A Tale of Two Courses; Teaching and Learning Undergraduate Abstract Algebra

Abstract

The abstract algebra course is an important point in the education of undergraduate mathematics majors and secondary mathematics teachers. Abstract algebra teachers have multiple goals for student learning, and the literature suggests that students have difficulty meeting these goals. Advisory reports have called for a move away from lecture toward investigation-based class sessions as a means of improving student understanding. Thus, it is appropriate to understand what is happening in the current teaching and associated learning of abstract algebra. The present study examined teaching and learning in two abstract algebra classrooms, one consciously using a lecture-based (i.e., deduction-theory-proof, or DTP) mode of instruction and the other an investigative approach. Instructional data was collected in classroom observations, and multiple written instruments and a set of interviews were used to evaluate student learning. Each instructor hoped students would develop a deep and connected knowledge base and attempted to create classroom environments where students were constantly engaged as a means of doing so. In the lecture class, writing proofs was the central activity of class meetings; nearly every class period included at least one proof. In the investigative class, the processes of computing and searching for patterns in various structures were emphasized.

At the end of the semester, students demonstrated mixed levels of proficiency. Generally, students did well on items that were relatively familiar, and poorly when the content or context was unfamiliar. In the DTP course, two students demonstrated significant proficiency with analytical argument; the remainder demonstrated mixed proficiency with proof and very little proficiency with other content. The students in the investigative class all seemed to develop similar levels of proficiency with the content, and demonstrated more willingness to explore unknown structures.

This study may prompt discussions about the relative importance of developing proof-proficiency, students' ability to formulate and investigate hypotheses, developing students' content knowledge, and students' ability to operate in and analyze novel structures.