The Role of Instrumental and Relational Understanding in Proofs about Group Isomorphisms

The ability to construct proofs is a crucial skill in advanced mathematics that most students lack. To investigate the causes of students' difficulty, we observed a small group of undergraduates and doctoral students constructing proofs about group isomorphisms. Undergraduates were able to construct very few proofs, despite having an understanding of mathematical logic and often possessing the instrumental knowledge needed to prove the propositions in our study. Doctoral students proved every proposition in our study. Our analysis reveals that doctoral students regularly used their relational understanding of group isomorphisms to guide their proof attempts, while undergraduates seldom did. We conclude that what one can prove solely using instrumental understanding is often limited, and using a relational understanding may be necessary to be an effective proof constructor.