem.
Once we established this condition, the problems we used were chosen using
factorial design with four factors or independent variables of the problems, which are:
First factor
The first factor, which we call A, includes whether one of the simple structures
that make up the two-step problem has an additive or multiplicative character. We
understand the additive structure here to include problems that are solved with one
addition or subtraction. Likewise, we understand by multiplicative structure problems
solved with one multiplication or division. The variable A refers to the double arithmetic
relationship present in the two-step problem and in this study takes two values,
corresponding to the possible combinations of a problem composed of two steps, a simple
additive structure and another multiplicative structure:
TME, vol10, nos.1&2, p .391
Al for an additive structure followed by a multiplicative structure
(+, ×).
A2 for a multiplicative structure followed by an additive structure (×, +).
Second factor
Since the two-step problems that compose the instrument we have used all contain
a simple scheme of comparison, we have limited the possible variants of these
comparison problems to two kinds, consistently worded comparison problems and
inconsistently worded comparison problems (Lewis & Mayer, 1987). Attending to these
two kinds of wording for comparison problems, we consider the variable to be the kind of
wording in the comparison, which we have called variable E and which takes two values:
El if the wording of the comparison is consistent.
E2 if the wording of the comparison is inconsistent.
El
Consistent wording
John
has
15
marbles
Peter has 3 times more marbles than John
How many marbles do they have
altogether?
E2 Inconsistent wording
Peter
has
15
marbles
Peter has 3 times more marbles than John
How many marbles do they have
altogether?
Third factor
Each of the simple relationships involved in a two-step problem can be of either
increase or decrease (Castro, et al., 1996; Castro, Rico, Castro, & Gutiérrez, 1994; Rico,
Castro, González, & Castro, 1994). We call R the variable that combines the two
possibilities in the double relationship. In this study, we will take into account two
values:
Castro & Frías-Zorilla
R1 for the relationship increase-increase (I I).
R2 for the relationship increase-decrease (I D).
From the point of view of direct translation based on key words, this variable
provides information most specifically about the arithmetic relationship that can be used.
Increase will refer to addition or multiplication and decrease to subtraction or division.
Fourth factor
The fourth factor is the variable, our main focus of attention. It includes the
number of nodes in the two-step problem. The number of nodes, which we call the
variable nodes (N), has two values in this study:
N1 for two-node problems.
N2 for one-node problems.
N1 two-node problems
N2 one-node problems
Mary has 15 trading cards. George has 3 Mary has 15 trading cards, and Paula has
times more trading cards than Mary. How 90 cards. George has 30 more cards than
many trading cards do George and Mary Mary. How many cards do George and
have between the two of them?
Paula have between the two of them?
Instrument and procedure
The instrument used in this experiment was a questionnaire with sixteen
problems. The sixteen problems correspond to the possible combinations that emerge
from crossing the four factors mentioned above in a factorial design. So as not to
overwhelm the study subjects with too many problems, we divided this set of sixteen
problems into two questionnaires of eight problems each, according to the following
distribution:
TME, vol10, nos.1&2, p .393
N1
E1
E2
N2
A1
A2
A1
A2
R1
Q1
Q2
Q1
Q2
R2
Q2
Q1
Q2
Q1
R1
Q1
Q2
Q1
Q2
R2
Q2
Q1
Q2
Q1
Q1 Questionnaire Nº 1 Q2 Questionnaire Nº 2
The problems in these questionnaires were solved by the children individually and
silently in the classroom using pen and paper. Each child was given a questionnaire at
random.
Results
The answers given by the subjects to the problems posed were evaluated as
correct or incorrect, taking into account the choice and execution of the operations, as
well as the expression of the result. We have classified a response as correct when the
subject has chosen the right two operations between the corresponding data and has
expressed the solution correctly, writing it in the space provided for the result the
expression of the relationship that each problem required according to the instructions
provided on the questionnaires. This circumstance occurred in different ways. The most
common was to perform two operations, executing the corresponding algorithms, and to
conclude with the full expression by answering the question posed in the problem.
However, we have also considered correct those answers in which this was done
implicitly. For example, given the problem:
Javier has 12 pairs of pants. Javier has 3 more shirts than pairs of pants.
How many ways can Javier combine pants and shirt?
Castro & Frías-Zorilla
Some subjects did one of the operations (12+3=15) mentally, so that the only
explicit operation that appears is 12×15=180. In cases like this, we have evaluated the
answer as correct, since we understand that student chose the two correct operations,
performed one as a mental calculation and the other as a written algorithm, and provided
the correct answer: “Javier can combine his shirts and pants in 180 different ways.” We
have also considered answers to be correct if the answer was expressed elliptically, for
example, “They can be combined in 180 ways.” In cases where students chose the
operations to be performed correctly and used the correct data but committed a
calculation error in the algorithm, we have considered the answer to be correct, even
though the result shows a quantity different from the correct one. In this case, we believe
that this kind of error does not affect the subject’s understanding of the problem.
The answers were evaluated as incorrect when one of the two operations to be
performed was not the correct one or the subject did not perform the operation with the
proper data. No response on one of the operations was also qualified as incorrect, since it
shows that the subject did not understand at least one of the two relationships in the
problem. No answer at all was also evaluated as incorrect.
The success rates at which the children in the study were able to translate each of
the questionnaire problems into its arithmetic representation are shown in Table 1 as
percentages. They range from 20% for the most difficult problem to 90% for the least
difficult. This result shows that some of the factors that define the problem influence their
difficulty. To highlight which variables have a significant influence, we have applied a
variance analysis to the four factors.
TME, vol10, nos.1&2, p .395
Table 1. Percentages of success in the questionnaire problems according to factors
N1 -two nodes N2 -one node
A1 +× A2 ×+ A1+× A2 ×+
Consistent
E1
R1 I I
37
80
36
90
R2 I D 34
57
33
58
Inconsistent R1 I I 28
E2
R2 I D 22
36
34
55
30
20
51
Using the success rate measured in percentages as a dependent variable, we have
applied variance analysis to detect whether the four factors defined in the study had a
significant effect on the success rate. The variance analysis applied to the data obtained
shows a significant effect on the following cases:
variable N number of nodes (F = 6.677, p=0.010). The percentage of success
on problems with one or two nodes is: two nodes-N1 (41%) and one nodes-N2
(63%).
variable R combinations of increase and decrease (F=20.982, p=0.000), with a
percentage of success on the combinations of: increase-increase (49%) and
decrease-increase (38%).
variable E or kind of wording (F=56.504, p=0.000): Consistent (61%) and
inconsistent (45%).
variable A combination of the additive and multiplicative relationships
(F=116.760, p= 0.000). The percentages of success on the combinations of
additive and multiplicative relationships used were: A1(+×) combination
(30%) and A2(×+) combination (57 %).
Castro & Frías-Zorilla
We interpret the marked difference in difficulty shown by combinations A1 and
A2 according to the restriction imposed, that is, that the problems be comparison (additive
or multiplicative) in the first step and either additive or Cartesian product combination in
the second step. In problems of the type +×, we use the additive comparison in the first
step and the Cartesian product in the second step. In problems ×+, we use the
multiplicative comparison in the first step and the additive combination in the second.
The presence of the Cartesian product in the simple scheme corresponding to the second
step of the problems seems to cause the difference in difficulty.
The only significant interaction effect influenced by the variable of node is N×A
(F=6.084, p=0.014). This interaction does not change the order of difficulty of the values
of the variable node, however, as can be seen in graphic 1.
Percentage
Interaction NxA
70%
60%
50%
40%
30%
20%
10%
0%
One node
Two nodes
+x
x+
Arithmetic Relationships
Graphic 1. Percentages of correct answers according to nodes and combinations of
arithmetic relationships
In graphic 1, we can see that the problems with two nodes are more difficult to
translate into a symbolic representation than the problems with one node for the two
combinations of arithmetic operations. We can conclude from this analysis that the
TME, vol10, nos.1&2, p .397
number of connections between the two relationships is a significant differentiating
characteristic in two-step problems. The percentages of success on one-node problems
(63%) and two-node problems (41%) show a significant difference in students’
performance between these two kinds of problem. This difference does not depend on the
other factors considered.
Error analysis
In written products, we found that in addition to typical errors already identified
in one-step problems (such as the additive error or the inversion error), the sample
subjects produced new errors in the two-step problems, errors that we identified as errors
belonging to the double structure itself. Since our goal is to characterize the issues that
differentiate the two-step problems, we will stick to the description of the errors specific
to the double structure.
Type 1 error: performing only one operation
This error is characterized by using only one operation to solve a two-step
problem. The operation may be either one of the two correct operations that should be
performed or the wrong operation for another reason. In all of the cases, the subjects do
not attempt to perform more operations but instead give as an answer the result of the
only operation that they have performed with the two pieces of information from the
problem. Most of the cases observed occurred in problems with two nodes (and only two
pieces of information). In a few cases, this kind of error occurred with a problem of only
one node (with three pieces of information). Table 2 shows examples of this kind of
error.
Castro & Frías-Zorilla
Table 2. Error in performing one operation
Problems
Example 1
Anne has 12 pairs of pants
Anne has 3 shirts fewer than pants.
How many ways can he combine
pants and shirts?
Example 2
John has 24 balls
John has 3 times fewer balls than
Peter. How many balls do they
have between the two of them?
Example 3
Anne has 48 trading cards
Mary has 4 times more trading
cards than Anne. How many do
they have between the two of
them?
Errors
Comments
Omits the
12 3 = 9
Result: He can combine operation
pants and shirts in 9
ways
Omits
the
24 + 3 = 27
Result: Between the two operation
of them, they have 27
balls
Omits the
48 × 4 = 192
Result: Between the two of operation
them, they have 192
trading cards
second
first
second
In the problems used in this study, the two relationships are ordered; the first one
is always a comparison and the second a combination. For this kind of error, we can
therefore distinguish the cases in which the subject forgot the first relationship from those
in which the subject forgot the second:
1. Forgetting the first relationship
In this case, subjects take the two pieces of information in the problem and
perform an operation without taking into account the first relationship in the context of
the problem. They focus their attention on the second relationship, which is the one in
which the problem’s question appears. Examples 1 and 2 in Table 2 illustrate this case.
2. Forgetting the second relationship
In this case, they take the two pieces of information in the problem and work with
them in the context of the first relationship stated in the problem, not taking into account
TME, vol10, nos.1&2, p .399
the second relationship. In the result, they answer the question in the problem that
corresponds to the second relationship although this value was obtained with the first.
Example 3 in Table 2 fits this type of error.
Type 2 error. Ordered data
This error is characterized by choosing the data for performing the relationships in
the same order in which they appear in the problem. In certain problems in our study, this
leads to an error in the two relationships in the problem. The students take the first two
pieces of information that appear in the word problem and perform the operation, then to
perform another operation on the result and the third piece of information, and finally,
with this result, to find the solution. An example of this error can be seen in the solution
given to the problem in Table 3.
Table 3. Error in ordered data
Problem
George has 18 shirts and 6 belts. George has 3 shirts more
than pairs of pants. How many ways can he combine pants
and belt?
Solution with Type 2 error
18 6 = 12; 12 × 3 = 36
Result: He can combine
pants and belt 36 ways
This way of acting indicates recognizing the two relationships in the problem,
even distinguishing between the two simple structures, one additive and the other
multiplicative. But the subjects do not associate the data and the relationships in each
structure correctly. This leads us to think that the choice of data is mechanical or
algorithmic and that order of presentation takes precedence over any other characteristic
of the problem. In many cases, we see that, if the correct order coincides with the order in
which the data are presented, the subjects give the correct response, but when the correct
order is different than the order in which the data are presented, students make mistakes.
Castro & Frías-Zorilla
These last two kinds of error, one operation and ordered data, occur in the same
subjects; that is, that for the two-node problems they commit the error of one operation
and for a one-node problem, that of ordered data.
Type 3 error. Repeating the unshared information
In the two-node problems, we saw an error that consisted of using twice the
unshared piece of information in the two simple structures that compose the two-step
problem, while using the shared piece of information only once. An example is shown in
Table 4.
Table 4. Error of confusing repeated information
Problem
Solution with Type 3 error
Lucia has 15 shirts. Lucia has 3 fewer 15 + 3 = 18; 18 × 3 = 36
shirts than pairs of pants. How many Result: She can combine shirts and
ways can she combine shirts and pants? pants in 36 different ways
The previous solution that contains the Type 3 error shows that the subjects have
recognized the two relationships and distinguished two structures, one additive and the
other multiplicative. Further, the repetition of one piece of information from the problem
in the calculations (in this case, the 3) seems to indicate that the subject recognizes that
he or she must use this piece of information twice. The error occurs in choosing the right
piece of information.
Type 4. Other errors
In this section, we include errors that do not fit any of those mentioned above,
cases in which it is difficult to know what motivated the subjects’ choice of operations.
Most of these cases occur in problems with one node in which the student only
TME, vol10, nos.1&2, p .401
recognizes as characteristic of the problem that there are always two or more operations
but chooses the operation and/or the data related to it arbitrarily or by chance.
The distribution of the four kinds of error described according to levels of 5th and
6th grade are shown in Table 5. Here, we differentiate two subtypes two subtypes for the
error one operation one type for the error ordered data and another for repeat unshared
datum, whereas in classifying the others we include the unclassifiable wrong answers in
the foregoing, as well as missing answers.
Table 5. Frequencies of each error at each level and total errors
Error
type
Subtype
5th grade
Frequency
Forgetting the first
relationship
16
One
operation Forgetting
the
second relationship 45
Ordered
data
22
Repeat
unshared
datum
6
Others
10
Total
99
%.
6th grade
Frequency
%.
16.16
10
11.2426
14%
45.45
52
58.4297
52%
22.22
14
15.7336
19%
6.06
10.10
9
4
89
10.1115
4.49 14
188
8%
7%
100%
Total
Frequency%
As can be seen in Table 5, all kinds of error detected appear in the two levels (5th
and 6th grades). Overall, the error in one operation has occurred with similar frequency at
both levels, but this is due to the fact that the two subtypes compensate for each other.
That is, students in 5th grade omit the first operation more frequently, whereas those in 6th
omit the second more frequently. The next most frequent error is that of ordered data,
which occurs with greater frequency in students in 5th grade than those in 6th.
Castro & Frías-Zorilla
Since we chose and identified the problems based on the four factors (N, E, R, A),
it is reasonable to attempt to relate the types of error defined to these factors. We
classified the association between the errors as belonging to two-step problems. The four
factors are shown in Table 6, which includes the distribution of frequencies for each of
the problems, according to the combination of factors.
Table 6. Frequency of errors in the combination of four factors
Factors
Type of error
One operation
N E R
E1
N1
E2
E1
N2
E2
Ordered Repeating
Others
data
the unshared
A Forgetting the Forgetting
information
the
first relationship second relationship
R1 A1
A2
R2 A1
A2
R1 A1
A2
R2 A1
A2
R1 A1
A2
R2 A1
A2
R1 A1
A2
R2 A1
A2
7
0
3
1
4
0
5
5
2
0
0
0
0
0
1
0
15
13
18
10
11
10
13
4
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
9
1
6
4
11
2
3
0
3
0
1
0
1
1
3
0
0
1
0
0
1
0
3
1
0
0
0
0
0
0
0
0
2
0
5
0
2
1
4
0
Conclusions
In this study, we have demonstrated a new characteristic associated with two-step
word problems: the number of connections between the two simple structures that
compose the problem, which we have called “node.” We have established a specific class
TME, vol10, nos.1&2, p .403
of two-step arithmetic word problems that contain only two known quantities in their
wording. We have shown that these problems have a common characteristic: they are
formed of additive and/or multiplicative structures connected by two nexus or nodes. Our
starting hypothesis is that the number of nodes affects the difficulty of translating the
wording of the problem into a mathematical representation. With a sample of students in
the last two grades of elementary school in Spain, we have confirmed this hypothesis, in
the sense that the two-step arithmetic word problems with two nodes are more difficult to
translate into arithmetic expressions than similar problems with one node. Further, we
have significant evidence that the result is not influenced by other variables that also
influence the difficulty of translating arithmetic expressions, such as whether the
relationship of comparison is expressed in consistent or inconsistent language or whether
the additive and multiplicative relationships are of increase or decrease. The result is also
independent of the combinations of additive and multiplicative structures that compose
the scheme of the two-step problem. Although there is significant interaction between the
factor node and the factor that represents the combinations of additive and multiplicative
structures, the analysis of this interaction shows that the order of difficulty in the two-step
problems remains the same.
Likewise, from an analysis of the errors committed by the children, we have
found that in addition to the errors already identified in one-step problems and reviewed
in the literature, there are patterns of error associated with two-step problems; that is,
errors that do not occur in one-step problems. We stress the presence of three of these:
performing only one operation, working with the data in the order in which they appear in
the statement of the problem, and using one piece of information twice, in the two
Castro & Frías-Zorilla
operations, when in reality it should be used only once in one operation. The error of
performing one operation occurred with greater frequency in the two-node problems,
whereas the error of working with the data in the order in which they appear occurred
more often in one-node problems. Therefore, the number of nodes is an issue that enables
us to differentiate between types of problems and to explain part of the difficulty that
two-step arithmetic word problems pose to children. When the students have to solve
word problems, the number of nodes in a two-step problem is shown to be a cognitive
variable that influences the problem-solving process.
The limitations of the study performed are related to the kind of problem, the
students’ level, and the research focus adopted. Within the different semantic categories
of the problems identified in the additive and multiplicative structure, our study imposed
the restriction that the first relationship stated in the problem corresponds to the semantic
category of additive or multiplicative comparison. Likewise, the second relationship
always corresponds to an additive combination or a multiplicative combination. These
conditions can mediate the results obtained in terms of difficulty, kind of error, and
frequency of error. The results obtained must also be restricted to the students’ level. In
our case, these are students at the end of their elementary education. The results cannot
therefore be extrapolated to lower levels, although similar results could emerge in the
first year of the next educational level, the first year of secondary education. Although the
methodology employed is valid for achieving the goal we proposed and the evidence
shows the representations that the students produce in response to the two-step word
problems, they are sensitive to the presence of one or two connections between the
relationships. This is already a significant result from the point of view of the
TME, vol10, nos.1&2, p .405
development of the school curriculum. This study could be continued by tackling from a
qualitative point of view the psychological reasons for the different student errors in
problems with one and two nodes.
Acknowledgements
This study was developed within the framework of the National I+D+I Plan through project EDU200911337, entitled “Modelling and representations in mathematics education”, financed by the Spanish
Ministry of Science and Technology with cofunding from FEDER.
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TME, vol10, nos.1&2, p.407
Trajectory of a problem: a study in Teacher Training
Alain Kuzniak, Bernard Parzysz & Laurent Vivier
Laboratoire de Didactique André Revuz, Université Paris Diderot, France
Abstract: Problems are frequently used in mathematics to introduce and convey new
notions and skills. Hence, teachers transform and adjust those problems to their students'
level. The present study focuses on this transformation process on the particular case of a
geometric problem posed by two teacher educators in one French Institute for Teacher
Training. The whole process is described as a trajectory of the problem through various
institutions from training center to secondary school and back. Before presenting the
notion of trajectory of the problem, some elements about a general theoretical frame
which refers to didactics of mathematics are presented.
Keywords: Geometry, open problem, problem situation, problem solving, teacher
training, technologies.
Introduction
The idea of grounding the teaching of mathematics on making students solve
problems is not new, especially in primary education. From the 1970’s on it has been
very popular in many countries, undoubtedly as a reaction to the abstract teaching given
during the so-called ‘modern math’ period. This pedagogical trend was variously
structured according to the country, and the use of problems for learning maths depends
to some extent on both cultural traditions and theoretical frames underlying teaching
which are specific of each country. We became aware of these differences on the
occasion of a joint research undertaken by a French team (from the LDAR, Paris-Diderot
University) and a Mexican team (from Cinvestav, Mexico-city). The scope of this study,
presented at the Cerme 7 Conference (Rzeszów, 2011) by Kuzniak, Parzysz, Santos and
Vivier (2011), was the question of the initial training of teachers to the use of
technologies for the teaching of maths. On the Mexican side, the implementation was
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos.1&2, pp.407-440
2013©The Author(s) & Dept of Mathematical Sciences-The University of Montana
Kuzniak, Parzysz & Vivier
based on the problem solving methodology, whereas on the French side the stress was put
on the notion of open problem, in connection with Brousseau’s Theory of Didactical
Situations (TDS).
In this article we shall present in detail our approach for this research, within a
training course for prospective mathematics secondary school teachers, with reference to
some of the theoretical frames used by our team, and especially the notions of open
problem and instrumental approach (sec. 1.1 and 1.2). Besides, the training course
situation here studied belongs to what can be described as a training homology strategy
(sec. 1.3). The problem at work is used to develop among pre-service teachers, not only
their mathematical knowledge, but also their didactical knowledge.
After having exposed an a priori analysis of the problem (sec. 2), we describe in
section 3 the work required from the students-teachers which is split into three steps.
Then, we expose and analyse the various transformations of the problem chosen for the
training.
Finally, in discussion section (4), we define a framework (sec. 4.1) intended to
describe and analyse what we call the trajectory of the problem, that is its global
evolution, from its use in the training course to its setting up in a regular classroom. We
conclude the section (§4.2 sq.) with remarks on some important points related to teacher
training.
1. Context and stake of the study
1.1 Problem solving in French context
As Artigue and Houdement (2007) underscore it, there does not exist a tradition
of education research on problem solving in French didactic research even if Polya and
Schoenfeld works are well known. This characteristic partly results form the influence of
TME, vol10, nos.1&2, p .409
the Theory of Didactical Situations (TDS in the following) initiated by Brousseau (see
Brousseau, 1997, for reference texts in English) and from the pedagogical approach
developed by the IREM (Institut de Recherche sur l'Enseignement des Mathématiques).
Both introduced two kinds of perspective on problem solving: problem situation and open
problem.
The notion of “problem situation” appeared in France in the 1980’s in
Brousseau’s TDS, which is based on a socio-constructivist conception of learning. A
problem situation is a learning approach aiming at fostering the acquisition of a new
knowledge by the students. Its setting up implies identifying previously their conceptions
by analysing their errors. On this basis the teacher conceives of and sets up a situation
presenting some specific features, namely:
be relevant for the cognitive objective aimed at;
have a meaning for the student;
allow him/her to begin the search for a solution;
be rich (in terms of mathematical and heuristic contents);
be possibly formulated within several conceptual “settings” (Douady, 1986) or
“semiotic registers” (Duval, 2006).
The notion of “open problem” was introduced at about the same time (Arsac et
al.1988, Arsac & Mante, 2007). In comparison with the problem situation, the aim of an
open problem is methodological rather more than cognitive. The students are induced to
implement processes of a scientific type, i.e. experimenting, formulate conjectures, test
them and validate them. The problem must belong to a conceptual domain in which
students are somewhat familiar with, the wording (statement) has to be short and induce
Kuzniak, Parzysz & Vivier
neither a solution nor a solving method. Here is an example taken from APMEP (1987):
What is the biggest product of two numbers which can be obtained by using once each of
the digits 1, 2, 3,…,9 to write these numbers?
In fact, open problem and problem situation refer to two complementary sides of a
mathematician’s work:
in the case of an open problem the question is to find a genuine and personal
solution, with one’s own means, the general solution can be out of reach of the
students (and possibly the teacher);
in the case of a problem situation the question is, starting from a specific
problem, to elaborate a more general knowledge (concept, process…) which is
intended to be institutionalised, socially acknowledged and mastered by all the
students.
The French official curricula for junior high school integrated recently − though
without naming them − these two practices:
If solving problems allows the emergence of new elements of knowledge, it
is also a privileged means to broaden its meaning and to foster its
mastery. For that, more open situations, in which the students must
autonomously appeal to their knowledge, play an important role. Their
treatments require initiative and imagination and can be achieved by
making use of different strategies, which must be made explicit and
compared, without necessarily privileging one of them. (BOEN 2008, page
10, our translation.)
The notion of research narrative (narration de recherche), which is explicitly
linked with those of open problem and problem situation, appeared in France some
twenty years ago, first at junior high school level, before being extended to senior high
and primary school (Bonafé et al. 2002). It involves asking the student to write an
account of the thought processes he/she has undertaken in order to solve a given problem,
TME, vol10, nos.1&2, p .411
pointing out his/her ideas, successes, failures, etc. The features of the problem are the
same as for an open problem, but its statement has often several questions and is such
that the student must be able to start a research, test his/her results and validate them.
And, if possible, different solutions can be considered.
1.2 Integration and influence of technologies
Pre-service teachers in maths are accustomed to solving mathematical problems
with specific software, mainly of the symbolic calculation or dynamic geometry types but
this does not mean that they are prepared to use them as future teachers. Research studies
into teaching in technological contexts (see Laborde, 2001) show that the students
(preservice teachers) do not have or have little knowledge of the teaching of mathematics,
that is to say, they are unaware of the development of mathematical notions in teaching
situations and they have difficulties in the use of software in a learning situation. This
makes it necessary to integrate specific work in the form of understanding teaching using
software into teacher training.
Specific studies on teacher training within a technological context (see Chacon
and Kuzniak, 2011) are few. And they show the need to go more deeply into processes
regarding proof and the structuring of different spaces of knowledge (teaching,
mathematical, instrumental) which a teacher must structure when using dynamic software
for geometric learning. Moreover future teachers have to be aware of secondary school
students difficulties related to instrumental knowledge.
1.3 Teacher training
Till the end of 2010, IUFMs, Instituts Universitaires de Formation des Maîtres
(French University Training Colleges), have been in charge of the formation of
preservice teachers. The IUFMs were accepting, after a first selection, maths graduate
Kuzniak, Parzysz & Vivier
students from any University (three years of study). During one year, students were
preparing a competitive examination with academic maths knowledge. The successful
candidates received a theoretical and practical education of one year (the “second year”)
in the Institute and were in charge of a class for six hours a week; they received a salary.
Nowadays, students need to have a master and pass competitive examination to become
teachers. Preservice secondary teachers could follow a master in teacher education (two
years) at University, they are not in charge of a class and are not paid during the second
year. Our experimentation was made in 2010 before the new system.
As it is well known, preservice teachers need a set of knowledge on maths and
teaching, usually described with the notion of Pedagogical Content Knowledge (PCK)
introduced by Shulman (1986) to complement subject content knowledge, and based on
this idea, various refinements have been made to describe knowledge that is really needed
to teach mathematics known as Mathematical Knowledge for Teaching (MKT). Teaching
mathematics is obviously connected to Mathematical Content Knowledge but also to
other ones that are not automatically owned by a specialist of mathematics and that are
more or less close to mathematics like history, epistemology, didactics, psychology or
pedagogy. This large set of knowledge is classified in two parts. The first one, that is
made explicit and structured clearly within the frame of didactical theories, constitutes
Didactical Content Knowledge. The second one, that is not explicitly written and
theorised, but exists in the professional action of each teacher is what is called “third
knowledge” (Houdement & Kuzniak, 2001). Within this framework, the question is how
to introduce and combine the various types of knowledge. And how to give to students
TME, vol10, nos.1&2, p .413
who are specialists of mathematics at university level, a level in school mathematics
which are often far away from the first one.
The combination between various types of knowledge can take different forms:
they can be suggested to or developed by the students; they can be juxtaposed or
connected; the connection can be explained or not…So we have distinguished various
strategies which differ concerning the explanation of knowledge, the combination
between them, the position they give to the students. Strategies also depend on the
knowledge considered as dominant and on the transposition made by the teacher trainer.
During our experimentation, we followed a strategy firstly based on homology
and then on transposition. That means that we first use the lack of knowledge of content
and teaching for the classroom of the preservice teachers as a pretext to build a learning
situation close to a conception of teaching favoured by French curriculum. The preservice
teachers, or student teachers, are considered similarly as maths students searching a
problem and supposed to analyse the teaching session to pinpoint elements of didactical
knowledge and the “third” knowledge. The strategies based on transposition favour
didactical knowledge. Then, we tried to know more about the phenomena of transposition
of knowledge that might be a bias in every teaching situation (Chevallard, 1985). Student
teachers are considered as teachers examining their own teaching way. We detail this
with the notion of problem trajectory for the training.
2. Presentation of the problem the folded square and a priori analyses
The problem we discussed in this paper is the core of a pre-service teachers’
training course that conveys didactical knowledge about problem use in the class. For this
reason, this problem was asked to fulfil several conditions:
To be an « open » problem easily integrated in the teachers' training process.
Kuzniak, Parzysz & Vivier
To allow the link between several semiotic registers (Duval, 2006) and the use
of various mathematical settings (Douady, 1986) related to French curriculum.
To be solved in different technological contexts especially those using
dynamic software.
To be open to a number of exploitation and transformation in class with pupils
and training session with future teachers. This point relates to our idea of
problems trajectory.
To these various constraints linked to a training context, we added one more
related to the context of a comparative study. For that, we chose a problem or a kind of
problems already given by other researchers using other theoretical approaches.
The problem posed to the students belongs to a kind of problems named “shopsign problems” as used in Artigue, Cazes and Vandebrouck (2011). In such problems,
with geometric support, two areas representing a shop-sign are determined by a point
situated within a square or a circle or a rectangle... Both areas change in function of the
position of the point in the square. These problems are introduced in a geometric setting
but to solve them, a change to algebra or calculus settings is generally required. Changes
of semiotic registers with algebraic or functional notations are also needed to get a
solution. The functions used are quadratic polynomial functions which allow a
mathematical treatment in synchronization with the secondary school curriculum.
By using dynamical geometric software as Geogebra, it is also possible to solve
such problems in a graphical setting by focusing on the covariation of areas without the
use of a functional or algebraic writing. It is indeed possible of drawing a graphical
representation of the phenomena studied without any algebraic writing of the function:
TME, vol10, nos.1&2, p .415
the curve is defined as a locus of points. The number of solutions that students can find
and understand is increased by the use of technological tools introducing an experimental
perspective in the implemented working space.
The problem was presented in a real context with material and not with a writing
in mathematical form: A square, cut in a bi-color sheet, is given to the students. And they
have to fold it along a diagonal and compare the areas of both visible parts of different
colours. Students are entirely in charge of the problem representation according to the
first step of the modelling circle in Blum and Leiss (2005) view. By doing that, we do not
favour any mathematical approaches and frames but to control the task effectively made
by student teachers and reach our training objectives on the use of technologies for
teaching, student teachers have been encouraged to use some software as it will be
detailed in sec 3.1. on problem trajectories.
The problem is not original and was used in French and Mexican contexts
(Kuzniak et al., 2011) with the following form, Mexican Task, which will give the reader
an easier access to the mathematical stake of the problem.
Mexican Task. A square piece of paper ABCD, the side of
which is l, has a white front side and a blue back side.
Corner A is folded over point A' on the diagonal line AC.
Where should point A' be located on this diagonal (or: how
far is A' from the folding line) in order to have the total
visible area half blue and half white?
D
A
A’
C
B
In this version, a figure is associated to the text and that orients and makes easier
the mathematical work of students. It is no more necessary to fold the square and the
problem for students is to find the mathematical expression of both areas: area A1 of the
blue triangle and area A2 of the white hexagon. Moreover, the side of the square is given
Kuzniak, Parzysz & Vivier
as a parameter l and the question is exactly on the place of point A' on the diagonal.
Visual adjustments are invalidated by calculations for the area of the triangle seems larger
than the other in the case of equality1. So, to solve the problem students need to reason on
an elaborate and high level.
Two great types of reasoning are expected:
In the first one, students need to determine an algebraic expression for each
area and solve a quadratic equation; in France, this approach is only possible
without help in grade 11.
In the second one, it is possible to reason in figural register. Indeed, the
drawing given in the text makes visible three ''useful'' areas, the two areas to
compare and a new area A3, equal to A1: the area of the triangle of vertex A
completing A1 to make the square of diagonal AA'. This new area does not
exist in the real folding since the triangle does not have a material existence in
this case. With the use of this new area, it is possible to find, almost without
any calculation, a solution of the problem. The drawing makes clear a
decomposition of the square ABCD which implies the equality 2A1 + A2 = l2
between the areas and in the case where A1=A2, we get 3A1= l2.
If x denotes the side of the square made by the two rectangle and isosceles
triangles, as A1= x²/2, then 3x²/2=l², hence x²=(2/3)l².
It should be noted that if we take the unknown d on the diagonal, d is the height of
one of the rectangle and isosceles triangles, then x²=2d² and so d2=l2/3. This way gives a
simple solution to the original problem posed by Carlson et Bloom (2005):
1
Let’s note that these invalidations are operational since the grade 6 (it has been noted with the
class of the student teacher STe).
TME, vol10, nos.1&2, p .417
A square piece of paper is white on the frontside and black on the backside
and has an area of 3 in². Corner A is folded over point A' which lies on the
diagonal AC so that the total visible area is half white and half black. How
far is A' from the folding line. (op. cit. p. 55)
In the case chosen by the authors, the area of the square is of three square inches
and we get immediately d²=1 and therefore d=1. This initial formulation of the problem is
really more complex than those used in our study with a real folding and material that
allow the student chose a more « natural » variable as the side or the diagonal or in
Mexican Task approach where a drawing and the variable are provided. The form used by
Carlson and Bloom is not geometric meaningful because it gives only the area of the
square. This probably explains much of the difficulties2 encountered by their students,
though advanced in mathematics.
The requested use of a software in the task posed in our study changes again the
nature of the task. The software – Geogebra – gives an area immediately to each of the
surfaces and, as mentioned, it allows – and to some extent encourages – the use of
graphics, without the need for an algebraic notation. One could represent graphically A1
and A2 in function of x (or d) and then solves the problem by considering the curve
intersection (see figure 1 in sec 3.2.1). It is also possible to solve the problem by drawing
the graph of point which coordinates are (A1,A2) – it is a straight line – and considering
the intersection with the line y=x.
With this first analysis, it is already clear that the same initial problem can be
transformed in different ways leading to very different tasks, depending on the support
and tools provided to students or preservice teachers and obviously on curriculum
2
In the adaptation of STb, described in section 3.2.1, the square has an of area 27 cm2 but
the square is given to students (within the Geogebra software).
Kuzniak, Parzysz & Vivier
content. These tasks may also depend strongly on institutional constraints integrated by
teachers and their trainers. This is the subject of the study presented in the third section.
3. Transformations of the problem for teacher training
In this section we study the various transformations of a single problem P0 inside
the French educational system through two institutions: a training center for teachers and
secondary school classes. More precisely, this study involves two groups of student
teachers and two teacher educators, named TEa and TEb in the following. The aim of the
research is to grasp the impact of an initial training of math secondary schoolteachers on
their actual teaching in a classroom: what remains of the training when these teachers are
back with their students with real constraints? Due to this aim, our study is not based on
Brousseau’s theory nor on problem solving but on a specific framework presented in
section 4.1. We suppose that the changes of institutions motivate and make necessary
some transformations, the study of which will enable to better understand some
constraints lying on teachers, together with some usual practices of the profession.
3.1 The transformations of the problem
In the training course involved in the present study we shall distinguish three
stages of transformations of problem P0. In this section we describe these stages.
Stage 1. First transformation: from problem P0 to problem P1
Problem P0 (section 2) required a first transformation in order to be given in the
initial training of secondary schoolteachers. The students are prospective math teachers
and the aim of the educators (TEa and TEb) is twofold: at the beginning it is a matter of
insuring that their students have well understood the problem with its educational
potential, the various ways for solving it and the possible difficulties of the solutions. In a
TME, vol10, nos.1&2, p .419
second time they will be asked to transform this problem in order to use it in their own
training classrooms.
Here is the form chosen for P1 by TEb, together with the working instructions
given to the student teachers (the form chosen by TEa was very close).
You have at your disposal a square of paper, one side of which is white and
the other is grey. A fold shows a diagonal of the square
A type of folding bringing a vertex of the square on this diagonal, like the one
performed on the enclosed square, is considered.
One intends to compare the white and grey areas obtained in that kind of
folding.
For both groups TEa and TEb, problem P1 was based on this ‘minimalist’
presentation making use of a model: TEa showed the student teachers the folding with a
material square and TEb decided upon sending the instructions with a material square by
mail.
The student teachers are asked to work on the problem and show their entire
solution process (Schoenfeld 1985).
This solution is complemented by a research
narrative (cf. section 1.1). It is during this research phase that the student teachers, here in
a ‘student’ position, had to use at least one technological tool3 to explore the problem
favouring experimental approach according to the French curriculum.
The choice of a problem as ‘bare’ as possible from the mathematical point of view
has also a didactical aim, conveyed by homology: encourage the prospective teachers to
use, on one side problems with an open question, and on the other side technologies for
solving them. By so doing the educators hoped that the student teachers would feel free to
operate their own choices, both from a mathematical point of view (cf. a priori analysis in
3
To be chosen among: spreadsheet, dynamic geometry software, calculator.
Kuzniak, Parzysz & Vivier
section 2) and as regards the actual modes of class implementation by integrating
technological tools (cf. section 1.2).
Stage 2. Second transformation: from problem P1 to problem P2
Again in the training center, the student teachers were asked to write down the
wording of a problem and to make explicit the modes of implementation for their
students in their classrooms. Actually, the students involved are also math teachers in a
secondary class (junior or senior high school). At this stage, the issue is not to pose the
problem in a class but, in the training center, to think about the form that the problem
could take if it were posed to a class. In that sense it may be considered as a virtual
problem P2 which marks the outcome of the work for TEa’s training group. This stage
could possibly have been carried on, but its existence and its control had not explicitly
been anticipated in the course specific for this group of training students. A description of
the work of TEa’s group is developed in Kuzniak, et al. (2011).
Stage 3. Third transformation: from problem P2 to problem P3
In TEb’s group, after a session of the ‘seminar’ type in which the students had to
expose their work in stages 1 and 2, they were asked to write down a problem P3, again
with making explicit its modes of implementation and its aim, and above all to actually
pose it to their own students. Then they had to present in the training center, again during
a session of the seminar type, and a posteriori analysis of problem P3 posed in their class,
illustrating it with their students’ writings. This shift from the training center (virtual
problem P2) to the classroom (real problem P3) supposes a sharper adaptation of the
problem to the trainee’s class, in particular because of the real constraints.
3.2. Description of complete trajectories developed by student teachers
We call the set of stages transforming problem P0 which has been exposed above
a trajectory of this problem. Of course, every student teacher develops his/her own
TME, vol10, nos.1&2, p .421
trajectory, which can even be trajectories because classroom is an important factor
influencing the transformations of a problem.
Below are described the complete trajectories of P0 elaborated by five student
teachers of the TEb group, named STa, STb,... STe. In fact, the differences between these
trajectories are essentially due to the mathematical aims linked with the teaching contents
of each class and with standard activities of textbooks at the different teaching levels. The
student teachers try to design and develop teaching activities which are as close as
possible to what we call suitable mathematical working space (Kuzniak 2011).
Hence, the aims of each problem are different according to the mathematical
contents aimed at. On the other hand, a teacher will only give his students a problem on
the condition that it fits well in the syllabus. For this reason it is necessary to supply the
student teachers with problems having strong potentialities and open to varied
adaptations. In the present case, problem P1 (cf. section 2), elaborated after discussion by
the teacher educators, is adequate and, as will be seen, might give rise to adaptations at
all secondary education levels. Another common characteristic that we noticed is that the
problem was always used to introduce a new knowledge and never an assessment of an
old knowledge.
3.2.1 Two pre-service teachers’ trajectories at grade 10
In this first case we consider two student teachers, STa and STb, teaching in
seconde grade (grade 10), which in France is the first course of senior high school. In
spite of different modes, essentially due to the real constraints of the two classes, the two
trajectories presented here are very close to each other. Such closeness can be explained
by the fact that the aims chosen, depending on the teaching program of the class, were
practically identical, that is, a global study of polynomial functions. Indeed, problem P1 is
Kuzniak, Parzysz & Vivier
close to a standard type of exercises which can be found in many textbooks at this
education level: a geometric statement followed by a modelling by a quadratic function
enabling to solve the initial problem.
Stage 1. Solving problem P1
STa and STb solved the problem in a similar manner and used the graphs of the
two functions defined by modelling the two areas (triangle and hexagon) generated by the
use of Geogebra software. The variable chosen, called x, is the length of the side of the
small square. The intersection point of the two curves gives an approximate solution: the
common measure of the areas is its ordinate while the measure of the side of the small
square in the case of equality is its abscissa. However, the use of Geogebra by the two
student teachers was very different:
STa constructed, in a same file, the square simulating the folding and drew the
two curves representing the areas as functions of the distance between the
folded vertex and a free point on the folding diagonal (cf. figure 1).
STb as well made a construction with Geogebra to simulate the folding (two
constructions were proposed) but functions are used in another file. She first
got the two algebraic functions then graphically represented them (cf. figure
2). In this case Geogebra was in fact used as a graphics software and not as a
dynamic geometry software.
TME, vol10, nos.1&2, p .423
Figure 1. Use of Geogebra by STa for the solving of P1
Figure 2. Use of Geogebra by STb for the solving of P1
Another difference between STa and STb appears in how each of them considers
the square length l with the software:
Kuzniak, Parzysz & Vivier
STa fixed the value of the square length to l = 3 cm though he received a 5 cm
length square by mail: he considered this value inadequate because it did not
allow a good representation of the two curves on the computer window, the
size of the objects being estimated too big. This last point shows that he has a
quite poor knowledge of the software since he modified the situation instead
of using the Geogebra potentiality to manage the mathematical situation.
STb did not fix the square length since the parameter l is managed by the
software through a cursor and the two functions introduced are defined using
this parameter. So the abscissa of the intersection point of the two curves
gives the searched value of x as a function of l.
Nevertheless, neither STa nor STb undertook a deeper exploration of the situation
within the software. They only gave approximate values4 of the solution:
STa wondered whether the same reasoning is still valid when the value of l −
that is the square size − is changed but it seems that he did not try answering
this dilemma.
STb did not try to search the link between the solution, which is the abscissa
xA of point A in figure 2, and the parameter l given by the cursor. Indeed, the
graph of function l → xA(l) could be easily obtained by considering the point
of coordinates (l,xA). Then, one can easily see that this graph is a straight line.
During the exploration of the possible solutions, the two student teachers did not
use any other software. Their researches within a paper and pencil environment are also
4
STa obtained the approximate value 2.46 for l = 3 cm; STb gave the approximate solution
values with 5 decimals. STb noticed that these approximate solutions were also approximate
values of 1/3 (for l =1) or 4/3 (for l=2, cf. figure 2). But this remark was without any consequence
on splitting the square area into three thirds: STb stuck to her approximate determination of x.
TME, vol10, nos.1&2, p .425
very close. The configuration studied is general and both use the l parameter to name the
side of the square and a variable (or unknown) x to name the side of the small square.
After calculation of the two areas as functions of x, the problem was solved in the case of
equality, with the answer
2 / 3 l accompanied by a justification for not considering the
negative root of the equation. The comparison of the areas was made by using the
extreme values x=0 and x=l, as well as an argument (implicit for STa) about continuous
functions.
On the other hand, a notable difference between STa and STb appeared in the
management of the geometric setting. Using properties of orthogonal symmetry STb
developed a detailed proof on the nature of the triangles which seem to be isosceles and
rectangle. STa apparently remained at a visual stage (of the GI type, see (Houdement &
Kuzniak, 1999)) since he did not make any remarks on the geometric configuration,
although he fully used it in his calculations.
Stages 2 and 3. Problems P2 (virtual) and P3 (real)
For both STa and STb these problems were integrated in the chapter on
polynomial functions of degree two.
For STa, the statement of the virtual problem P2 is identical to P1 (with the
exception of the length of the side of the square which is fixed to 5 cm) with the use of
Geogebra in half-classes. Though the precision "the length of the side is not given" can
be noticed, the statements of the real problem P3 and P2 are almost identical (and so is the
case for P1). However, P3’s implementation modes are very different. It is finally given as
homework, the choice being left to students to send a Geogebra file by Internet or to give
back a paper-and-pencil work. Contrary to P2, the use of the software is not required.
Sending works by electronic mail had already been used in the year but none of SPa’s
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students chose this option for this work, and finally all of them achieved a paper-andpencil work (presumably using calculator).
In both problems, P2 and P3, STa encouraged his students to make the folding by
themselves. However, in P2 the square was given whereas in P3 the square had to be
constructed by the students themselves: therefore they had to choose the length of the
side.
For STb problem P2 is close to P1 but, with the addition of specific questions, it
became a closed problem. The l side was fixed to 6 cm and only the case of equality was
asked; the actual folding was required (for this a bi-colored square on which a diagonal
had been drawn was given to every student); a question asked to prove the existence of an
isosceles rectangle triangle; notations for geometric points and the variable x were
provided and use of Geogebra was considered – in half-classes – to represent the two
curves and thus allow a graphical resolution of the problem (let’s notice that this type of
task has already been asked in this class).
Although if in P3 there is no question about the nature of the triangle, STb
mentioned that the nature of this triangle would be assumed. Finally l was fixed to 3 3
(more or less like in problem P0, although STb did not know of it) and the question was
then more open, no procedure was imposed anymore, the students had the choice
between Geogebra software and paper-and-pencil environment. Two questions were
asked: one on the case of equality and the second on the comparison of areas. The
students, by groups of three, had to cut out a square. Two different aids had been
prepared by STb: for students who choose Geogebra (the square 3 3 size was already
constructed) a hint indicates some Geogebra tools, and for those who chose the paper-
TME, vol10, nos.1&2, p .427
and-pencil environment several possibilities for choosing the unknown, or variable x,
were given (this help was not immediately provided and was limited to cases of
blockage).
3.2.2 A pre-service teacher’s trajectory at grade 8
One student teacher, STc, was in charge of a grade 8 class. At this level, two
mathematical contents, obviously in relation with the syllabus, were considered:
mathematical proof in geometry (chosen by STc) and algebraic calculation.
Stage 1. Solving problem P1
STc produced a long research, exploring various points of view on the problem,
remaining mostly in a geometrical setting. He produced proofs using geometrical tools
and notions: isometric triangles, intercept theorem (known in France as the théorème de
Thalès), orthogonal symmetry, Pythagoras’ theorem, perpendicular bisector, square,
bisector, sum of angles of a triangle. He chose a variable x on the diagonal (he
instinctively did not consider the side of the square) and calculated the areas but he could
not solve the problem.
In his research on problem P1, STc made a clear distinction between geometrical
paradigms GI and GII (Houdement & Kuzniak, 1999) which constitutes one of the stakes
of the teaching of geometry at junior secondary school. An attempt to cut out figures for
determining areas (especially for the hexagon) was also noticed but STc concluded that it
was impossible to find a solution without using the above mentioned geometry tools.
He also used the Geogebra software to simulate the folding and visualize the
hexagonal area by a curve, using sizes measured by the software (length and area). Like
for STa, the value of l leads to a curve that does not fit well in the graphical window. But
instead of modifying the value of l, STc divided the ordinates of the points by 10. He
Kuzniak, Parzysz & Vivier
stopped when seeing that he got a parabola as his calculations had shown him. He did not
solve the problem, neither with the software (contrary to STa and STb), nor by using the
notion of function (the curve shows only that there is a parabola).
Stages 2 and 3. Problems P2 (virtual) and P3 (real)
STc did not produce a virtual problem P2 (this comes probably from an omission
or misunderstanding of the statement), and in his real problem for his class he put the
stress on the teaching of proof. The problem P3 he proposed was stated only in a paperand-pencil environment, and there are multiple reasons for this:
he points out constraints in the use of the computer room;
he thinks that his students are not able to use a software for making a
conjecture without being guided and he wants to keep the character open of
the problem;
he thinks his grade 8 class is a ‘good’ one.
He then considers a paper-and-pencil work in small groups, planned for two
sessions. The problem P3 he poses asks to cut out a 6 cm sided square, with the students
achieving actual folding, and includes only one question: "How to achieve this folding so
that the grey area is equal to the white area?"
The aim is twofold, as it can be noticed in the planned institutionalization: proof
of the fact that the hexagon is obtained by removing a small square and calculation of the
position giving equal areas. Besides, after the first session a student proposed to cut out
the square into three figures having the same area (hexagon and two isosceles rectangle
triangles) but without being able to justify it. Then STc adjusted his plans and thought of
proposing a solution based on the areas: the area of the small square must be equal to the
two thirds of the total area, and therefore the side of the small square (which gives the
TME, vol10, nos.1&2, p .429
solution) is
2 / 3 × 6 cm or
24 cm. However, in the class reality, the aspects linked to
geometrical proof were hardly tackled during the session.
3.2.3 Two pre-service teachers’ trajectories at grade 6
At grade 6 level, the mathematical notions that the students know do not allow the
use of the previous mathematical supports (functions, algebraic calculation, geometrical
proof). It seems that the calculation of areas of polygonal figures is the only possible
mathematical support at this level. Thus, it is not surprising that this very content
constitutes the choice of both student teachers, STd and STe, who are considered in this
section.
Stage 1. Solving problem P1
STe used Geogebra for modelling the folding. A visual adjustment with the
measures of the two areas allowed him reducing the gap between them in order to solve
the problem in an approximate way. Then, in order to make a conjecture, STe tried
searching for a notable value, the approximate solution could be an approximation of it.
His attempts were not successful in spite of two constructions depending on whether the
mobile point is on the side of the square or on the diagonal – these lengths being, in each
case, fixed to 10 cm for making the research of a conjecture easier.
Then STe shifted to paper-and-pencil environment. After fixing the length of the
square to 1, he produced two calculations of the solution by taking two unknowns,
respectively the side x of the small square, and 1–x. For STe, it is explicit that equal areas
corresponds to cutting out the square into three thirds, but the general comparison of
areas is not taken in account.
In her research for a solution, STd started with working in a paper-and-pencil
environment; she named x the length of the side of the small square and l the length of the
Kuzniak, Parzysz & Vivier
side of the initial one, calculated the two areas and solved the problem of their equality.
Let us remark that she wrote, without justification, that the comparison of areas is solved
with the help of the equality case. The comparison of areas was made with respect to the
value l/√1.5. Then STd carried out the folding with her square: "I measured l (7,3cm), I
did the calculation, which gave x=5.96". STd found that, visually, there seemed to be a
little difference between the areas and she thought that it is due to an optical illusion. She
then gave a construction of the folding with the help of the Geogebra software. STd
regrets that this only provides an approximate value of the solution, like the ones
obtained with a square of paper: measures and area calculations.
Stages 2 and 3. Problems P2 (virtual) and P3 (real)
STe proposed a statement of the virtual problem P2 identical to P1’s, but he fixed
the side of the square to l=12 cm. The scenario he considered includes three steps:
an initiation, during about 20 min, in a session that involved an actual folding
of a particular square, a statement of the problem and first attempts of
solution;
a second stage, in the computer room, to determine an approximate solution
with the help of Geogebra;
a last stage, working in pairs, aiming to justify the solution found with the
help of a cutting out of the square (this last step being not explicit).
He proposed a ‘dressing’ of the problem in order to make it more concrete for his
students: a square field inherited by three brothers has to be divided between them. The
eldest receives the total big square minus a small square (situated in ‘a corner’), this
remaining small square being shared between the two others. The question is: "do the
three brothers have equitable parts?". This dressing, not taken up in problem P1, changes
TME, vol10, nos.1&2, p .431
significantly the problem because it turns it onto cutting the initial square into three
polygons of equal areas. There is not folding anymore and nothing is said on how the
small square is shared between the two younger brothers (nor even if it is equitable).
The real problem P3 took up this idea of contextualisation, but remains closer to
problem P1: a firm wants to make a logo defined by the folding of a square of side 12 cm
and the constraint of equality of the two areas. STe also took up the idea of three phases,
only slightly modified:
1. a first activity, on paper, to understand the problem;
2. a second activity, with Geogebra (construction and research are very guided),
to find out an approximate value, which is quite suitable for the realization of
the logo;
3. here the justification was replaced by a actual construction of the logo on
paper, using this approximate value (this third step was planned in the same
session than point 2).
The student teacher STd proposed a problem P2 taking up problem P1 and
modifying the question in the same way as STe: "How has the black corner to be folded
so that it has the same area than the white surface?" The possibilities for using calculator
as well as the Geogebra software were mentioned (under the condition of not asking to
draw the diagram, judged too complex for this level). In particular, STe planned to have
the students work in groups of four in a computer room and let them choose their
environment.
The real problem P3, differs notably from P1 by the fact that one the interest is
only in the equality (like P2) and especially the fact that an approximate value is
Kuzniak, Parzysz & Vivier
explicitly asked: "Determine as precisely as possible a folding of this type, so that the
white part and the colourful part have the same area". The work was organised in groups
of three students in the computer room, with a possibility to use Geogebra or only paper
and pencil. Each group was provided a square of paper, the size of which was 3 cm, 4 cm,
5 cm or 6 cm (STd explicitly adjusted this choice of the didactical variable: multiple of 3
or not).
4. Discussion
4.1 About trajectories
In this section we propose an original frame to organize and analyse the emerging
trajectories to deal with the problem, like those which have been set out in section 3. The
aim of this frame is to take into account various dimensions of a problem (institution and
persons involved, goal(s) aimed at) and study the nature and the dynamics of the changes
which take place through the successive ‘moves’ of this problem from one institution to
another.
At the start there is a problem, not necessarily mathematical, coming from an
institution I, that may involve an everyday life or any domain of knowledge. Then there
are several didactical institutions I1, I2,… in which successive alternative forms
(‘avatars’) of the initial problem will show up. In each institution Ik (k ≥ 1) one or several
individuals Tk in a ‘teacher’ (or ‘educator’) position, as well as individuals Sk in a
‘student’ (or ‘trainee’) position, will be distinguished.
These institutions will be concatenated between them in the following way: the
problem was introduced in Ik under the Pk avatar by Tk who poses it to the Sk with a given
purpose. Then one of the Sks, who in institution is in a ‘teacher’ position (Tk+1 = Sk),
TME, vol10, nos.1&2, p .433
poses the problem to his/her students Sk+1s under the avatar Pk+1, with a purpose which is
generally different (figure 3).
Ik
Tk
Tk+1
Pk
Sk
Ik+1
Pk+1
Sk+1
Figure 3. Concatenation of institutions
Of course this process can possibly be carried on from an institution to another
(I1, I2,… , In), depending on the involved individuals. The succession of stages − and
hence of avatars of the problem − constitutes the trajectory of the problem.
Example (figure 4).
Stage 1. In a training center for teachers (institution I1) a math educator finds a
problem written in everyday language in a magazine. He/she thinks that it could well give
rise to a geometrical activity for his/her trainees. Then he/she transforms it into a
geometrical wording and, within the training curriculum, asks the trainees to search ‘all
possible solutions’ of the problem, regardless to the classroom level. (mathematical a
priori analysis).
Stage 2. Again within the training curriculum (institution I2=I1), the teacher
educator asks his/her trainees to transform the wording into a new one that could be
posed as a research problem to a class of a given level (didactical a priori analysis).
Stage 3. Back to his/her school (institution I3), each trainee undertakes posing the
problem in his/her class. For that he/she transforms again the wording according to this
Kuzniak, Parzysz & Vivier
particular class and poses it by asking his/her students to use their knowledge to find a
solution to the problem.
Stage 4. Back to the training center (institution I4=I1), the educator asks the
trainees, gathered in groups according to the level of their classes, to work out for that
level a new formulation of the wording, in order to make it a research problem taking into
account the implementation that they could observe in their own classes (a posteriori
analysis).
training
center
educator
training
center
classrooms
educator
trainee 1
avatar 3.1
training
center
educator
students
avatar 1
trainees
trainees
trainee 2
trainees
avatar 3.2
avatars 2
students
avatars 4
…
stage 1
stage 2
stage 3
stage 4
Figure 4 : Examples of trajectories of a problem
The first three stages correspond to the example of training constituting the study
of section 3: I1=I2 is the training center and I3 is one of the secondary school classes.
Stage 4 could not be achieved during the training. It is nevertheless important, either
being put into play in the training center or not, because it marks the start of a cycle of
transformation of the problem taking into account the feedbacks from the students. This
is a central component of the profession of teacher.
TME, vol10, nos.1&2, p .435
Moreover, one may quite consider conceiving trajectories in which other modes
of transmission of problems intervene. For instance think of a continued training instead
of an initial one or a debate between teachers of a same secondary school.
4.2 On training
During the first session dedicated to presentation of the problem P1, the teacher
educator TEb made an unsuccessful attempt to orientate the trajectories by encouraging
the teacher students to think about the use of spreadsheet in the class. However, as we
saw it in the class of STd, sixth grade students could generate values tables close to what
they could get faster with spreadsheet. We could observe the teachers’ difficulties to
integrate spreadsheet in their actual practices despite an important focus during the
training. It could suggest a training underperforming, but this opinion needs to be
qualified because it seems that spreadsheet, according various studies, is a tool especially
difficult to integrate into lessons by teachers. Indeed, Haspekian (2005) mentions some
specific problems on spreadsheet instrumentation or teaching of particular notions related
to spreadsheet (such as delicate and complex notion of cell) which do not exist or not
under the same form in maths knowledge at this grade: that can interfere negativity with
the teaching of algebra. Teachers can be aware of these difficulties and avoid the use of
spreadsheet in class despite the official demand from educative institution. The
interpretation is confirmed by the experiment of TEa. One group of teacher students had
to prepare a session using spreadsheet. Convinced of the impossibility of using
spreadsheet in their own class, they prepared a session dedicated to the teaching of
algorithms without any actual adaptation to the level of their students. They argued that
the use of a spreadsheet needs too much time and knowledge which is not of
mathematical nature.
Kuzniak, Parzysz & Vivier
Open problems and problem-situations with a-didactic potential are largely
favored by the training in teacher training institution I1, especially to encourage student
teachers to not only ask problems with closed questions to their students. As the problems
P3, posed in class, were generally open, we can conclude that prospective teachers were
aware of this mathematics education complexity. This is perhaps due to the training based
on homology that we gave to the students and which postulates that teachers students will
reproduce the form of the teaching they received during their training in I1 .
It should also be noted that the virtual problem P2 does not provide a lot of
information on the actual course in class, except to check that changes of the
mathematical support could only be possible in the class in front of school students (see
STc). Even when student teachers know they will have to manage the problem with their
students, the real constraints of the class do not seem to be taken into account before they
are involved in real teaching scenarios with their own students. This leads to significant
differences between laboratory work in I2 towards I3 and the actual work in I3 and could
suggest that the training on problems prepared in I2 is not representative and far away
from the reality of class teaching - even if this work remains interesting for training. That
too should lead teachers educators to complete the training by requiring prospective
teachers to engage into an actual implementation in a class with an a posteriori analysis.
This demand can also show them that, first, it is possible to implement in I3 the
requirement made in I1 and, then, that the demand of the training institution is not
opposite to the demand of school institution as some students think of it.
4.3 On the choice of the specific technological context
All prospective teachers have chosen to use Geogebra software to approach their
problem research in response to the demand of using a technological context. This sole
TME, vol10, nos.1&2, p .437
choice of Geogebra could be explained by some factors. First, training in I2 favors this
software which is widely used in French secondary school system. On other hand,
Geogebra which is a multi-purpose software is well adapted to the problem: P1 is
generally seen as a geometrical problem and therefore the use of a dynamic geometry
software is somehow natural, and for grade 10 the problem is also connected to functions
as modeling tools and the use of Geogebra to make graphics is well suited.
For the problem posed in class, three different environments are employed for
solving it: a paper and pencil environment or Geogebra (STa, STd), only Geogebra (STb,
STe) and no use of software (STc). Moreover, there are few mentions of the use of a
calculator (STd is the unique teacher who speaks explicitly about it) while school
students use it widely. Perhaps, this lack of allusion to calculator is due to the fact that
teachers do not perceive it as a technological environment (despite the instructions see
sec 3.1) and they think of a computer. It is also possible that its use is now considered
transparent and routine for prospective teachers and they feel no need to mention it.
4.4 On the folding
All prospective teachers keep the idea of the folding to present the problem to
their students. Probably this anchoring to the real world supports the devolution of the
problem as the attitude of STe suggests it: he left aside the idea of folding in the virtual
problem P2, but it takes again this idea when he poses the real problem P3 to his students
in class.
However, the folding is not easy to define as we can see it in I1 where the teacher
educators had been obliged to mention other geometric terms than the area like square
and diagonal and vertices. The diagonal could be also drawn (an even marked by a fold
as STe did it). Other ways are possible: STc pointed out the vertex to fold on the diagonal
Kuzniak, Parzysz & Vivier
by coloring the good corner to use; STd has not defined the folding with enough
precision and students did not well understand the instructions so that STd added some
comments during the session in class; STa and STb made an unequivocal coding of the
figure (like in the Mexican task in sec. 2).
4.5 On the problem
It is indeed a problem with a high potential that can be addressed at all levels of
secondary education. The student teachers have all agreed without hesitation to pose it
with their own transformations to their classes and students and, according to their
comments, the students were interested in solving the problem P3.
Many
adjustments
were
made
especially
concerning
modalities
of
implementation. But despite the diversity of educational levels where the problem was
given, the core of the mathematical problem stays stable with few changes. Among the
changes, we can note essentially: the value of l (except for STa) and the research of the
equality (except for student teachers teaching in grade 10, STa and STb). The biggest
adjustment was made by STd, who introduced the concept of precision of the solution.
By and large, the problem P1 did the job.
We can conclude that the transformations of the problem P1 to give it in class are
simultaneous oriented by the researches of the mathematical solution and by the official
syllabus of the grades involved in the teaching. It would be interesting to know what will
be the use of the problem by the teachers some years later and how the trajectory of the
problem continues evolving. We intend to make an interview with the prospective
teachers involved in this study in the future. Another point of interest is the impact of
such problems on school students and some material need to be used to precise this
crucial point.
TME, vol10, nos.1&2, p .439
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TME, vol10, nos.1&2, p.441
Primary school teacher’s noticing of students’ mathematical thinking in
problem solving
Ceneida Fernández1
Salvador Llinares2
Julia Valls 3
Universidad de Alicante, Spain
Abstract: Professional noticing of students’ mathematical thinking in problem solving
involves the identification of noteworthy mathematical ideas of students’ mathematical
thinking and its interpretation to make decisions in the teaching of mathematics. The goal
of this study is to begin to characterize pre-service primary school teachers’ noticing of
students’ mathematical thinking when students solve tasks that involve proportional and
non-proportional reasoning. From the analysis of how pre-service primary school
teachers notice students’ mathematical thinking, we have identified an initial framework
with four levels of development. This framework indicates a possible trajectory in the
development of primary teachers’ professional noticing.
Keywords: Professional noticing, proportional reasoning, pre-service primary teacher’s
learning, classroom artifacts.
INTRODUCTION
Teachers and problem solving: the role of understanding the students’
mathematical thinking
Solving problem is a relevant task in mathematics teaching. However, teachers
need to understand the students’ thinking in order to manage problem solving situations
in classroom. Teachers’ abilities to identify the mathematical key aspects in the students’
thinking during problem solving are important to performance teaching for
1
[email protected]
2
[email protected]
3
[email protected]
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos.1&2, pp.441-468
2013©The Author(s) & Dept of Mathematical Sciences-The University of Montana
Fernández, Llinares & Valls
understanding. The development of these abilities to interpret students’ thinking may
allow teachers to make appropriate instructional decisions, for instance, the selection and
design of mathematical tasks in problem solving activities (Chamberlin, 2005).
Although the analysis of students’ thinking is highlighted as one of the central
tasks of mathematics teaching, identifying the mathematical ideas inherent in the
strategies that a student used during the mathematical problem solving could be difficult
for the teacher. However, teachers need to know how students understand the
mathematical concepts in order to help them to improve their mathematical
understanding (Schifter, 2001; Steinberg, Empson, & Carpenter, 2004). This approach is
based on listening to and learning from students (Crespo, 2000) since, in this case, the
teacher has to make decisions in which students’ thinking is central.
Identifying the possible strategies used by students in problem solving allows
teachers to interpret why a particular problem could be difficult and also to pose
problems considering the characteristics of students’ thinking. On the other hand, if
teachers understand the mathematical ideas associated with problems in each particular
mathematical domain, they may be able to interpret the mathematical understanding of
students appropriately. This knowledge could help teachers to know which characteristics
make problems difficult for students and why (Franke & Kazemi, 2001).
Considering these previous reflections about the relevant role of students’
thinking in mathematics teaching, an important goal in some mathematics teachers
programs is the development of teachers’ ability to interpret students’ mathematical
thinking (Eisenhart, Fisher, Schack, Tassel, & Thomas, 2010). Some mathematics teacher
education programs have reported findings that support this approach but have also
TME, vol10, nos.1&2, p .443
reported that the development of this expertise is a challenge (Llinares & Krainer, 2006).
The findings in these studies have pointed out that the more or less success of programs
depends on how pre-service teachers understand the mathematical ideas in the
mathematical problems and the students’ mathematical thinking activated in the problem
solving activities (Norton, McCloskey, & Hudson, 2011; van Es & Sherin, 2002; Wallach
& Even, 2005).
Teachers’ professional noticing of students’ mathematical thinking
Research on mathematics teacher development underlines the importance of the
development of pre-service teachers’ professional noticing in the teaching of mathematics
(Jacobs, Lamb, & Philipp, 2010; Mason, 2002; van Es & Sherin, 2002). Researchers and
mathematics teacher educators consider the noticing construct as a way to understand
how teachers make sense of complex situations in classrooms (Sherin, Jacobs, & Philipp,
2010). Particularly, Mason (2002) introduces the idea of awareness to characterize the
ability of noticing as a consequence of structuring the teacher’s attention about relevant
teaching events. A particular focus implies the identification of key aspects of students’
mathematical thinking and its interpretation to make decisions in the teaching of
mathematics (Jacobs et al., 2010). Previous researches have indicated the relevance of
pre-service teachers’ interpretations of students’ mathematical thinking to determine the
quality of the teaching of mathematics (Callejo, Valls, & Llinares, 2010; Chamberlin,
2005; Crespo, 2000; Sherin, 2001). Therefore, the necessity that pre-service teachers base
their decisions on students’ understandings underlines the importance to characterize and
understand the development of this skill (Hiebert, Morris, Berk, & Jansen, 2007). This
fact justifies the necessity to focus our attention on how pre-service teachers identify and
Fernández, Llinares & Valls
interpret students’ mathematical thinking in different mathematical domains (Hines &
McMahon, 2005; Lobato, Hawley, Druken, & Jacobson, 2011).
Previous research on how students solve problems in specific mathematical
domains has provided useful knowledge about the development of student’s
mathematical thinking in these domains that could be used in the study of the
development of the noticing skill. One of these mathematical domains is the transition
from students’ additive to multiplicative thinking in the context of the proportional
reasoning. Multiplicative structures in the domain of natural numbers that come from the
expressions a × b = c, have some aspects in common with additive structures, such as the
multiplication as a repeated addition, but also have their own specificity that is not
reducible to additive aspects (Clark & Kamii, 1996; Lamon, 2007; Fernández & Llinares,
2012-b). For example, tasks that involve the meaning of ratio such as: John has traveled
by car 45 km in 38 minutes, how many km will he travel in 27 minutes? However, a
characteristic of this transition is the difficulty that students of different ages (primary
and secondary school students) encounter to differentiate multiplicative from additive
situations. This difficulty is manifested in students who over-use incorrect additive
methods on multiplicative situations (Hart, 1988; Misailidou & Williams, 2003;
Tourniaire & Pulos, 1985), and who over-use incorrect multiplicative methods on
additive situations (Fernández & Llinares, 2011; Fernández, Llinares, Van Dooren, De
Bock, & Verschaffel, 2011-a, 2011-b; Van Dooren, De Bock, Janssens & Verschaffel,
2008). These previous researches have provided results that underline key ideas in the
transition from additive to multiplicative structures. These ideas have allowed us the
TME, vol10, nos.1&2, p .445
opportunity to design instruments to analyze pre-service teachers’ professional noticing
of students’ mathematical thinking.
The aim of this study is to characterize pre-service teachers’ noticing of students’
mathematical thinking in the domain of the transition from additive to multiplicative
thinking, particularly, in the context of the proportional reasoning. Therefore, we are
going to characterize how pre-service primary school teachers interpret students’
mathematical thinking when they are analyzing the student’s written work in
mathematical tasks. Research questions are:
Which aspects of students’ mathematical thinking do pre-service teachers
identify in multiplicative and additive situations?
How do pre-service teachers interpret the aspects of involved students’
mathematical thinking?
When we tried to answer these two questions, an additional result emerged: a
framework with different levels that describes pre-service primary school teachers’
noticing of students’ mathematical thinking in the domain of students’ transition from
additive to multiplicative thinking in the context of proportional reasoning. In this sense,
pre-service teachers’ interpretations of the student’s written work in the mathematical
tasks help us to identify how they interpret the information about the way in which
students have solved the problems. So, in this case, we hypothesized that students’
solutions to the problems could help pre-service teachers interpret how students are
thinking about the given situations.
Fernández, Llinares & Valls
METHODS AND PROCEDURES
Participants
The participants in this study were 39 pre-service primary school teachers that
were enrolled in the last semester of their training program. The three years of teacher
education program offers a combination of university-based coursed and school-based
practice. Pre-service teachers take foundational courses in education and method courses
in different areas such as mathematics, language and social science, and a 12-week
school teaching practicum. These pre-service teachers had still not made teaching
practices at schools, but they had finished a mathematics method course in their first year
of the training program (90 hours). This mathematics method course is focused on
numerical sense, operations and modes of representation and, particularly, it has
approximately 9 hours focused on the idea of ratio as an interpretation of rational
numbers. We considered that characterizing pre-service teachers’ noticing of students’
mathematical thinking in problem solving could provide information about the
development of pre-service teachers’ learning during the teaching practices.
Instrument
Pre-service teachers had to examine six students answers to four problems (Figure
1), two proportional problems (modelled by the function f(x) = ax, a≠0) (problems 2 and
4) and two non-proportional problems with an additive structure (modelled by the
function f(x) = x+b, b≠0) (problems 1 and 3). Additive and proportional situations differ
on the type of relationship between quantities. For example, in Peter and Tom’s problem
(problem 1) the relationship between Peter’s and Tom’s number of boxes can be
expressed through an addition: Tom’s laps = Peter’s laps + 60 (the difference between
TME, vol10, nos.1&2, p .447
quantities remains constant). On the other hand, in Rachel and John’s problem (problem
2), the relationship between the number of flowers that Rachel and John have planted can
be expressed through a multiplication: John plants 3 times more flowers than Rachel (60
= 20 × 3). The first problem is an additive situation while the second situation is a
proportional one. These differences among proportional and additive situations are
considered in the problems with the sentences “they started together” or “Peter starter
later/David started earlier” and “John plants faster/Laura pastes slower” or “they go
equally fast”.
The students’ answers show different correct strategies used in proportional
situations (the use of internal ratios, the use of external ratios, the building-up strategy,
the unit rate and the rule of three as correct strategies) but they were used incorrectly in
the additive problems. On the other hand, the additive strategy was used as correct
strategy in additive problems but as incorrect strategy in proportional ones.
Pre-service teachers had to examine a total of 24 students’ answers (four problems
× six students) and respond to the next three issues related to the relevant aspects of the
professional noticing of students’ mathematical thinking skill (Jacobs et al., 2010):
“Please, describe in detail what you think each student did in response to each
problem” (related to pre-service teachers’ expertise in attending to students’
strategies).
“Please, indicate what you learn about students’ understandings related to the
comprehension of the different mathematic concepts implicated” (related to
pre-service teachers’ expertise in interpreting students’ understanding).
Fernández, Llinares & Valls
“If you were a teacher of these students, what would you do next?” (that is,
documenting pre-service teachers’ expertise in deciding how to respond on the
basis of students’ understandings).
The six students’ answers to the four problems were selected taking into account
previous research on proportional reasoning. We focus our attention on the research
findings that describe different profiles of primary and secondary school students when
they solve proportional and non-proportional problems (Fernández & Llinares, 2012-a;
Van Dooren, De Bock & Verschaffel, 2010). These students’ profiles are:
students who solve proportional and additive problems proportionality,
students who solve proportional and additive problems additively,
students who solve both type of problems correctly, and
students who solve problems with integer ratios using proportionality
(regardless the type of problem) and solve problems with non-integer ratios
using additive strategies.
So, four out of six students’ answers corresponded with one of these profiles and
the other two students’ answers used methods without sense. These last two students’
answers were included as buffer answers. Furthermore, to avoid those results were
affected by other specific variables of the test, problems and students’ answers order was
varied. So, 20 different versions of the test were designed.
TME, vol10, nos.1&2, p .449
Figure 1. Problems and students’ answers used in the test
Analysis
Pre-service teachers’ answers were analyzed by three researchers. From a
preliminary analysis of a sample of pre-service teachers’ answers, we generated an initial
Fernández, Llinares & Valls
set of rubrics to make visible aspects to characterize the professional noticing of students’
mathematical thinking in the context of proportional reasoning. These initial rubrics were
refined as the analysis was progressing. Finally, we generated four-level descriptors
which were applied to all pre-service teachers’ answers:
Level 1. Proportional from additive problems are not discriminated
Level 2. Discriminate proportional from additive problems without identifying
the mathematical elements.
Level 3. Discriminate proportional from additive problems identifying the
mathematical elements but without identifying students’ profiles.
Level 4. Discriminate proportional from additive problems identifying the
mathematical elements and the students’ profiles.
Therefore, firstly, we classified pre-service teachers in two groups: pre-service
teachers who discriminated proportional and additive situations, and pre-service teachers
who did not discriminate both situations.
Secondly, focused on pre-service teachers who discriminated both situations, we
analyzed if they discriminated the situations identifying the mathematical elements that
characterize proportional and additive situations and if they were able to identify
students’ profiles. This second stage of the analysis tried to identify the quality of preservice teachers’ interpretations considering whether they have used specific mathematics
elements to justify their interpretations. To do this, we took into account the
mathematical elements of proportional and additive situations (Table 1) and the strategies
used by students (Table 2). So, we analyzed if pre-service teachers identified the
TME, vol10, nos.1&2, p .451
strategies and integrated the mathematical elements in their written text produced
(relating the characteristics of the problem and the strategy) when they answered the task.
We analyzed all pre-service teachers’ answers but three out of thirty-nine preservice teachers were not classified in one of these levels because their answers were
incomplete.
Table 1. Mathematical elements of the situations
Proportional situation f(x) = ax, a ≠ 0
Additive situation f(x) = x + b, b ≠ 0
The function passes through origin “they The function does not pass though
origin “they started later or earlier”
started together”
The value of the slope changes “someone goes The value of the slope remains
constant “They go equally fast”
faster or slower”
External ratios are constant (f(x)/x = a) and The difference between relationed
internal ratios are invariant (a/b = f(a)/f(b))
quantities remains constant
f(x)-x = b
Table 2. Students’ strategies used to solve the problems
Proportional situations
Additive situations
The use of external ratios Additive strategy
The use of internal ratios
Unit-rate
Building-up strategies
Rule of three algorithm
Fernández, Llinares & Valls
Results
In this section, we present the characterization of the different levels in the
development of pre-service teachers’ noticing of students’ mathematical thinking skill in
the mathematical domain of proportionality.
Level 1. Proportional from additive problems are not discriminated (25 out of 39
pre-service teachers).
In this level we classified pre-service teachers who did not discriminate
proportional from additive situations. These pre-service teachers considered
that all the problems were proportional (so proportional methods were the
correct strategies to solve all these problems), or
that all the problems were additive (so additive methods were the correct
strategies to solve all these problems).
For example, a pre-service teacher gave the next argument in the answer of
student 5 to problem 2 (proportional situation) (Figure 1): “This answer is correct. The
student has found out by how much Rachel goes from 4 to 20 and repeated the process
with John”. This pre-service teacher identified the multiplicative relationship between
quantities used by the student 5 to solve the problem, but this pre-service teacher said in
the answer of student 5 to problem 3 (additive situation): “This answer is correct. The
student has found out the multiplicative relationship between 12 and 48 and then has
multiplied 24 by this number”. In this case, the preservice teachers did not recognize the
additive character of the situation.
Another pre-service teacher gave the next argument to the answer of student 4 to
problem 3 (additive situation): “The answer is correct. The student has obtained the
TME, vol10, nos.1&2, p .453
difference between the dolls manufactured by David and Ann. Afterwards, the student has
added 48 that are the dolls manufactured by Ann later”. However, when this pre-service
teacher interpreted the answer of student 5 to problem 2 (proportional situation), he did it
erroneously “This student has used a correct method and has obtained a correct result.
Firstly, the student has computed the difference between the flowers planted by John and
Rachel and has obtained 8 flowers. After, taking into account this difference, the student
has added this number (8 flowers) to the 20 flowers planted by Rachel obtaining how
many flowers has John planted”.
So, both pre-service teachers did not discriminate proportional from additive
situations. Pre-service teachers in this level focus their attention on superficial features of
the situations and show a lack of mathematical knowledge. As a consequence, their
interpretations of students’ answers mainly rely on the description of the operations
carried out and not on the meanings.
Level 2. Discriminate proportional from additive problems without identifying the
mathematical elements (2 out of 39 pre-service teachers).
We classified in this level pre-service teachers who discriminated proportional
from additive situations but did not justify the difference between problems taking into
account the mathematical elements of the situations. Therefore, these pre-service teachers
identified the correctness of the strategies used by students in each type of problem
(relating the situation with the strategy used by the student) but without justifying why
the strategy is correct or incorrect taking into account the characteristics of the situations.
Fernández, Llinares & Valls
For example, a pre-service teacher indicated in the answer of the student 1 to
problem 1 (Figure 1): “The answer is correct. The student has determined how many
boxes has Peter loaded, is that, the difference between the boxes loaded by Peter at the
end (60 boxes) and the boxes loaded by Peter initially (40 boxes). So, this difference (20
boxes) is also the number of boxes loaded by Tom. So, 100 + 20 = 120”.
Pre-service teachers in this level only describe the operations carried out by
students, but, in this case, the descriptions are related to the correctness of the strategy in
each type of problem (subject matter knowledge).
Level 3. Discriminate proportional from additive problems identifying the
mathematical elements but without identifying students’ profiles (6 out of 39 pre-service
teachers).
In this level we classified pre-service teachers who discriminated proportional
from additive situations justifying the difference between situations taking into account
some mathematical elements of the situations. However, these pre-service teachers were
not able to identify students’ profiles.
For example, a pre-service teacher indicated in the answer of the student 1 to
problem 1 (Figure 1): “The answer is correct. This student has computed the difference
between the boxes loaded by Peter initially and later (20 boxes). As the problem said that
the two people loaded equally fast but Peter started earlier, 20 are also the boxes loaded
by Tom. So the student has added 20 boxes to the boxes loaded for Tom”. This preservice teacher justified the difference between situations with the mathematical elements
TME, vol10, nos.1&2, p .455
of the situations, in that case, mentioning two characteristics of the additive situations:
“They loaded equally fast but someone started earlier”.
However, these pre-service teachers did not identify students’ profiles because
they did not relate globally the behavior of each student to the four problems. For
example, the pre-service teacher mentioned above identified the behavior of student 3
(student who solve both type of problems correctly): “this student has solved all the
problems correctly” but this pre-service teacher were not able to identify the behavior of
student 4 (student who solve all the problems additively) since he said “this student has
solved problems 1 and 3 (the same type) incorrectly and problems 2 and 4 (other type of
problem) correctly” neither the behavior of student 5 (student who solve all the problems
using proportionality)
“this student has solved problems 1 and 3 (the same type)
incorrectly and problems 2 and 4 (other type of problem) correctly” because he/she did
not identify that the student used the same strategy regardless the type of problem.
Level 4. Pre-service teachers who discriminate proportional from additive
problems identifying the mathematical elements of the situations and the students’
profiles (3 out of 39 pre-service teachers).
In this level we classified pre-service teachers who discriminated proportional
from additive problems justifying the difference between problems taking into account
the mathematical elements of the situations and identifying the students’ profiles.
For example, a pre-service teacher indicated in the answer of the student 1 to
problem 1 (Figure 1): “The student has obtained the difference between the two Peter’s
quantities and used it to obtain the number of boxes loaded by Tom. The answer is
Fernández, Llinares & Valls
correct because the two people loaded equally fast and the difference has to be the
same”.
This pre-service teacher was able to identify students’ profiles. In that way, this
pre-service teacher indicated in relation to the answers of student 3 (student who solve all
problems correctly) “this student know the correct methods and apply them in the both
type of problems”, in relation to the answers of student 4 (student who solve all problems
additively) “This student only do correctly the problems where the speed is the same but
someone starts earlier or later. This student apply the same method to the both type of
problems” and in relation to the answers of student 5 (student who solve all problems
using proportionality) “this student only do correctly problems where the speed is not the
same. This student always applies the same method to all the problems”. Pre-service
teachers in this level are able to relate strategies within and across problems in order to
see students’ overall performance to a certain type of problem focusing on a relation of
relations.
DISCUSSION
Initially, the goal of this research was to characterize what pre-service teachers
know about students’ mathematical thinking in the context of proportional and nonproportional problem solving before their teaching practices. However, the design of the
test allows us to characterize a trajectory of the development of teachers professional
noticing of students’ mathematical thinking. In the identified trajectory, pre-service
teachers moved from the non-recognition of the characteristics of the situations towards
the identification of the characteristics of the situations and the strategies used by
students and the recognition of students’ profiles when solving problems. This last level
TME, vol10, nos.1&2, p .457
shows pre-service teachers’ willingness and ability to analyze students’ mathematical
thinking in relation to the additive and multiplicative situations.
The development of a framework to characterize pre-service teachers professional
noticing of students’ mathematical thinking
Results show the difficulty of pre-service teachers to identify the relevant aspects
of students’ mathematical thinking in relation to the students’ transition from additive to
multiplicative thinking. This difficulty is manifested by pre-service teachers’ difficulty in
differentiate proportional from non-proportional situations (25 out of 39). This finding
indicates a weakness in their own subject-matter knowledge about multiplicative and
additive situations. Identifying the mathematical elements of additive and multiplicative
situations is the first step to interpret properly students’ mathematical thinking during the
problem solving.
On the other hand, although some pre-service teachers could recognize the
difference between both situations, they had difficulties in justifying why students’
answers were or were not correct taking into account the mathematical elements of the
situations. Furthermore, they had difficulties in interpreting globally all students’
answers. This result shows the complex knowledge that pre-service teachers have to use
to identify and interpret the way in which students solve the problems.
Another relevant result is the characterization of pre-service teachers’
development of professional noticing of students’ mathematical thinking. A framework
consisted of four levels characterizing the development of this skill has been built. The
transition from level 1 to 2 is determined when pre-service teachers are capable of
analyzing the characteristics of situations to discriminate both types of problems. In level
Fernández, Llinares & Valls
2, pre-service teachers focus on the correctness of students’ answers and tend to accept
students’ correct answers as evidence of understanding without making specific
inferences about what or how students were or were not understand. The transition from
level 2 to 3 is determined when pre-service teachers are capable to relate students’
strategies with the characteristics of the problems justifying through the mathematical
elements if the strategy is correct or incorrect. That is to say, pre-service teachers look
beyond the surface of the student’s answer. Finally, the transition from level 3 to 4 is
determined when pre-service teachers are able to see student’s overall performance to a
certain type of problem. That is to say, pre-service teachers are able to relate strategies
within and across problems in order to see how those strategies are related to other
groups of problems. In this case, pre-service teachers display a greater attention towards
the meaning of students’ mathematical thinking rather than towards some surface
features. Finally, the fact that some pre-service teacher focus on individual answers rather
that characterizing the students’ profiles could be related with the design of the task. For
further researches, it is necessary to formulate more specific questions that address preservice teachers to examine all the answers provided by each student to the four problems
as a whole.
The different levels identified and the transition between them show how preservice teachers professional noticing of students’ mathematical thinking is developed
and therefore, it allows us to begin to understand pre-service teachers learning (Figure 2).
The key elements in this framework are how pre-service teacher use the evidence
(students actions/operations) to describe what or how the student is thinking, and how
they generate an explanation of what the student knows or thinks providing or not
TME, vol10, nos.1&2, p .459
evidence to support the explanation. The characteristics of this framework are similar to
rubrics in the description of how pre-service teacher build a model of student thinking in
a context of prediction assessments (Norton et al., 2011).
Figure 2. A framework to characterize pre-service teachers’ professional noticing of
students’ mathematical thinking in the context of proportionality
The different levels in the framework support the idea that the subject-matter
knowledge is necessary for teaching, but it is not a sufficient condition because teachers
need to interpret the students’ behavior in problem solving situations using their
understanding of mathematical knowledge (Crespo, 2000). Constructing a model for
learning to notice students’ thinking, such as the framework presented, implies to focus
on the organized knowledge about problems and on the range of strategies used by
students to solve the problems (Franke & Kazemi, 2001).
In a previous research, van Es (2010) also provided a framework for learning to
notice the student thinking articulating two central features of noticing: what teachers
notice and how teachers notice. Van Es generated this framework using meetings with
Fernández, Llinares & Valls
seven elementary school teachers in which each teacher shared clips from his or her own
classroom and discussed aspects of the lesson. Although van Es study and our research
use different evidences and come from different contexts, it is possible to identify some
features that provide insights about the noticing construct and its development. One of the
relevant aspects showed in the two researches is how teachers or pre-service teachers go
from a baseline to extent the noticing skill indicating how teachers/pre-service teachers
go from noticing superficial aspects to consider the connections between different
relevant aspects and meanings. However, there are also differences between the two
frameworks: the role played by the mathematical content knowledge in the noticing skill
and how it is integrated (as we have shown in the translation from one level to the next).
This framework should be considered as an initial approach to the
characterization of the development of noticing. However, it points out two additional
aspects that we should be considered. Firstly, the emergence of this framework is linked
to a specific type of problems. Therefore, it is necessary more researches using different
types of problems to extend and to validate this framework and this approach. Secondly,
in the context of mathematics teacher education programs we could complement the
written test (the questionnaire) with students’ interviews.
Teacher education, problem solving and the development of the teacher’s noticing
skill
A goal in mathematics teacher education is the development of pre-service
teachers’ ability to model the student’s thinking and to use evidences from the students’
behavior when solving problems to construct this model (Norton et al., 2011). However,
if pre-service teachers have a lack of content knowledge in solving the mathematical
TME, vol10, nos.1&2, p .461
tasks, they could have difficulties in building an appropriate model of the students’
mathematical thinking. This is the case of pre-service teachers who did not differentiate
the additive and multiplicative relations in the situations proposed in our study. As a
consequence, the first level in the development of teachers’ professional noticing of
students’ mathematical thinking is defined by the understanding of the mathematical
knowledge. So, an aspect of pre-service teacher’s content knowledge for teaching in the
context of multiplicative and additive situations is related to the discrimination between
proportional and non-proportional relationships. It is possible that the lack of knowledge
that pre-service teachers have about proportionality may be due to the way in which
proportionality is often taught at schools in which there is an over-use of missing-value
problems and an overemphasis on routine solving processes (De Bock, Van Dooren,
Janssens, & Verschaffel, 2007).
Since the proportionality is more than a four-term relation, in order to pre-service
teachers could develop a professional noticing, it is necessary that they extend their
understanding and consider other features of proportionality such as straight line graphs
passing through the origin and the constant slope of such graphs identified with the
coefficient of proportionality when it is adopting a functional approach. The differences
between proportional and non-proportional situations should be another feature. Whether
a good problem solver in a given domain is one who knows the connections between the
different mathematical parts, a teacher who wants to interpret the students’ mathematical
thinking during a problem solving situation in the classroom also needs to know the
mathematical structure of the domain. In this case a lack of pre-service teacher’s content
knowledge could limit his/her ability to model the student thinking. In this way, this
Fernández, Llinares & Valls
study examines pre-service teachers’ capacities needed to make sense students’ thinking
about proportionality.
If teacher education programs require pre-service teachers to notice students’
mathematical thinking in problem solving contexts then we should make an effort to
document what is what prospective teachers notice in different mathematical domains
and how the development of this skill could be characterized. Previous studies in initial
mathematics teacher programs have reported improvements in noticing, going from a
descriptive and evaluative noticing towards a more analytic and interpretative one
(Crespo, 2000; Norton et al., 2011; van Es & Sherin, 2002). Furthermore, some studies
underlined the benefits of teachers’ discussions about students’ written work. In a
previous experience, seven prospective secondary school mathematics teachers solved the
task proposed in this study and discussed it in an on-line debate (Fernández, Llinares, &
Valls, 2012). Although, initially, prospective teachers had difficulties attending and
interpreting the students’ mathematical thinking in the domain of multiplicative and
additive structures, when prospective teachers with a lower level of noticing interacted
with other with a higher level of noticing in an on-line discussion, they changed their
interpretations to reach mutual understanding. This process led prospective teachers with
a lower level of noticing to develop a new understanding of students’ mathematical
thinking. From these preliminaries findings, we hypothesized that teachers could develop
ways to elicit and listen to students’ mathematical thinking when they focus their
discussion on the students’ written work. In this sense, focusing on students’ written
work remains an instrument for relating mathematics knowledge and students’
mathematical thinking.
TME, vol10, nos.1&2, p .463
Our findings also provide additional information for the design of materials in teacher
training programs that take into account the characteristics of pre-service teachers’
learning and their understanding of proportionality (Ben-Chaim, Keret, & Ilany, 2007). In
this sense, the instrument used in this research could be adapted as teaching material to
create opportunities for the learning of pre-service teachers. These opportunities of
learning should be focused on the development of pre-service teachers’ skills to identify
and interpret student’s written work. In fact, a characteristic of our research instrument is
that it is based on the details of students thinking and it is elaborated from the research
based on students’ understanding of additive and multiplicative structures (Fernández &
Llinares, 2012-a; Van Dooren et al., 2010). So, firstly, pre-service teachers could solve
the different problems and discuss on the possible different answers. Secondly, they
could share the interpretations of students’ solutions to the problem discussing on the
mathematical understanding of each strategy and how particular strategies were elicited.
Acknowledgements
1.
The research reported here has been financed in part by Ministerio de
Educación y Ciencia, Dirección General de Investigación, Spain, under Grants
no. EDU2008-04583, nº EDU2011- 27288 and in part by the Universidad de
Alicante (Spain) under proyecto emergente nº GRE10-10.
2.
An earlier version of this paper was presented as a research report at XV
SEIEM- Ciudad Real, Spain. September 2011.
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A Proposal for a Problem-Driven Mathematics Curriculum Framework
Judith S. Zawojewski
Illinois Institute of Technology, Chicago, IL
Marta T. Magiera
Marquette University, Milwaukee, WI
Richard Lesh
Indiana University, Bloomington, IN
Abstract: A framework for a problem-driven mathematics curriculum is proposed,
grounded in the assumption that students learn mathematics while engaged in complex
problem-solving activity. The framework is envisioned as a dynamic technologicallydriven multi-dimensional representation that can highlight the nature of the curriculum
(e.g., revealing the relationship among modeling, conceptual, and procedural knowledge),
can be used for programmatic, classroom and individual assessment, and can be easily
revised to reflect ongoing changes in disciplinary knowledge development and important
applications of mathematics. The discussion prompts ideas and questions for future
development of the envisioned software needed to enact such a framework.
Keywords: Problem-based Mathematics, Curriculum frameworks, Mathematical
Modeling, Model-Eliciting Activities.
Introduction
Curriculum frameworks are commonly organized around categories of
mathematical topics (e.g., number, geometry), such as in the new Common Core School
Mathematics Standards (NGA & CCSSO, 2011) and the National Council of Teachers of
Mathematics (NCTM) standards documents (1989, 2000) for the United States (U.S.).
Oftentimes, to convey the nature of mathematics teaching and learning, the content topics
are cross-referenced with other types of mathematical behaviors, such as the “process
standards” (e.g., problem solving, reasoning and proof) of the NCTM documents, and the
“practices” (e.g., model with mathematics, attend to precision) of the CCSSM document.
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos.1&2, pp.469-506
2013©The Author(s) &Dept of Mathematical Sciences-The University of Montana
Zawojewski, Magiera & Lesh
Another approach is to formulate mathematics curriculum frameworks based on
assumptions about learning mathematics, such as the Dutch curriculum framework
described by van den Heuvel-Panhuizen (2003) (e.g., informal to formal, situated to
generalized, individual to social). The developers of mathematics curriculum frameworks
choose their organization and structure in order to communicate a mathematics
curriculum to broad audiences (e.g., teachers, administrators, parents, students). The
choices for content and the representation of curricula made by the framework
developers, in turn, convey a distinctive perspective on mathematics curriculum,
accompanied by inevitable (some intended, some unintended) consequences when users
of the framework transform the represented curriculum into prescriptions for classroom
experiences and assessment. A proposal for framing and representing a problem-driven
mathematics curriculum is described in this article. The proposal envisions a framework
that grows out of Lesh and colleague’s work on models-and-modeling, which has focused
on using modeling problems as sites for revealing and assessing students’ thinking (e.g.,
Lesh, Cramer, Doerr, Post & Zawojewski, 2003), and more recently by Richard Lesh to
teach data modeling (personal conversation, Dec. 21, 2012). The proposal also envisions
a representational system that builds on a one originally posed by Lesh, Lamon, Gong
and Post (1992), and is particularly poignant today because technology is now available
that could carry out the proposal.
Why an Alternative Framework?
Assumptions about Curriculum Frameworks
TME, vol10, nos.1&2, p .471
Curriculum frameworks convey a view of mathematics learning to stake holders
in education, influencing the full range of mathematics education activity—from
implementation to assessment. For example, the two foundational NCTM curriculum
documents (1989, 2000) contributed to a huge shift in views of mathematics curriculum
in the U. S. Prior to the publication of these documents, schools, districts and state
curriculum guides predominantly listed expected mathematical competencies by grade
level, commonly referred to as scope and sequence documents. The NCTM standards
documents introduced a process dimension (problem solving, reasoning, connections,
communication) in addition to the common practice of describing mathematics
competencies and performance expectations. Further, discussions about the mathematical
processes and expected mathematical performances were embedded in the context of
illustrative problems, teaching and learning scenarios, and ways of thinking about
mathematics. These standards documents impacted not only state curriculum standards,
but also resulted in the development of the now-famous NSF curricula (described in
Hirsch, 2007a). Research on the standard-based curricula suggests that students using
these curricula demonstrate enhanced learning of mathematical reasoning and problem
solving (Hirsch, 2007b).
The new Common Core State Standards in Mathematics (NGA & CCSSO, 2011),
adopted by 45 of the United States and 3 territories, lists mathematical learning
objectives, or standards, organized by grade level, and is accompanied by a completely
separate discussion of eight mathematical practices. There is no discussion in the
document to help the practitioner envision what the implementation of the intended
curriculum will look like—leaving the accomplished curriculum more dependent on
Zawojewski, Magiera & Lesh
professional development and local school culture to fill in the picture. One advantage to
the separation of mathematics competencies from the mathematical practices may be to
avoid representing the mathematics curriculum as an array, which can inadvertently
convey a view of mathematics curriculum as disaggregated into bits and pieces
represented by each cell.
Consider, for example, the Surveys of Enacted Curriculum (SEC) (Porter, 2002),
which are intended to drive assessment of student performance. The SEC is organized in
a two-dimensional framework of cognitive demand (memorize, perform procedures,
demonstrate understanding, conjecture/generalize/prove, and solve non-routine problems)
vs. disciplinary topics (e.g., functions, data analysis, rational expressions). It divides the
(K-12) mathematics topics dimension into 19 general categories, each of which is then
divided into 4 to 19 smaller mathematical topics. “Thus, for mathematics, there are 1,085
distinct types of content contained in categories represented by the cells” (Porter,
McMaken, Hwant, & Yang, 2011, p. 104). Porter’s fine-grained representation of
curriculum is intended to ensure coverage of mathematical topics and types of cognitive
demand while minimizing gaps and overlaps. However, such a representation may lead to
an enacted curriculum prescribed by the “pieces” (i.e., the cells), and if educators are
prompted to “teach to the test” an unintended emphasis on disconnected mathematics
education may result. Further, once a framework like this is codified by formal external
assessments, the mathematics content becomes more difficult to revise in response to the
needs of evolving fields of science, engineering and technology.
An alternative may be found in the Dutch mathematics curriculum, rooted in
Realistic Mathematics Education (RME) learning theory, initially developed by the well-
TME, vol10, nos.1&2, p .473
respected Dutch mathematics educator, Freudenthal (1991), and continued at the
Freudenthal institute today. The work in RME portrays a vision of mathematics as a
human activity that combines learning and problem solving as a simultaneous activity.
Smith & Smith (2007) describe the three dimensions around which the RME-based
mathematics curriculum framework is organized: informal to formal; situated to
generalized; and individual to social. In practice, RME emphasizes curriculum designed
to encourage students’ development via progressive mathematization. van den HeuvelPanhuizen (2003) describes progressive mathematization as the growth of an individual’s
mathematical knowledge from informal and connected to the local context, to an
increasing understanding of solutions designed to reach some level of schematization
(making shortcuts, discovering connections between concepts and strategies, making use
of these new findings in a new way), and finally to an increasing understanding of formal
mathematical systems.
The work on such progressive mathematization is growing (e.g., hypothetical
learning trajectories as described by Clements & Sarama, 2004a; 2004b). But, questions
have been raised by Lesh and Doerr (in press): Do all students optimally learn along a
particular normalized path (learning line, learning trajectory)? Do all students learn the
“end product” in the same way? Likely not. Rather than describing a particular learning
objective or standard as a goal for learning, they use Vygotsky’s (1978) “zone of
proximal development” to describe particular goals for students’ learning as regions
around those goals that are individualistic and dependent on a variety of interacting
factors. Such might include the scaffolding provided by the teacher, the language that the
student has and the teacher uses, and the technology or manipulatives that may or may
Zawojewski, Magiera & Lesh
not be available during the learning episode. Further, Lesh and Doerr, using Piaget’s
(1928, 1950) notion of decalage, describe how apparent learning of an objective may
mask the partial development of an idea when “operational thinking” for one concept
may occur years earlier or later than comparable levels of “operational thinking” for
another closely related concept (Lesh & Sriraman, 2005). Lesh and Doerr emphasize that
individuals learn in different ways and develop their understandings along different paths.
They argue that intended “final products” (i.e., identified as standards or learning
objectives) are likely to be in intermediate stages of development in most students, and
open to revision and modification as they encounter new situations for which they need to
form a mathematical interpretation.
Assumptions about Mathematics Learning
Lesh and Zawojewski (2007) refer to the work of various theorists and researchers
(e.g., Lester & Charles, 2003; Lester & Kehle, 2003; Schoen & Charles, 2003; Silver,
1985; Stein, Boaler, & Silver, 2003) to establish a close relationship between the
development of mathematical understandings and mathematical problem-solving. Their
perspective on learning “treats problem solving as important to developing an
understanding of any given mathematical concept or process . . . . [and]. . . the study of
problem solving needs to happen in the context of learning mathematics . . .” (p. 765). In
particular, Lesh and Harel (2003), and Lesh and Zawojewski’s (1992) description of
“local concept development” highlight the simultaneous increase in an understanding of a
specific problem situation and the development of one’s mathematization of the problem.
“[S]tudents begin these type of learning/problem-solving experiences by developing
[local] conceptual systems (i.e., models) for making sense of real-life situations where it
TME, vol10, nos.1&2, p .475
is necessary to create, revise or adapt a mathematical way of thinking” (Lesh &
Zawojewski, 2007, p 783).
What is meant by local concept development and learning? Consider the Grant
Elementary School Reading Certificate activity described in Figure 1, in which students
are asked to create a set of “rules for awarding certificates” (i.e., a decision model). As
described in Figure 1, the students generate a variety of models as an answer to this
problem, and their answers provide windows to their mathematical thinking and
learning—their local concept development.
Grant Elementary School Reading Certificates Problem1
In this activity, third grade students are asked to create and apply a set of decision rules for
awarding certificates to readers who read a lot and who read challenging books. The students
are given sample sets of individual reader’s accomplishments, each presented in a table
including the title of each book read, the number of pages for each book, and the difficulty
level of the book (labeled as easy, medium, hard). The tension between the two criteria for
earning a certificate (reading a lot of books and reading challenging material) was intentional,
in order to enhance the potential for various reasonable models to be developed.
Summary of Group #1 Response:
Students should read either 10 books, or more than 1000 pages.
At least 2 of the books read should be hard books.
This group clearly communicates the decision rules (i.e., model) and takes into account both
required conditions: reading many books, and reading challenging books. Readers can readily
apply the rules to the given data sets. For example, in one data set, the reader had read 5 books
(two of which were hard), and a total of 722 pages. Given the clarity of the decision rules, a
reader can figure what he or she needs to do to earn a certificate. In this illustrative case, one
way for the reader to earn a certificate is to read 5 more books (even if they are all easy).
Another way is to pick one long book that has at least 279 pages.
Summary of Group #2 Response:
A student gets 1 point per page for easy-to-read books.
A student gets 2 points per page for hard-to-read books.
A student has to earn 1000 points to get a reading certificate.
This set of decision rules is clearly communicated, and a reader could easily apply the decision
rules and self-assess. However, a reader could earn a reading certificate award by reading only
easy books, not meeting the criteria that readers must read both hard and easy books.
Therefore, the set of rules does not meet the requirements for a “good” set of rules.
Figure 1. Two Illustrations of Local Concept Development
1 This activity, in full, can be found in Yeatts, C. L., Battista, M. T., Mayberry, S.,
Thompson, D., & Zawojewski, J. S. (2004). Navigating through problem solving and
reasoning in grade 3. Reston, VA: National Council of Teachers of Mathematics.
Zawojewski, Magiera & Lesh
The mathematical goals of the activity are three-fold. First, each group of problem
solvers is expected to generate a mathematical model, meaning they must develop a
procedure or algorithm that meets the criteria given—that those earning a certificate must
read a lot of books and read challenging books. In the generation of a model, many
students engage in other types of mathematical knowledge development, such as
quantifying qualitative information and differentially weighing and/or rank ordering
factors. Each of the two responses described in Figure 1 represents different locally
developed concepts, which are represented in the groups’ model (i.e., a set of rules). Note
that the first response meets the criteria, whereas the second does not. Note, also, how in
each case, the model developed is situated in the context of the problem, and is also
dependent on the knowledge that individuals bring to the group—about mathematics,
about reading programs, about meaning of “challenging books” and meaning of “reading
a lot.” A second goal is for students to practice basic skills, such as recognizing the need
for and carrying out calculations, and comparing and ordering numbers. These take place
as the students test their proposed models, and in the full activity, students are given
further sets of data to conduct additional tests of the model they have generated. A third
area of learning is generalization, which is driven by the design of the problem. In
particular, a good response to this problem is one in which the model produced is reusable (reliably produces the same results for a given set of data), share-able (the decision
rules are clearly and precisely communicated to all of the students, the teachers, and the
parents, resulting in reliable application of the model across users), and modifiable
(rationales and assumptions on which the model is built are articulated so others can
make intelligent adjustments for new situations). Without assumptions or rationales
TME, vol10, nos.1&2, p .477
explained, intelligent modification of models can be quite difficult, if not impossible.
Notice that neither of the two sample responses in Figure 1 meets the modifiability
criteria for generalization, but they have addressed the re-usability and the share-ability
criteria for generalization.
Over the years, Lesh and colleagues have reported on the local concepts
developed by small groups of students as they engage in various problems, such as the
one described in Figure 1. They indicate that individual students often pose initially
primitive solutions, and as a result of social interactions, challenges, testing and revision,
their initial solutions typically move toward a consensus model that is more stable. The
learning of mathematics is described as an iterative process of expressing, testing and
revising one’s conceptual model. In particular, by using mathematical modeling as a way
to think about mathematics learning, Lesh and Doerr (2003) describe a move away from
behaviorist views on mathematics learning based on industrial age hardware metaphors in
which the whole is viewed simply as a sum of the parts and involving simple causal
relationships. Their perspective on mathematics learning also moves beyond softwarebased information processing metaphors, which involve layers of recursive interactions
leading at times to emergent phenomena at higher levels that are not directly derived
from the characteristics of lower levels. Instead, they align their models-and-modeling
perspective on mathematics learning with a biology-based “wetware” metaphor, in which
“neurochemical interactions . . . involv[e] logics that are ‘fuzzy,’ partially redundant, and
partly inconsistent and unstable” (Zawojewski, Hjalmarson, Bowman, & Lesh, 2008, p.
4). Assumed is that students arrive to school with dynamic mathematical conceptual
systems already in place, that these conceptual systems are active and evolving before,
Zawojewski, Magiera & Lesh
during and after problem solving and learning episodes, and that students must be
motivated to engage in experiences by intellectual need (Harel, 2007) in order to learn.
Thus, even when two students in a group may appear to have the same end product
knowledge on one task, changing the task slightly, but keeping it mathematically
isomorphic with the original, often reveals that the two students are thinking about the
intended mathematical ideas in significantly different ways (Lesh, Behr, & Post, 1987;
Lesh, Landau, & Hamilton, 1983).
What is the role of the small group in learning? Social aspects of acquiring
knowledge from communities have been characterized in society over the decades (e.g.,
Mead, 1962, 1977, Thayer, 1982), and more recent work describes learning in
communities of practice in various trades and occupations (Greeno, 2003; Boaler, 2000;
Wenger & Snyder, 2000; Lave & Wenger, 1991; Wenger, 1998). These situations of
social learning are characterized by the presence of a teacher, tutor, or mentor who
models, teaches and collaborates with novices while engaged in the specific context of
practice, rather than in a classroom. Other social aspects of learning have also been
documented in situations where there is no teacher/tutor/mentor available. For example,
researchers have documented successful collaborations among groups of diverse experts,
where any needed leadership emerges flexibly from within the group in response to
emerging challenges and opportunities (Cook & Yanow, 1993; Wenger, 2000; Wenger &
Snyder, 2000; Yanow, 2000). Both perspectives on social aspects of learning are based
on the assumption that all members of a group bring some understanding to the table, that
the knowledge each brings is idiosyncratic, that the knowledge elicited by the problem is
specific to the context, and that local concept development takes place among the group
TME, vol10, nos.1&2, p .479
members while simultaneously each individual in the group is adapting and modifying
one’s own understanding.
Social aspects of problem solving and learning are also related to the development
of representational fluency, because interactions among collaborators require
representations be used to communicate. When presenting initial solution ideas to peers, a
problem solver typically describes one’s own model using spoken words, written
narratives, diagrams, graphs, dynamic action (e.g., gestures or using geometric software),
tables, and other representations. The interpreting peer, who works to make sense of these
representations, may request clarification, an additional explanation, or may point out
inconsistencies, misrepresentations or other flaws. The peers, thus, iteratively negotiate a
consensus meaning. Lesh and Zawojewski (2007) describe various social mechanisms
that can elicit the use of representations, leading to the development of representational
fluency, including: problem solvers making explanations to each other; groups or
individuals keeping track of ideas they have tried; problem solvers making quick
reference notes for new ideas to try as they continue in a current line of thinking; and
problem solvers documenting their current line of reasoning when they must temporarily
disrupt the work. These types of mechanisms, based largely on communication with
others and oneself, provide the need to generate and use representations, and develop
representational fluency.
Toward an Alternative Framework
Given the assumptions about learning grounded in problem solving, a number of
challenges face the development of a framework for a problem-driven mathematics
curriculum. How can a curriculum framework feature problem-solving activity as the
Zawojewski, Magiera & Lesh
center of learning, while national and state standards documents highlight specific
mathematical content as the central feature? How can a curriculum framework
accommodate both the multi-topic nature of realistic mathematics problems and the pure
mathematical nature of other mathematics investigations? How can a framework be
represented to convey the complexities implied by the previous questions, yet be practical
in meeting practical classroom needs? How can a framework be represented to inform the
static nature of various standards documents, while also being responsive to changing
societal needs and demands?
Envisioning a Curriculum Framework and It’s Representation
What is Meant by a Problem-Driven Framework?
The development of problem-driven mathematics text series gained momentum in
the U.S. in response to the 1989 NCTM Curriculum and Evaluation Standards for School
Mathematics. In general, the NSF-funded texts (described in Hirsch, 2007a) are
comprised of units of study organized around applied problems or mathematical themes.
In many cases, these curricula use mathematical problems to launch and motivate
learning sequences that progress toward development of understanding and proficiency
for specified mathematical goals. For example, two of the design principles for
developing the Mathematics in Context text series, which is based on the Dutch RME, are
that the starting point of any instructional sequences “should involve situations that are
experientially real to students” and “should . . . be justifiable by the potential end point of
a learning sequence” (Web & Meyer, 2007, p. 82). The commitment to an experiential
basis reflects the commitment to problem-solving as a means to learning, while the well-
TME, vol10, nos.1&2, p .481
defined mathematical end points correspond to a commitment to a curriculum framework
organized around specific mathematical standards or learning objectives. In contrast, the
problem-driven curriculum, Mathematics: Modeling Our World, described by Garfunkel
(2007), is characterized by using mathematical models as end points. The dilemma for
the Mathematics: Modeling Our World development team was coordinating the
mathematics content naturally emerging from their model-based problem-driven
curriculum with a standard mathematics topic driven curriculum framework. Garfunkel
describes how the team grappled with the need to “cover” the scope and sequence of the
required curriculum:
“[W]e believed (and still believe) that if we could not find, for a particular
mathematics topic, a real problem to be modeled, that that topic would not
be included in our curriculum.. . . Instead of ‘strands’ as they are usually
defined we chose to organize curriculum around modeling themes such as
Risk, Fairness, Optimization. We made an explicit decision . . . not to
create a grid with boxes for mathematical and application topics. Instead,
within the themes we chose areas and problems that we believed would
carry a good deal of the secondary school curriculum. . . . For example, it
was decided that one of the major mathematical themes of Course 1 was to
be Linearity, so that each of the units in the course had to carry material
leading to a deepening understanding not only of linear functions and
equations, but also of the underlying concept of linearity.” (pp.161-162).
Garfunkel’s dilemma illuminates a fundamental mismatch between a curriculum
framework that identifies a list of specific mathematical learning objectives or standards
as outcomes, and the development of a curriculum framework driven by problem solving,
and in particular, modeling. The “coverage” issue seems to force the enacted problemdriven curriculum to be a mix of problem-driven units accompanied by a collection of
gap fillers to address missed content objectives. Thus, while Mathematics: Modeling Our
World began the journey toward a problem-driven curriculum, it was challenged by the
Zawojewski, Magiera & Lesh
coverage constraint, speaking to the question about what content should be included in a
mathematics curriculum framework.
As a result, questions are raised about envisioning a problem-driven curriculum
framework. Should a problem-driven curriculum framework have as final goals students’
deep understanding of mathematical ideas that support certain types of problems, models
or themes, or to demonstrate abilities about certain big mathematical ideas that were
initiated in problem-solving settings? If the goal of a problem-driven curriculum is to
cover certain mathematical models or themes, should the designers of a curriculum cover
only those areas that naturally emerge in modeling or problem-solving work? If, on the
other hand, the goal of a problem-driven curriculum framework is to accomplish certain
big mathematical ideas, is the power of learning those ideas through problem solving to
some extent defeated?
What is the Nature of Mathematics Content in a Problem-Driven Framework?
This larger question raises at least three issues about what mathematics to include
in a problem-driven framework: What type of problems will the curriculum framework
accommodate? What are the boundaries on the mathematics content to be covered? And
how does the curricular framework adapt to evolving societal, scientific and
technological needs concerning what mathematics is important?
A problem-driven curriculum framework would need to incorporate pure
mathematical investigations, real-world applications, and modeling problems, among
others. Whereas some problems nicely map onto a single mathematical big idea, others,
especially applied and modeling problems emphasize multiple mathematical big ideas—
TME, vol10, nos.1&2, p .483
adding to the complexity of developing such a framework. Consider, for example, the
Aluminum Crystal Size2 MEA, included in Figure 2, as an illustration.
Aluminum Crystal Size Problem Description
The activity is situated in the context of the manufacture of softball bats that would resist denting, but also
won’t break. In materials science, one learns that the larger the typical size of crystals in a metal, the more
prone to bending, and the smaller the typical size the more brittle the metal. A problem was posed that had
two purposes. The first was to motivate the problem solver to quantify crystal size. The second was to
establish a context where a client needs a procedure to quantifying crystal size as part of their quality
control. The client in the problem “hires” the problem solver to create a way to measure, or quantify,
aluminum crystal size using two-dimensional images, such as the ones here:
The images are given in different scales, making visual comparison of crystals in the three samples
difficult. Therefore, the mathematical procedure would need to take scale into account.
A number of different approaches typically emerge, including:
Draw a rectangular region to designate a sample within each image. Calculate the area of the
rectangle in which the crystals are enclosed. Count the number of crystals in a rectangle drawn.
Compute the average area per crystal. Compare samples.
Select a sample of crystals within each image and estimate the area of each crystal (e.g., by
measuring the distance across the widest part of a crystal, and the length of the distance
perpendicular to that widest part, and then finding the product of those lengths). Compute the
average area per crystal. Compare samples.
Figure 2. Aluminum Crystal Size Problem
In the Aluminum Crystal Size Problem, multiple big ideas in mathematics are
relevant to producing a good solution. Spatial reasoning is important as the problem
solver needs to figure out ways to quantify regions that are not consistently shaped nor
consistently sized, yet must be considered collectively as a “class” tending toward a
2
For the full activity, see: Diefes-Dux, H. A., Bowman, K. J., Zawojewski, J. &
Hjalmarson, M. (2006). Quantifying aluminum crystal size part 1: The model-eliciting
activity. Journal of STEM Education and Research (7) 1&2, 51-63.
Hjalmarson, M., Diefes-Dux, H. A., Bowman, K. J. & Zawojewski, J. S. (2006).
Quantifying aluminum crystal size part 2: The Model-Development Sequence. Journal
of STEM Education and Research h (7) 1&2, 64-73.
Zawojewski, Magiera & Lesh
certain size. The problem solver must also consider what parts of the regions to use in the
quantification. Measurement is another big idea addressed, since a definition of crystal
size needs to be generated and mathematized. Proportional reasoning is needed because
the micrographs are all shown to different scales, which needs to be accounted for in the
development of the mathematical model for crystal size. Sampling is important when
deciding what regions of the micrograph to use to determine the size of the crystals in the
full image; a good solution will incorporate a method for selecting samples to include in
the mathematical model. Measures of centrality are likely to emerge because quantifiable
characteristics of the various crystals need to be summarized in some way to come up
with a single measurement of crystal size. Finally, mathematical modeling is the
centerpiece of the activity. If the Aluminum Crystal Size activity is used as the
centerpiece of a unit of study, the problem context drives what mathematical topics are
encountered. A framework, then, is needed to help make decisions about which topics to
investigate more deeply, whether to stay within the problem context in those
investigations, and whether or when to incorporate other more conventional lessons or
purely mathematical investigations on the conventional topics.
The second consideration concerns the boundaries of mathematics curricular
topics. For example, an economics problem may require designing a mathematical model
that optimizes costs while producing the highest quality possible. An engineering
problem may have ethical ramifications, where the “best” possible mathematical solution
to attain cost-effectiveness may not meet equity considerations. A problem may lend
itself to an elegant mathematical solution that uses cutting-edge technology, but the
solution may not work with the commonly available technology. In the real world, when
TME, vol10, nos.1&2, p .485
clients want quantitative-based solutions that are cost-effective yet most powerful,
thoroughly but quickly produced, and usable by a wide audience yet secure from abusers,
the mathematical and non-mathematical considerations are inseparable. A collaboration
of engineering educators have grappled with such an issue in the context of engineering
education, where the goal has been to teach foundational engineering principles through
mathematical modeling problems that carry competing constraints when considering
ethical components (e.g., Yildirim, Shuman, Besterfield-Sacre, 2010).
The content of mathematics curriculum needs to be an entity that can evolve, and
can be flexible and nimble as problems faced in the workplace and society evolve—the
third consideration. To illustrate, two hundred years ago the computational algorithms
needed for bookkeeper’s math were appropriately the main focus of school mathematics
content. Now-a-days, research on current professional use of mathematics in fields such
as engineering (e.g., Ginsburg, 2003, 2006), health sciences (Hoyles, Noss, & Pozi, 2001;
Noss, Holyes & Pozi, 2002) and finance (Noss & Hoyles, 1996) reports an increasing
need for students to develop or adapt mathematical models to solve novel problems and
to flexibly interpret and generate representations. Zawojewski, Hjalmarson, Bowman, &
Lesh (2008) indicate that “the real world uses of mathematics are described [in the
studies referenced above] as often requiring that mathematical knowledge be created or
reconstituted for the local [problem] situation and that content knowledge be integrated
across various mathematics topics and across disciplines” (p. 3).
Zawojewski, Magiera & Lesh
A Proposal for an Alternative Problem-Driven Curriculum Framework
Major dilemmas of constructing on over-arching curriculum framework were
illuminated using the two problem driven curriculum frameworks described above. But,
even when considered together, the RME and Mathematics: Modeling Our World do not
necessarily accommodate all aspects of important mathematics to be learned. In
particular, the RME framework is driven by problem-solving launches followed by a
sequence of activities and instruction that lead to an increased understanding of formal
mathematical systems. Garfunkel’s Mathematics: Modeling Our World is organized
around themes such as risk, fairness, optimization and linearity, each representing
important areas of mathematics associated with formal mathematical modeling. Both
generally aim toward formal mathematical goals, but do not have as end goals
mathematics deeply embedded within broad contextual situations and areas such as ethics
or equity. The alternative proposed here is based on a notion of model-development
sequences that broadens the one described by Lesh, Cramer, Doerr, Post, & Zawojewski
(2003). Like RME and Garfunkels’ curricula’s development, the underlying assumption
is that powerful learning of mathematics emerges from students’ mathematization of
problematic situations. Going beyond RME and Garfunkel, a problem-driven
mathematics curriculum framework built around model-development sequences has the
potential to incorporate both formal mathematical big ideas/models and real world messy
models that are intertwined with non-mathematical constraints.
Lesh, Cramer et al. (2003) describe model-development sequences as beginning
with model-eliciting activities (MEAs), which are instantiated in the two problems
presented so far (Figures 1 & 2). The main characteristic of a MEA is that the problem
TME, vol10, nos.1&2, p .487
requires students to create a mathematical model in response to the task posed, which
could be extended to the production of smaller parts of formal mathematical systems.
MEAs have traditionally been designed using six specific design principles (Lesh,
Hoover, Hole, Kelly, & Post, 2000) to devise “authentic” contexts, involving a client with
a specified need for a mathematical model that facilitates making a decision, making a
prediction, or explaining a reoccurring type of event in a system. Following the initial
MEA are planned model-exploration activities (MXAs), which vary from comparing and
contrasting trial models posed by peers in a class, to more conventional meaning-based
instruction on various mathematical aspects of the model. For example, the Aluminum
Crystal Size problem may be followed up with a lesson on the role and power of random
sampling for making inferences, or an opportunity for students to compare and contrast
their procedures for determining typical crystal size in micrograph samples. Similarly,
one of the authors interviewed a teacher who enacted an MXA activity with her third
grade students who had completed the Grant Avenue Reading Certificate Problem
(Figure 1). After the teacher asked the third grade students to present their rules to each
other, she asked students to identify similarities and differences among the sets of rules,
and probed students perceptions of the pros and cons. By asking questions about what
aspects of the situation each set of rules attended to and ignored, and how the choice of
variables influenced the impacts the outcomes, she was teaching foundational ideas of
modeling. For example, Group 2’s response (Figure 1) ignores the number of books in
the data—depending only on the number of pages to represent “reading a lot.” Group 1
(Figure1), on the other hand, used all three types of data (number of pages, number of
books and the rating of easy/medium/hard). Even though Group 2’s response did not fully
Zawojewski, Magiera & Lesh
meet the criteria articulated in the problem, the use of page numbers only, and not the
number of books, is defendable as an indicator of amount of reading. Helping the
students articulate rationales for their decisions supports the development of an initial
understanding that models are systems that represent larger systems, and inevitably
capture some features of the original system, while ignoring other aspects.
A model-development sequence closes with a model-adaptation or modelapplication activity (MAA). To illustrate the power of a MAA, consider the full sequence
of activities that has been used in the first-year engineering course (with students fresh
out of high school) at Purdue University3. The opening Nano Roughness4 MEA, (see
Figure 3) is “set in the context of manufacturing hip-joint replacements where the
roughness of the surface determines how well the joint replacement moves and wears
within the hip socket” (Hjalmarson, et al., p. 41). Given digital images of the molecular
surface of different samples of metal, students were asked to create a procedure for
quantifying the roughness of each sample, which resulted in a variety of models. The
subsequent MXA introduced students to a conventional engineering model for
quantifying roughness, the average maximum profile (AMP) method, and then asked
them to compare their model for quantifying roughness to the conventional engineering
model. The goal for this MXA was to enable students to identify and understand trade-
3
Purdue’s first-year engineering program has been using MEAs and model-development
sequences for the past 9 years with approximately1500 student per year in West
Lafayette, Indiana, USA (Hjalmarson, Diefes-Dux & Moore, 2008).
4
The full activity can be found in J. S. Zawojewski, H. A. Diefes-Dux, & K. J. Bowman
(Eds.), (2008). Models and modeling in engineering education: Designing experiences
for all students (pp. 317-322). Sense: Rotterdam, The Netherlands. The lead author of the
activity is Tamara J. Moore.
TME, vol10, nos.1&2, p .489
offs between models, and to identify and understand rationales and assumptions
underlying different models.
Nano Roughness MEA
This goal of this activity is to produce a procedure to quantify roughness of metal surfaces at a
microscopic level. Students are given atomic force microscope (AFM) images, similar to the one
below, of three different samples of metal surfaces. At the atomic level, the lighter parts of the image
represent higher surface, and the darker parts of the image represent lower surface. The gray scale
indicator, to the side, provides information about the height of the surface. To motivate the problem
situation, the students learn that the company, who is their client, specializes in biomedical
applications of nanotechnology. They are planning to produce synthetic diamond coatings for use in
orthopedic and biomedical implants, and need to have a way to quantify roughness of the coating
surfaces. Given three top-view images of gold samples (illustrated in the one sample below), the
modelers are asked to develop a procedure for quantifying the roughness of the material so the
procedure could be applied to measure roughness in other types of metal samples.
Sample of an AFM image of gold surface (AFM data courtesy of Purdue University Nanoscale Physics Lab)
Figure 3. Nano Roughness MEA Description
The model-development sequence closes with a Model-Adaptation Activity
(MAA) that requires students to adapt either their model for measuring roughness, or the
conventional model, to a new situation. To do the work in the Purdue example, students
were given a raw data set that had been used to produce a sample digital image. These
raw data had been gathered by using an atomic force microscope (AFM), which uses a
nano-scale probe dragged along the surface of the metal sample in lines at regular
Zawojewski, Magiera & Lesh
intervals, measuring the relative heights along the bumps of the molecules. The students
were asked to generate, using MATLAB, a cross sectional view of any line segment
drawn on an image of the gold surface. In particular, they produced graph-like products
that portrayed the relative heights of the bumps and valleys for any line segment drawn
on an image. The mathematical learning goals for this MAA were to conduct 2dimensional array manipulations of the data and to incorporate statistical reliability
considerations into the process. Broader learning goals for the Nano Roughness problem
include programming and fundamental engineering principles—illustrating how
mathematics learned may be embedded and intertwined to what traditionally has been
considered non-mathematical topics.
While MEAs, and their accompanying model-development sequences have
traditionally been tied to authentic realistic modeling contexts, the basic concept of
eliciting a mathematical model can be broadened to incorporate the more traditional
modeling work, such as described by Garfunkel (2007). The model-development
sequence framework can also be envisioned to include the elicitation of aspects of formal
mathematical systems, such as what is the aim in RME. In other words, modeldevelopment sequences have a great deal of potential to serve as an umbrella framework
that provides a way to unify problem-driven curricula frameworks, especially when
considering the flexibility of Learning Progress Maps (LPMs), which is proposed as a
possible way to represent problem-driven curriculum frameworks.
Envisioning a Representational System for Problem-Driven Curriculum
TME, vol10, nos.1&2, p .491
Using a metaphor of topographical maps, Learning Progress Maps (LPMs) can be
thought of as a dynamic representation of mathematics curriculum and students’ learning
(Lesh, Lamon, et al., 1992; Lesh, unpublished manuscript). Lesh’s goal in developing this
concept has been to help teachers readily answer practical classroom questions such as:
What concepts do my students still need to address in this unit I am teaching? Which
topics would be strategic to address next? What are concepts or topic areas where my
students appear to require more experience? Which students are having difficulty with
specific concepts, and which have demonstrated learning in those areas? Single score
results from large-scale measures do not provide useful information for these questions,
whereas item-by-item information for every student might be overwhelming to use as an
everyday tool to make decisions about classroom instruction. Portfolio assessment is
difficult to define and standardize, let alone use for day-to-day classroom decisions about
instruction. On the other hand, good teachers do develop their own personal methods to
keep track of individual students’ progress in a variety of ways, although their systems
are idiosyncratic to the teacher, often very detailed, and usually perceived by others as
too time consuming to maintain.
How Might a Learning Progress Map (LPM) Represent a Problem-driven Curriculum?
Consider a hypothetical topographical map representing a curriculum organized
around mathematical big ideas, important mathematical models, or formal mathematical
systems, presented in Figure 4. Lesh, Lamon, et al. (1992) describe the mountains of the
landscape as corresponding to the “big mathematical ideas” of a given course (6 to 10 big
ideas in this case), and the surrounding terrain of foothills and valleys as depicting facts
and skills related to the big ideas. Using the topographical maps metaphor, one can think
Zawojewski, Magiera & Lesh
about the height of the mountain as representing the relative importance of big
mathematical ideas in the course, while relationships among the big ideas can be
expressed by the proximity of the mountains to each other. The tops of the mountains
would represent deep understanding of the big idea, abilities supporting the big ideas can
be represented on the sides of the mountains, and associated tool skills (e.g.,
manipulations, skills, facts) can be represented by the regions of the surrounding foothills
and valleys.
Figure 4. Representation of Big Ideas, Supporting Abilities, and Tools in a Course
A top-down view of the topographical curriculum map (illustrated in Figure 5),
might delineate the interplay of the big mathematical ideas, supporting abilities and tool
skills to be “covered” in the given course, in a way analogous to a traditional scope-andsequence document.
TME, vol10, nos.1&2, p .493
Figure 5. Top Down View of Curriculum Scope and Sequence
On a LPM, problem-solving, modeling, deep insights into a designated
mathematical big idea, and higher-order mathematical thinking about the idea would be
designated in regions on the tops of the mountains. Thus, problems that involve multiple
big mathematical ideas, such as the Nano Roughness MEA, could be represented by
multiple mountains (e.g., 3-d geometry, proportions, sampling, measurement,
mathematical models). The height of the mountains, and the arrangement of the regions
around them, would represent the relative importance of the major mathematical areas
with respect to the MEA. Supporting concepts and procedures would correspond to the
sides of the mountain, and needed skills and facts would correspond to the adjoining
valleys around each mountain. For example, in the Nano Roughness MEA, the fluent
interpretation and manipulation of the scales would be an important component of
proportional reasoning, and thus represented on the sides of a proportional reasoning
mountain. The valleys nearby each mountain would represent the automatic skills and
concepts that might be thought of as the tools of the trade for that big idea, such as
masterful and precise computation or algebraic manipulation. Another illustration might
be the linearity theme of Mathematics: Modeling out World, as described by Garfunkel
(2007). Linearity might be the name of the mountain, and the idea of representing linear
expressions in various forms (as narratives of situations, as tables, as graphs) may each
correspond to regions along the side of that mountain, and fluent manipulation of linear
equations might be represented in the valley nearby the mountain.
The potential flexibility of the proposed representational system is greatly enabled
by the power of technology. For example, given that the concept of linearity is a major
theme in Course 1 of Mathematics: Modeling out World, linearity may be important to a
Zawojewski, Magiera & Lesh
number of units, and emerge in a variety of contexts. In the mountain representational
system for each unit, the theme of linearity may be represented as an overlay of a
particular colors or textures (e.g., striping, dotting) on all terrains. Further, theoretical
perspectives on learning may also be represented using different intensities of colors to
illustrate the three dimensions in RME, or an activity’s classification in modeling
sequences i.e., MEAs, MXAs and MAAs. The envisioned representation of a curriculum
framework could provide teachers with the opportunity to manipulate the map, providing
varied views of the curriculum. For example, a teacher may want to see how linearity
emerges across chapters within a course by viewing any and all mountains that represent
linearity across chapters. While one can imagine many useful scenarios of manipulation,
the greatest potential for LPMs, however, is probably representing students’ progress
through the curriculum.
How Might a Learning Progress Map (LPM) Serve Assessment?
In a problem-driven curriculum framework, assessment of big ideas and models
would be supported by LPMs which are envisioned as providing manipulable
representations of students' attained curriculum. Specific assessment data can be used to
“fill in” appropriate regions of a LPM for a particular student in a course. Since in a
problem-driven curriculum, the students’ mathematical experiences begin in problemsolving environments, and supporting skills may be learned or mastered later and at
various levels, record-keeping is potentially very challenging. Planning assessment points
to correspond with particular regions of the map would be a strategy for input points that
would in turn help keep track of accomplishments by individuals, while also potentially
providing a visual picture that organizes the assessment data for the individual students.
TME, vol10, nos.1&2, p .495
Assessment data points can be drawn from students’ responses to problem-solving
or modeling activities and used to guide subsequent instructional activity. To illustrate,
consider the work of Diefes-Dux and colleagues, who have been very active in
documenting students’ modeling performance on iterations of revised solutions to MEAs.
They have developed systematic ways to evaluate the development of mathematical
models that students generate (e.g., Carnes, Diefes-Dux, & Cardella, 2011; Diefes-Dux,
Zawojewski, & Hjalmarson, 2010). Their assessment rubric (Diefes-Dux, Zawojewski,
Hjalmarson, & Cardella, 2012) that addresses four general characteristics of the models is
made into task-specific versions for each MEA. Their recent work has focused on the
challenge of identifying and implementing feedback to students with a goal of prompting
students to rethink and revise their solution model to be more powerful and efficient
(personal conversation with Diefes-Dux, January 17, 2012). One can imagine that this
line of research would be enhanced with the proposed framework and representational
system. For example, Diefes-Dux and colleagues’ evaluate the generalizability of
students’ models based on three criteria. Assessment of a model’s “re-usability”
documents the stability of the model over its independent applications; that is, whenever
the model is re-applied to a given data set the model will produce the same results each
time. Assessment of model’s “share-ability” documents whether the model is
communicated well enough so that other users can apply the model independently and
reliably. Finally, the assessment of the model’s “adaptability” focuses on the articulation
of critical rationales and assumptions on which the model is constructed, so that an
external user would be able to intelligently modify the model for new, somewhat
different, circumstances. These three dimensions could be easily represented and
Zawojewski, Magiera & Lesh
manipulated in the envisioned framework to look for patterns and trends in students’
series of revised models.
In a problem-driven curriculum framework, assessment of students’ performance
on concepts, skills, and procedures that support big ideas and models can be facilitated by
LPMs and guided by available mathematics education research. For example, in the
Ongoing Assessment Project (OGAP), Petit and colleagues (e.g., Petit, Laird, & Marsden,
2010) examined all available mathematics education research in selected domains,
identified important benchmarks and “trouble spots” of understanding, and targeted those
specific concepts and skills for the development of assessment items and activities. They
have completed the work on fractions, multiplication and proportions. Such assessment
items can be used as data points in the side regions and valleys of mountains
corresponding to the big mathematical ideas. Further, in conjunction with the growing
body of research on learning progressions (e.g., Clements, 2004), assessment points that
have been embedded in the learning trajectories can become benchmarks that are
carefully placed to track general progress as students eventually abstract from their
variety of situations to generalized mathematical ideas.
How Might a Learning Progress Map (LPM) be Used to Inform Practice and Programs?
The envisioned dynamic LPMs would provide a means for teachers to quickly and
easily identify information relevant to day-to-day questions for teaching and students’
learning. For example: What concepts do my students still need to address? Which topics
would be strategic to address next? Which students are (or are not) having difficulty with
specific concepts? Using a keystroke, summarized students’ assessment data could
displayed on the LPM, providing opportunities for nimble decision-making about
TME, vol10, nos.1&2, p .497
classroom practice. By illuminating the whole class’s attainment on the LPM curriculum,
teachers would be able to see what yet needs to be addressed in the course, and what may
need some reteaching. Profiles of individual students’ attainment could help teachers plan
to group students for differentiated instructional experiences. LPMs could, for example,
help teachers to form problem-solving groups by identifying students with a variety of
expertise relevant to the problem. Individual profiles, when displayed side-by-side, could
also inform teachers’ decisions about students access to limited resources (e.g., volunteer
tutors, particular technological assistance, advanced placement coursework).
Self-assessment could become a major component of classroom experience.
Students could use their own individual profiles to self-assess their own progress, and
perhaps even select problems through which they can address their own areas of need. In
an advanced version of LPM, where the curriculum topics are linked to appropriate
problems, perhaps students could select a context they like to think about (e.g., sports,
health care), and be assigned an appropriate problem from the targeted area of need. By
integrating an assessment system with the curricular map, LPMs could be used as a tool
to guide students’ selection of problems that have the potential to move them forward
mathematically.
Professional development and program evaluation can also be enhanced through
LPMs. Lesh, Lamon, et al., (1992) describe a variety of program level assessments that
could be accomplished by dynamic LPMs. For example, a summary class attainment map
that looks like the one in Figure 5, suggests instruction that is highly skill-based, and thus
provides an opportunity to for a teacher to confront one’s own (perhaps unconscious)
assumption that problem solving and deep conceptual understanding can only be
Zawojewski, Magiera & Lesh
addressed after all of the “basics skills” have been accomplished. On the other hand, a
summary class attainment map that looks like the one illustrated in Figure 6 might
suggest that a teacher is effectively implementing a problem-driven curriculum, given
that the attained map illustrates splashes and spreads from multiple points near the tops of
the mountains, and oozing downward to the sides of mountains and surrounding valleys.
Fig 5: LPM (green) in Skill-based Attainment by Students
Fig 6: A LPM (green) in Multi-level Attainment by Students
Reflections
The envisioned problem-driven mathematics curriculum framework supported by
a dynamic representational system, LPM, seems feasible. Given the potential of today’s
technologies, design research (Kelly, Lesh, Baek, 2008) methodologies could be used to
TME, vol10, nos.1&2, p .499
simultaneously build, study and revise theoretical, pedagogical, and practical
considerations of a problem-driven curriculum framework and its representational
system. The LPM could be manipulated and revised quickly and easily in response to
various changing conditions, such as changes in what constitutes important mathematics,
changes in important problem context, changes in new content-driven state standards, and
changes in interdisciplinary and social considerations. While the representational system
has yet to be actualized, many aspects of problem-driven curricular frameworks are
already under research and development. Imagining future work that links technologydriven LPMs and problem-driven curriculum frameworks brings a variety research
questions and potential issues for investigation.
Given that problem-driven mathematics curriculum frameworks are grounded in
the assumption that students learn mathematics while engaged in complex problemsolving activity, a question arises about how LPMs could be used to represent such
curricula. What would a LPM look like for a course, or a unit of study? What will be
identified as the “big ideas” or mountains around which the mathematical terrain is
developed? What variables need to be represented in the LPM, beyond content topics?
What needs to be fixed and what needs to be flexible in the software? These are only a
few of the questions that need to be answered in interdisciplinary teams of mathematics
educators, curriculum developers, assessment experts, and software developers in a
design process.
How can LPMs be used to identify when, and the extent to which, problem-based
instruction supports the given problem-based curriculum? Collaborative research and
development would be needed to design software to display an image, such as the one in
Zawojewski, Magiera & Lesh
Figure 5 that represents successful implementation of problem-based instruction. The
design of the software would require the identification of variables and development of
models to show the splashes and spreads from multiple points near the tops of the
mountains, oozing downward, and eventually filling in the valleys. The needed data
include the curriculum specifications, student assessment data, and teacher input about
experiences implemented. The goal would be to provide real-time information to teachers
and their support personnel concerning what students are learning, and to use that
information to adjust instructional strategies to align with those appropriate for problembased learning.
How might LPMs assist classroom teachers in their enactment of a problemdriven curriculum, yet help to keep an eye on “content coverage” as potentially required
by other stake holders? To support implementation of problem-driven curricula in
environments that are driven by standards and emphasize content coverage, teachers’
need to have tools that help them traverse the challenges of real world implementation.
The envisioned LPMs must have embedded in them the ability to manipulate the
representations so that teachers can easily check on “content coverage” while teaching a
problem-based curriculum. Further, they need to be able to easily check on individual
student progress in order to plan for reasonable differentiation. Challenges in
implementing a problem-based curriculum must be addressed by well-designed LPMs
that are easily used by teachers to inform their questions and issues.
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