Validating Proof Comprehension Tests in Mathematics

智力价值：讲授高等数学的主要手段是展示数学证明。然而，无论是在课堂上，还是在数学教育研究中，很少有人对学生对这些证明的理解进行评估。本项目的目标是开发一种学生理论证明的综合评价模式。具体地说，该项目致力于在过渡到证明的过程中为三个证明生成和验证证明理解测试。该项目分为三个阶段。第一阶段是利用Mejia-Ramos模型编制三个证明题的开放式证明理解评价题。一小部分数学专业的本科生在半结构化面试中回答了这些问题，他们的回答进入了创建一个多项选择评估题库的过程。在第二阶段，大样本的数学专业学生完成了三个证明的选择题测试。使用他们的回答，与测试中其他项目高度相关的项目通过因子分析的形式被删除，以产生针对这些证明的多项选择测试的简化版本。最后，对一小部分数学专业本科生进行了访谈，他们完成了开放式和短式多项选择题的评估测试。这一阶段旨在验证多项选择题测试是考生理解三个证明的有效指标。这一过程的结果预计是三个简短的、多项选择的证明理解测试，这是学生对正在研究的证明的理解的有效指标。广泛的影响：需要对STEM学生理解他们所读的证明的程度进行更好的评估，以改善大学层面的证明教与学。一些大学生不能很好地理解数学证明，阻碍了他们进入STEM学科。在未来数学教师的特殊情况下，这种失败阻碍了他们获得有效教授数学所需的内容知识。证明理解测试的产生满足了数学家、数学专业学生和数学教育研究人员的迫切需求。对于数学教师来说，对学生对证明的理解程度的不完全评估意味着他们既不会收到关于课堂总体质量的反馈，也不会收到关于学生可能感到困惑的特定证明领域的反馈。对于数学专业的学生来说，对他们被要求完成的证明的理解能力的评估会鼓励他们投入时间学习这些证明；有意义的评估项目会将他们的注意力引导到他们通常不会考虑的证明的重要方面。有效的证明理解测试对本科数学教育的研究人员也很有用，因为它允许他们衡量与证明呈现相关的不同教学技术的有效性，并系统地解决学生对证明的哪些方面感到困惑的问题。

Intellectual Merit: The primary means for teaching advanced mathematics 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. The goal of this project is to develop a student comprehension assessment model of theoretical proofs. Specifically, the project is engaged in generating and validating proof comprehension tests for three proofs in a transition-to-proof course.The project has three phases. The first stage is developing open-ended proof comprehension assessment questions for three proofs using the Mejia-Ramos model. A small sample of undergraduate mathematics majors answers these questions in a semi-structured interview and their responses go into the creation a large repository of multiple-choice assessment items. In the second stage a large sample of mathematics majors completes these multiple-choice tests for the three proofs. Using their responses, items that correlate highly with other items in the tests are dropped using a form of factor analysis to produce a reduced version of the multiple-choice tests for these proofs. Finally, a small sample of undergraduate mathematics majors is interviewed as they complete both the open-ended and the short multiple-choice versions of the assessment tests. This stage seeks to verify that the multiple-choice tests are a valid indicator of the test taker's understanding of the three proofs. The result of this process is expected to be three short, multiple-choice proof comprehension tests that are valid indicators of students' understanding of the proofs being studied.Broader impact: Better assessments of the extent to which STEM students comprehend the proofs they read are needed to improve the teaching and learning of proof at the university level. The failure of some university students to successfully understand mathematical proofs prevents them from entering STEM disciplines. In the particular case of prospective teachers of mathematics, this failure prevents them from gaining the content knowledge they need to teach mathematics effectively. The generation of proof comprehension tests serves urgent needs for mathematicians, mathematics majors, and mathematics education researchers. For mathematics faculty, incomplete assessment of students' comprehension of proofs means they do not receive feedback both on the general quality of their lectures and on specific areas of proofs that students may have found confusing. For mathematics majors, assessment of their comprehension of the proofs they are asked to complete encourages them to invest time in studying these proofs; and meaningful assessment items direct their attention to important aspects of the proof that they may not ordinarily consider. Valid proof comprehension tests are also useful to researchers in undergraduate mathematics education by allowing them to measure the effectiveness of different teaching techniques related to proof presentation and systematically address questions about what aspects of proof students find confusing.