MARK ASIALA, ANNE BROWN, JENNIFER KLEIMAN and DAVID MATHEWS THE DEVELOPMENT OF STUDENTS’ UNDERSTANDING OF PERMUTATIONS AND SYMMETRIES 1. INTRODUCTION This paper reports on a study of the nature of abstract algebra students’ understanding of permutations and symmetries. Until recently, scholarly literature dealt with permutations, finite symmetry groups, and frieze groups in elementary education (Gill, 1993; Petreshane, 1982), secondary education (McLeay & Abas, 1991; Ellis-Davies, 1986; Okolica & Mac- rina, 1992; Shilgalis, 1992), and undergraduate education (Gallian, 1990), but only on the level of instructional suggestion. Now the growing body of research literature concerning the learning of collegiate mathematics contains a few studies focusing on abstract algebra concepts (Dubinsky, Dautermann, Leron & Zazkis, 1994; Zazkis & Dubinsky, 1996; Zazkis, Dubinsky & Dautermann, 1996). To that literature, our study will add a narrowly focused analysis of how students successfully learned about symmetries of a regular polygon and permutations of finite sets. The study was carried out according to a very specific framework for research and curriculum development that is used to investigate the learn- ing of collegiate mathematics. 1 In this report we will present results that suggest that the theoretical perspective (referred to below as the APOS Theory) in this framework is useful for understanding the mental construc- tions made by students learning about permutations and symmetries, and serves to increase our understanding of how learning about permutations and symmetries might take place. We will also see that students experienc- ing instruction based on this theoretical perspective successfully performed mathematical tasks that require an understanding of permutations. Finally, we present a base of information which sheds light on the epistemology and pedagogy associated with permutations and symmetries. 1 This framework is being used and developed by RUMEC-1, a subgroup of the larger Research in Undergraduate Mathematics Education Community (RUMEC). International Journal of Computers for Mathematical Learning 3: 13–43, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.