Validating Proof Comprehension Tests in Mathematics

智力优势：高等数学教学的主要手段是通过数学证明的呈现。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都很少得到评估。本课题的目标是建立一个学生对理论证明的理解评估模型。具体地说，该项目在“过渡到证明”课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模型为三个证明开发开放式证明理解评估题。一小部分数学专业的本科生在一次半结构化的访谈中回答了这些问题，他们的回答将被用于创建一个大型的选择题评估库。在第二阶段，大量的数学专业学生完成三个证明的选择题测试。根据他们的回答，使用因子分析的形式删除与测试中其他项目高度相关的项目，以产生这些证明的多项选择测试的简化版本。最后，我们采访了一小部分数学专业的本科生，他们同时完成了开放式和短选择题的评估测试。这一阶段的目的是验证选择题是测试者对三个证明理解程度的有效指标。这一过程的结果预计将是三个简短的，多项选择的证明理解测试，这是学生理解所研究的证明的有效指标。更广泛的影响：需要更好地评估STEM学生理解他们所阅读的证明的程度，以改善大学水平的证明教学。一些大学生不能成功地理解数学证明，这阻碍了他们进入STEM学科。在未来数学教师的特殊情况下，这种失败使他们无法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者的迫切需求。对于数学教师来说，对学生对证明理解的不完整评估意味着他们没有收到关于讲课总体质量的反馈，也没有收到关于学生可能感到困惑的证明的特定领域的反馈。对于数学专业的学生来说，评估他们对被要求完成的证明的理解程度可以鼓励他们投入时间来研究这些证明；有意义的评估项目将他们的注意力引向他们通常不会考虑的证据的重要方面。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，它允许他们衡量与证明展示相关的不同教学技术的有效性，并系统地解决有关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.