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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. 

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. 

Mathematical Knowledge for Teaching Proof (MKT-P) 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 MKT-P. The present study examines qualitative differences in feedback given by STEM majors, in-service and pre-service 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 MKT-P, making the results of this study significant empirical support for the construct of MKT-P as a type of knowledge that is unique to teachers. 

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. 

In this paper, we offer a novel framework for analyzing the Opportunities for Reasoning-and 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 curricular-based mathematical discourse and as participation in the meta-discourse 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, curricular-based 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. 

Mathematical Knowledge for Teaching Proof (MKT-P) 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 MKT-P. The present study examines qualitative differences in feedback given by STEM majors, in-service and pre-service 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 MKT-P, making the results of this study significant empirical support for the construct of MKT-P as a type of knowledge that is unique to teachers. 

It has been suggested that integrating reasoning and proof in mathematics teaching requires a special type of teacher knowledge - Mathematical Knowledge for Teaching Proof (MKT-P). Yet, several important questions about the nature of MKT-P remain open, specifically, whether MKT-P is a type of knowledge specific to teachers, and whether MKT-P can be improved through intervention. We explored these questions by comparing performance on an MKT-P questionnaire of in-service secondary mathematics teachers, undergraduate STEM majors, and pre-service secondary mathematics teachers. The latter group completed the questionnaire twice- before and after participating in a capstone course, Mathematical Reasoning and Proving for Secondary Teachers. Our data suggest that MKT-P is indeed a special kind of knowledge specific to teachers and it can be improved through interventions. 

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. 

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. 

Future mathematics teachers must be able to interpret a wide range of mathematical statements, in particular conditional statements. Literature shows that even when students are familiar with conditional statements and equivalence to the contrapositive, identifying other equivalent and non-equivalent forms can be challenging. As a part of a larger grant to enhance and study prospective secondary teachers’ (PSTs’) mathematical knowledge for teaching proof, we analyzed data from 26 PSTs working on a task that required rewriting a conditional statement in several different forms and then determining those that were equivalent to the original statement. We identified three key strategies used to make sense of the various forms of conditional statements and to identify equivalent and non-equivalent forms: meaning making, comparing truth-values and comparing to known syntactic forms. The PSTs relied both on semantic meaning of the statements and on their formal logical knowledge to make their judgments.