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
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Abstract 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: preand postquestionnaires, PSTs’ responses to the inclass activities, their lesson plans, reflections on lesson enactment, and selfreport. We discuss design principles that supported PSTs’ learning and their applicability beyond the study context. 
Karunakaran, S. ; Higgins, A. (Ed.)Mathematical Knowledge for Teaching Proof (MKTP) has been recognized as an important component of fostering student engagement with mathematical reasoning and proof. This study is one component of a larger study aimed at exploring the nature of MKTP. The present study examines qualitative differences in feedback given by STEM majors, inservice and preservice secondary mathematics teachers on hypothetical students’ arguments. The results explicate key distinctions in the feedback provided by these groups, indicating that this is a learnable skill. Feedback is cast as a component of MKTP, making the results of this study significant empirical support for the construct of MKTP as a type of knowledge that is unique to teachers.more » « less

Karunakaran, S. ; Higgins, A. (Ed.)Mathematical Knowledge for Teaching Proof (MKTP) has been recognized as an important component of fostering student engagement with mathematical reasoning and proof. This study is one component of a larger study aimed at exploring the nature of MKTP. The present study examines qualitative differences in feedback given by STEM majors, inservice and preservice secondary mathematics teachers on hypothetical students’ arguments. The results explicate key distinctions in the feedback provided by these groups, indicating that this is a learnable skill. Feedback is cast as a component of MKTP, making the results of this study significant empirical support for the construct of MKTP as a type of knowledge that is unique to teachers.more » « less

Olanoff, D. ; Johnson, K. ; Spitzer, S. (Ed.)It has been suggested that integrating reasoning and proof in mathematics teaching requires a special type of teacher knowledge  Mathematical Knowledge for Teaching Proof (MKTP). Yet, several important questions about the nature of MKTP remain open, specifically, whether MKTP is a type of knowledge specific to teachers, and whether MKTP can be improved through intervention. We explored these questions by comparing performance on an MKTP questionnaire of inservice secondary mathematics teachers, undergraduate STEM majors, and preservice secondary mathematics teachers. The latter group completed the questionnaire twice before and after participating in a capstone course, Mathematical Reasoning and Proving for Secondary Teachers. Our data suggest that MKTP is indeed a special kind of knowledge specific to teachers and it can be improved through interventions.more » « less

The benefits of using video in teacher education as a tool for reflection and for developing professional expertise have long been recognized. Recent introduction of 360 video technology holds promise to extend these benefits as it allows prospective teachers to reflect on their own performance by considering the classroom from multiple perspectives. This study examined nine prospective secondary teachers’ (PSTs) noticing and selfreflection on the 360 recordings of their own teaching. The PSTs, enrolled in a capstone course Mathematical Reasoning and Proving for Secondary Teachers, taught a prooforiented lesson to small groups of students in local schools while capturing their teaching with 360 video cameras. We analyzed the PSTs’ written comments on their video and reflection reports to identify the categories of noticing afforded by the 360 technology as well as the instances of PSTs’ learning. The results point to the powerful potential of 360 videos for supporting PSTs’ selfreflection and professional growth.more » « less

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.more » « less

Karunakaran, S. S. ; Reed, Z. ; Higgins, A. (Ed.)Future mathematics teachers must be able to interpret a wide range of mathematical statements, in particular conditional statements. Literature shows that even when students are familiar with conditional statements and equivalence to the contrapositive, identifying other equivalent and nonequivalent forms can be challenging. As a part of a larger grant to enhance and study prospective secondary teachers’ (PSTs’) mathematical knowledge for teaching proof, we analyzed data from 26 PSTs working on a task that required rewriting a conditional statement in several different forms and then determining those that were equivalent to the original statement. We identified three key strategies used to make sense of the various forms of conditional statements and to identify equivalent and nonequivalent forms: meaning making, comparing truthvalues and comparing to known syntactic forms. The PSTs relied both on semantic meaning of the statements and on their formal logical knowledge to make their judgments.more » « less

We describe an instructional module aimed to enhance prospective secondary teachers’ (PSTs’) subject matter knowledge of indirect reasoning. We focus on one activity in which PSTs had to compare and contrast proof by contradiction and proof by contrapositive. These types of proofs have been shown to be challenging to students at all levels and teachers alike, yet there has been little research on how to support learners in developing this knowledge. Data analysis of 11 PSTs, points to learning opportunities afforded by the module and the PSTs’ challenges with indirect reasoning.more » « less

For reasoning and proving to become a reality in mathematics classrooms, preservice teachers (PSTs) must develop knowledge and skills for creating lessons that engage students in proofrelated activities. Supporting PSTs in this process was among the goals of a capstone course: Mathematical Reasoning and Proving for Secondary Teachers. During the course, the PSTs designed and implemented in local schools four lessons that integrated within the regular secondary curriculum one of the four proof themes discussed in the course: quantification and the role of examples in proving, conditional statements, direct proof and argument evaluation, and indirect reasoning. In this paper we report on the analysis of 60 PSTs’ lesson plans in terms of opportunities for students to learn about the proof themes, pedagogical features of the lessons and cognitive demand of the proofrelated tasks.more » « less

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.more » « less