Analogical structure sense: A case study of students’ analogical reasoning between groups and rings

Analogical reasoning is an important mathematical process for undergraduate students. However, it is unclear how students understand analogies that are presented to them, and more importantly, how students understand and create their own analogies. In this paper, we present a case study of four students as they reason analogically about several structures in abstract algebra. In particular, we expand on the notion of structure sense to include a wider range of structures in advanced mathematics and attend to each students’ analogical structure sense associated with each structure. Findings suggest that although students may possess a strong structure sense for group-theoretic structures, it is not necessarily the case that they possess a comparatively strong analogical sense of structure for ring-theoretic structures. In addition, those students with weaker senses of structure for group-theoretic structures are still able express productive reasoning about ring-theoretic analogies. Implications for future research and instructional practice are discussed.