> < ^ Date: Thu, 06 Jan 1994 16:54:00 -0600
^ From: Larson <larson@stolaf.edu >
> ^ Subject: Computing in Beginning Undergraduate Algebra

We have been experimenting with using computer software in
almost all of our undergraduate mathematics courses. In particular,
I have focused on using GAP in our beginning course in abstract
algebra. As a preface to explaining what we have settled on,
let me begin by describing our beginning our algebra course.

All mathematics majors are required to take one term of abstract
algebra. One reason for this is that it is one of only two
courses in which primary emphasis is placed on developing
"mathematical maturity." That is to say, we have
regarded our beginning undergraduate algebra course as
a theoretical course where students are expected to learn
to write proofs precisely and accurately, to think and reason
logically, and to gain an appreciation for generalization and
abstraction.

The course introduces groups, rings, and fields, with lots of
examples, and covers the standard theorems (e.g., Lagrange's Theorem,
Fundamental Homomorphism Theorem for groups and rings, Fundamental
Theorem of Field Theory). The content is a slightly abridged
version of the first 25 chapters of Joseph A. Gallian's
``Contemporary Abstract Algebra'', Third Edition, 1994,
D. C. Heath and Company, Lexingon, Massachusetts.

Such a full agenda leaves little time for extended computer
projects. Nevertheless, with careful planning we
have supplemented this theoretical approach with
four computer projects, using GAP (for groups) and MAPLE (for
rings and fields). About one class period is devoted to each
project. These examples show the value of the
computer in pedagogy, understanding, research, and application.

In the first project the computer is used to solve various sliding
block puzzles. Specifically, students specify sets of generators
and the computer calculates their subgroups. Students might wonder
how the computer is able to do this, and it could lead,
although not in this course, to further study of generators and
relations, and the Todd-Coxeter algorithm.

The second project helps students understand the Fundamental Theorem
of Finite Abelian Groups. Specifically, for arbitrary \$n\$, students
use the computer to help them identify the group of units
modulo \$n\$ as a direct sum of cyclic groups of prime power order,
and to construct its cycle graph.

In the third project, students ask questions such as ``What
proportion of the elements of a group have property \$P\$?''
The computer grinds out data and calculates the proportion for
any desired group. This, in turn, leads to conjectures and
possible proofs.

The fourth project uses the computer to calculate
``encoding'' polynomials for multiple error correcting BCH codes.
The computer is also used to detect and correct
errors in received messages. Among other things, this project
reinforces the importance of finite field algebra.

These projects show students the scope and power of computer
software, and hopefully motivates them to want to continue
their studies in mathematics and algebra in particular.

I am currently preparing versions of these projects for
distribution by e-mail (in plain TEX). If interested in
getting a copy by e-mail, my address is

larson@stolaf.edu

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