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1. 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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2. A Sociocultural Perspective on Beginning Teachers Enacting Reasoning and Proving Practices

   Weingarden, M. ; Buchbinder, O. ( January 2022 , Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)

   Karunakaran, S. S. ; Higgins, A. (Ed.)

   The critical role of teachers in supporting student engagement with reasoning and proving has long been recognized (Nardi & Knuth, 2017; NCTM, 2014). While some studies examined how prospective secondary teachers (PSTs) develop dispositions and teaching practices that promote student engagement with reasoning and proving (e.g., Buchbinder & McCrone, 2020; Conner, 2007), very little is known about long-term development of proof-related practices of beginning teachers and what factors affect this development (Stylianides et al., 2017). During the supervised teaching experiences, interns often encounter tensions between balancing their commitments to the university and cooperating teacher, while also developing their own teaching styles (Bieda et al., 2015; Smagorinsky et al., 2004; Wang et al., 2008). Our study examines how sociocultural contexts of the teacher preparation program and of the internship school, supported or inhibited proof-related teaching practices of beginning secondary mathematics teachers. In particular, this study aims to understand the observed gap between proof-related teaching practices of one such teacher, Olive, in two settings: as a PST in a capstone course Mathematical Reasoning and Proving for Secondary Teachers (Buchbinder & McCrone, 2020) and as an intern in a high-school classroom. We utilize activity theory (Leont’ev, 1979) and Engeström’s (1987) model of an activity system to examine how the various components of the system: teacher (subject), teaching (object), the tasks (tools), the curriculum and the expected teaching style (rules), the cooperating teacher (community) and their involvement during the teaching (division of labor) interact with each other and affect the opportunities provided to students to engage with reasoning and proving (outcome). The analysis of four lessons from each setting, lesson plans, reflections and interviews, showed that as a PST, Olive engaged students with reasoning and proving through productive proof-related teaching practices and rich tasks that involved conjecturing, justifying, proving and evaluating arguments. In a sharp contrast, as an intern, Olive had to follow her school’s rigid curriculum and expectations, and to adhere to her cooperating teacher’s teaching style. As a result, in her lessons as an intern students received limited opportunities for reasoning and proving. Olive expressed dissatisfaction with this type of teaching and her desire to enact more proof-oriented practices. Our results show that the sociocultural components of the activity system (rules, community and division of labor), which were backgrounded in Olive’s teaching experience as a PST but prominent in her internship experience, influenced the outcome of engaging students with reasoning and proving. We discuss the importance of these sociocultural aspects as we examine how Olive navigated the tensions between the proof-related teaching practices she adopted in the capstone course and her teaching style during the internship. We highlight the importance of teacher educators considering the sociocultural aspects of teaching in supporting beginning teachers developing proof-related teaching practices. 

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3. Preservice Secondary Mathematics Teachers’ Reflective Noticing from 360-degree Video Recordings of Their Own Teaching.

   Buchbinder, O. ; Brisard, S. ; Butler, R. ; McCrone, S. ( January 2021 , Journal of technology and teacher education)

   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 self-reflection on the 360 recordings of their own teaching. The PSTs, enrolled in a capstone course Mathematical Reasoning and Proving for Secondary Teachers, taught a proof-oriented 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’ self-reflection and professional growth. 

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4. Learning to Teach Reasoning and Proof in an Online Setting: The Case of Nancy.

   Liu, J. ; Buchbinder, O. ( January 2022 , Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)

   Karunakaran, S. S. ; Higgins, A. (Ed.)

   Preparing prospective secondary teachers (PSTs) to teach mathematics with a focus on reasoning and proving is an important goal for teacher education programs. A capstone course, Mathematical Reasoning and Proving for Secondary Teachers, was designed to address this goal. One component of the course was a school-based experience in which the PSTs designed and taught four proof-oriented lessons in local schools, video recorded these lessons, and reflected on them. In this paper, we focus on one PST – Nancy, who took the course in Fall 2020 during the pandemic, when the school-based experience moved online. We analyzed how Nancy’s Mathematical Knowledge for Teaching Proof (MKT-P) evolved through her attempts to teach proof online and through repeated cycles of reflection. 

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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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