Dear Forum members,
I want to thank Joe Kinkaid for his letter to the Forum about using
GAP in teaching a 'Modern Algebra' course, the very explicit material
about the course given on the WWW pages quoted have been particularly
helpful to understand what was done. As I have said in previous
letters to the Forum, I think by far more material on the use of a
system such as GAP in teaching should be produced and made available.
Let me first comment on a few remarks in Joe's letter:
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
I am not so surprised, most students nowadays bring along even from
highschool time some practice of handling a computer and know some
basics about computer languages, so they should not find the language
of GAP intimidating - we of the older generation may just project our
own lack of such preknowledge onto the kids here. On the other hand
students who have chosen math as their first topic will be more
intrigued by the possibility to really handle abstract structures.
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.
I do agree wholeheartedly.
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.
I do agree again, making a system such as GAP as familiar a tool as a
pocket calculator and hence starting to use it early seems to be
right. In fact some of the first exercises given on Joe's WWW pages
concern just handling lists in GAP and that clearly needs no
mathematical background but can on the other hand already be used for
'Discrete Structures'. Thus in particular one can come to groups etc
in the 'Modern Algebra' more directly.
Our course system is different of course, but we do encourage our
first year students to use Maple as well as GAP for Linear Algebra,
which is our first course, already.
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.
Agreed again. May I suggest two closely related topics that provide
'real' groups: crystallographic groups and finite reflection
groups. For the second topic the book by L.C. Grove and C.T Benson is
a very nice introduction. We use it in our first year, second term, as
a text for voluntary 'proseminars' for very interested students, with
the aim of really going to the classification of finite Coxeter
groups. That would be by far too much for a course on 'Modern Algebra'
as Joe describes, but chapter 2 (only running up to page 27)
introduces the two- and three dimensional groups and that already
provides infinitely many examples including (binary) tetrahedral
octahedral and icosahedral group. It also gives rise to many examples
of isomorphic groups coming in different geometric meanings and thus
gives excellent motivation for understanding that crucial notion that
students often have difficulties to grasp, because in reality they see
almost always just one specimen from each isomorphism class.
On page 27 of that book the students moreover have seen a famous
example that mathematics (here algebra of course) allows to *predict*
all possible symmetries of any crystal that might be found in the
wilderness provided a crystal is a substance with a lattice of
molecules. That can be made a great experience, and a system such as
GAP can be seen as instrumental in really looking at the subgroup
relations of these symmetry groups.
Another approach to crystallographis groups is via studying
2-dimensional wallpaper designs. This way you can even bring in
infinite groups of affine mappings, the space groups (Bieberbach
groups). Fortunately the new GAP share package CrystGAP allows to
handle these quite well.
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.
Textbooks often do work for many pages with these small examples
because by hand calculations you could not go much further up. I have
seen a text that uses a very primitive system for groups and then just
does the same. I think that is wrong. For all my prejudice in favour
of GAP, I think one should still make the students do the difference
of Z6 and S3 by hand, (in fact that's what I do) but then indeed use
GAP for bigger examples. And for the purpose of a "Modern Algebra'
course A5 coming as the icosahedral group can already be considered as
'big'. I do not know a textbook that does this - maybe somebody is
challenged to write it?
I have at present two students working for their 'Staatsexamen',that
is the qualification for becoming (German) highschool teachers in
math. For this they have to write a thesis and the topic that I have
given them is to develop some sets of exercises on various aspects of
groups. The theses are in German and so far there are only drafts,
which therefore I do not wanty to distribute generally yet. However if
somebody is interested, and writes to me, I will make the files
available to him or her. There is a vague plan to translate these into
English and rework them, but do not take this remark as a promise.
I would appreciate any comments or advice anyone has regarding this class.
Hope that some of the above is useful for somebody. Kind regards,