Assessing the comprehension of mathematical proofs at the undergraduate level

Juan Pablo Mejia Ramos
Associate Professor
Rutgers University

Need. Comprehending mathematical proofs is central in advanced mathematics courses taken by students pursuing undergraduate programs in STEM disciplines and prospective mathematics teachers. In these courses, textbooks and lecturers focus on presenting formal definitions of new concepts and proving theorems about those concepts. Underlying this widely used pedagogical format is the assumption that students can learn by studying the proofs that their professors and textbooks present. However, research reveals that even mathematics majors have great difficulty doing so, which leaves students unable to capitalize on the instruction in their advanced mathematics courses. At the center of this issue is our poor understanding of the construct of proof comprehension, how to assess it in the classroom in a meaningful way, and how it relates to other measures of learning and competence at the advanced undergraduate level. Guiding questions and Outcomes. This project expends prior theoretical and empirical investigations on the notion of proof comprehension; investigations that have focused on what it means to comprehend a mathematical proof at the advanced undergraduate mathematics level, and how to assess such comprehension. In a previous exploratory project, we demonstrated how a theoretical model of proof comprehension could be used to develop valid and reliable proof comprehension tests. The current project capitalizes on lessons learned from those initial explorations to develop and validate a larger battery of proof comprehension tests for an undergraduate course in real analysis, identify factors that predict students’ ability to comprehend proofs, and investigate the extent to which students’ proof comprehension ability predicts general course performance. Broader Impact. The impact of these results and anticipated deliverables (the proof comprehension tests themselves) goes beyond the research field of education. They directly serve the needs of both mathematics instructors (who, research shows, desire better ways of evaluating their students’ understanding of the proofs that they present) and undergraduate students in advanced mathematics courses, for whom the study of meaningful proof comprehension assessments directs their attention to important aspects of proof and helps them see proof as a tool for studying and learning mathematics. In the case of prospective mathematics teachers, such a shift in perspective regarding the notion of mathematical proof would have a positive impact on the mathematics education of school students everywhere.


Keith Weber, Rutgers University, New Brunswick, NJ; Drew Gitomer, Rutgers University, New Brunswick, NJ; Kathleen Melhuish, Texas State University, San Marcos, TX; Kristen Lew, Texas State University, San Marcos, TX