Teaching and Learning Group Theory

algebra is an important course in the undergraduate mathematics curriculum. For some undergraduates, abstract algebra is the first mathematics course in which they must move beyond learning templates and procedures for solving common classes of problems (Dubinsky, Dautermann, Leron, and Zazkis, 1994). For most undergraduates, this course is one of their earliest experiences in coping with the difficult notions of mathematical abstraction and formal proof. Empirical research studies attest to students’ difficulties in abstract algebra; these studies have shown that many students do not understand fundamental concepts in group theory (e.g., Leron, Hazzan, and Zazkis, 1995; Asiala, Dubinsky, Mathews, Morics, and Oktac, 1997) and have difficulty writing proofs in a group theoretic context (e.g., Selden and Selden, 1987; Selden & Selden, this volume; Hart, 1994; Weber, 2001; Harel & Brown, this volume) after completing an abstract algebra course. The purpose of this chapter is to use the research literature to illustrate some of undergraduates’ difficulties in group theory, the primary content area in most abstract algebra courses, and to describe alternative forms of pedagogy that may be useful in overcoming these difficulties. Undergraduates’ Difficulties in Group Theory In this section, we present four episodes that illustrate undergraduates’ difficulties in understanding and reasoning about concepts in group theory. We will begin each section by presenting an excerpt from a clinical interview in which undergraduates were asked to describe a concept in group theory or complete a group theoretic task. We will then use these transcripts as a basis to discuss general difficulties that undergraduates have in group theory and to state some of the reasons that undergraduates have these difficulties. Episode 1. Students’ Understanding Of Cosets And Normality Asiala et al. (1997) interviewed a group of 31 students who participated in an experimental abstract algebra course. After the course, some students met with researchers for interviews in which they discussed central group theoretic concepts. Below, one student, Carla, was asked how she would determine if K (a 4-element subgroup of S4) was a normal subgroup of S4. Carla: OK, you just have to check if when you multiplied two elements together if it is the same that you did them in the reverse order. Like if you take (1234) and compose it with (1432)[...] see if you get the same thing as when you take (1432) and compose it with (1234).