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.