By Mark Asiala, Anne Brown, Jennifer Kleiman, and David
Mathews
This paper is one in a series of studies by members of the "Research in Undergraduate Mathematics Education Community", or RUMEC, concerning the nature and development of college students' mathematical knowledge. The present paper examines how abstract algebra students might come to understand permutations of a finite set and symmetries of a regular polygon . We give initial theoretical analyses of what it could mean to understand permutations and symmetries, expressed in terms of the Action-Process-Object-Schema epistemological framework. We describe an instructional treatment designed to help foster the formation of mental constructions postulated by the theoretical analysis, and discuss the results of interviews and performance on examinations. These results suggest that our pedagogical approach was reasonably effective in helping students to develop strong conceptions of permutations and symmetries. Based on the data collected as part of this study, we propose revised epistemological analyses of permutations and symmetries. Finally, we give pedagogical suggestions and present several open questions relating to the subject.