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  1. 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 (fourmore »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.« less
  2. Karunakaran, S. S. ; Reed, Z. ; Higgins, A. (Ed.)
    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 usedmore »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.« less
  3. We describe an instructional module aimed to enhance prospective secondary teachers’ (PSTs’) subject matter knowledge of indirect reasoning. We focus on one activity in which PSTs had to compare and contrast proof by contradiction and proof by contrapositive. These types of proofs have been shown to be challenging to students at all levels and teachers alike, yet there has been little research on how to support learners in developing this knowledge. Data analysis of 11 PSTs, points to learning opportunities afforded by the module and the PSTs’ challenges with indirect reasoning.
  4. For reasoning and proving to become a reality in mathematics classrooms, pre-service teachers (PSTs) must develop knowledge and skills for creating lessons that engage students in proof-related activities. Supporting PSTs in this process was among the goals of a capstone course: Mathematical Reasoning and Proving for Secondary Teachers. During the course, the PSTs designed and implemented in local schools four lessons that integrated within the regular secondary curriculum one of the four proof themes discussed in the course: quantification and the role of examples in proving, conditional statements, direct proof and argument evaluation, and indirect reasoning. In this paper wemore »report on the analysis of 60 PSTs’ lesson plans in terms of opportunities for students to learn about the proof themes, pedagogical features of the lessons and cognitive demand of the proof-related tasks.« less
  5. We use a curriculum design framework to analyze how prospective secondary teachers (PSTs) designed and implemented in local schools, lessons that integrate ongoing mathematical topics with one of the four proof themes addressed in the capstone course Mathematical Reasoning and Proving for Secondary Teachers. In this paper we focus on lessons developed around the conditional statements proof theme. We examine the ways in which PSTs integrated conditional statements in their lesson plans, how these lessons were implemented in classrooms, and the challenges PSTs encountered in these processes. Our results suggest that even when PSTs designed rich lesson plans, they oftenmore »struggled to adjust their language to the students’ level and to maintain the cognitive demand of the tasks. We conclude by discussing possible supports for PSTs’ learning in these areas.« less
  6. For reasoning and proof to become a reality in mathematics classrooms, it is important to prepare teachers who have knowledge and skills to integrate reasoning and proving in their teaching. Aiming to enhance prospective secondary teachers’ (PSTs) content and pedagogical knowledge related to proof, we designed and studied a capstone course Mathematical Reasoning and Proving for Secondary Teachers. This paper describes the structure of the course and illustrates how PSTs’ interacted with its different components. The PSTs first strengthened their content knowledge, then developed and taught in local schools a lesson incorporating proof components. Initial data analyses show gains inmore »PSTs’ knowledge for teaching proof and dispositions towards proving, following their participation in the course.« less
  7. The design-based research approach was used to develop and study a novel capstone course: Mathematical Reasoning and Proving for Secondary Teachers. The course aimed to enhance prospective secondary teachers’ (PSTs) content and pedagogical knowledge by emphasizing reasoning and proving as an overarching approach for teaching mathematics at all levels. The course focused on four proof-themes: quantified statements, conditional statements, direct proof and indirect reasoning. The PSTs strengthened their own knowledge of these themes, and then developed and taught in local schools a lesson incorporating the proof-theme within an ongoing mathematical topic. Analysis of the first-year data shows enhancements of PSTs’more »content and pedagogical knowledge specific to proving.« less