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Re: Mathematica as a New Approach to Teaching Maths
*To*: mathgroup at smc.vnet.net
*Subject*: [mg127521] Re: Mathematica as a New Approach to Teaching Maths
*From*: Andrzej Kozlowski <akozlowski at gmail.com>
*Date*: Wed, 1 Aug 2012 04:55:50 -0400 (EDT)
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Although I agree with a lot of what you write, I would like to point out a couple of exceptions.
Firstly, even following this forum for a while should make you realise how important "lyrics" such as existence theorems and even more so non-existence theorem are for practical computations even with Mathematica. As evidence I can cite numerous posts to the forum from people who attempted to find solutions to problems (explicit solutions of certain equations, Laplace transforms etc) where knowing a suitable non-existence theorem would have solved them a great deal of time and trouble and perhaps even directed them to some more worth-while problem.
Another issue, which you seem to ignore, is the huge importance of geometry and geometrical thinking, which can led to enormous simplifications in computation or even to solutions of problems where computational approaches are hopeless. This happens not just in mathematics but a great deal in physics particularly in the work of relativists (such as Roger Penrose, Stephen Hawking, Edward Witten etc).
Andrzej Kozlowski
On 31 Jul 2012, at 04:14, Alexei Boulbitch wrote:
>
> 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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