An Initial Framework for Analyzing Students' Reasoning with Equivalence Across Mathematical Domains

The concept of equivalence is foundational in mathematics and is pervasive in the K-16 curriculum. Though much research has focused on equivalence, nearly all of it is domain-specific, and it is therefore unclear how students' reasoning about equivalence in one domain might influence their reasoning about it in another, if at all. This highlights a need for increased theoretical unity and coherence. In this theoretical paper, we propose an initial framework for analyzing students' reasoning about equivalence across domains. We use the framework to highlight commonalities amongst the ways in which equivalence is interpreted with respect to fractions, K-12 algebra, modular arithmetic, and linear algebra. We demonstrate the framework's strength as an analytical tool by using it to conduct detailed analyses of student data from already-published studies in combinatorics and abstract algebra. We conclude by suggesting ways in which this framework lays a rich foundation for future research.