Undergraduate Mathematics Education Students' Understanding of Proof by Contradiction and Proof by Contraposition
type: CreativeWork
description: 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…
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…
author: ., Hapipi
language: eng
keywords: Understanding proof, proof by contradiction (PBCd), proof by contraposition (PBCp), general concept of proof (GCP), negating competence, engagement, convincing and explanatory powers
license: https://www.repository.cam.ac.uk/bitstreams/d5410349-8df1-44de-b814-86bdc9c24ced/download
license: http://purl.org/NET/rdflicense/allrightsreserved
date published: 2025-04-27
date modified: 2026-07-22T22:24:03Z
provider: Cambridge University
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Cambridge

PropertyValue
property ID: DOI
value: 10.17863/CAM.121731