"CAN A PROOF AND A COUNTEREXAMPLE COEXIST?" A STUDY OF STUDENTS' CONCEPTIONS ABOUT PROOF

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. In this article, we investigate whether high-attaining secondary students have the misconception that it is possible to have a poof and a counterexample for the same mathematical statement. The data consisted of 57 student surveys augmented by follow-up interviews with 28 students. While analysis of the survey data alone offered considerable evidence for the existence of the misconception among several students in our sample, subsequent analysis with the inclusion of the interview data showed no evidence of the misconception. Implications for methodology and research are discussed in light of these findings.