An Abstract Algebra Story

Abstract

We agree. And we think there's a fairly wide consensus on this among experienced abstract algebra instructors, and an even wider one among experienced students. Statement: There's little the conscientious math professor can do about it. The stuff is simply too hard for most students. Students are not well-prepared and they are unwilling to make the effort to learn this very difficult material. We disagree. But we suspect that many experienced abstract algebra instructors hold such beliefs. This is especially true for some excellent instructors: Their lectures are truly masterpieces, surely you can't improve much on that; so if the students still fail, that's too bad, but it can't really be helped. We claim that, far from being an immutable fact of nature resulting from inadequacies of the student, this failure is, at least in part, an artifact of a too narrowly conceived view of instruction. In fact, replacing the lecture method with constructive, interactive methods involving computer activities and cooperative learn- ing, can change radically the amount of meaningful learning achieved by average students. In this paper we would like to paint a picture of such an alternative approach, which we and others have been developing and using in our classes over the last several years. We are painfully aware of the limitations inherent in any attempt to give such a description by means of the written text only. It would have been much better if you could actually visit our classes and observe the dynamics of the students' interactions with both the computer and their peers. By way of compro- mise, we will try to simulate such a visit by organizing our paper around several classroom and some commentary on the events depicted in each scenario. As a matter of principle, we have tried to make the scenarios as realistic as space limitation permits.