Can a proof and a counterexample coexist? Students' conceptions about the relationship between proof and refutation.

Despite the importance of proof and refutation in students' mathematical education, students' conceptions about the relationship between proof and refutation have not been the explicit focus of research thus far. Nevertheless, the combined consideration of findings from different studies suggests that some students believe it is possible to have a proof and a counterexample for the same assertion. In this article, we investigated the possible existence of this misconception among high-attaining secondary students. We used survey data from 57 students and follow-up interview data with 28 of them. Our analysis of the survey data alone offered considerable evidence for the existence of the misconception among several students. Yet, our analysis of data from the interviews, which aimed to test the tentative conclusions we had drawn from the survey data analysis, showed no evidence of the misconception. Implications for methodology and research are discussed in the light of these findings. [ABSTRACT FROM AUTHOR]