Undergraduate Students' Reasoning about Equivalence in Multiple Mathematical Domains: Exploration and Theory-Building

Equivalence, the idea that two objects can be considered “the same” in some way, is one of the most important concepts in mathematics. Learners from elementary through graduate school encounter equivalence in many ways across many mathematical topics. Unfortunately, students can have difficulties in understanding ideas of equivalence in more advanced mathematics courses.

One possible challenge is that equivalence is often treated as a new concept each time it is introduced within or across courses. In addition, students’ ways of thinking about this fundamental idea are not yet well understood. To begin to fill this knowledge gap, this project aims to develop a theory about how students reason with equivalence across two mathematical disciplines: combinatorics and abstract algebra.

To gather data on which to base the theory, the project will examine the current body of literature on equivalence, analyze textbooks, and conduct interviews with mathematicians and students. The theory that emerges from this research will help researchers and educators better understand different ways to reason about equivalence across mathematical domains.

This work also may have long-term benefits: such a theory could inform the design of curricular materials to help students at all levels see instances of equivalence in a more consistent, linked fashion.

Equivalence is one of the most fundamental, far-reaching concepts in all of mathematics and an essential component of the K-16 mathematics curriculum. Its importance is particularly evident at the postsecondary level, where equivalence manifests and plays a key role in virtually every domain from calculus to abstract algebra. Despite its prevalence and importance, undergraduate students can be challenged to understand instances of equivalence, especially if similar concepts are introduced in a disconnected way.

Moreover, characterizations of equivalence in research are often implicit or domain-specific, speaking to the need for cognitive models that might prove useful within and across mathematical disciplines. This project will work toward a crosscutting theory of equivalence that could be applied in multiple contexts. Focusing on the domains of combinatorics and abstract algebra, the project’s primary research questions are: (1) What is entailed in undergraduate students’ ways of thinking about equivalence within the domains of abstract algebra and combinatorics? (2) What is entailed in undergraduate students’ ways of thinking about equivalence across these domains?

To answer these questions, the project will leverage existing literature, textbook analysis, and interviews with mathematicians to develop an initial theory and then rigorously refine that theory via sequences of exploratory and targeted task-based clinical interviews with students, focusing on abstract algebra in Year 1, combinatorics in Year 2, and both domains in Year 3. This project is funded by the EHR Core Research (ECR) program, which supports work that advances fundamental research on STEM learning and learning environments, broadening participation in STEM, and STEM workforce development.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.