Abstract: The general metric space is a fundamental structure in the field of real analysis. Success in real analysis can have substantial implications for both undergraduate and graduate mathematics students, as the course can be an indicator of future graduate success. Despite its importance, we know very little about how students understand fundamental concepts in real analysis. To explore student understanding, I conducted two paired teaching experiments (Steffe & Thompson) in which undergraduate students reinvented (Freudenthal, 1991) the general definition of a metric space, focusing on the metric function. In this talk, I will discuss meaningful results that emerged from the teaching experiment data. By exploring particular episodes of student activity, I will demonstrate different ways that students might come to understand the metric function, and give preliminary suggestions for facilitating productive understandings of the metric function based on underlying constructivist perspectives.
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