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Re: Mathematica as a New Approach to Teaching Maths
*To*: mathgroup at smc.vnet.net
*Subject*: [mg127515] Re: Mathematica as a New Approach to Teaching Maths
*From*: Alexei Boulbitch <Alexei.Boulbitch at iee.lu>
*Date*: Thu, 2 Aug 2012 04:45:21 -0400 (EDT)
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Dear Andrzej, dear Community,
There is an old joke we used to tell in my student years. It is as follows.
The same mathematical equation is given to a physicist and mathematician.
A week later the physicist comes with a solution, while mathematician comes
and reports to have proved that a solution exists. When the physicist have
heard this he told: "If I thought that it does not have one, I wouldn't bother myself to solve it".
At this point everybody laughed.
Now let me switch to a serious way. Thank you Andrzej for having made this note. This will enable me to formulate few more things and to make some previous statements clearer. It is related to a very serious problem, and I feel it necessary to discuss it. I hate philosophy, but to answer I have to start with a little bit of philosophy.
In physics we have another arguments to prove the existence. I would call them experimental. Equations that we solve come from experimental facts. It is a common practice that physicists formulate and solve their equations well before mathematicians turn their attention to these equation. A good example would be, say, the Ginsburg-Landau equation, or the Gell-Man and Low equation. There are also other examples. Equation may be formulated incorrectly by some researcher, but if it is of importance, it will be very soon corrected by community, since many scientists typically attack important problems at a time. A wrong solution may be found or inexistent solution may be searched for, but it will be also corrected for the same reason. But as soon as (using such united efforts) the problem has been formulated correctly, it has a solution, since it describes an observable phenomenon also established experimentally by the united efforts. A good example is the high Tc superconductivi
ty (http
://en.wikipedia.org/wiki/High-temperature_superconductivity) as the example of a correct experimentally established problem and of the "cold fusion" (http://en.wikipedia.org/wiki/Cold_fusion ) as of the incorrect one.
David Park has yesterday let me know about a nice article written by V.I. Arnol'd. I was astonished and delighted to find in his paper the point of view that mathematical statements (such as axioms) are experimental facts. This is true and evident, of course, but that is not how mathematicians use to look at it. And this is quite close to what I think about physical level of the existence proofs. You can find his article in English translation here:
http://pauli.uni-muenster.de/~munsteg/arnold.html
David, let me thank you once more for this article.
You did get me not quite correct, Andrzej: I understand the general use of the so-called, "mathematical lyrics", though I personally never proved any existence theorem. I am sure that it is very good to have a solid knowledge of some mathematical fact, such as i.e. that some equation has no solution . In fact this knowledge, if any, gives one a more solid ground to hope that a new problem (a freshly formulated one) may correctly reflect the nature. Only "may", not "is correctly reflecting", since the existence of a mathematical solution alone does not yet guarantee the correctness of the model.
However, it can only very rarely be used in our practical life. The reason is that only few of us have an outstanding luck to formulate a completely new problem or a new equation. One needs to be of a calibre of Schr=F6dinger or Landau or equally great, and simultaneously, so-to-say, your area should be mature enough for such a new equation. I should admit that it is not my case and, probably, today not that for the area of physics, where I work. Most of us solve equations that are already known, and only need slight modifications. This, of course, also gives rise to the question of the existence of solution, but they are typically coped with the way I described above.
During my uni years in USSR I had to learn an incredible amount of philosophy of Marxism-Leninism and a huge amount of mathematical theorems of existence. My teachers in the both subjects pretended that I will constantly need the both. Now having completed the most part of my professional life, I see that I have (almost) never used the both. And no of my classmates did, except for few who have chosen to become functionaries of the Communist Party(those few heavily used Marxist lexicon learned at philosophy lectures). So I conclude that the valuable time of myself and of most of my classmates has been wasted.
Apart from emotions, this took place for the simple reason that all of us were stubbornly taught things that were necessary for only few of us. What I think could be done with it, I have already written in my previous post. In short: to Caesar what is Caesar's, to God what is God's. Those whose aim is Mathematics itself, should learn Mathematics. Those who need to apply Mathematics should only become aware of most important mathematical facts, the level of this awareness being different for different classes of people: physicists, engineer, biologist etc. They should instead heavily learn to operate mathematics. This is the attitude of Landau that I cited in the previous post. However, it should be done on the contemporary level, which is today different from that only 15 years ago. This is more or less close to the position of Konrad Wolfram, as I understood it during his talk at the IMS 2012.
There is one serious obstacle for that, however. It is the attitude of pedagogical authorities. Indeed, assume that a child is trained at school to use Mathematica or whatever else in his everyday life, and is not at all (or not enough) trained to make calculations by hand. What he will do during exams, if computer will not be admitted? Will such a person be admitted to the uni after flushing the school examination for that reason?
It is not an abstract question for me. My younger son is 10 at present, and I am going to teach him to use Mathematica very soon. As soon as he starts to have algebra at school. I need to find the way of how to manage that, since to teach him something extra to his school course in parallel to the regular homework may be a heavy burden.
I guess that to solve this problem (not on my personal level, but on a more general one) the pedagogical community should first be convinced to at least make such an experiment with clearly stated conditions of passing the examination with the computer equipped by Mathematica.
Thank you, Andrzej, once more for your comment,
Best regards, Alexei
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
-----Original Message-----
From: Andrzej Kozlowski [mailto:akozlowski at gmail.com]
Sent: Tuesday, July 31, 2012 10:00 PM
To: Alexei Boulbitch
Cc: mathgroup at smc.vnet.net
Subject: [mg127515] Re: Mathematica as a New Approach to Teaching Maths
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 com=
putation or even to solutions of problems where computational approaches ar=
e hopeless. This happens not just in mathematics but a great deal in physic=
s particularly in the work of relativists (such as Roger Penrose, Stephen H=
awking, 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 lette=
r is rather well known in Russia, but, I guess, is absolutely unknown outsi=
de, in part due to its language: it is written in Russian, and partially du=
e to political difficulties of the time when it has been written.
>
> This letter claims, in particular, that mathematicians load students by w=
hat Landau ironically called "the mathematical lyrics" instead of teaching =
them to get to the point of their calculations. The lyrics for him were mul=
tiple theorems together with their proofs heavily inserted into the course,=
especially the multiple existence theorems. Nothing changed , however, sin=
ce the Landau letter has been written and made public, and when 40 years la=
ter I was taught the university mathematics, I had to learn an impressive a=
mount of such a lyrics, which I almost NEVER then used during all my life i=
n theoretical physics. In contrast I was very poorly taught to calculate, a=
nd 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 i=
deas of the talk Konrad Wolfram gave on occasion of the Mathematica Symposi=
um 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 a=
pproach can be applied on the new level, as we all here understand.
>
> Fred Simmons in his paper (http://alexandria.tue.nl/openaccess/Metis21784=
5.pdf ) states in particular that in some cases a straightforward applicati=
on of Mathematica functions may be not enough and some "deeper" understandi=
ng of mathematical properties staying behind are necessary to go to the suc=
cessful end. Here the term "deeper" (I believe) should be taken again in th=
e sense of having an idea of how this could be calculated "by hand", and wi=
th this knowledge to be able to make it "on screen". I think the argumentat=
ion of Prof. Simmons in the paper was also in favour of such type of a deep=
ening. However, one may understand the word "deeper" also in the "lyric" se=
nse. In my own practice I still met no case where any deeper knowledge of t=
heunderlying "lyrics" was necessary, but often need to go deeper in the fir=
st sense.
>
> However, one statement seems to contradict the other, at least to some ex=
tent. This contradiction, however, may be solved by letting mathematicians=
teach mathematics going "in depth" in any sense. Instead one can move Mat=
hematica-based teaching to other courses that are (i) heavily based on math=
ematics, but are (ii) calculation (rather than proof)-oriented. One course =
fitting to these requirements is, of course, physics, and to go this way on=
e might start with the Experimental Physics taught during the very first se=
mester. Some minimal Mathematica knowledge might be introduced in its very =
beginning, and one may go on gradually introducing Mathematica functions an=
d 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 Mathem=
atica 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" sess=
ions. The latter are important since the step by step calculations done slo=
wly 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 stud=
ents, but only to some of them. Say, to those adhering to Physics or Engine=
ering. But one should start with something. May the analogous approach be d=
one say, with chemistry or biology courses? I do not know, I never taught a=
ny 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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