Abstract Algebra and the Conversation of Humankind

Abstract

: Peer review is especially difficult to facilitate in advanced mathematical writing. Typically, only someone with an appropriate level of disciplinary knowledge can understand the workings of a mathematical proof, for example, let alone provide useful feedback to a novice proof-writer. This presents a challenge to writing programs and writing centers charged with supporting writing throughout the curriculum. In this article, we discuss our efforts to support student proof-writing in an advanced abstract algebra course, in which students are expected to write their own sophisticated proofs of challenging mathematical propositions. Building primarily on the work of Ken Bruffee, we assert that math proofs are a form of normal discourse. Bruffee (1984) contends that collaborative learning is an especially good way for students to practice normal discourse with an audience of knowledgeable peers. In such an arrangement, the student, teacher, and peer reviewer each make different contributions to the learning experience. The peer reviewer, in our case, is a trained undergraduate writing consultant. Our analysis of teaching and learning artifacts, formal and informal student evaluations of the course, and transcripts of a student focus group, leads us to conclude that the collaboration has two observable outcomes: first, we get a higher percentage of student-written proofs that demonstrate an understanding of threshold concepts in abstract algebra; and second, students learn to communicate better and become members of the mathematical discourse community. We contend that these two are recursive