On Learning Fundamental Concepts of Group Theory

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 speciic group theory issues. We make here only some quite general observations about learning these speciic topics, the complex nature of \understanding", and the role of errors and misconceptions in light of an action-process-object-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 eeects on learning these topics to some future time when more extensive research can provide a more solid foundation for the design of speciic pedagogies.