CAREER:Understanding and Teaching the Processes Used to Comprehend Mathematical Arguments and Proofs

在以改革为导向的K-12数学课堂和上层大学课堂中，都非常重视证明和证明的活动。特别是，在这些教室里，我们希望学生通过参加和批评别人的数学论点来学习新的数学。为了实现这些目标，学生需要能够理解他人的数学论点，并确定数学论点是否有效。研究表明，学生和数学教师往往缺乏这种能力。我们对一个人应该用什么样的认知过程来确定一个数学论证是否有效，或者通过阅读一个数学论证来构建理解，知之甚少。本建议的目的是通过以下一系列研究来解决这两个问题：（a）一项专家-新手研究，观察数学专业本科生和数学家阅读和评价一系列数学论证，其结果将为（B）开发教学技术和课程材料，使用设计研究范式向学生传授理解数学论证所需的知识和策略，和（c）一个对照实验，其中（B）中开发的教学技术将在一组学生中实施，这些学生在毕业后的表现，测试将与对照组的表现进行比较。智力优点。拟议的工作将提供深入了解的数学实践的阅读和评估数学论点，并将阐明为什么学生发现这些活动的问题。建议的工作将产生文件有效的教学方法，教本科生的知识和技能，有效地阅读和评估数学论点。更广泛的影响。这些结果将引起经常教授以证明为导向的课程的数学教授和其他科学学科（如物理学）教师的兴趣，在这些学科中，论证起着关键作用。这些结果也将是有用的教育工作者谁准备未来的数学教师。这些研究的结果将在数学家和数学教育的学术会议上发表，这些结果将发表在同行评审的期刊上。教育目标。我将为数学教授举办研讨会，他们正在寻找新的方法来教授以证明为导向的数学课程，

In both reform-oriented K-12 mathematics classrooms and in upper-level university classrooms, a significant emphasis is placed on the activities of justifying and proving. In particular, in these classrooms, we expect students to learn new mathematics by attending to and critiquing the mathematical arguments of others. For these goals to be achieved, students need to be able to understand others' mathematical arguments and to determine whether or not a mathematical argument is valid. Research has shown that students and mathematics teachers frequently lack this ability. We know little about what cognitive processes a person should use to determine whether a mathematical argument is valid or to construct understanding from reading a mathematical argument. The purpose of this proposal is to address both of these issues with the following series of studies:(a) An expert-novice study in which undergraduate mathematics majors and mathematicians are observed as they read and evaluate a collection of mathematical arguments, the results of which will inform(b) The development of pedagogical techniques and curricula materials using a design research paradigm to teach students the knowledge and strategies they need to comprehend mathematical arguments, and(c) A controlled experiment in which the pedagogical techniques developed in (b) will be implemented with a group of students and these students' performance on a post-test will be compared to a control group's performance.Intellectual merit.The proposed work will provide insight into the mathematical practices of reading and evaluating mathematical arguments and will delineate why students find these activities problematic. The proposed work will yield documented effective teaching methods to teach undergraduates the knowledge and skills to read and evaluate mathematical arguments effectively. Broader impact.These results will be of interest to mathematics professors who regularly teach proof-oriented classes and teachers of other scientific disciplines, such as physics, where argumentation plays a pivotal role. These results will also be useful for educators who prepare future teachers of mathematics. The results of these studies will be presented at academic conferences for both mathematicians and mathematics educations, and these results will be published in peer-reviewed journals.Educational goals.I will give workshops for mathematics professors who are looking for new ways to teach proof-oriented mathematics courses and for