In this article, we focus on a group of 39 prospective elementary (grades K-6) teachers who had rich experiences with proof, and we examine their ability to construct proofs and evaluate their own constructions. We claim that the combined “construction–evaluation” activity helps illuminate certain aspects of prospective teachers’ and presumably other individuals’ understanding of proof that tend to defy scrutiny when individuals are asked to evaluate given arguments. For example, some prospective teachers in our study provided empirical arguments to mathematical statements, while being aware that their constructions were invalid. Thus, although these constructions considered alone could have been taken as evidence of an empirical conception of proof, the additional consideration of prospective teachers’ evaluations of their own constructions overruled this interpretation and suggested a good understanding of the distinction between proofs and empirical arguments. We offer a possible account of our findings, and we discuss implications for research and instruction.
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