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1. Opportunities to Engage Secondary Students in Proof Generated by Pre-service Teachers

   Buchbinder, O ; McCrone, S. ( May 2019 , Research in Undergraduate Mathematics Education)

   For reasoning and proving to become a reality in mathematics classrooms, pre-service teachers (PSTs) must develop knowledge and skills for creating lessons that engage students in proof-related 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 proof-related tasks. 

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2. Preparing prospective secondary teachers to teach mathematical reasoning and proof: the case of the role of examples in proving

   https://doi.org/10.1007/s11858-023-01493-4  

   Buchbinder, Orly ; McCrone, Sharon ( May 2023 , ZDM – Mathematics Education)

   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 courseMathematical Reasoning and Proving for Secondary Teachersand 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.

    

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3. Taking proof into secondary classrooms – supporting future mathematics teachers

   Buchbinder, O ; McCrone, S. ( November 2018 , Proceedings of the 40th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA))

   For reasoning and proof to become a reality in mathematics classrooms, it is important to prepare teachers who have knowledge and skills to integrate reasoning and proving in their teaching. Aiming to enhance prospective secondary teachers’ (PSTs) content and pedagogical knowledge related to proof, we designed and studied a capstone course Mathematical Reasoning and Proving for Secondary Teachers. This paper describes the structure of the course and illustrates how PSTs’ interacted with its different components. The PSTs first strengthened their content knowledge, then developed and taught in local schools a lesson incorporating proof components. Initial data analyses show gains in PSTs’ knowledge for teaching proof and dispositions towards proving, following their participation in the course. 

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4. Prospective teachers enacting proof tasks in secondary mathematics classrooms.

   Buchbinder, O. ; McCrone, S. ( January 2019 , Congress of the European Society for Research in Mathematics Education)

   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. 

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5. Preservice teachers learning to teach proof through classroom implementation: Successes and challenges

   Buchbinder, O. ; McCrone, S. ( June 2020 , The Journal of mathematical behavior)

   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. 

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