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

Abstract

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.

Keywords:

• conceptions
• counterexample
• proof
• surveys
• interviews
• validity

Acknowledgements

The data reported in this article were collected and analysed with the support of a grant from the Economic Social and Research Council (ESRC) to the first author (RES-000-22-2536). The opinions expressed in the article are those of the authors, and do not necessarily reflect the position, policies, or endorsement of the ESRC. Part of an earlier version of the article was presented at, and will be published in the proceedings of, the Sixth Congress of the European Society for Research in Mathematics Education (CERME 6), in Lyon, France, in January 2009.

Notes

1. The use of the term ‘proof’ is restricted in this article to valid arguments that establish the truth of mathematical assertions.

2. Our use of the term ‘misconception’ should not be interpreted as lack of appreciation of students' views about mathematics. Its use is intended to designate views that deviate from conventional views widely accepted nowadays in the field of mathematics.

3. Toward the end of the school year, the two teachers took more responsibility for planning the lesson sequences and the first author assumed more of a supportive role. This pattern of collaboration continued during the following school year (the two teachers taught the same groups of students in Year 11).

4. Although the survey we used was based in part on a section of a longer survey developed by the Longitudinal Proof Project (Küchemann and Hoyles 2001–03), the investigation of students' conceptions about whether it is possible to have a proof and a counterexample for the same assertion was not an apparent intention of the developers of the longer survey. The methods we used for collecting and analysing the survey data were our own.