Developing and Validating Proof Comprehension Tests in Real Analysis

教授在高等数学课程中向学生传达数学的主要方式是通过数学证明的演示。然而，无论是在课堂上，还是在数学教育研究中，学生对这些证明的理解很少被评估。 这种评估空白的部分原因是缺乏有效的评估工具来衡量这种理解。这个项目将开发和验证本科真实的分析课程的八个可靠的证明理解测试。证明理解测验的生成是数学教师、高等数学学生和数学教育研究者的迫切需要。对于数学教师来说，评估学生的证明理解能力将提高对他们讲座有效性的反馈，或者揭示学生感到困惑的证明方面。对于高等数学课程的学生来说，知道他们将根据他们对证明的理解进行评估，这将鼓励他们学习这些证明，并可以将他们的注意力引导到他们通常不会考虑的证明方面。对于未来的数学教师来说，这种评估可以帮助他们将证明视为研究和学习数学的工具，这可能对他们的教学产生积极的影响。最后，一些数学教育研究者已经开始研究不同证明呈现模式的有效性，开发有效的工具来评估学生对证明的理解将进一步促进这一领域的研究。在本项目中，作者将开发并验证简短的多项选择测试，以可靠地评估读者在真实的分析中对八个证明的理解。对于每个证明，作者将首先使用他们的理论模型生成开放式证明理解评估问题，观察12名数学专业学生单独回答这些问题，并使用他们的回答创建一个大型多项选择评估项目库。他们将在12个数学专业的学生中试行这些项目，然后让200名学生完成8项选择题测试。对于每一个长测试，学生的反应的心理测量分析将允许作者选择一个子集的项目，提供的信息相当于长测试提供的信息。他们将通过要求12名数学专业的学生完成开放式和简短的多项选择测试，并将他们的分数与其他指标的表现进行比较，来验证这些简短的测试。这一阶段的目的是验证多项选择题测试是考生对证明理解的有效指标。本研究旨在探讨学生的证明理解能力（及其与其他能力的关系），以及对所开发测试的心理测量学特性的研究，将对数学教育中证明理解的结构的研究做出重大贡献。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

The primary way that professors convey mathematics to students in advanced mathematics courses is through the presentation of mathematical proofs. However, both in the classroom and in mathematics education research, students' understanding of these proofs is rarely assessed. This assessment void is partly due to the lack of valid assessment tools to measure such understanding. This project will develop and validate eight reliable proof comprehension tests for an undergraduate real analysis course. The generation of proof comprehension tests serves urgent needs for undergraduate mathematics instructors, students in advanced mathematics courses, and mathematics education researchers. For mathematics instructors, the ability to assess students' proof comprehension would improve feedback on the effectiveness of their lectures, or reveal aspects of proofs that students find confusing. For students in advanced mathematics courses, knowing they will be assessed on their comprehension of proofs will encourage them to study these proofs and can direct their attention to aspects of proof that they may not ordinarily consider. For prospective mathematics teachers, this assessment could help them see proof as a tool for studying and learning mathematics, which could have a positive impact on their teaching. Finally, some mathematics education researchers have begun to study the efficacy of different modes of proof presentation, and developing valid instruments to assess students' comprehension of proofs would further facilitate research in this area.In this project, the authors will develop and validate short, multiple-choice tests to reliably assess a reader's comprehension of eight proofs in real analysis. For each proof, the authors will first generate open-ended proof comprehension assessment questions using their theoretical model, observe 12 mathematics majors answering these questions individually, and use their responses to create a large repository of multiple-choice assessment items. They will pilot these items with 12 mathematics majors, before having 200 students complete the eight multiple-choice tests. For each long test, psychometric analyses of student responses will allow the authors to select a subset of items that provide information equivalent to the information provided by the long test. They will validate these short tests by asking 12 mathematics majors to complete both the open-ended and the short multiple-choice versions of the tests, and by comparing their scores with performance in other measures. The aim at this stage will be to verify that the multiple-choice tests are valid indicators of the test taker's understanding of the proofs. This study of students' proof comprehension (and its relationship to other competencies), as well as the study of the psychometric properties of the developed tests, will make a significant contribution to the study of the construct of proof comprehension in mathematics education.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.