Papers by Anna Sierpinska
Routledge eBooks, Jan 11, 2013
Mathematics education library, Jun 14, 2006
Springer eBooks, 1996
... International Handbook of Mathematics Education. 755. 784 C 1996 Kluwer Academic Publishers. ... more ... International Handbook of Mathematics Education. 755. 784 C 1996 Kluwer Academic Publishers. Printed in the Netherlands ... At first sight it seems obvious to reduce numeracy to'ele-mentary'skills or abilities, and in practice numeracy is often understood in this waythat is. ...
Anna Sierpinska & Jeremy Kilpatrick. ... But, as Vergnaud rightly points out, in mathemat... more Anna Sierpinska & Jeremy Kilpatrick. ... But, as Vergnaud rightly points out, in mathematics education research, the practice/theory issue arises in two contexts: in mathematics (a practical activity versus a theoretical body of knowledge), and in mathematics education (teaching ...
Didactica Mathematicae, Dec 30, 2019
Didactica Mathematicae, 1990
Didactica Mathematicae, 1988
Kluwer Academic Publishers eBooks, Dec 23, 2005
Educational Studies in Mathematics, Nov 1, 2004
It is for the first time in its history that ESM publishes a series of "papers" made not only of ... more It is for the first time in its history that ESM publishes a series of "papers" made not only of words and static pictures on paper, but also of animated and interactive graphical representations, and motion pictures with sound ("video-clips") of mechanical devices and of students, teachers and researchers working with them. This issue of ESM makes the word "paper" sound completely inadequate. In its beginnings, ESM gave voice mainly to researchers in mathematics education. But then authors started to illustrate their papers with verbatim transcripts of conversations involving students, teachers and researchers. The research subjects' voice could thus be "heard", albeit only in a metaphorical way. With the video-papers, the subjects' voice is now literally heard. Moreover, the reader is also a "viewer", able not only to hear the words being exchanged but also to observe the gestures, the actions, and the background context of the conversations. What was previously hidden from the reader is now there for everyone to see and hear. The view and the perspective were chosen by the author, but.. . the wings of the theatre are open. This changes the relationship of the reader to the data, and, therefore, also the relationship between the author and the reader. With so much access to the original data, the reader may become a participant in the research and co-construct an interpretation of the data with the author. I hope the readers-viewers of this "video-Special Issue" of ESM will make the most of this unprecedented opportunity. The technology of this "Video Special Issue", as we have come to call it among the editors, is not yet perfect. Certainly, the images and the sounds could be better. But the ideas brought forth through this multi-media experience and collaboration of students, teachers and researchers open many new horizons for teaching, learning and doing research in mathematics education.
Revue des sciences de l'éducation, 1996
Tous droits réservés © Revue des sciences de l'éducation, 1996 Ce document est protégé par la loi... more Tous droits réservés © Revue des sciences de l'éducation, 1996 Ce document est protégé par la loi sur le droit d'auteur. L'utilisation des services d'Érudit (y compris la reproduction) est assujettie à sa politique d'utilisation que vous pouvez consulter en ligne.
Journal for Research in Mathematics Education, Mar 1, 1999
Page 1. Review Coming of Age in Academe Mathematics Education as a Research Domain: A Search for ... more Page 1. Review Coming of Age in Academe Mathematics Education as a Research Domain: A Search for Identity. (1998). Anna Sierpinska and Jeremy Kilpatrick (Eds.). Dordrecht, The Netherlands: Kluwer. xiii + 576 pp. ISBN 0-7923-4599-1 (hb., 2 vols.) $250. ...
Journal for Research in Mathematics Education, Mar 1, 1999
Theories That Gyre and Gimble in the Wabe Mathematics Education as a Research Domain: A Searchfor... more Theories That Gyre and Gimble in the Wabe Mathematics Education as a Research Domain: A Searchfor Identity. (1998). Anna Sierpinska and Jeremy Kilpatrick (Eds.). Dordrecht, The Netherlands: Kluwer. xiii + 576 pp. ISBN 0-7923-4599-1 (hb., 2 vols.) $250. ISBN 0-7923-4600-9 (pb., 2 vols.
Neste capitulo procuramos interligar os principais argumentos oferecidos nos capitulos anteriores... more Neste capitulo procuramos interligar os principais argumentos oferecidos nos capitulos anteriores e na ICMI Study Conference de modo a proporcionar ao leitor uma visao retrospectiva da indubitavelme nte complexa questao, ‘O que e investigacao em Educacao Matematica e o que sao os seus resultados?’. A esta questao, sentimos necessidade de juntar uma outra: O que significa ser um i nvestigador em Educacao Matematica? Temos uma identidade comum? Por todo o livro ecoa o tema que nenhuma destas questoes, nem outras que dela resultam, pode ter uma resposta definitiva. Tal como as questoes de investigacao em Educacao Matematica, as questoes sobre o proprio campo podem ser exploradas mas nao ha forma de as resolver de uma vez por todas do modo, por exemplo, como se prova um teorema matematico. Cada geracao deve recolocar a si mesma a questao do que e, afinal, a Educacao Matematica. Embora os capitulos precedentes demonstrem a nossa incapacidade para fornecer respostas definitivas, eles salientam e elaboram os progressos que fizemos na clarificacao dos significados das questoes. Como indicou o Documento para Discussao e reafirmou a Conferencia, o objectivo deste ICMI Study era identificar “perspectivas, objectivos, questoes de investigacao e formas de abordar os problemas” de modo a que pudesse ocorrer um confronto produtivo de perspectivas e pudesse crescer a nossa compreensao mutua. Deste modo, este capitulo nao representa um consenso acabado e final sobre o nosso campo. Em vez disso, aponta para direccoes onde podemos continuar a procura sem fim de quem somos e de quem queremos ser.
Educational Studies in Mathematics, 2005
Focus on learning problems in mathematics, 1993
Educational Studies in Mathematics, Nov 1, 1987
The article presents a report on four 45 minute sessions with a group of 17 year old humanities s... more The article presents a report on four 45 minute sessions with a group of 17 year old humanities students. These sessions were the first of a series organised with the aim of exploring the possibilities of elaborating didactical situations that would help the students overcome epistemological obstacles related to limits. Students' attitudes pertinent to the development of the notion of limit, as well as changes of these attitudes, are described and analysed.
New ICMI studies series, 1998
In this chapter, we attempt to draw together the principal arguments made in the preceding chapte... more In this chapter, we attempt to draw together the principal arguments made in the preceding chapters and at the ICMI Study Conference so as to provide the reader with a retrospective view of the clearly very complex issue, ‘What is research in mathematics education, and what are its results?’. To that question, we are compelled to add: What does it mean to be a researcher in mathematics education? Do we have a common identity?
Uploads
Papers by Anna Sierpinska
I taught this course in the fall of 2012 to students enrolled in the Master in the Teaching of Mathematics program at Concordia University.
Based on my own experience of teaching such a course in the Department of Mathematics and Statistics of Concordia University in Montreal, the document contains information about the purpose of the course as it was presented to the participants in the “Course outlines”, the forms of assessment used, lists of references to literature on difficulties in different domains of mathematics, examples of lectures and classroom activities and examples of topics on which participants chose to do their research.
It seems that the rationale of all these courses rests on the assumption that school presents a distorted view of mathematics, and that this view has to be quickly rectified when a student enters university or, more generally, endeavors to become an educated and reflective adult. For example, the online Stanford “Introduction to Mathematical Thinking” (designed by Keith Devlin) highlights the procedural and routine character of school mathematics. In “An introduction to mathematics”, written over a century ago, Whitehead stresses the difference in the kind of questions that are considered important in school mathematics as compared to the mathematics done by mathematicians. For example, in the chapter on “Variables”, he says that both use letter variables and algebraic expressions, but the focus on “solving” equations and inequalities in school leads students to believe that determining the values of the unknowns is the most important mathematical problem. For a mathematician, on the other hand, the problem of determining the values of the unknowns is a subordinate one to the question of relations among two or more variables and whether certain algebraic (or more generally, analytic) statements about these variables hold true for some or for any values of the variables and in what domains.
When the course MAST 217 – Introduction to Mathematical Thinking, offered at Concordia University, was first conceived (in mid-1990s), one of its “fathers”, W.P. Byers, wanted it to be a course for math specialization students, to prepare them for ways of thinking characteristic of the research mathematicians that some of them would hopefully become. But the course ended up being institutionalized as optional for specialization students. It was aimed mainly at math major students with no more than 12 credits (4 courses) in undergraduate mathematics. At the beginning, the course was not compulsory even for these students. So classes were usually small, with 20-30 students, some math majors, some from other departments (philosophy, computer science, etc.). The mathematical background of the students in the course was varied and not quite predictable from term to term and even from one section of the course to another. But this was not a problem because there was no fixed and detailed syllabus “to cover” in the course; it was not necessarily to be an “introduction to set theory with elements of logic and abstract algebra.” The instructors adjusted their choice of problems, topics and level of mathematical discourse to the group of students that happened to be registered in their section. The communication style in the classroom was discussion based, interactive, with the teacher building on students’ own experience. In class, the teacher would propose a problem, students would try to solve it, and then their attempts would be discussed in small groups and eventually in the whole class, with the teacher pointing to mathematical laws implicitly used or violated in the students’ solutions. A loose collection of problems for initiating such discussions was written up in the form of a “course pack” and made available to students and instructors in the university bookstore. The form of assessment was likewise left to the discretion of the instructor. There were no common midterms or common final examinations even if there was more than one section of the course offered in a term. I taught this course many times between 1995 and 2018. Sometimes I assessed students’ progress based on portfolios, sometimes I gave projects; sometimes there was a final examination and sometimes there was not.
Sometime in the first decade of the 21st century, the course became compulsory in the “BSc or BA Major in Mathematics and Statistics program”. The result was a dramatic increase in the number of students registered in the course: from 20-30 students in the 1990s to around 60 in 2010s. Assessing students’ progress based on portfolios or projects became very difficult. The interactive communication style in the classroom that characterized the course at the beginning also became more difficult to obtain, especially the whole class, teacher-led, discussion phase. But, even in the 2018-19 Undergraduate Calendar, the course was still described as a “problem-solving” course.
There is still no fixed syllabus. Each instructor has their own way of teaching the course. This freedom is very appealing, but there is a downside to it. An instructor may be doing very interesting things in the course, but if they keep their experience to themselves and don’t share it with other instructors much practical knowledge about teaching the course is lost.
Fortunately, in the Department of Mathematics and Statistics at Concordia University, with its tradition of research in mathematics education, not all of this knowledge is lost. One instructor of the MAST 217 course, Dr. Nadia Hardy, made her teaching available as an object of study to a graduate student. The student, Erin Murray, wrote an MSc thesis based on her observations in the section of MAST 217 that her supervisor taught (the thesis is posted on Spectrum, on the Concordia Libraries webpage). Nadia Hardy’s course used a radical problem-based approach, with students working on different problems at their own pace, and a varied panoply of assessment tools.
The present document describes how I taught the course, mainly in the winter term of 2018, with some remarks on my teaching in 2016 and 2017. My approach was much more traditional than Nadia’s, with a lot more lecture time and assessment based on homework assignments, a midterm and a final examination.
This document is not a report on a study in mathematics education. I just describe what I did in the course, the lectures I gave, the activities and exercises I offered students to work on in class, the homework I assigned, the practice exercises I suggested, and the questions the student were given at the midterm and final examinations. I also share some reflections that came to me after the course; these reflections are called “Notes” in the text.
Some information about students’ work is offered: a few reactions in class to an exercise or a question I posed; an analysis of students’ responses to two problems in the final examination that provides evidence of persistent difficulties with the negation of implication; and, in an Appendix, a summary of types of students’ responses to the “manually graded” homework assignments. The Appendix was co-written with the teaching assistant assigned to my course, Vinicius Gontijo-Lauar, who marked the “manually graded” part of the homework assignments. Vinicius developed grading rubrics for marking the assignments and most of these are presented in the Appendix, together with descriptions of types of students’ solutions.
There was also the “electronically graded” part of homework assignments made of multiple choice questions that were programmed on a computer (in Moodle). In the document, the electronically graded assignments are presented in the form of ordinary text, not in computer code. The code versions of the assignments are available upon request.
The course lasted 13 weeks and the class met twice a week, for 1h15mn each time. There was one midterm test, held in class, and a 3 hours final examination after the end of classes.
The document is organized chronologically, from week 1 to week 13, and the final examination. The last chapter contains some comments on the results of the final examination. This is followed by a list of references. After the list of references there is an alphabetical Index of more important mathematical terms introduced in the document, facilitating the use of the document as a resource.
I post this document on the web, as another account of the practice of teaching an “Introduction to Mathematical Thinking” course. I believe that any description of a teaching practice is useful. For if a potential instructor reads the description, even if they do not like the approach, and choose to do things differently, their practice will already be better informed and justified than if they did those things without being aware of the choices they made.
The organization of the material is based on a theory of levels of geometric reasoning proposed by a pair of Dutch mathematics educators, Peter van Hiele and Dina van Hiele-Geldof, in the 1950s (van Hiele, 1959) . In the text, you will find an explanation of what it means to reason at each level, and you will study examples of activities that are intended to help students to progress to a higher level of reasoning.
The first chapter gives a brief outline of the van Hiele levels: Visualization, Analysis, Informal Deduction, Deduction and Rigor. Also discussed are the assumptions about what is geometry and why to teach it to elementary school children, underlying the van Hiele theory. These assumptions are not to be seen as universally accepted among mathematics educators. Alternative points of view exist and some of them are presented in the chapter. If the van Hiele levels have been chosen as an organizing theory for this text, it is mainly for practical reasons and not because we believe that the van Hiele levels theory is “the correct one”. Two such practical reasons are sketched below.
The presentation of the levels as a whole affords an opportunity to draw a broad picture of geometry as a mathematical theory. Elementary teachers will usually not study geometry as a mathematical theory at more advanced levels, but awareness of the existence of such theory and a sense of its nature is accessible to them and can broaden their knowledge of humanity’s cultural heritage. Some elementary students might be interested in studying mathematics in the future, and teachers should be able to respond to their questions by directing them to relevant sources of information.
Chapters II and III are focused on transitions from Visualization to Analysis (II) and from Analysis to Informal Deduction (III). Activities intended to foster these transitions are situated in the context of the fundamental geometric constructions (e.g., drawing a circle, a segment of a given length, a triangle with given sides and angle, etc.) and properties of polygons, especially different types of quadrilaterals.
The study of transitions from each van Hiele level to the next gives future teachers an opportunity simultaneously to engage with geometric knowledge and reasoning, difficulties in learning geometry, and instructional tools for identifying the difficulties and fostering progress in learning geometry. Given the restricted time for mathematics in elementary teacher preparation programs, such condensation of the three main aspects of geometry teaching – content knowledge, knowledge of learning and knowledge of teaching – is an opportunity not to be lightly dismissed.
Chapter IV is devoted to thinking about geometric transformations (translations, reflections, rotations) at the Analysis level mainly. We devote a separate chapter to this topic because of its specific difficulty and the importance it has in the elementary school curriculum.
The last chapter in this text is a collection of problems, divided into 5 thematic sections. The first three (construction tasks, sorting tasks, geometric transformations) serve to consolidate the knowledge exposed in Chapters II-IV. The last two (lengths and areas, and solid geometry) offer directions of extending this knowledge to new geometric contexts.
The organization of the material is based on a theory of levels of geometric reasoning proposed by a pair of Dutch mathematics educators, Peter van Hiele and Dina van Hiele-Geldof, in the 1950s (van Hiele, 1959) . In the text, you will find an explanation of what it means to reason at each level, and you will study examples of activities that are intended to help students to progress to a higher level of reasoning.
The first chapter gives a brief outline of the van Hiele levels: Visualization, Analysis, Informal Deduction, Deduction and Rigor. Also discussed are the assumptions about what is geometry and why to teach it to elementary school children, underlying the van Hiele theory. These assumptions are not to be seen as universally accepted among mathematics educators. Alternative points of view exist and some of them are presented in the chapter. If the van Hiele levels have been chosen as an organizing theory for this text, it is mainly for practical reasons and not because we believe that the van Hiele levels theory is “the correct one”. Two such practical reasons are sketched below.
The presentation of the levels as a whole affords an opportunity to draw a broad picture of geometry as a mathematical theory. Elementary teachers will usually not study geometry as a mathematical theory at more advanced levels, but awareness of the existence of such theory and a sense of its nature is accessible to them and can broaden their knowledge of humanity’s cultural heritage. Some elementary students might be interested in studying mathematics in the future, and teachers should be able to respond to their questions by directing them to relevant sources of information.
Chapters II and III are focused on transitions from Visualization to Analysis (II) and from Analysis to Informal Deduction (III). Activities intended to foster these transitions are situated in the context of the fundamental geometric constructions (e.g., drawing a circle, a segment of a given length, a triangle with given sides and angle, etc.) and properties of polygons, especially different types of quadrilaterals.
The study of transitions from each van Hiele level to the next gives future teachers an opportunity simultaneously to engage with geometric knowledge and reasoning, difficulties in learning geometry, and instructional tools for identifying the difficulties and fostering progress in learning geometry. Given the restricted time for mathematics in elementary teacher preparation programs, such condensation of the three main aspects of geometry teaching – content knowledge, knowledge of learning and knowledge of teaching – is an opportunity not to be lightly dismissed.
Chapter IV is devoted to thinking about geometric transformations (translations, reflections, rotations) at the Analysis level mainly. We devote a separate chapter to this topic because of its specific difficulty and the importance it has in the elementary school curriculum.
The last chapter in this text is a collection of problems, divided into 5 thematic sections. The first three (construction tasks, sorting tasks, geometric transformations) serve to consolidate the knowledge exposed in Chapters II-IV. The last two (lengths and areas, and solid geometry) offer directions of extending this knowledge to new geometric contexts.
Knowledge presented in the text is meant as a foundation for the “Specialized Content Knowledge” (Ball, Thames, & Phelps, 2008) for the teaching of fractions in elementary schools. It is intended as teachers’ knowledge, not knowledge that could be imparted, as is, to schoolchildren. It is neither the mathematics of school textbooks for children, nor the mathematics of academic textbooks for future mathematicians.
Our intention is to look at the content to be taught in elementary school about fractions from the point of view of mathematics itself. We will strive to clarify the meanings of the fundamental concepts (fraction of a quantity, ratio, proportion, etc.) by formulating definitions and using precise language; we will justify techniques and rules of operations on fractions by reasoning; we will analyze the structure of problems about fractions and classify them into types. In that sense, it is a mathematical knowledge base for making informed decisions in planning lessons, responding to students’ questions and preparing exercises for them. We will leave aside, for the time being, the issues of the psychology of teaching and learning fractions: how children learn fractions, what difficulties they experience and what teachers can do to help them overcome these difficulties. These issues will be dealt with in the course Teaching Mathematics III.
In Part I of the course notes we focus on fractions as a way to represent the multiplicative relationship between two quantities (lengths, areas, volumes, weights, capacities, market values, etc.). We view fractions as an answer to the question of how many times one quantity is bigger or smaller than another. We explicitly define what it means for one quantity to be such and such fraction of another quantity. In concrete problems we identify and name the quantities that are being compared and the units with which they are measured. The comparison may require change of unit or conversion from one unit to another. We become aware of these operations, analyze them and learn how to write them. In brief, in this course, we make explicit what is often left implicit in teaching fractions to children.
In Part I of the course notes, we will talk mainly about quantities. So, for example, instead of talking about abstract numbers, e.g., 3/4 and 1/8 , and calculating their sum, 3/4+ 1/8 = 7/8 , we will be talking about, say, Tom using up 3/4 of a 120-gallon water tank, meaning the quantity of 90 gallons, and Jerry using up 1/8 of the same water tank and asking how much of the tank they used together. We will be adding quantities, not abstract numbers, expressing this operation in terms of numbers of units (gallons in this case), as
3/4 of 120 Gal + 1/8 of 120 Gal = 7/8 of 120 Gal
or as 90 Gal + 15 Gal = 105 Gal.
In this text, the concept of fraction of a quantity will be derived from the practices of measurement, particularly the operations of change of unit and unit conversion. If we want to measure some quantity, the existing units of measurement may not represent it precisely enough, and we have to change those units into some finer ones. For example, if we measure the length of some tiny piece of metal and find that it measures 1 mm and a little bit and find that 3 such “bits” cover 2 mm, then we will say that 1 bit = 2/3 mm. This is a “conversion equation.”
The approach to fractions through measurement – we call it the Measurement Approach – proposed in this text, is different from the more common Division Approach to fractions through division of numbers, modeled by division of things (e.g., apples or pizzas), or graphical representations of things. We explain the differences between the Measurement and the Division approaches to fractions in more detail in Chapter I.
Chapter II focuses on the practical and theoretical foundations of the Measurement Approach. It starts by explaining how the notion of a fraction of a quantity could be derived from the historical practices of measurement. Then the meaning of the phrase “one quantity is such and such fraction of another quantity” is defined in two ways. One is inspired by barter exchanges where integer multiples of two quantities are equated with each other. The other is based on measuring each of the quantities independently with a common unit and then comparing those measurements multiplicatively.
Because a fraction represents a multiplicative relation between two quantities, the operation of multiplication requires special attention. When dealing with whole numbers of quantity only, understanding multiplication as repeated addition was enough. But this meaning cannot be applied to taking, say, three-fifths times some quantity, and we need to redefine the operation of multiplication so that it covers multiplication of both a quantity by a whole number and by a fraction.
The definitions introduced in the second chapter constitute the conceptual basis for the subsequent chapters in which a whole arithmetic of fractions of quantities has been developed. The various operations on fractions of quantities are studied, such as calculating the fraction of a fraction of a quantity; addition of two fractions of a quantity; their subtraction; division of a quantity by a fraction and division of a quantity by a quantity. The names of these operations include terms known from the arithmetic of whole abstract numbers (e.g., 2 × 3) or whole numbers of quantities (e.g., 2 × 3 kg), such as “addition”, “subtraction”, “multiplication” and “division”, but the meaning of these terms is extended or modified to apply to the domain of fractions of quantities.
The purpose of this elaborate development is to provide future elementary teachers with a coherent theory that fulfills a twofold purpose. On one hand, it may help them develop a mature understanding of fractions, beyond the understanding they have acquired in school (Ma, 1999). On the other hand, it may equip them with the practical tools and flexibility to operate in a variety of contexts and within, likely, very different developments of the topic of fractions, which they may encounter in their future practice.
Throughout the text, understanding of fractions of quantities and operations with them is fostered by means of examples, exercises and end-of-chapter “Problem sets.” Chapters VII (Proportional quantities and proportional reasoning) and VIII (Comparison of fractions of quantities) offer an opportunity to consolidate this understanding in solving somewhat more complex problems.
NOTE: This is a slightly revised and corrected version of the course notes published in 2014 by Concordia University and used as a text in the course EDUC 387 Teaching Mathematics II taught in the Winter 2015 term. An earlier and rather different version of these course notes – using only one definition of fraction of a quantity, and authored by Anna Sierpinska only – has been published in 2013, and used as a text in EDUC 387 Teaching Mathematics II in the Winter of 2014.
In this second part, in Chapter I, we abstract the notion of fraction from the context of quantities and conceptualize it as part of an abstract number system – the rational numbers system – embedded in an even larger system of real numbers. One aim here is to situate the theory of fractions of quantities within a broader mathematical horizon. Another is that we need the notion of real number to understand the notion of ratio, which we introduce in Chapter II, and distinguish it from fraction. The notion of percent, studied in Chapter III, is closely related to ratio, and therefore also requires a broadening of the concept of number.
Note: This is a slightly revised – mainly corrected – version of course notes under the same title published in 2014.
We developed our theory and methodology along with our study of data from interviews with 14 students who achieved high grades in a first linear algebra and some of whom obtained high grades also in a second linear algebra course. We also studied the final examination questions in these courses from the point of view of the level of theoretical thinking that was, in principle, needed to obtain full marks.
The report has five chapters and is preceded by an introduction, where the research question is formulated and put in context.
In Chapter I we describe our model of theoretical thinking. We argue that practical and theoretical thinking are deeply intertwined in the development of scientific knowledge, and we posit that theoretical thinking develops against practical thinking, which thereby also constitutes a condition for the existence and significance of theoretical thinking. We then justify why we think that theoretical thinking is necessary for understanding linear algebra. The chapter concludes with a summary of the postulated features of theoretical thinking. We name the main categories of these features "reflective", "systemic" and "analytic" thinking.
In Chapter II we present our interviews with a group of 14 students who were selected for having achieved high grades in a section of a first linear algebra course (they were all taught by the same instructor). Six of these students achieved high grades also in a second linear algebra course. We explain our design of the seven interview questions and our method of coding students' responses: the code [1, 0] is used to represent theoretical behavior, [0, 1] represents practical behavior and [1, 1] represents a mixture of theoretical and practical behavior. Six questions in the interview had mathematical content and the seventh question was meant to incite the students to reveal their beliefs about, among others, truth, mathematical knowledge, and reasons for taking mathematics courses.
In describing students' behavior in each question we explain how we related this behavior with our model of theoretical thinking. This leads us to an operationalization of the model in terms of features of students' behavior; we identify 18 "theoretical behavior" features
In Chapter III we present the contents of the linear algebra courses taken by the interviewed students and we analyze in detail the final examinations given in these courses. We look at the examination questions from the point of view of the relevance of theoretical thinking in their solution. We point to ways of reformulating the questions so that theoretical thinking becomes more relevant in their solution.
Chapter IV contains a quantitative evaluation of the students' theoretical behavior. We treat the numerical coding introduced in Chapter II as scores and we define three categories of statistical indices as functions of these scores: expectation, tendency and capability, which we apply to the group as a whole (group indices) and to individual students (individual indices). We use these indices to examine the relations between the students' disposition in the sense of expectation, tendency and capability, and their grades in the courses. We also apply the group indices to dress a profile of theoretical thinking strengths and weaknesses in the whole group of students, compared with the profile of the subgroup of students who achieved high grades in both courses. These profiles are formulated in terms of rankings of the 18 theoretical behavior features according to the indices of expectation, tendency and capability. We conclude with a comparison of the students' disposition to theoretical thinking and the relevance of this kind of thinking in solving the examination questions.
In Chapter V we formulate some more general conclusions from the study and, looking back at our research methodology, we reflect on the empirical value of our findings.
The report closes with some remarks on the theme of the idea of a university. We propose that the development of theoretical thinking is an important part of the mission of the university and that there is much room for improvement in mathematics courses in contributing to fulfill this mission.