What do prospective mathematics teachers mean by “definitions can be proved”?

Abstract : The research reported here is part of an ongoing study in which prospective middle school mathematics teachers’ conceptions of definition are investigated through their responses to semi-structured interview questions about defining quadrilaterals. Here we present findings from their responses to a subset of the interview questions, with the purpose of understanding what they mean by the expression “definitions can be proved”- an expression commonly referenced, and considered as erroneous in the research literature. Analysis of the responses, through using thematic coding and Toulmin’s (1958) scheme, revealed that participants attributed two different meanings to the phrase: (1) proving the claim that a written definition accurately designates an intended concept and (2) proving the concept being defined (erroneous). Based on our findings, we point to a reconsideration of the phenomenon by the research community.