Orly Buchbinder | University of New Hampshire (original) (raw)

Papers by Orly Buchbinder

Research paper thumbnail of Instructional Technology Tetrahedron and Network Visualization: Conceptualizing Online Teaching Through a Lens of Reflective Noticing

Digital Experiences in Mathematics Education, 2024

Online learning and teaching, accelerated by the global pandemic and rapid advancement of digital... more Online learning and teaching, accelerated by the global pandemic and rapid
advancement of digital technology, require novel conceptual and analytical tools to understand better the evolving nature of online teaching. Drawing on the classical model of the instructional triangle and previous attempts to extend it, we propose the Instructional Technology Tetrahedron (ITT)—a conceptual framework that integrates technology into the instructional triangle to represent the role of technology, as a learning tool and a mediator between teachers, students, and content. Combining the ITT framework with network visualization strategies allowed for representing the intensity of interactions within the tetrahedron. We illustrate the affordances of the ITT framework by analyzing reflective noticing patterns of three prospective secondary teachers (PSTs) who reflected on the video recordings of their own online teaching, with each PST teaching four online lessons to groups of high-school students. We demonstrate the utility of the ITT framework to characterize individual noticing patterns, in a particular lesson and across time, and to support a variety of cross-case comparisons. The discussion sheds light on the broader implications of
the ITT framework.

Research paper thumbnail of Guiding principles for teaching mathematics via reasoning and proving

HAL (Le Centre pour la Communication Scientifique Directe), Feb 2, 2022

Against the backdrop of policy documents and educational researchers' vision of proof as an essen... more Against the backdrop of policy documents and educational researchers' vision of proof as an essential component of teaching mathematics across content areas and grade levels, teaching of reasoning and proof in mathematics classrooms remains an elusive goal. Teachers and the type of teaching they enact in classrooms are crucial for achieving this goal. This theoretical paper builds on the concept of proof-based teaching and suggest a set of guiding principles for what we call teaching mathematics via reasoning and proving. These principles were developed as a part of a multi-year design based project, and implemented in an undergraduate course Mathematical Reasoning and Proving for Secondary Teachers. We illustrate these principles using examples from proof-oriented lessons plan developed by prospective secondary teachers.

Research paper thumbnail of Examining the role of examples in proving processes through a cognitive lens: the case of triangular numbers

Zdm – Mathematics Education, Feb 18, 2011

In this paper, we analyze the role of examples in the proving process. The context chosen for thi... more In this paper, we analyze the role of examples in the proving process. The context chosen for this study was finding a general rule for triangular numbers. The aim of this paper is to show that examples are effective for the construction of a proof when they allow cognitive unity and structural continuity between argumentation and proof. Continuity in the structure is possible if the inductive argumentation is based on process pattern generalization (PPG), but this is not the case if a generalization is made on the results. Moreover, the PPG favors the development of generic examples that support cognitive unity and structural continuity between the argumentation and proof. The cognitive analysis presented in this paper is performed through Toulmin's model.

Research paper thumbnail of Erratum to: What Can You Infer from This Example? Applications of Online, Rich-Media Tasks for Enhancing Pre-service Teachers’ Knowledge of the Roles of Examples in Proving

Mathematics education in the digital era, 2017

In the originally published version of the chapter, Figure 4 in Chapter "What Can You Infer from ... more In the originally published version of the chapter, Figure 4 in Chapter "What Can You Infer from This Example? Applications of Online, Rich-Media Tasks for Enhancing Pre-service Teachers' Knowledge of the Roles of Examples in Proving" was portrait oriented and downsized, hence the text was virtually unreadable. The erratum of the book has been updated with the change.

Research paper thumbnail of On Convincing Power of Counterexamples

The Mathematics Enthusiast, Feb 1, 2024

Despite plethora of research that attends to the convincing power of different types of proofs, r... more Despite plethora of research that attends to the convincing power of different types of proofs, research related to the convincing power of counterexamples is rather slim. In this paper we examine how prospective and practicing secondary school mathematics teachers respond to different types of counterexamples. The counterexamples were presented as products of students' arguments, and the participants were asked to evaluate their correctness and comment on them. The counterexamples varied according to mathematical topic: algebra or geometry, and their explicitness. However, as we analyzed the data, we discovered that these distinctions were insufficient to explain why teachers accepted some counterexamples, but rejected others, with seemingly similar features. As we analyze the participants' perceived transparency of different counterexamples, we employ various theoretical approaches that can advance our understanding of teachers' conceptions of conviction with respect to counterexamples.

Research paper thumbnail of Supporting prospective secondary mathematics teachers in creating instructional explanations through video based experience

Research paper thumbnail of Systematic Exploration of Examples as Proof: Analysis with Four Theoretical Frameworks

ICME-13 monographs, 2018

This chapter offers a multi-layered analysis of one specific category of students’ example-based ... more This chapter offers a multi-layered analysis of one specific category of students’ example-based reasoning , which has received little attention in research literature so far: systematic exploration of examples. It involves dividing a conjecture’s domain into disjoint sub-domains and testing a single example in each sub-domain. I apply four theoretical frameworks to analyze student data: The Mathematical-logical framework for the interplay between examples and proof, Proof schemes framework, Transfer-in-pieces framework, and the Theory of instructional situations . Taken together, these frameworks allow to examine the data from mathematical, cognitive and social perspectives, thus broadening and deepening the insights into students thinking about the relationship between examples and proving. Implications for teaching and learning of proof in school mathematics are discussed.

Research paper thumbnail of Pre-service teachers' construction of algebraic proof through exploration of math-tricks

This paper contributes to the ongoing effort to create rich learning opportunities for prospectiv... more This paper contributes to the ongoing effort to create rich learning opportunities for prospective teachers to engage with reasoning and proving. Twenty elementary and middle school pre-service teachers completed individual projects in which they explored "math-tricks"unconventional computational algorithms-as a part of an undergraduate proof course. Our findings suggest that the task evoked uncertainty with respect to why the tricks work and motivation to resolve the uncertainty by means of algebraic proof. We discuss the potential of this task to create rich opportunities for prospective teachers to conduct explorations, construct algebraic proofs and reflect on their experience from learner's perspective.

Research paper thumbnail of Who is Right? – What Students’ and Prospective Teachers’ Responses to Scripted Dialog Reveal About Their Conceptions of Proof

Advances in mathematics education, Oct 31, 2017

This chapter explores the potential of using scripted student responses, embedded in a task title... more This chapter explores the potential of using scripted student responses, embedded in a task titled Who is right?, as a tool to diagnose argumentation and proof-related conceptions of high-school students and pre-service mathematics teachers (PSTs). The data, collected in two separate studies, were examined for evidence of participants’ conceptions of the role of examples in proving and refuting universal statements. Additional analysis explored what types of criteria are used by the high-school students and the PSTs when evaluating scripted arguments, as well as whether participants were consistent in their evaluations across the collection of arguments. The data revealed that, when evaluating scripted arguments, high-school students used mainly mathematical criteria and strived to maintain consistency in their evaluations across the collection of arguments. On the contrary, PSTs applied both mathematical and pedagogical considerations in their evaluations, thus judging multiple, and even contradictory arguments as correct.

Research paper thumbnail of What Can You Infer from This Example? Applications of Online, Rich-Media Tasks for Enhancing Pre-service Teachers’ Knowledge of the Roles of Examples in Proving

Mathematics education in the digital era, Oct 13, 2016

There is a consensus among mathematics educators that in order to provide students with rich lear... more There is a consensus among mathematics educators that in order to provide students with rich learning opportunities to engage with reasoning and proving, prospective teachers must develop a strong knowledge base of mathematics, pedagogy and student epistemology. In this chapter we report on the design of a technology-based task “What can you infer from this example?” that addressed the content and pedagogical knowledge of the status of examples in proving of pre-service teachers (PSTs). The task, originally designed and implemented with high-school students, was modified for PSTs and expanded to involve multiple components, including scenarios of non-descript cartoon characters to represent student data. The task was administered through LessonSketch, an online interactive digital platform, to 4 cohorts of PSTs in Israel and the US, across 4 semesters. In this chapter we focus on theoretical and empirical considerations that guided our task design to provide rich learning opportunities for PSTs to enhance their content and pedagogical knowledge of the interplay between examples and proving, and address some of the challenges involved in the task implementation. We discuss the crucial role of technology in supporting PST learning and provide an emergent framework for developing instructional tasks that foster PSTs’ engagement with proving.

Research paper thumbnail of Prospective teachers enacting proof tasks in secondary mathematics classrooms

HAL (Le Centre pour la Communication Scientifique Directe), Feb 6, 2019

We use a curriculum design framework to analyze how prospective secondary teachers (PSTs) designe... more We use a curriculum design framework to analyze how prospective secondary teachers (PSTs) designed and implemented in local schools, lessons that integrate ongoing mathematical topics with one of the four proof themes addressed in the capstone course Mathematical Reasoning and Proving for Secondary Teachers. In this paper we focus on lessons developed around the conditional statements proof theme. We examine the ways in which PSTs integrated conditional statements in their lesson plans, how these lessons were implemented in classrooms, and the challenges PSTs encountered in these processes. Our results suggest that even when PSTs designed rich lesson plans, they often struggled to adjust their language to the students' level and to maintain the cognitive demand of the tasks. We conclude by discussing possible supports for PSTs' learning in these areas.

Research paper thumbnail of Preservice teachers learning to teach proof through classroom implementation: Successes and challenges

The Journal of Mathematical Behavior, Jun 1, 2020

Abstract Proof and reasoning are central to learning mathematics with understanding. Yet proof is... more Abstract Proof and reasoning are central to learning mathematics with understanding. Yet proof is seen as challenging to teach and to learn. In a capstone course for preservice teachers, we developed instructional modules that guided prospective secondary mathematics teachers (PSTs) through a cycle of learning about the logical aspects of proof, then planning and implementing lessons in secondary classrooms that integrate these aspects with traditional mathematics curriculum in the United States. In this paper we highlight our framework on mathematical knowledge for teaching proof and focus on some of the logical aspects of proof that are seen as particularly challenging (four proof themes). We analyze 60 lesson plans, video recordings of a subset of 13 enacted lessons, and the PSTs’ self- reported data to shed light on how the PSTs planned and enacted lessons that integrate these proof themes. The results provide insights into successes and challenges the PSTs encountered in this process and illustrate potential pathways for preparing PSTs to enact reasoning and proof in secondary classrooms. We also highlight the design principles for supporting the development of PSTs’ mathematical knowledge for teaching proof.

Research paper thumbnail of Guided discovery of the nine-point circle theorem and its proof

International Journal of Mathematical Education in Science and Technology, Aug 30, 2017

The main purpose of the paper is to present a new proof of the two celebrated theorems: one is "P... more The main purpose of the paper is to present a new proof of the two celebrated theorems: one is "Ptolemy's Theorem" which explains the relation between the sides and diagonals of a cyclic quadrilateral and another is "Nine Point Circle Theorem" which states that in any arbitrary triangle the three midpoints of the sides, the three feet of altitudes, the three midpoints of line segments formed by joining the vertices and Orthocenter, total nine points are concyclic. Our new proof is based on a metric relation of circumcenter.

Research paper thumbnail of Preparing prospective secondary teachers to teach mathematical reasoning and proof: the case of the role of examples in proving

Zdm – Mathematics Education, May 17, 2023

Mathematics teacher education programs in the United States are charged with preparing prospectiv... more Mathematics teacher education programs in the United States are charged with preparing prospective secondary teachers (PSTs) to teach reasoning and proving across grade levels and mathematical topics. Although most programs require a course on proof, PSTs often perceive it as disconnected from their future classroom practice. Our design research project developed a capstone course Mathematical Reasoning and Proving for Secondary Teachers and systematically studied its effect on PSTs' content and pedagogical knowledge specific to proof. This paper focuses on one course module-Quantification and the Role of Examples in Proving, a topic which poses persistent difficulties to students and teachers alike. The analysis suggests that after the course, PSTs' content and pedagogical knowledge of the role of examples in proving increased. We provide evidence from multiple data sources: pre-and post-questionnaires, PSTs' responses to the in-class activities, their lesson plans, reflections on lesson enactment, and self-report. We discuss design principles that supported PSTs' learning and their applicability beyond the study context.

Research paper thumbnail of Making university mathematics matter for secondary teacher preparation

Zdm – Mathematics Education, May 24, 2023

Internationally, questions about the perceived utility of university mathematics for teaching sch... more Internationally, questions about the perceived utility of university mathematics for teaching school mathematics pose an ongoing challenge for secondary mathematics teacher education. This special issue is dedicated to exploring this topic and related issues in the preparation of secondary mathematics teachers-by which we mean teachers of students with ages, approximately, of 12-18 years. This article introduces this theme and provides a semi-systematic survey of recent related literature, which we use to elaborate and situate important theoretical distinctions around the problems, challenges, and solutions of university mathematics in relation to teacher education. As part of the special issue, we have gathered articles from different countries that elaborate theoretical and empirical approaches, which, collectively, describe different ways to strengthen university mathematics with respect to the aims of secondary teacher education. This survey paper serves to lay out the theoretical groundwork for the collection of articles in the issue.

Research paper thumbnail of Advancing reasoning and proof in secondary mathematics classrooms: instructional modules for supporting teachers

Research paper thumbnail of Constrasting social and sociomathematical norms of two groups of students in a postsecondary Precalculus class

This paper characterizes the engagement of two groups of students in a Precalculus course at a fo... more This paper characterizes the engagement of two groups of students in a Precalculus course at a fouryear public university. A set of "Multiple Solutions Activities" was designed for the course to expose groups of students to alternative solution methods, allowing instructors to explicitly negotiate productive norms to foster students' flexible knowledge. Over the duration of the semester, the groups developed contrasting social and sociomathematical norms. One group's norms seem to be particularly influenced by students' experience taking the same course the prior semester in a more traditional format.

Research paper thumbnail of Mathematical Reasoning and Proving for Prospective Secondary Teachers

The design-based research approach was used to develop and study a novel capstone course: Mathema... more The design-based research approach was used to develop and study a novel capstone course: Mathematical Reasoning and Proving for Secondary Teachers. The course aimed to enhance prospective secondary teachers' (PSTs) content and pedagogical knowledge by emphasizing reasoning and proving as an overarching approach for teaching mathematics at all levels. The course focused on four proof-themes: quantified statements, conditional statements, direct proof and indirect reasoning. The PSTs strengthened their own knowledge of these themes, and then developed and taught in local schools a lesson incorporating the proof-theme within an ongoing mathematical topic. Analysis of the first-year data shows enhancements of PSTs' content and pedagogical knowledge specific to proving.

Research paper thumbnail of Representations of Practice in Teacher Education and Research—Spotlights on Different Approaches

ICME-13 monographs, 2018

Representations of practice provide an opportunity to refer to teachers’ professional environment... more Representations of practice provide an opportunity to refer to teachers’ professional environment both when designing tasks for teacher education or professional development, and when investigating aspects of teacher expertise. This volume amalgamates contributions by the members of the discussion group on representations of practice, which took place during ICME 13. The discussion group sought to collect experiences with different forms of representations of practice in pre-service and in-service teacher professional development settings, and of the use of representations of practice for researching into aspects of teacher expertise and its development. In this introductory chapter we provide an overview of different approaches to representing practice, and address key methodological issues that came up in the monograph’s chapters and in the discussion group’s meetings. We suggest four key questions along which such approaches can be discussed.

Research paper thumbnail of Supporting classroom implementation of proof-oriented tasks: Lessons from teacher researcher collaboration

HAL (Le Centre pour la Communication Scientifique Directe), Feb 1, 2017

This paper reports on a professional development (PD) which aimed to support secondary teachers i... more This paper reports on a professional development (PD) which aimed to support secondary teachers in incorporating argumentation and proof-oriented tasks in their classrooms. The teachers interacted with researcher-developed models of proving tasks in a variety of ways, including modifying the tasks to their classrooms contexts, implementing the tasks, sharing and reflecting on the experiences. In the process of modifying proof-oriented tasks by teachers some of the original researcher-intended goals were lost, while other unexpected affordances emerged. This raises important questions regarding modes of teacher-researcher collaborations around proof-oriented classroom interventions, and their potential effectiveness.

Research paper thumbnail of Instructional Technology Tetrahedron and Network Visualization: Conceptualizing Online Teaching Through a Lens of Reflective Noticing

Digital Experiences in Mathematics Education, 2024

Online learning and teaching, accelerated by the global pandemic and rapid advancement of digital... more Online learning and teaching, accelerated by the global pandemic and rapid
advancement of digital technology, require novel conceptual and analytical tools to understand better the evolving nature of online teaching. Drawing on the classical model of the instructional triangle and previous attempts to extend it, we propose the Instructional Technology Tetrahedron (ITT)—a conceptual framework that integrates technology into the instructional triangle to represent the role of technology, as a learning tool and a mediator between teachers, students, and content. Combining the ITT framework with network visualization strategies allowed for representing the intensity of interactions within the tetrahedron. We illustrate the affordances of the ITT framework by analyzing reflective noticing patterns of three prospective secondary teachers (PSTs) who reflected on the video recordings of their own online teaching, with each PST teaching four online lessons to groups of high-school students. We demonstrate the utility of the ITT framework to characterize individual noticing patterns, in a particular lesson and across time, and to support a variety of cross-case comparisons. The discussion sheds light on the broader implications of
the ITT framework.

Research paper thumbnail of Guiding principles for teaching mathematics via reasoning and proving

HAL (Le Centre pour la Communication Scientifique Directe), Feb 2, 2022

Against the backdrop of policy documents and educational researchers' vision of proof as an essen... more Against the backdrop of policy documents and educational researchers' vision of proof as an essential component of teaching mathematics across content areas and grade levels, teaching of reasoning and proof in mathematics classrooms remains an elusive goal. Teachers and the type of teaching they enact in classrooms are crucial for achieving this goal. This theoretical paper builds on the concept of proof-based teaching and suggest a set of guiding principles for what we call teaching mathematics via reasoning and proving. These principles were developed as a part of a multi-year design based project, and implemented in an undergraduate course Mathematical Reasoning and Proving for Secondary Teachers. We illustrate these principles using examples from proof-oriented lessons plan developed by prospective secondary teachers.

Research paper thumbnail of Examining the role of examples in proving processes through a cognitive lens: the case of triangular numbers

Zdm – Mathematics Education, Feb 18, 2011

In this paper, we analyze the role of examples in the proving process. The context chosen for thi... more In this paper, we analyze the role of examples in the proving process. The context chosen for this study was finding a general rule for triangular numbers. The aim of this paper is to show that examples are effective for the construction of a proof when they allow cognitive unity and structural continuity between argumentation and proof. Continuity in the structure is possible if the inductive argumentation is based on process pattern generalization (PPG), but this is not the case if a generalization is made on the results. Moreover, the PPG favors the development of generic examples that support cognitive unity and structural continuity between the argumentation and proof. The cognitive analysis presented in this paper is performed through Toulmin's model.

Research paper thumbnail of Erratum to: What Can You Infer from This Example? Applications of Online, Rich-Media Tasks for Enhancing Pre-service Teachers’ Knowledge of the Roles of Examples in Proving

Mathematics education in the digital era, 2017

In the originally published version of the chapter, Figure 4 in Chapter "What Can You Infer from ... more In the originally published version of the chapter, Figure 4 in Chapter "What Can You Infer from This Example? Applications of Online, Rich-Media Tasks for Enhancing Pre-service Teachers' Knowledge of the Roles of Examples in Proving" was portrait oriented and downsized, hence the text was virtually unreadable. The erratum of the book has been updated with the change.

Research paper thumbnail of On Convincing Power of Counterexamples

The Mathematics Enthusiast, Feb 1, 2024

Despite plethora of research that attends to the convincing power of different types of proofs, r... more Despite plethora of research that attends to the convincing power of different types of proofs, research related to the convincing power of counterexamples is rather slim. In this paper we examine how prospective and practicing secondary school mathematics teachers respond to different types of counterexamples. The counterexamples were presented as products of students' arguments, and the participants were asked to evaluate their correctness and comment on them. The counterexamples varied according to mathematical topic: algebra or geometry, and their explicitness. However, as we analyzed the data, we discovered that these distinctions were insufficient to explain why teachers accepted some counterexamples, but rejected others, with seemingly similar features. As we analyze the participants' perceived transparency of different counterexamples, we employ various theoretical approaches that can advance our understanding of teachers' conceptions of conviction with respect to counterexamples.

Research paper thumbnail of Supporting prospective secondary mathematics teachers in creating instructional explanations through video based experience

Research paper thumbnail of Systematic Exploration of Examples as Proof: Analysis with Four Theoretical Frameworks

ICME-13 monographs, 2018

This chapter offers a multi-layered analysis of one specific category of students’ example-based ... more This chapter offers a multi-layered analysis of one specific category of students’ example-based reasoning , which has received little attention in research literature so far: systematic exploration of examples. It involves dividing a conjecture’s domain into disjoint sub-domains and testing a single example in each sub-domain. I apply four theoretical frameworks to analyze student data: The Mathematical-logical framework for the interplay between examples and proof, Proof schemes framework, Transfer-in-pieces framework, and the Theory of instructional situations . Taken together, these frameworks allow to examine the data from mathematical, cognitive and social perspectives, thus broadening and deepening the insights into students thinking about the relationship between examples and proving. Implications for teaching and learning of proof in school mathematics are discussed.

Research paper thumbnail of Pre-service teachers' construction of algebraic proof through exploration of math-tricks

This paper contributes to the ongoing effort to create rich learning opportunities for prospectiv... more This paper contributes to the ongoing effort to create rich learning opportunities for prospective teachers to engage with reasoning and proving. Twenty elementary and middle school pre-service teachers completed individual projects in which they explored "math-tricks"unconventional computational algorithms-as a part of an undergraduate proof course. Our findings suggest that the task evoked uncertainty with respect to why the tricks work and motivation to resolve the uncertainty by means of algebraic proof. We discuss the potential of this task to create rich opportunities for prospective teachers to conduct explorations, construct algebraic proofs and reflect on their experience from learner's perspective.

Research paper thumbnail of Who is Right? – What Students’ and Prospective Teachers’ Responses to Scripted Dialog Reveal About Their Conceptions of Proof

Advances in mathematics education, Oct 31, 2017

This chapter explores the potential of using scripted student responses, embedded in a task title... more This chapter explores the potential of using scripted student responses, embedded in a task titled Who is right?, as a tool to diagnose argumentation and proof-related conceptions of high-school students and pre-service mathematics teachers (PSTs). The data, collected in two separate studies, were examined for evidence of participants’ conceptions of the role of examples in proving and refuting universal statements. Additional analysis explored what types of criteria are used by the high-school students and the PSTs when evaluating scripted arguments, as well as whether participants were consistent in their evaluations across the collection of arguments. The data revealed that, when evaluating scripted arguments, high-school students used mainly mathematical criteria and strived to maintain consistency in their evaluations across the collection of arguments. On the contrary, PSTs applied both mathematical and pedagogical considerations in their evaluations, thus judging multiple, and even contradictory arguments as correct.

Research paper thumbnail of What Can You Infer from This Example? Applications of Online, Rich-Media Tasks for Enhancing Pre-service Teachers’ Knowledge of the Roles of Examples in Proving

Mathematics education in the digital era, Oct 13, 2016

There is a consensus among mathematics educators that in order to provide students with rich lear... more There is a consensus among mathematics educators that in order to provide students with rich learning opportunities to engage with reasoning and proving, prospective teachers must develop a strong knowledge base of mathematics, pedagogy and student epistemology. In this chapter we report on the design of a technology-based task “What can you infer from this example?” that addressed the content and pedagogical knowledge of the status of examples in proving of pre-service teachers (PSTs). The task, originally designed and implemented with high-school students, was modified for PSTs and expanded to involve multiple components, including scenarios of non-descript cartoon characters to represent student data. The task was administered through LessonSketch, an online interactive digital platform, to 4 cohorts of PSTs in Israel and the US, across 4 semesters. In this chapter we focus on theoretical and empirical considerations that guided our task design to provide rich learning opportunities for PSTs to enhance their content and pedagogical knowledge of the interplay between examples and proving, and address some of the challenges involved in the task implementation. We discuss the crucial role of technology in supporting PST learning and provide an emergent framework for developing instructional tasks that foster PSTs’ engagement with proving.

Research paper thumbnail of Prospective teachers enacting proof tasks in secondary mathematics classrooms

HAL (Le Centre pour la Communication Scientifique Directe), Feb 6, 2019

We use a curriculum design framework to analyze how prospective secondary teachers (PSTs) designe... more We use a curriculum design framework to analyze how prospective secondary teachers (PSTs) designed and implemented in local schools, lessons that integrate ongoing mathematical topics with one of the four proof themes addressed in the capstone course Mathematical Reasoning and Proving for Secondary Teachers. In this paper we focus on lessons developed around the conditional statements proof theme. We examine the ways in which PSTs integrated conditional statements in their lesson plans, how these lessons were implemented in classrooms, and the challenges PSTs encountered in these processes. Our results suggest that even when PSTs designed rich lesson plans, they often struggled to adjust their language to the students' level and to maintain the cognitive demand of the tasks. We conclude by discussing possible supports for PSTs' learning in these areas.

Research paper thumbnail of Preservice teachers learning to teach proof through classroom implementation: Successes and challenges

The Journal of Mathematical Behavior, Jun 1, 2020

Abstract Proof and reasoning are central to learning mathematics with understanding. Yet proof is... more Abstract Proof and reasoning are central to learning mathematics with understanding. Yet proof is seen as challenging to teach and to learn. In a capstone course for preservice teachers, we developed instructional modules that guided prospective secondary mathematics teachers (PSTs) through a cycle of learning about the logical aspects of proof, then planning and implementing lessons in secondary classrooms that integrate these aspects with traditional mathematics curriculum in the United States. In this paper we highlight our framework on mathematical knowledge for teaching proof and focus on some of the logical aspects of proof that are seen as particularly challenging (four proof themes). We analyze 60 lesson plans, video recordings of a subset of 13 enacted lessons, and the PSTs’ self- reported data to shed light on how the PSTs planned and enacted lessons that integrate these proof themes. The results provide insights into successes and challenges the PSTs encountered in this process and illustrate potential pathways for preparing PSTs to enact reasoning and proof in secondary classrooms. We also highlight the design principles for supporting the development of PSTs’ mathematical knowledge for teaching proof.

Research paper thumbnail of Guided discovery of the nine-point circle theorem and its proof

International Journal of Mathematical Education in Science and Technology, Aug 30, 2017

The main purpose of the paper is to present a new proof of the two celebrated theorems: one is "P... more The main purpose of the paper is to present a new proof of the two celebrated theorems: one is "Ptolemy's Theorem" which explains the relation between the sides and diagonals of a cyclic quadrilateral and another is "Nine Point Circle Theorem" which states that in any arbitrary triangle the three midpoints of the sides, the three feet of altitudes, the three midpoints of line segments formed by joining the vertices and Orthocenter, total nine points are concyclic. Our new proof is based on a metric relation of circumcenter.

Research paper thumbnail of Preparing prospective secondary teachers to teach mathematical reasoning and proof: the case of the role of examples in proving

Zdm – Mathematics Education, May 17, 2023

Mathematics teacher education programs in the United States are charged with preparing prospectiv... more Mathematics teacher education programs in the United States are charged with preparing prospective secondary teachers (PSTs) to teach reasoning and proving across grade levels and mathematical topics. Although most programs require a course on proof, PSTs often perceive it as disconnected from their future classroom practice. Our design research project developed a capstone course Mathematical Reasoning and Proving for Secondary Teachers and systematically studied its effect on PSTs' content and pedagogical knowledge specific to proof. This paper focuses on one course module-Quantification and the Role of Examples in Proving, a topic which poses persistent difficulties to students and teachers alike. The analysis suggests that after the course, PSTs' content and pedagogical knowledge of the role of examples in proving increased. We provide evidence from multiple data sources: pre-and post-questionnaires, PSTs' responses to the in-class activities, their lesson plans, reflections on lesson enactment, and self-report. We discuss design principles that supported PSTs' learning and their applicability beyond the study context.

Research paper thumbnail of Making university mathematics matter for secondary teacher preparation

Zdm – Mathematics Education, May 24, 2023

Internationally, questions about the perceived utility of university mathematics for teaching sch... more Internationally, questions about the perceived utility of university mathematics for teaching school mathematics pose an ongoing challenge for secondary mathematics teacher education. This special issue is dedicated to exploring this topic and related issues in the preparation of secondary mathematics teachers-by which we mean teachers of students with ages, approximately, of 12-18 years. This article introduces this theme and provides a semi-systematic survey of recent related literature, which we use to elaborate and situate important theoretical distinctions around the problems, challenges, and solutions of university mathematics in relation to teacher education. As part of the special issue, we have gathered articles from different countries that elaborate theoretical and empirical approaches, which, collectively, describe different ways to strengthen university mathematics with respect to the aims of secondary teacher education. This survey paper serves to lay out the theoretical groundwork for the collection of articles in the issue.

Research paper thumbnail of Advancing reasoning and proof in secondary mathematics classrooms: instructional modules for supporting teachers

Research paper thumbnail of Constrasting social and sociomathematical norms of two groups of students in a postsecondary Precalculus class

This paper characterizes the engagement of two groups of students in a Precalculus course at a fo... more This paper characterizes the engagement of two groups of students in a Precalculus course at a fouryear public university. A set of "Multiple Solutions Activities" was designed for the course to expose groups of students to alternative solution methods, allowing instructors to explicitly negotiate productive norms to foster students' flexible knowledge. Over the duration of the semester, the groups developed contrasting social and sociomathematical norms. One group's norms seem to be particularly influenced by students' experience taking the same course the prior semester in a more traditional format.

Research paper thumbnail of Mathematical Reasoning and Proving for Prospective Secondary Teachers

The design-based research approach was used to develop and study a novel capstone course: Mathema... more The design-based research approach was used to develop and study a novel capstone course: Mathematical Reasoning and Proving for Secondary Teachers. The course aimed to enhance prospective secondary teachers' (PSTs) content and pedagogical knowledge by emphasizing reasoning and proving as an overarching approach for teaching mathematics at all levels. The course focused on four proof-themes: quantified statements, conditional statements, direct proof and indirect reasoning. The PSTs strengthened their own knowledge of these themes, and then developed and taught in local schools a lesson incorporating the proof-theme within an ongoing mathematical topic. Analysis of the first-year data shows enhancements of PSTs' content and pedagogical knowledge specific to proving.

Research paper thumbnail of Representations of Practice in Teacher Education and Research—Spotlights on Different Approaches

ICME-13 monographs, 2018

Representations of practice provide an opportunity to refer to teachers’ professional environment... more Representations of practice provide an opportunity to refer to teachers’ professional environment both when designing tasks for teacher education or professional development, and when investigating aspects of teacher expertise. This volume amalgamates contributions by the members of the discussion group on representations of practice, which took place during ICME 13. The discussion group sought to collect experiences with different forms of representations of practice in pre-service and in-service teacher professional development settings, and of the use of representations of practice for researching into aspects of teacher expertise and its development. In this introductory chapter we provide an overview of different approaches to representing practice, and address key methodological issues that came up in the monograph’s chapters and in the discussion group’s meetings. We suggest four key questions along which such approaches can be discussed.

Research paper thumbnail of Supporting classroom implementation of proof-oriented tasks: Lessons from teacher researcher collaboration

HAL (Le Centre pour la Communication Scientifique Directe), Feb 1, 2017

This paper reports on a professional development (PD) which aimed to support secondary teachers i... more This paper reports on a professional development (PD) which aimed to support secondary teachers in incorporating argumentation and proof-oriented tasks in their classrooms. The teachers interacted with researcher-developed models of proving tasks in a variety of ways, including modifying the tasks to their classrooms contexts, implementing the tasks, sharing and reflecting on the experiences. In the process of modifying proof-oriented tasks by teachers some of the original researcher-intended goals were lost, while other unexpected affordances emerged. This raises important questions regarding modes of teacher-researcher collaborations around proof-oriented classroom interventions, and their potential effectiveness.