RE: [Axiom-mail] tracking the solver

[Bill Page]

On December 7, 2005 4:39 PM Jurgis Pralgauskis wrote:
> ...
> can I get step by step output (as we are taught in calculus
> studies) instead of just the final answer?

One thing that you should keep in mind is that for many (most?)
mathematical problems, the type of symbolic manipulation that
is done by the computer to solve the problem is quite different
than the usual approach used by a human being. For example,
how often have you used Grobner basis methods when solving
an equation by hand? Similarly the symbolic integration
algorithm is quite unlike the methods typically taught in
calculus studies.

So understanding Axiom's solution method, step by step will
certainly teach you about doing symbolic mathematics by
computer, but I have some doubts about it's relevance to
learning mathematics in general.

Systems such as Maple and Mathematica usually attempt to
address this issue by implementing additional ways of solving
certain problem and also a set of semi-manual methods for
manipulating expressions (sometimes called "student methods")
which are good for learning but are seldom as efficient as
the usual computer algebra algorithms. Certainly this approach
is also possible in Axiom, but as far as I know, very little
work of this kind has been done yet.

> 
> or is it better to ask such questions in axiom-developer
> mailing list? thanks in advance

I think axiom-math is the right place to ask this kind of
question.

> 
> ps.: the idea behind this is that I want to show our
> lecturers, that swotting the formulas or methods is no
> big use in XXI age.

What is the word "swotting"?

If you mean that symbolic computer algebra systems are now
useful for a very wide range of applications and that they
can solve many problems that would be (nearly) impossible
or at least quite impractical by hand, then I agree. But I
also think that these systems have a long way to go before
they can be presented as "expert mathematicians" in the sense
of an artificial intelligence. All of these systems, including
the most expensive commercial systems, should come with a
large sign to put on your desk that says: "Use at your own
risk!" ;)

> The only thing you have to think is to look through the
> formulas and find the pattern, according to which solving
> goes on. The two first semesters of higher math is mainly
> swotting the methods and no big imagination training -  I'd
> say, why are we taught to invent bicycles?
> 

Well, why are you even taught how to manually manipulate
the "symbolic" decimal notation for integer addition and
multiplication? I think the answer at least in part, is that
it is very difficult to teach people how to "think" (because
mostly we do not know how we think) so instead we teach them
how to calculate and hope that some of them - mostly by
accident - might also start to think. ;)

BTW, very little of the calculation that Axiom does is based
on "finding patterns".

Regards,
Bill Page.