Most edited books in mathematics education fall somewhere along the continuum anchored by studies... more Most edited books in mathematics education fall somewhere along the continuum anchored by studies of different topics with a common theoretical framework on the one end, and studies of a common topic from different theoretical frameworks on the other end. Lehrer and Chazan’s new book, Designing Learning Environments for Understanding Geometry and Space can be positioned closer to the latter end of this continuum. All of the chapters, in one way or another, focus on learning environments for studying geometry. However, the perspectives range from largely psychological in focus (e.g., chapters by Koedinger, Dennis and Confrey, Pegg and Davey, and Olive) to more socially-focused (e.g., Lehrer, Jacobson, et al; Clements, et al; Middleton and Corbett; and Renninger, Weimar, and Klotz). Such a diversity opens the field of geometry education to a variety of readers interests. In this review, I address the following questions: 1) How is the book organized? 2) What are some of the major them...
Research has shown that undergraduate students benefit from seeing examples of mathematics applie... more Research has shown that undergraduate students benefit from seeing examples of mathematics applied to real-world situations. This article describes three different paradigms for how math and discipline partner faculty worked together to create mathematical activities that illustrate applications of the topics being studied in precalculus and calculus. All three examples are discussed within the framework of PDSA cycles to describe the process by which the teams collaborated to plan, enact, study, and refine their lessons. Findings discuss both the difficulties of creating integrated activities (differences in terms and definitions between mathematics and science faculty, different foregrounding of mathematics versus science among faculty), and the value of the resultant lessons, such as increased level of student engagement, higher cognitive demand, and the role that relevant applications can play in piquing student interest in STEM.
Number 12 in NCTM's Journal for Research in Mathematics Education monograph series, this publ... more Number 12 in NCTM's Journal for Research in Mathematics Education monograph series, this publication describes a design experiment that focused on linear measurement and was conducted in a first-grade class. Its chapters explore three themes that concern design researchers: the relation between the development of instructional designs and the analysis of students' learning, the relation between communal classroom processes and individual students' reasoning and the role of tools in supporting development.
Research has shown that undergraduate students benefit from seeing examples of mathematics applie... more Research has shown that undergraduate students benefit from seeing examples of mathematics applied to real-world situations. This article describes three different paradigms for how math and discipline partner faculty worked together to create mathematical activities that illustrate applications of the topics being studied in precalculus and calculus. All three examples are discussed within the framework of PDSA cycles to describe the process by which the teams collaborated to plan, enact, study, and refine their lessons. Findings discuss both the difficulties of creating integrated activities (differences in terms and definitions between mathematics and science faculty, different foregrounding of mathematics versus science among faculty), and the value of the resultant lessons, such as increased level of student engagement, higher cognitive demand, and the role that relevant applications can play in piquing student interest in STEM.
This paper reports on efforts to design a computer-based instructional unit for 7th grade student... more This paper reports on efforts to design a computer-based instructional unit for 7th grade students that would not only address state standards but would also provide some groundwork for the students ' subsequent formal study of algebra. SimCalc Mathworlds software was used to explore the same motion from different perspectives. The overall objective is to build on what students know and to provide learning experience on which they build. Instructional sequences implemented and students ' comments regarding what they thought they had experienced during the instructional sequences are included. (KHR) Reproductions supplied by EDRS are the best that can be made from the original document. a71-
DISCUSSIONS of the role of algorithms in the elementary school curriculum give rise to conflictin... more DISCUSSIONS of the role of algorithms in the elementary school curriculum give rise to conflicting notions of how algorithms should be approached in the classroom. These views range from encouraging students to invent their own algorithms with minimal guidance to teaching students to perform traditional algorithms. In this paper, we will discuss an approach that eschews both these extremes. This approach values students ’ construction of nonstandard algorithms. However, it also emphasizes the es-sential role of the teacher and of instructional activities in supporting the development of students ’ numerical reasoning. In addition, this approach highlights the importance of discussions in which students justify their algorithms. It therefore treats students’ development of increasingly sophisticated algorithms as a means for conceptual learn-ing. In our view, an approach of this type is consistent with reform recommendations, like those of the National Council of Teachers of Mathemat...
ABSTRACT. We analyze the interrelations between prospective and practicing teachers’ learning of ... more ABSTRACT. We analyze the interrelations between prospective and practicing teachers’ learning of the mathematics of change and the development of their emerging under-standing of effective mathematics teaching. The participants in our study, who were all interested in teaching secondary mathematics, were mathematics majors who had signifi-cant formal knowledge of the fundamental concepts of calculus prior to taking our courses, but who often experienced and expressed procedural orientations toward the teaching of mathematics. To address this difficulty, we developed novel computer-based activities to challenge the participants ’ mathematical understandings and required them to use tech-nology during short teaching episodes they conducted with younger students. To analyze our participants ’ understandings, we developed a framework that juxtaposes the roles of the participants as students and teachers, and their understanding of mathematics and of pedagogical strategies. Our analysis ...
In this article we assess the effectiveness of an Internet-based mathematics-mentoring program th... more In this article we assess the effectiveness of an Internet-based mathematics-mentoring program that was designed to enhance prospective teachers' ability to communicate with middle school students about mathematics. The mentoring involved reading solutions that students submitted to the Math Forum's online Problem of the Week, and writing feedback to those students after first practicing mentoring techniques in an online discussion forum. This report reports the results of a study held in two courses: one in the fall of 2000 (n=19 mentors), and the other in the spring of 2001 (n=10 mentors). We use quantitative data to document improvement trends in mentoring quality, and qualitative data to examine the program's success from the perspective of the prospective teachers. The results indicate that many of the prospective teachers learned to guide students' problem solving with questions and hints and learned about the range and types of solution methods that students u...
Number 12 in NCTM's Journal for Research in Mathematics Education monograph series, this publ... more Number 12 in NCTM's Journal for Research in Mathematics Education monograph series, this publication describes a design experiment that focused on linear measurement and was conducted in a first-grade class. Its chapters explore three themes that concern design researchers: the relation between the development of instructional designs and the analysis of students' learning, the relation between communal classroom processes and individual students' reasoning and the role of tools in supporting development.
This paper discusses the following pedagogical questions and technological solutions that evolved... more This paper discusses the following pedagogical questions and technological solutions that evolved as three multimedia case studies for preservice teachers were developed: (1) What tools can be garnered to encourage preservice teachers to look beyond superficial aspects of a classroom to observe the complexities of the teaching and learning process? (2) How can preservice teachers be engaged in the process of teacher reflection? (3) How can this case study be made different from a video by exploiting the use of multimedia? (4) How can preservice teachers be engaged in the mathematics of the lesson and encourage them to anticipate possible student strategies? (5) How can instructors be supported who are planning to use the CD-ROM in their classrooms? and (6) What technological boundaries will need to be overcome? The design goal of the project was to create a means for helping preservice teachers observe the complexity of the teaching and learning process through the use of innovative...
The Journal of Computers in Mathematics and Science Teaching, 2012
Online video sharing websites have had a profound impact on the ways that students learn about th... more Online video sharing websites have had a profound impact on the ways that students learn about the world, about the culture, and about mathematics. However, we would like to challenge McLuhan's assertion that changing the media has necessarily changed the message. In fact, the authors posit that the majority of online videos are seen as particularly helpful to students because they do not change the overall approach to teaching mathematics or the implicit message that mathematics is a set of unrelated tasks to memorize. The goal in this set of iterative research and design cycles was to explore potential forms that the complement to procedural videos might take. This article describes the results of interviews with pre-service teachers who had taken an online survey that comprised a typical procedural video, a conceptual video, and a dynamic graph applet to support students' explorations of multiple approaches to the idea of function inverses. The analysis focuses on the discursive practices that emerged as preservice math teachers explored each of the online resources. Our conclusion discusses four recommendations for creating further disruptive technologies.
Most edited books in mathematics education fall somewhere along the continuum anchored by studies... more Most edited books in mathematics education fall somewhere along the continuum anchored by studies of different topics with a common theoretical framework on the one end, and studies of a common topic from different theoretical frameworks on the other end. Lehrer and Chazan’s new book, Designing Learning Environments for Understanding Geometry and Space can be positioned closer to the latter end of this continuum. All of the chapters, in one way or another, focus on learning environments for studying geometry. However, the perspectives range from largely psychological in focus (e.g., chapters by Koedinger, Dennis and Confrey, Pegg and Davey, and Olive) to more socially-focused (e.g., Lehrer, Jacobson, et al; Clements, et al; Middleton and Corbett; and Renninger, Weimar, and Klotz). Such a diversity opens the field of geometry education to a variety of readers interests. In this review, I address the following questions: 1) How is the book organized? 2) What are some of the major them...
Research has shown that undergraduate students benefit from seeing examples of mathematics applie... more Research has shown that undergraduate students benefit from seeing examples of mathematics applied to real-world situations. This article describes three different paradigms for how math and discipline partner faculty worked together to create mathematical activities that illustrate applications of the topics being studied in precalculus and calculus. All three examples are discussed within the framework of PDSA cycles to describe the process by which the teams collaborated to plan, enact, study, and refine their lessons. Findings discuss both the difficulties of creating integrated activities (differences in terms and definitions between mathematics and science faculty, different foregrounding of mathematics versus science among faculty), and the value of the resultant lessons, such as increased level of student engagement, higher cognitive demand, and the role that relevant applications can play in piquing student interest in STEM.
Number 12 in NCTM's Journal for Research in Mathematics Education monograph series, this publ... more Number 12 in NCTM's Journal for Research in Mathematics Education monograph series, this publication describes a design experiment that focused on linear measurement and was conducted in a first-grade class. Its chapters explore three themes that concern design researchers: the relation between the development of instructional designs and the analysis of students' learning, the relation between communal classroom processes and individual students' reasoning and the role of tools in supporting development.
Research has shown that undergraduate students benefit from seeing examples of mathematics applie... more Research has shown that undergraduate students benefit from seeing examples of mathematics applied to real-world situations. This article describes three different paradigms for how math and discipline partner faculty worked together to create mathematical activities that illustrate applications of the topics being studied in precalculus and calculus. All three examples are discussed within the framework of PDSA cycles to describe the process by which the teams collaborated to plan, enact, study, and refine their lessons. Findings discuss both the difficulties of creating integrated activities (differences in terms and definitions between mathematics and science faculty, different foregrounding of mathematics versus science among faculty), and the value of the resultant lessons, such as increased level of student engagement, higher cognitive demand, and the role that relevant applications can play in piquing student interest in STEM.
This paper reports on efforts to design a computer-based instructional unit for 7th grade student... more This paper reports on efforts to design a computer-based instructional unit for 7th grade students that would not only address state standards but would also provide some groundwork for the students ' subsequent formal study of algebra. SimCalc Mathworlds software was used to explore the same motion from different perspectives. The overall objective is to build on what students know and to provide learning experience on which they build. Instructional sequences implemented and students ' comments regarding what they thought they had experienced during the instructional sequences are included. (KHR) Reproductions supplied by EDRS are the best that can be made from the original document. a71-
DISCUSSIONS of the role of algorithms in the elementary school curriculum give rise to conflictin... more DISCUSSIONS of the role of algorithms in the elementary school curriculum give rise to conflicting notions of how algorithms should be approached in the classroom. These views range from encouraging students to invent their own algorithms with minimal guidance to teaching students to perform traditional algorithms. In this paper, we will discuss an approach that eschews both these extremes. This approach values students ’ construction of nonstandard algorithms. However, it also emphasizes the es-sential role of the teacher and of instructional activities in supporting the development of students ’ numerical reasoning. In addition, this approach highlights the importance of discussions in which students justify their algorithms. It therefore treats students’ development of increasingly sophisticated algorithms as a means for conceptual learn-ing. In our view, an approach of this type is consistent with reform recommendations, like those of the National Council of Teachers of Mathemat...
ABSTRACT. We analyze the interrelations between prospective and practicing teachers’ learning of ... more ABSTRACT. We analyze the interrelations between prospective and practicing teachers’ learning of the mathematics of change and the development of their emerging under-standing of effective mathematics teaching. The participants in our study, who were all interested in teaching secondary mathematics, were mathematics majors who had signifi-cant formal knowledge of the fundamental concepts of calculus prior to taking our courses, but who often experienced and expressed procedural orientations toward the teaching of mathematics. To address this difficulty, we developed novel computer-based activities to challenge the participants ’ mathematical understandings and required them to use tech-nology during short teaching episodes they conducted with younger students. To analyze our participants ’ understandings, we developed a framework that juxtaposes the roles of the participants as students and teachers, and their understanding of mathematics and of pedagogical strategies. Our analysis ...
In this article we assess the effectiveness of an Internet-based mathematics-mentoring program th... more In this article we assess the effectiveness of an Internet-based mathematics-mentoring program that was designed to enhance prospective teachers' ability to communicate with middle school students about mathematics. The mentoring involved reading solutions that students submitted to the Math Forum's online Problem of the Week, and writing feedback to those students after first practicing mentoring techniques in an online discussion forum. This report reports the results of a study held in two courses: one in the fall of 2000 (n=19 mentors), and the other in the spring of 2001 (n=10 mentors). We use quantitative data to document improvement trends in mentoring quality, and qualitative data to examine the program's success from the perspective of the prospective teachers. The results indicate that many of the prospective teachers learned to guide students' problem solving with questions and hints and learned about the range and types of solution methods that students u...
Number 12 in NCTM's Journal for Research in Mathematics Education monograph series, this publ... more Number 12 in NCTM's Journal for Research in Mathematics Education monograph series, this publication describes a design experiment that focused on linear measurement and was conducted in a first-grade class. Its chapters explore three themes that concern design researchers: the relation between the development of instructional designs and the analysis of students' learning, the relation between communal classroom processes and individual students' reasoning and the role of tools in supporting development.
This paper discusses the following pedagogical questions and technological solutions that evolved... more This paper discusses the following pedagogical questions and technological solutions that evolved as three multimedia case studies for preservice teachers were developed: (1) What tools can be garnered to encourage preservice teachers to look beyond superficial aspects of a classroom to observe the complexities of the teaching and learning process? (2) How can preservice teachers be engaged in the process of teacher reflection? (3) How can this case study be made different from a video by exploiting the use of multimedia? (4) How can preservice teachers be engaged in the mathematics of the lesson and encourage them to anticipate possible student strategies? (5) How can instructors be supported who are planning to use the CD-ROM in their classrooms? and (6) What technological boundaries will need to be overcome? The design goal of the project was to create a means for helping preservice teachers observe the complexity of the teaching and learning process through the use of innovative...
The Journal of Computers in Mathematics and Science Teaching, 2012
Online video sharing websites have had a profound impact on the ways that students learn about th... more Online video sharing websites have had a profound impact on the ways that students learn about the world, about the culture, and about mathematics. However, we would like to challenge McLuhan's assertion that changing the media has necessarily changed the message. In fact, the authors posit that the majority of online videos are seen as particularly helpful to students because they do not change the overall approach to teaching mathematics or the implicit message that mathematics is a set of unrelated tasks to memorize. The goal in this set of iterative research and design cycles was to explore potential forms that the complement to procedural videos might take. This article describes the results of interviews with pre-service teachers who had taken an online survey that comprised a typical procedural video, a conceptual video, and a dynamic graph applet to support students' explorations of multiple approaches to the idea of function inverses. The analysis focuses on the discursive practices that emerged as preservice math teachers explored each of the online resources. Our conclusion discusses four recommendations for creating further disruptive technologies.
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