Examining the concept of inverse: Theory-building via a standalone literature review

Inverse is a critical topic throughout the K–16 mathematics curriculum where students encounter the notion of mathematical inverse across many contexts. The literature base on inverses is substantial, yet context-specific and compartmentalized. That is, extant research examines students’ reasoning with inverses within specific algebraic contexts. It is currently unclear what might be involved in productively reasoning with inverses across algebraic contexts, and whether the specific ways of reasoning from the literature can be abstracted to more general ways of reasoning about inverse. To address this issue, we conducted a standalone literature review to explicate and exemplify three cross-context ways of reasoning that, we hypothesize, can support students’ productive engagement with inverses in a variety of algebraic contexts: inverse as an undoing, inverse as a manipulated element, and inverse as a coordination of the binary operation, identity, and set. Findings also include explicating affordances and constraints for each of these ways of reasoning. Finally, we reflect on when and how standalone literature reviews can serve the purpose of unifying fragmented and obscured insights about key mathematical ideas.