Advanced mathematics is seen as an integral component of secondary teacher preparation, and thus most secondary teacher preparation programs require their students to complete an array of advanced mathematics courses. In recent years, though, researchers have questioned the utility of proposed connections between advanced and secondary mathematics. It is simply not clear in many cases-to researchers, teacher educators, and teachers themselves-exactly how advanced mathematics content is related to secondary content. In this paper, we propose using a conceptual analysis-a form of theory in which one explicitly describes ways of reasoning about a particular mathematical idea-to address this issue. Specifically, we use conceptual analyses for the foundational notions of equivalence and inverse to illustrate how the ways of reasoning needed to support productive engagement with tasks in advanced mathematics can mirror and reinforce those that are similarly productive in school mathematics. To do so, we propose conceptual analyses for the key concepts of equivalence and inverse and show how researchers can use these conceptual analyses to identify connections to school mathematics in advanced mathematical tasks that might otherwise be obscured and overlooked. We conclude by suggesting ways in which conceptual analyses might be productively used by both teacher educators and future teachers.

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.