Undergraduate Mathematics Education Students' Understanding of Proof by Contradiction and Proof by Contraposition

Abstract

Proof by contradiction (PBCd) and proof by contraposition (PBCp) are essential yet challenging for students, particularly undergraduate mathematics students, and are often perceived as less convincing and explanatory than direct proof (DP). While several factors have been identified as contributing to students’ difficulties and perceptions, further research in this area is needed. For instance, although understanding negation is a recognised prerequisite for both of these proof methods, no studies have systematically examined its impact on students’ understanding thereof. Similarly, although DP is generally considered easier to understand than indirect proof methods, few studies have compared students’ understanding of PBCd/PBCp to their understanding of DP for the same statements. Furthermore, while student engagement is believed to play a crucial role in learning, its influence on students’ understanding of these proof methods has not been studied. Moreover, the convincing and explanatory powers of these proofs remains inconclusive, as scholars disagree, for example, on whether indirect proofs can be explanatory. In this study I aimed to address these gaps by investigating undergraduate mathematics education students’ understanding of PBCd and PBCp. Specifically, I employed a mixed-method research design to investigate students’ understanding of these proof methods in relation to five aspects: (1) their understanding of the general concept of proof (GCP), (2) their competence in negating statements, (3) their engagement with statements prior to them being presented with proofs for these statements, (4) their understanding of these proof methods compared to DP for the same statements, and (5) their perceptions of the convincing and explanatory powers of these proof methods again relative to DP. This study involved 87 undergraduate mathematics education students in their fifth semester at the Department of Mathematics Education, University of Mataram, Indonesia. All 87 participants completed survey instruments, and 12 of them were interviewed. The survey instruments, developed based on relevant theoretical perspectives, were reviewed by experts before being piloted twice. Meanwhile, the interviews aimed to clarify and deepen the researcher’s understanding of participants’ responses to the surveys. The interview data served to explain, support, or contrast the survey findings. Therefore, the interview questions and the analysis of interview data were conducted in relation to the survey findings. This study found that students’ understanding of the general concept of proof (GCP) significantly influences their understanding of PBCd and PBCp. However, the contribution of this factor was relatively small. Moreover, when combined with other significant factors, its influence on students’ understanding of PBCd and PBCp weakened. On the other hand, students’ competence in negating statements had a strong association with their understanding of both of these proof methods. In particular, students’ competence in negating single statements and statements with quantifiers significantly impacted their understanding of PBCd, while their competence in negating conditional statements did not. In contrast, students’ understanding of PBCp was influenced by their competence in negating single and conditional statements, but not statements involving quantifiers. Additionally, students’ competence in negating statements was found to affect their ability to identify contradictions within PBCd and the modular structure of PBCp. Although no significant difference was observed between students’ understanding of PBCd and DP for the same statements, a significant difference was found between their understanding of PBCp and DP. Furthermore, students’ engagement with statements significantly influenced their understanding of both PBCd and PBCp (medium effect sizes). Regarding the proofs’ convincing and explanatory powers, students found PBCd to be just as convincing and explanatory as DP. However, while students found both PBCp and DP convincing and explanatory, they perceived PBCp to be less so than DP. In addition to their contribution to research knowledge about students’ understanding of indirect proof methods, this study advances theoretical frameworks for understanding, assessing, and investigating students’ difficulties with these proof methods. Moreover, the study’s findings have important implications for teaching PBCd and PBCp, especially about negating competence, which is crucial for understanding these proof methods. In particular, the study provides guidance on structuring negation-related content and highlights key areas for university instructors to focus on, such as negating single statements and those involving quantifiers or conditionals. Additionally, the findings underscore the role of student engagement with statements prior to students’ exposure to proofs for these statements. Finally, a methodological contribution of the study is the development of several instruments, notably one for measuring negating competence. Future research can further develop these instruments and test them in new contexts.