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Re: if using Mathematica to solve an algebraic problem

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  • Subject: [mg108980] Re: if using Mathematica to solve an algebraic problem
  • From: "David Park" <djmpark at comcast.net>
  • Date: Fri, 9 Apr 2010 03:31:11 -0400 (EDT)

Sometimes I find it difficult to understand these discussions.

For example, Richard's: "There is of course the possibility that something
really useful will be developed that will make it possible to teach all students
everything they need to know." What kind of something would that be, and in
what way would it make it possible? It seems like a rather vague but
expansive goal.

Similarly, David suggests that Mathematica might be "terribly dangerous"
with the possibility of "becoming skilled in answering questions through
Mathematica, rather than actually learning the subject!" (Not always bad. Is
there anything wrong at becoming skilled at driving to various locations in
your city without actually learning how the internal combustion engine
works?)

It's all a matter of how one uses Mathematica. WRI has left plenty of
opportunities for using Mathematica in a way that does engage the
mathematics. The problem is that the default tendency for new users is to
use it as a super graphical calculator or as a programming language aimed at
obtaining a result for some specific calculation. Fine, that's one way of
using Mathematica and there is nothing wrong with it.

But another way to use Mathematica is to try to set up the rules or axioms
for some subject matter and then practice using them to carry out various
derivations or prove various theorems. Mathematica may do the dog work but
you have to decide the steps and see why various axioms are necessary. There
might be various ways to do this for teaching. Should the teacher provide
the axioms and the student just uses them? Or should you start with a vague
discussion of some subject and have the students and teacher together
develop the axioms, something like the Math Circles?

There is a lot that can be done and there are some places where WRI could
make things easier for this approach. (For example I find that SyntaxForm in
InterpretationBox often causes crashes and this causes problems in trying to
implement manipulation of unevaluated nested sums or integrals. There is not
enough ability for users to set automatic Precedence for their own defined
functions.) But there is no reason why things can't be set up to actually
think about and do math, and not just grind out calculations.


David Park
djmpark at comcast.net
http://home.comcast.net/~djmpark/  







From: David Bailey [mailto:dave at removedbailey.co.uk] 


Richard Fateman wrote:
> Yet.  There is of course the possibility that something really useful 
> will be developed that will make it
> possible to teach all students everything they need to know.  
> Experiments in "computer aided instruction"
> have not been all that successful, although there are a few (e.g. there 
> is a statistics course at Berkeley
> with computer technology in learning and testing;  there are many ways 
> of making computer programming
> "more automated" that help teach the subject.)

It is many years since I was doing calculus courses myself - all we had 
then were "scientific calculators", which were amazingly accurate 
compared to slide rules, but I must say, I would have hated the idea of 
"computer aided instruction". Learning - at least for me - had to be a 
somewhat disorganised process punctuated with the occasional "Aha!".

I also think that if Mathematica had been available to me back then, I 
would have felt it was a bit like an addictive substance - very 
interesting in small doses, but also terribly dangerous. There would 
have always been the possibility of becoming skilled in answering 
questions through Mathematica, rather than actually learning the subject!

David Bailey
http://www.dbaileyconsultancy.co.uk




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