Definitional ambiguity: a case of continuous function

Abstract

Definitions are an integral aspect of mathematics. In particular, they form the backbone of deductive reasoning and facilitate precision in mathematical communication. However, when an agreed-upon definition is not established, its ability to serve these purposes can be called into question. While ambiguity can be productive, the existence of multiple non-equivalent definitions for the same term can make the truth value of certain mathematical statements unclear. In this study, we asked mathematics educators to determine the truth of a definitionally ambiguous mathematical claim. Based on their responses, we identified several factors that influenced the teachers’ choice of definitions. Finally, we consider the pedagogical implications of employing such a task in teacher preparation programs.