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Hi,

This is mostly about using GAP in teaching, but at the bottom I have a

question about writing *.g files.

I used GAP to help teach Math 306 -- Modern Algebra this spring semester

and I wanted to share my experiences with you. The eight students in this

class are sophomores, juniors and seniors with a variety of backgrounds. In

case you would care to see it, the entire syllabus is available through the

home page for the class at http://www.peru.edu/~kincaid/math306. As it says

there, we used Hillman and Alexanderson's _Abstract Algebra_ _A First

Undergraduate Course_ as the textbook. I am not familiar with Gallian's

textbook, though I have seen mention of it on this list.

In a letter to GAP-forum in December of 1993, Bill Haloupek of University

of Wisconsin-Stout commented that GAP is pretty intimidating to

undergraduate students. I found this to be generally true as well. We teach

this class for both Math majors and CS majors and I thought the CS majors

would take to GAP more quickly than the math majors did. I was surprised to

learn that this was wrong. My best students were those with strong

mathematics backgrounds and some computer science background. Those who

were weaker in mathematics, even with good computer science backgrounds,

found GAP difficult to work with. I hasten to add that with only eight

students in the class, my experience may not be indicative of the general

situation.

I don't think GAP _has_ to be intimidating. I believe it could be very

easily accessible if it were introduced to the students properly. I am not

sure how to do that, but I do have an idea for next year. See below.

The class met on Tuesdays and Thursdays. On Tuesdays, we met in a classroom

and discussed theorems and examples. On most Thursdays, we met in the

computer lab and the students worked through handouts I had written showing

how to use GAP to solve problems in the appropriate sections of the

textbook. Since I was learning GAP while I was writing those handouts, I am

painfully aware that there are many improvements which could be made. You

can find my handouts on-line through the class home page URL given above.

We had a midterm exam and a final exam. Both were held in the computer lab

and the students were encouraged to use GAP to solve the exercises on the

exam. The review sheet for the midterm exam and GAP macros to compute an

exam key for each of the tests (each student took a different exam) are

also available through that page.

This fall, I will be teaching Discrete Structures and next spring, I will

teach Modern Algebra again. I intend to introduce GAP to the students in

Discrete Structures (GAP was not used the previous time D.S. was taught)

and then build on that in Modern Algebra. Hopefully, by introducing GAP as

a programming language for solving discrete math problems first it will be

easier to introduce the group theoretic functions second. Also, between

now and then I will revise the presentations in Modern Algebra using what

I learned from this semester.

What did I learn this semester? Well, I'm not sure. One observation which

I'm still refining for myself is that students at this level need

motivation for the ideas beyond the search for beauty, truth and

understanding. The next time I teach this course, I hope to include more

motivation at the beginning semester for why we have different algebraic

structures in the first place.

Another observation regards the interaction between the textbook and GAP

(minimal actually). I find that students usually respond at or just below

the level I expect them to regardless of where I expect them to respond at.

We spent _far_ too much time this semester talking about the differences

between Z6 and S3 because the textbook spends a lot of time on groups that

small. I wanted to use GAP so that we could do more interesting examples

and calculations beyond S4 and A5, but I was following the text waiting for

it to take me there. Well, the textbook starts small and works up. I think

it would be better to start with a larger group and work down. Any

conjectures or ideas could be worked through on GAP or with smaller groups,

but we would need a proof to make any conclusion. I realize these ideas are

rough, but I'm still working them out for myself.

I would appreciate any comments or advice anyone has regarding this class.

Finally, I am trying to write my first GAP library routines. At the bottom

of the GAP page from the Math 306 web page given above is a link to a file

"grpring.g" which contains some functions I have written trying to work

with group rings. As I write these, I am trying to use the standard

formatting used in the other library files, but I don't understand all of

the coding in them. For example, the second column after a comment

character ('#') is often a letter. What do those letters mean? Is there

anyone out there who can give a quick overview of the style used in those

files? Are there any optimization considerations I should make note of

early in the creation process?

Has anyone ever tried to define group rings this way before?

I am continually working on that file and it barely does anything now, so

if you do take a minute and look it over, you may want to look it over

again a minute later to see what's new.

Thanks for your help.

Joe

-----

Joseph Kincaid | Mathematics is the alphabet with which God

kincaid@pscosf.peru.edu | has written the universe.

kincaid@hilbert.math.ksu.edu | -- Galileo

http://www.peru.edu/~kincaid | (except he said it in Italian, of course.)

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