How (not) to teach math
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- Published on 2012. december 11. kedd, 12:55
Arnold's thoughts are in line with those of my own
(The following are quoted from the first section of the article Vladimir I. Arnold "Topological problems of the theory of wave propagation, Russian Math. Surveys 51:1(1996), 1-47")
An apologia for Applied Mathematics
(Presented at the opening of the Congress on Industrial and Applied Mathematics at Hamburg, 3 July 1995.)
A common (though commonly suppressed) opinion both of pure mathematicians and theoretical physicists concerning 'industrial and applied' mathematics is that it consists of a mafia of weak thinkers, unable to produce any important scientific results, but simply exploiting the achievements of pure mathematicians of past generations, and that the members of this mafia are more interested in cash than in science and are hopelessly corrupted by this.
"They are so modest" a pure mathematician once said "that they do not hope to achieve anything in a direct honest way; they distance themselves from mathematicians simply to avoid honest competition".
I do not think that this characterization of applied mathematics was completely deserved. The achievements of Galileo devoted to business evoke no less admiration than the results of the pure philosopher Pascal.
The difference between pure and applied mathematics is not scientific but only social. A pure mathematician is paid for uncovering new mathematical facts. An applied mathematician is paid for the solution of quite specific problems.
Example. Columbus began by making a purely applied study, trying to find the way to China, and he was being paid for this. The end of his voyage is reminiscent of a fact of pure mathematics. Note that the immediate direct benefit to the Spanish economy of the discoveries of Columbus was far less than that from the coastal navigation of ordinary captains.
Mayakovsky has well described the difference between pure and applied mathematics in "How to make poetry". "The man who found that twice two is four was a great mathematician even if he found it out by counting cigarette stubs. Those who now calculate by the same formula much greater things, for example locomotives, are not mathematicians at all."
The theory of algebraic curves over finite fields has now become applied mathematics, financed by the CIA, the KGB and other similar organizations. Fermat's problem would also be applied if its solution were of monetary value. Many mathematicians of the twentieth century have warned of the dangers of dividing mathematics into parts. Hermann Weyl has written: "In our time the angel of topology and the devil of abstract algebra are fighting for every mathematical domain" (H. Weyl, "Invariants", Duke Math. J., 5 (1939), 500.).
In the first half of the century the devil was winning. After Lagrange, who banished pictures from mathematics, the algebraists and axiomatisers arrived— first Hilbert and then Bourbaki.
Example. Define the product of natural numbers by means of the algorithm of multiplying 'by columns'. Then the commutativity of multiplication is a theorem that is hard to prove. Afterwards one may force school children (or students) to study the formal proof of this theorem, terrorizing them and raising to an unparallelled height the authority not only of teachers but of their sciences also.
Generations of mathematicians were taught by this method, not having any contact at all with any other kind of mathematics. As a result they cannot understand any other science and enthusiastically busy themselves with tedious details of generalizations of achievements of their teachers of little interest.
Hilbert espoused the democratic principle according to which each system of axioms had an equal right to be studied, while the value of a mathematical achievement was determined only by its difficulty, as in alpine rock-climbing. As a result, there came about a divorce of 'pure' mathematics from all sciences, a system of mathematical education, criminal against those taught, and the image of mathematics in the common mind was of a dangerous parasitic sect on the body of science and technology, consisting of priests of a dying religion like the druids.
Landau said: "Why is it that mathematicians add prime numbers? Prime numbers were created to be multiplied" (the theorem on the representation of any sufficiently large odd number as the sum of three prime numbers is considered to be one of the greatest achievements of mathematics).
Taking revenge on humiliation undergone at school, the administrators in most countries, like pigs under an oak tree, are busying themselves now, after the abatement of warlike preparations, in redoubling their efforts to annihilate mathematics, especially 'pure' mathematics.
The USA government recently discovered that 85% of mathematicians, already existing or under training, especially those that are 'pure', are not essential to the country. Without star wars neither supercolliders nor mathematicians are required. Different projects are under consideration on how to reduce the number of mathematicians sevenfold. American specialists reckon that this will take ten years.
Unfortunately one has to acknowledge that it is 'pure' mathematicians who have been entirely responsible for creating with their own hands the general opinion that has been described. The axiomatic-deductive method, having led to the banishment of all examples (and especially the motives for introducing definitions) in the teaching of mathematics at every level is above all responsible for this.
R. Feynman clearly described this method of teaching ('leading to a state of self-propagating pseudo-education') in his book "Surely you're joking! Mr Feynman" (Surely you're joking, Mr. Feynman!, Bantam Books, Toronto-New York 1986, pp. 191-198.), describing physics teaching in Brazil in the first postwar years. Reading Feynman I quite often realized with embarrassment how close this 'Brazilian' method of teaching physics was to our teaching of mathematics.
R. Feynman gave this example. The lecturer declares that the moment of inertia of a material point relative to an axis is the product of the mass by the square of the distance from the axis. The students write down the definition. It seems that all is well. But Feynman explains that such teaching is completely unacceptable.
It is necessary to explain that a weight fastened to a door near its hinge hardly affects the opening of the door while one fastened to the handle interferes with it strongly. Without such an example the definition is meaningless - it only helps one to answer the examination question 'formulate the definition of moment of inertia'. Those who wish may find examples, illustrating the helplessness of students taught in this way, in Feynman's book.
Here are some examples from my experience of teaching in Paris. At the time of a written exam a student of the fourth course said to me: "I have forgotten my calculator—tell me, please, is 4/7 greater than or less than one?" Four-sevenths was the number on which depended the convergence of the integral, governing the behaviour of the dynamical system under study. This was a good student, but here it was simple fractions that he had evidently studied in the French version of the 'Brazilian' method!
Students of the École Normale in Paris have asked me: "Why do you call the ring of formal power series local? Does it really satisfy the axioms of a local ring?" For the non-specialist I must explain that this question is analogous to the question 'why do you call a circle a conic section?' These were the best students of mathematics in France. Clearly, some criminal algebraist had taught them the axioms of a ring (and even those of a local ring), without providing them with any examples (and, in particular, without explaining the origin of the term 'local').
After a mistake in the initial conditions the phase curve of a Hamiltonian system on the plane appeared not as the separatrix of a saddle but as a closed curve. This led to the following solution of the problem on determining the limit of the solution as t tended to infinity: 'by Theorem 45—in a written exam one can appeal to any source—there exists $T>0$ such that the value of $\phi(T)$ of the solution at the moment $T$ is equal to the initial value'. After this it was quite strictly proved by induction (using a uniqueness theorem), that $\phi(nT)=\phi(0)$ for any arbitrary integer $n$ .
From all this there comes the irreproachable result: 'the limit is equal to the initial value'! From the point of view of deductive-axiomatic mathematics in all these considerations there is not a single mistake: the error lies only in the formulation of the problem.
It is clear, however, that the writer does not understand anything, but only knows how to prove. The absurdity and even the criminality of a system of education, leading clearly intelligent people to such a state, seems to me to be obvious. For 'applied' work such 'knowledge' is useless and even dangerous (the consequences may take on the character of a Chernobyl disaster).
The aim of a mathematical lecture should be not the logical derivation of some incomprehensible assertions from others (equally incomprehensible): it is necessary to explain to the audience what the discussion is about and to teach them to use not only the results presented, but—and this is major—the methods and the ideas.