Investigating and Improving Undergraduate Proof Comprehension

Abstract

Undergraduate mathematics students see a lot of written proofs. But how much do they learn from them? Perhaps not as much as we would like; every professor knows that students struggle to make sense of the proofs presented in lectures and textbooks. Of course, written proofs are only one resource for learning; students also attend lectures and work independently or with support on problems. But because mathematics majors are expected to learn much of their mathemat-ics by studying proofs, it is important that we understand how to support them in reading and understanding mathematical arguments. This observation was the starting point for the research reported in this article. Our work uses psychological research methods to generate and analyze empirical evidence on mathematical thinking, in this case via experimental studies of teaching interventions and quantitative analyses of eye-movement data. What follows is a chrono-logical account of three stages in our attempts to better understand students' mathematical reading processes and to support students in learning to read effectively. In the first stage, we designed resources we called e-Proofs to support students in under-standing specific written proofs. These e-Proofs conformed to typical guidelines for multimedia learning resources, and students experienced them as useful. But a more rigorous test of their efficacy revealed that students who studied an e-Proof did not learn more than students who had simply studied a printed proof and in fact retained their knowledge less well. This led us to suspect that e-Proofs made learning feel easier, but as a consequence resulted in shallower engagement and therefore poorer learning. At the second stage we sought insight into pos-sible underlying reasons for this effect by using eye-movement data to study the mechanisms of mathematical reading. We asked undergraduate students and mathematicians to read purported proofs and found that experts paid more atten-tion to the words and made significantly more back-and-forth eye movements of a type consistent with attempts to infer possible justifications for mathematical claims. This result is in line with the idea that mathematical experts make active efforts to identify logical relationships within a proof and that effective guidance might therefore be needed to teach students to do the same thing.