I had some experience of teaching group theory that I like to tell you
I used to give to my students to compute (even by hand when no computer was
possible) concrete examples (full computations): I think that this is the
right way to make them ASSIMILATE the theoretical concepts. But, of course,
the task is horrible when the groups have order about (more or less) than
30 elements, while many elementary concepts cannot be illustrated by
so small groups.
More over, I found that the students are really exited if they have to
"guess" what group they have while this group is defined by an abstract
presentation. The only thing that they discourage them to continue
investigations is of course a long, perhaps endless, hand calculus.
That is why I made a few years ago (before finding GAP or handling
any symbolic system) a "small" program for investigation "small groups"
(I call it Hijara=little stones, to make people remember they learned
the first things in arithmetics with little stones). These groups may be
permutation groups, abstract groups, "modular groups" (=units of Z/nZ)
or matrix groups with coefficients in some Z/nZ (matrices are not yet
Now with such system, one can do elementary investigations in reasonable
time, on groups with some hundred elements and main elementary concepts
may be illustrated. For bigger groups, the program take a very long time
(some hours for groups of order = some thousands ).
The membership test is very expensive...and my implementation is
not optimized at all.
The fact that some groups to investigate are abstract make them return to
the theory for solving problem of how to handle that group. For them,
it is an "open question". There is no standard way to solve problems of
this nature, and all theoretical tools are potentially useful for
solving that problem.
The solution may be given later by the professor who is assumed to know
what group it is.
So the philosophy is that the computer as well as the book have the same
chance to make students thinking about what they are studying. We have only
to find the adequate pedagogy for using computer and any other tool
for communication. More over, the students need always at some moments...
professors...and this is not a discovery.
M. Taoufik Karkar