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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. 

Building on the theory of practical rationality, we explore how three beginning secondary mathematics teachers reconcile competing professional obligations, namely: disciplinary, individual, and institutional obligations. As these teachers transitioned from supervised teaching to teaching their own classrooms, they reconciled competing obligations and developed their own ideas about mathematics teaching and learning. The analysis revealed that it was only institutional obligation that conflicted with either disciplinary, or individual obligation, or with teachers’ own teaching preferences. No other two obligations appeared to clash. The conflict with institutional obligation was reconciled in favor of institutional obligation in less than 30% of instances. In the vast majority of cases, another obligation took precedence. 

Online learning and teaching, accelerated by the global pandemic and rapid advancement of digital technology, require novel conceptual and analytical tools to understand better the evolving nature of online teaching. Drawing on the classical model of the instructional triangle and previous attempts to extend it, we propose theInstructional Technology Tetrahedron(ITT)—a conceptual framework that integrates technology into the instructional triangle to represent the role of technology, as a learning tool and a mediator between teachers, students, and content. Combining the ITT framework with network visualization strategies allowed for representing the intensity of interactions within the tetrahedron. We illustrate the affordances of the ITT framework by analyzing reflective noticing patterns of three prospective secondary teachers (PSTs) who reflected on the video recordings of their own online teaching, with each PST teaching four online lessons to groups of high-school students. We demonstrate the utility of the ITT framework to characterize individual noticing patterns, in a particular lesson and across time, and to support a variety of cross-case comparisons. The discussion sheds light on the broader implications of the ITT framework. 

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. 

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. 

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. 

This study presents how the commognitive-based 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. 

We follow a beginning mathematics teacher, Olive, from the university-based 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. 

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. 

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.