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1. 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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2. Teacher learning to teach mathematics via reasoning and proving: a discursive analysis of lesson plans modifications

   https://doi.org/10.3389/feduc.2023.1154531  

   Weingarden, Merav; Buchbinder, Orly (August 2023, Frontiers in Education)

   Christiansen, I (Ed.)

   Despite the importance of reasoning and proving in mathematics and mathematics education, little is known about how future teachers become proficient in integrating reasoning and proving in their teaching practices. In this article, we characterize this aspect of prospective secondary mathematics teachers’ (PSTs’) professional learning by drawing upon the commognitive theory. We offer a triple-layer conceptualization of (student)learning,teaching, andlearning to teachmathematics via reasoning and proving by focusing on the discourses students participate in (learning), the opportunities for reasoning and proving afforded to them (teaching), and how PSTs design and enrich such opportunities (learning to teach). We explore PSTs’ pedagogical discourse anchored in the lesson plans they designed, enacted, and modified as part of their participation in a university-based course:Mathematical Reasoning and Proving for Secondary Teachers. We identified four types of discursive modifications: structural, mathematical, reasoning-based, and logic-based. We describe how the potential opportunities for reasoning and proving afforded to students by these lesson plans changed as a result of these modifications. Based on our triple-layered conceptualization we illustrate how the lesson modifications and the resulting alterations to student learning opportunities can be used to characterize PSTs’ professional learning. We discuss the affordances of theorizing teacher practices with the same theoretical lens (grounded in commognition) to inquire student learning and teacher learning, and how lesson plans, as a proxy of teaching practices, can be used as a methodological tool to better understand PSTs’ professional learning. 

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3. Beginning teacher’s trajectory of identity formation in the figured worlds of reform and traditional instruction

   Ben-Dor, Naama; Buchbinder, Orly (January 2025, Proceedings of the 46th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA))

   Kosko, K; Caniglia, J; Courtney, S; Zolfaghari, M; Morris, G (Ed.)

   Transitioning from a teacher education program to autonomous teaching is a complex process, fraught with challenges. This transition involves developing identities and teaching practices that allow novice teachers to reconcile the reformed teaching world of their teacher preparation program with the more traditional world of school teaching. In this paper, we follow the identity formation of one beginning teacher, Olive, by examining her narratives about her pedagogical actions as she transitions from being a pre-service teacher (PST) to being an intern (INT) to becoming a new teacher (NT). As PST, Olive’s narratives about her current and desired actions aligned with reform actions; as INT, a gap opened between her current traditional actions and desired reform actions; and as NT, the gap narrowed as she modified her desired narratives to more traditional ones. We discuss our findings and their scientific significance. 

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4. How novice teachers recontextualize the teaching of mathematics via reasoning proving – a dual case study

   Buchbinder, O (January 2024, 13th Congress of the European Society for Research in Mathematics Education)

   Drijvers, P; Csapodi, C; Palmér, H; Gosztonyi, K; Kónya, E (Ed.)

   This study is part of a larger project exploring how beginning teachers learn to teach mathematics via reasoning and proving. The study followed two beginning secondary mathematics teachers for two years. First, as students in a capstone course in which they learned to integrate reasoning and proving into teaching mathematics, and then as full-time interns in secondary schools. The culminating part of the internship was an action research / inquiry project devoted to reasoning and proving. This exploratory multi-case study examined how conducting such an inquiry project affected interns’ discourses and practices for teaching mathematics via reasoning and proving. The results show that both beginning teachers successfully recontextualized what they learned in the capstone course in their inquiry projects. Yet, there were substantial differences between the two interns, which affected their conclusions about continuing integrating reasoning and proving in their classrooms. 

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5. 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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