Abstract 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.

中文翻译：

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职前教师通过课堂实施学习教授证明：成功与挑战

摘要 证明和推理是通过理解学习数学的核心。然而，证据被视为教与学的挑战。在面向职前教师的顶峰课程中，我们开发了教学模块，引导准中学数学教师 (PST) 通过一个循环学习证明的逻辑方面，然后在中学课堂中规划和实施将这些方面与传统数学课程相结合的课程美国。在本文中，我们强调了我们用于教学证明的数学知识框架，并重点关注证明的一些被视为特别具有挑战性的逻辑方面（四个证明主题）。我们分析了 60 个课程计划、13 个已制定课程的子集的视频记录，以及 PST 的自我报告数据，以阐明 PST 如何计划和制定整合这些证明主题的课程。结果提供了对 PST 在此过程中遇到的成功和挑战的见解，并说明了 PST 准备在中学课堂上进行推理和证明的潜在途径。我们还强调了支持开发 PST 用于教学证明的数学知识的设计原则。