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Sriraman, B (Ed.)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.more » « lessFree, publiclyaccessible full text available February 1, 2025

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 fulltime interns in secondary schools. The culminating part of the internship was an action research / inquiry project devoted to reasoning and proving. This exploratory multicase 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.more » « lessFree, publiclyaccessible full text available January 1, 2025

Ayalon, M ; Koichu, B ; Leikin, R ; Rubel, L ; Tabach, M (Ed.)This study presents how the commognitivebased Opportunities for Reasoning and Proving (ORP) Framework, developed for research purposes to analyze mathematical tasks, was applied as a learning tool for teachers. Seven novice secondary teachers, who participated in a professional learning community around integrating reasoning and proving, were introduced to the ORP Framework and engaged in a sorting tasks activity. We show how the ORP Framework helped teachers to focus on the ORP embedded in tasks, to attend to student mathematical work, and to communicate about ORP coherently and unambiguously. We discuss the affordances of using a framework, which relies on the operationalized discursive language of commognition, to promote teachers’ communication around reasoning and proving.more » « less

Ayalon, M ; Koichu, B ; Leikin, R ; Rubel, L ; Tabach, M (Ed.)We follow a beginning mathematics teacher, Olive, from the universitybased course Mathematical Reasoning and Proving for Secondary Teachers through the supervised internship where Olive taught in her cooperating teacher’s classroom. By drawing upon Activity Theory, we compare her teaching within the two teaching settings, and we examine the opportunities for reasoning and proving she provided to her students in each teaching setting. As a prospective teacher, Olive provided her students opportunities for reasoning and proving. During the internship, these opportunities initially diminished due to institutional and contextual constraints. However, Olive gradually carved out unique paths to engage students with reasoning and proving as her teaching independence increased.more » « less

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 longterm development of proofrelated 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 proofrelated teaching practices of beginning secondary mathematics teachers. In particular, this study aims to understand the observed gap between proofrelated 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 highschool 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 proofrelated 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 prooforiented 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 proofrelated 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 proofrelated teaching practices.more » « less

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 schoolbased experience in which the PSTs designed and taught four prooforiented 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 schoolbased experience moved online. We analyzed how Nancy’s Mathematical Knowledge for Teaching Proof (MKTP) evolved through her attempts to teach proof online and through repeated cycles of reflection.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

Lischka, A. E. ; Dyer, E. B. ; Jones, R. S. ; Lovett, J. N. ; Strayer, J. ; Drown, S. (Ed.)In this paper, we offer a novel framework for analyzing the Opportunities for Reasoningand Proving (ORP) in mathematical tasks. By drawing upon some tenets of the commognitive framework, we conceptualize learning and teaching mathematics via reasoning and proving both as enacting reasoning processes (e.g., conjecturing, justifying) in the curricularbased mathematical discourse and as participation in the metadiscourse about proof, which is focused on the aspects of deductive reasoning. By cluster analysis performed on 106 tasks designed by prospective secondary teachers, we identify four types of tasks corresponding to four types of ORP: limited ORP, curricularbased reasoning ORP, logic related ORP, and fully integrated ORP. We discuss these ORP and the contribution of this framework in light of preparing beginning teachers to integrate reasoning and proving in secondary mathematics classrooms.more » « less

Lischka, A. E. ; Dyer, E. B. ; Jones, R. S. ; Lovett, J. N. ; Strayer, J. ; Drown, S. (Ed.)The rapid move to online teaching brought about by the global pandemic highlighted the need for the educational research community to develop new conceptual tools for characterizing these environments. In this paper, we propose a conceptual framework Instructional Technology Triangle (ITT) which extends the instructional triangle of teachers, students, and content to include technology as a mediating mechanism. We use the ITT framework to analyze noticing patterns in the written reflection of a prospective secondary teacher, Nancy, who, over the course of one semester taught online four lessons integrating reasoning and proof . The fluctuations in Nancy’s noticing patterns, in particular, with respect to technology, shed light on her trajectory of learning to teach online and the role of reflective noticing in this process. We discuss implications for teacher preparation and professional development.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