Exploring pre-service mathematics teachers’ intuitive strategies for solving open equivalence relations problems

Abstract

This article contributes to the body of research on equivalence relations by delineating the levels of sophistication and efficiency in the intuitive strategies employed by pre-service teachers (PSTs). Using an inductive approach, we analysed 1102 responses to two open equivalence relations problems, drawn from a 27-item baseline assessment, completed by 551 first-year PSTs across three cohorts (2022–2024). From the analysis, we began teasing out a conceptual framework that could explain the reasoning behind the responses. To authenticate the conceptual framework, we needed to unearth the strategies employed by PSTs, so we conducted follow-up task-based interviews with nine selected participants. These interviews provided insights into the reasoning behind the PSTs’ written responses and helped illuminate the developmental progression in their understanding of reasoning when solving equivalence relations problem. In particular, our analysis of the interviews revealed three distinct stages of sophistication (quantitative, additive, and multiplicative) and four levels of efficiency in PSTs’ reasoning when solving equivalence relations problems. The findings show that PSTs encountered significant difficulties with solving open equivalence relations problems, with 46% correct responses: 14% of which involved quantity-based reasoning, while 17% and 15% were additive and multiplicative-based reasoning. Interestingly, in 2023, multiplicative-based reasoning (which is the most sophisticated of the three types of reasoning), had the highest responses at 21.1%, compared to 11.4% in 2022 and 11.2% in 2024. Moreover, the strategy that involves working with a large equivalence difference (which strategy is the least efficient) was found to be the predominant strategy in the PSTs’ workings within the additive-based reasoning. Thus, these interview findings provided deeper insights into the patterns observed in the written responses regarding PSTs’ intuitive strategies.
Contribution: These findings provide insights into a possible teaching framework that delineates the stages of sophistication that students go through in the development of their knowledge of open equivalence relations problems. By introducing and classifying open equivalence relations problems as belonging to the Comparative Relational category of equivalence tasks, this study elaborates on the highest category of equivalence tasks. Also, this study offers a possible analytical framework for analysing levels of efficiency in students’ reasoning when solving open equivalence relations problems. Consequently, the findings have implications for teacher training programmes and intervention studies with an interest in building sophistication and efficiency into PSTs’ strategies for solving open equivalence relations problems.