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Re: Mathematica as a New Approach to Teaching Maths

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  • Subject: [mg127529] Re: Mathematica as a New Approach to Teaching Maths
  • From: "djmpark" <djmpark at comcast.net>
  • Date: Wed, 1 Aug 2012 04:58:31 -0400 (EDT)
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MathGroup followers might also be interested in the following piece by V.I.
Arnold:

http://pauli.uni-muenster.de/~munsteg/arnold.html   

I like this part toward the end: "Every working mathematician knows that if
one does not control oneself (best of all by examples), then after some ten
pages half of all the signs in formulae will be wrong and twos will find
their way from denominators into numerators."

Doing examples and avoiding such errors might be something Mathematica would
help with.

There is no reason one can't do things step-by-step with Mathematica. It's
certainly the best way when first approaching a new topic.


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




From: Alexei Boulbitch [mailto:Alexei.Boulbitch at iee.lu] 



There is a letter written in thirties by the great Russian theorist, Lev
Landau, to the rector of a technical university, where Landau taught at the
time. The letter discussed  useful and useless (or even evil) parts of the
content of mathematics curriculum for physicists and engineers. This letter
is rather well known in Russia, but, I guess, is absolutely unknown outside,
in part due to its language: it is written in Russian, and partially due to
political difficulties of the time when it has been written.

This letter claims, in particular, that mathematicians load students by what
Landau ironically called "the mathematical lyrics" instead of teaching them
to get to the point of their calculations. The lyrics for him were multiple
theorems together with their proofs heavily inserted into the course,
especially the multiple existence theorems. Nothing changed , however, since
the Landau letter has been written and made public, and when 40 years later
I was taught the university mathematics, I had to learn an impressive amount
of such a lyrics, which I almost NEVER then used during all my life in
theoretical physics. In contrast I was very poorly taught to calculate, and
if I can do it now, it is in spite, rather than due to these mathematics
courses.

Reading the Landau letter it is amazing, how much in common it has with
ideas of the talk Konrad Wolfram gave on occasion of the Mathematica
Symposium at London this summer. Of course, at Landau time no computers (let
alone, the computer algebra) were available. Now with all this at hand,
Landau approach can be applied on the new level, as we all here understand.

Fred Simmons in his paper
(http://alexandria.tue.nl/openaccess/Metis217845.pdf ) states in particular
that in some cases a straightforward application of Mathematica functions
may be not enough and some "deeper" understanding of mathematical properties
staying behind are necessary to go to the successful end. Here the term
"deeper" (I believe) should be taken again in the sense of having an idea of
how this could be calculated "by hand", and with this knowledge to be able
to make it "on screen". I think the argumentation of Prof. Simmons in the
paper was also in favour of such type of a deepening. However, one may
understand the word "deeper" also in the "lyric" sense. In my own practice I
still met no case where any deeper knowledge of theunderlying "lyrics" was
necessary, but often need to go deeper in the first sense.

However, one statement seems to contradict the other, at least to some
extent.  This contradiction, however, may be solved by letting
mathematicians teach mathematics going  "in depth" in any sense. Instead one
can move Mathematica-based teaching to other courses that are (i) heavily
based on mathematics, but are (ii) calculation (rather than proof)-oriented.
One course fitting to these requirements is, of course, physics, and to go
this way one might start with the Experimental Physics taught during the
very first semester. Some minimal Mathematica knowledge might be introduced
in its very beginning, and one may go on gradually introducing Mathematica
functions and ideas "on demand". No need to say that problems and tests to
the courses should be done in Mathematica. That may be the way around. About
the end of the Experimental Physics courses the students should be able to
use Mathematica themselves to solve their problems and should have such a
habit.

I would like to especially stress that as much as I see, the massive use of
Mathematica during lectures should not exclude the "talk and chalk"
sessions. The latter are important since the step by step calculations done
slowly in front of students have a potential to demystify science, and this
is important.

Of course, in order to be able to realize that program one needs a class
equipped with computers and Mathematica license, and also home licenses for
the students involved, as =A9er=FDch Jakub notes in his post.
Finally, this approach is, of course, not applicable to all types of
students, but only to some of them. Say, to those adhering to Physics or
Engineering. But one should start with something. May the analogous approach
be done say, with chemistry or biology courses? I do not know, I never
taught any of these. Somebody else may be able to  answer this question.




Alexei BOULBITCH, Dr., habil.
IEE S.A.
ZAE Weiergewan,
11, rue Edmond Reuter,
L-5326 Contern, LUXEMBOURG

Office phone :  +352-2454-2566
Office fax:       +352-2454-3566
mobile phone:  +49 151 52 40 66 44

e-mail: alexei.boulbitch at iee.lu<mailto:alexei.boulbitch at iee.lu>




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