Re: Mathematica as a New Approach to Teaching Maths
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- Subject: [mg127527] Re: Mathematica as a New Approach to Teaching Maths
- From: Andrzej Kozlowski <akozlowski at gmail.com>
- Date: Thu, 2 Aug 2012 04:49:21 -0400 (EDT)
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I think you have somewhat misunderstood my point about non-existence etc. I agree that physicists usually have good intuition about the existence or non-existence of the objects they try to compute. Your joke refers to it but it really misses the point. The point is that physicists use mathematics and quite often use mathematics that they do not understand perfectly. In such cases, they quite often try to do things that can't be done with these tools and sometimes can be done with others, and that is not something for which physical intuition is sufficient. Take a simple but actually quite profound example. It is usually easy to decide the existence of (say, real) solutions of polynomial equations by "empirical means"- e.g. by plotting a graph. If numerical computations were all physicists ever did that would be enough, but actually they are quite often interested in "explicit solutions". The fact that there can be no explicit formula in terms of radicals for roots of a general polynomial can not be established by any experiment and yet it is very useful to know, if not for other reasons than the one that it tells you that you should not look for one and use other methods instead. In fact analogous things can be done for differential equations and are of great use. The algorithm that Mathematica uses to compute many complicated integrals makes use of Galois theory, which itself is an excellent example of "mathematical lyrics". Also, everybody should remember that group theory, which arose partly from Galois theory, was also famously once declared by a famous physicist (James Hopwood Jeans) to be part of "mathematical lyrics" which would never have any conceivable use in physics. As for Arnold's article, I have quoted it myself on this forum (I have the Russian original which is much longer and contains other interesting points not included in the English translation). You can also take a look at my "obituary" of Arnold that I posted here: http://forums.wolfram.com/mathgroup/archive/2010/Jun/msg00067.html By the way, Arnold was an enthusiast of teaching Abel's theorem (on the impossibility of solving polynomial equations of degree 5 and higher) to high-school children and he produced a complete proof that starts from nothing and includes everything that needs to be known http://forums.wolfram.com/mathgroup/archive/2009/Apr/msg00639.html Andrzej On 1 Aug 2012, at 10:57, Alexei Boulbitch wrote: > 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 = superconductivity = (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: Re: 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 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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