Development of students' understanding of cosets, normality, and quotient groups

This paper reports on a continuing development of an abstract algebra course that was first implemented in the summer of 1990. This course was designed to address discrepancies between how students learn and how they were traditionally being taught. Based on results from the first implementation, pedagogical changes were made, including increased computer programming activities and other exercises which were designed to give the students the opportunity to build experience to draw on in order to construct understanding of the topics in class. A second experimental course was run. To assess the impact of these methods, and to continue to better understand how students go about learning, test results from the latter class and interviews with students from both experimental courses and a lecture-based class were analyzed. The students in the second experimental course demonstrated a deep understanding of the title concepts, especially cosets and normality. We discuss the details of the revised experimental course; the epistemological theory behind its design; and the framework used to analyze the results. We demonstrate through examples from interviews and test results the applicability of this analysis to the data, and the strides made by the students in comparison with the students from the lecture-based course, and with the students from the first experimental course. We hope to illustrate difficulties students face in learning abstract algebra, and to discuss instructional strategies to help students overcome these difficulties.