Students’ engagement with statements and their understanding of corresponding proofs by contradiction and contraposition

Engagement has been shown to influence students’ understanding and achievement, yet little is known about how engagement with statements relates to students’ understanding of corresponding proofs. In this paper, we take a step towards addressing this gap by investigating whether engaging with statements is associated with better understanding of proofs by contradiction and contraposition. Eighty-seven Indonesian undergraduate mathematics education students, organized in four groups, participated in the study. Two groups (n = 43) were randomly selected to first engage with the statements—reflecting on their meaning, judging their truth, and attempting to prove or disprove them—before responding to an instrument designed to measure their understanding of given proofs for the same statements; the other two groups (n = 44) proceeded directly to the instrument. There were six statements overall, three proved by contradiction and three by contraposition. We found that students who engaged with the statements achieved significantly higher mean scores on measures of proof understanding than those who did not, with a medium effect size. Regression analyses confirmed engagement as a significant predictor of students’ understanding for both proof methods, and individual semi-structured interviews with 12 participants provided qualitative support for these findings. Overall, our findings identify engagement with statements as a meaningful factor in students’ proof understanding and suggests that selectively incorporating an engagement stage before introducing key proofs may enhance undergraduate mathematics instruction beyond the typical Definition-Theorem-Proof (DTP) approach.