Learning binary operations, groups, and subgroups

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 binary operations, groups, and subgroups. We give preliminary theoretical analyses of what it could mean to understand these topics, 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 binary operations, groups, and subgroups. Based on the data collected as part of this study, we proposed revised epistemological analyses of these topics, and give some pedagogical suggestions related to these topics.