Developing and Validating Proof Comprehension Tests in Real Analysis

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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.