2.1 - Wrestling with Finite Groups; Abstract Algebra need not be a Passive Sport

Summary

Introduction

Abstract algebra is fundamentally abstract. (Perhaps this is stating the obvious.) But must the teaching and learning of abstract algebra take place in an entirely passive environment? Certainly not.

In the 2001 spring semester at the University of Sioux Falls, I structured an abstract algebra course around a single open-ended research project. The project was not an application or assessment of material already covered in class. This central project was intended to motivate and introduce much of the actual course content throughout the entire semester.

The class project called for students to catalog the central product structures of all finite 2-groups through order 32, and to articulate and prove a general statement about the central product structures of all abelian 2-groups. Because the class was small (six students), all of the students were expected to carry out the project as a single collaborative group.

The experiment appears to have been quite successful and could transfer well to other advanced mathematics courses, especially if the class size is small. I suggest that there are at least three key ingredients that should be present in the design and administration of your own project-driven course: 1) the semester project should be sufficiently open-ended to allow the students an authentic research and discovery experience; 2) the project topic should be rich enough to trigger many of the concepts that are essential to the course's syllabus; and 3) the instructor should be prepared to be flexible and responsive; reading word for word from a detailed set of lecture notes would defeat the very purpose of a project-driven course.