Exploring grade 11 learners’ functional understanding of proof in relation to argumentation in selected high schools.

Abstract

Research has established that understanding the functions of proof in mathematics and argumentation ability provide learners with a firm foundation for constructing proofs. Yet, little is known about the extent to which learners appreciate the functions of proof and whether an association between functional understanding of proof and argumentation ability exists. Guided by van Hiele’s and Toulmin’s theories, this study utilised a sequential explanatory design to randomly select three schools from a cluster grouping of ten Dinaledi high schools in the Pinetown district. Three survey questionnaires, Learners’ Functional Understanding of Proof (LFUP), self-efficacy scale, and Argumentation Framework for Euclidean Geometry (AFEG), were administered to a sample of 135 Grade 11 learners to measure their understanding of the functions of proof and argumentation ability, and to explore the relationship between argumentation ability and functional understanding of proof. Then, Presh N (pseudonym)—a female learner who obtained the highest LFUP score despite attending a historically under-resourced township school—was purposively selected from the larger sample. In addition to her responses on the questionnaires, a semistructured interview, and a standard proof-related task served as data sources to explain the origins of her functional understanding of proof. Statistical analyses were conducted on data obtained from questionnaires while pattern matching method was used to analyse the interview data. The analyses revealed that learners held hybrid functional understanding of proof, the quality of their argumentation was poor, the relationship between functional understanding of proof and argumentation ability was weak and statistically significant, and the collectivist culture and the teacher were the two factors which largely accounted for Presh N’s informed beliefs about the functions of proof. In addition, although she constructed a deductive proof, she did not perform the inductive segment prior to formally proving the proposition. The recommendation that Euclidean geometry curriculum needs to be revamped for the purpose of making functional understanding of proof and argumentation explicit and assessable content has implications for two constituencies. Instructional practices in high schools and methods modules at higher education institutions need to include these exploratory activities (functional understanding of proof and argumentation) prior to engaging in the final step of formal proof construction. Future research initiatives need to blend close-ended items with open-ended questions to enhance insights into learners’ functional understanding of proof. This study not only provides high school teachers and researchers with a single, reliable tool to assess functional understanding of proof but also proposes a model for studying factors affecting functional understanding of proof. Overall, the results of this study are offered as a contribution to the field’s growing understanding of learners’ activities prior to constructing proofs.