Geometric Refutations of Prospective Secondary Mathematics Teachers

Abstract

Increasing students’ exposure to mathematical reasoning (Lannin, Ellis, & Elliot, 2011) has been a constant call from mathematics education reform efforts. However, studies have raised concerns about whether teachers are prepared to teach reasoning. This paper explores one aspect of reasoning, refutations, by examining the geometric refutations of thirteen pre-service secondary teachers (PSTs). Findings suggest that the PSTs did struggle to create culturally accepted refutation, but their results were less concerning than previous studies suggest. Primary challenges include that the PSTs separate the work they do for the different subprocesses of reasoning, reason differently depending on the subprocess, and favor non-culturally accepted means of refutations. Additionally, when the PSTs created culturally accepted refutations, they often used specific counterexamples, which are less pedagogically powerful than general counterexamples (Zaslavsky & Peled, 1996). However, the PSTs in this study did provide general counterexamples more often than the participants of other studies (Potari, Zachariades, & Zaslavsky, 2009). Suggestions for ways to improve PSTs understanding of reasoning are provided.