The Pedagogical Examples of Groups and Rings That Algebraists Think Are Most Important in an Introductory Course

Abstract

This article reports on an exploratory study designed to investigate the reasoning behind algebraists’ selection of examples. Variation theory provided a lens to analyze their collections of examples. Our findings include the classes of examples of groups and rings that algebraists believe to be most pedagogically useful. Chief among their selection criteria was that these examples illustrate not only the concept at hand but also lay the foundation for more abstract constructions. Additionally, we found that the algebraists, tending to think in terms of classes of examples, used a relatively small number in their own teaching and research.

Résumé

Cet article donne le compte-rendu d’une étude préliminaire conçue dans le but d’analyser le raisonnement qui sous-tend le choix des exemples chez les algébristes. Grâce à la théorie de la variation, nous sommes en mesure d’analyser les séries d’exemples. Nos résultats incluent les classes d’exemples posés en termes d’ensembles et de cercles que les algébristes estiment les plus utiles sur le plan pédagogique. Parmi leurs critères de sélection, le plus important est le fait que ces exemples illustrent non seulement le concept dont il est question, ils constituent également les fondements de constructions plus abstraites. De plus, nous avons constaté que les algébristes, qui ont tendance à penser en termes de classes d’exemples, en citent un nombre relativement peu élevé dans leur propre enseignement et dans leur recherche.

Notes

¹In fact, there is stark and conclusive evidence to the contrary. Larsen's (2013) innovative group theory curriculum, for example, shows a strong preference for the dihedral (symmetry) groups in its instructional tasks and has been shown to produce marked conceptual learning gains (see Larsen, Johnson, & Bartlo, 2013). We also note that, mathematically speaking, Cayley's theorem suggests that the permutation groups alone sufficiently exemplify the concept of group.

²Many first-semester courses only cover group theory; thus, instructors might not think rings are appropriate for such a class. Dr. E noted that he teaches ring theory in a second-semester class.