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Establishing and leveraging equivalence is a central practice in mathematics. Though there have been many studies of students’ uses of equivalence, much of the research thus far has been domain-specific, and the literature generally lacks coherence within and across mathematical domains. In this theoretical paper, we propose an initial unifying framework for capturing the different ways that students might establish equivalence. Using constructs born out of the K-12 literature, we discuss how this framework can be applied to student reasoning in undergraduate settings. We do so by presenting the results of conceptual analyses of students’ possible uses of equivalence when thinking about vectors, isomorphisms and homeomorphisms, and single- variable limits. We then conclude with a detailed analysis of student data from combinatorics that identifies productive aspects of their uses of equivalence when constructing permutations. 

In this paper we analyze common solutions that students often produce to isomorphic tasks involving proportional situations. We highlight some key distinctions across the tasks and between the different equations students write within each task to help elaborate the different interpretations of equivalence at play: numerical, transformational, and descriptive. We use this opportunity to further explore the value of operationalizing these interpretations in both research and instruction.