Definitions play a central role in the discipline of mathematics and in teaching and learning mathematics (Vinner 1991). Knowledge of mathematical definitions is an important component of Subject Matter Knowledge for elementary teachers (Ball, Thames, and Phelps, 2008). Teachers’ conceptions of mathematical definitions may affect how they engage in or engage their students in the process of defining (Johnson, Blume, Shimizu, Graysay & Konnova, 2014). However, to date, little research has focused on US teachers’ conceptions of mathematical definitions and even less has examined this issue at the elementary level.In this study, 24 preservice elementary teachers who were in their senior year were interviewed about their understanding of what constitutes a mathematical definition, the distinction between good and bad definitions, and their understanding of the roles of mathematical definitions in mathematics. The interviews were audiotaped and transcribed. Data were analyzed for correctness and general themes were examined using theoretical perspectives found in the literature (e.g., Van Dormolen & Zaslavsky, 2003). This study compares the conceptions of preservice elementary teachers whose teaching major was mathematics (PSTs-M) with the conceptions of preservice teachers whose teaching major was not mathematics (PSTs-N) in order to reveal whether advanced mathematical training contributes to preservice elementary teachers’ understanding of mathematical definitions. The study results indicate that preservice elementary teachers understand quite a few of the necessary and preferred features of mathematical definitions which are suggested in the literature (e.g., precise). However, they also demonstrated limited understanding of how to distinguish a necessary feature from a preferred feature (e.g., noncircularity). Moreover, preservice elementary teachers had a few misconceptions about mathematical definitions. The most common misconceptions include that mathematical definitions are expected to be written in the format “A is B” and that mathematical definitions could be used to name a property/procedure/theorem. Throughout the whole interview, only one preservice elementary teacher demonstrated the idea that no mathematical definition needed to be proved. PSTs’ misconceptions of mathematical definitions contributed to their incorrect understanding of the relationship between mathematical definitions and proofs. Also, PSTs-M’s idea of mathematical definitions as a meta level concept was not that different from the idea of PSTs-N. The results of this study suggest that current instruction in mathematics courses for preservice elementary teachers is not very effective in preventing the formation of misconceptions regarding mathematical definitions. The finding that PSTs-M’s and PSTs-N's conceptions of mathematical definitions were similarly problematic provides further evidence that simply going through advanced training in mathematics does not automatically improve preservice elementary teachers’ understanding of mathematical definitions. This indicates that the nature and roles of mathematical definitions need to be addressed more directly and more often in teacher education programs and that assessment needs to be designed as well.
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