On learning fundamental concepts of group theory

Abstract

ABSTRACT. The research reported in this paper explores the nature of student knowledge about group theory, and how an individual may develop an understanding of certain topics in this domain. As part of a long-term research and development project in learning and teaching undergraduate mathematics, this report is one of a series of papers on the abstract algebra component of that project.The observations discussed here were collected during a six-week summer workshop where 24 high school teachers took a course in Abstract Algebra as part of their work. By comparing written samples, and student interviews with our own theoretical analysis, we attempt to describe ways in which these individuals seemed to be approaching the concepts of group, subgroup, coset, normality, and quotient group. The general pattern of learning that we infer here illustrates an action-process-object-schema framework for addressing these specific group theory issues. We make here only some quite general observations about learning these specific topics, the complex nature of "understanding", and the role of errors and misconceptions in light of an action-process-schema framework. Seen as research questions for further exploration, we expect these observations to inform our continuing investigations and those of other researchers.We end the paper with a brief discussion of some pedagogical suggestions arising out of our considerations. We defer, however, a full consideration of instructional strategies and their effects on learning these topics to some future time when more extensive research can provide a more solid foundation for the design of specific pedagogies.

Accommodation in the formal world of mathematical thinking

“…In addition to a lack of information about teaching practices in abstract algebra, there is considerable evidence documenting student difficulty with the subject's most basic concepts (e.g. [12,13]). For the purpose of this paper, we focus on both the mathematician and the student's journals on four lectures leading up to the proof of an elegant theory.…”

The Design and Validation of a Group Theory Concept Inventory

“…For many students, this course is the first time concepts are to be reasoned about formally based on their properties (Hazzan, 1999). Dubinsky, Dautermann, Leron and Zazkis (1994) noted, "mathematics faculty and students generally consider it to be one of the most troublesome undergraduate subjects" (p. 268). While this statement is largely accepted, student understanding in the course has not been empirically evaluated on a large scale.…”

“…Annie Bergman aided me in a great deal of analysis throughout the process. I also could not have (Dubinsky et al, 1994) (Leron & Hazzan, 1996) Figure 1. How conceptual understanding interferes with proving.…”

Mathematical Explanatory Strategies Employed by Prospective Secondary Teachers

“…While a great deal of research has been conducted on (a) justification and argumentation (e.g., Harel and Sowder 2007;Weber and Alcock 2004) and on (b) the learning and understanding of abstract algebra (e.g., Dubinsky et al 1994;Findell 2001;Hazzan 1999;Leron et al 1995), most of the relevant research focused on students' conceptions of and production of proofs and on difficulties students have with particular concepts in group theory. Thus, research offers some understanding of the acquisition of abstract algebra knowledge but offers little regarding prospective teachers' transfer of this knowledge to secondary school teaching.…”

“…1 This study focuses on the links between abstract algebra concepts and school mathematics because (a) abstract algebra is a course requirement for most prospective secondary mathematics teachers and (b) abstract algebra is generally the first undergraduate course that requires students to grapple with concepts presented structurally (Dubinsky et al 1994). …”