Written: May 5, 2005.

Truly great mathematicians (and scientists, of course) are also great teachers. Hence it it is not surprising that two of my heroes, representing completely different kinds of mathematics, have similar teaching philosophies.

The first one, due to my colleague and hero Israel Gelfand, I will call the Gelfand Principle, which asserts that whenever you state a new concept, definition, or theorem, (and better still, right before you do) give the SIMPLEST possible non-trivial example. For example, suppose you want to teach the commutative rule for addition, then 0+4=4+0 is a bad example, since it also illustrates another rule (that 0 is neutral). Also 1+1=1+1 is a bad example, since it illustrates the rule a=a. But 2+1=1+2 is an excellent example, since you can actually prove it: 2+1=(1+1)+1=1+1+1=1+(1+1)=1+2. It is a much better example than 987+1989=1989+987.

The Gelfand Principle should also be used in research articles. It is much easier to follow a new definition or theorem after a simple example is first given. Even proofs would be easier to follow if they are first spelled out concretely for a special case.

The other principle is closely related but of an even greater scope. It is due to another hero of mine, Bruno Buchberger, of Grobner bases fame, the pioneering giant of Computer Algebra. Buchberger, like Gelfand, is also very much involved in pedagogy, and his brainchild THEOREMA, developed with his RISC-Linz team, is a great computer-assisted learning tool for humans.

Many traditional mathematicians dislike computers, and even amongst those that don't mind them, most of them want to ban them from the classroom. For example, counter-calculus-reformists George Andrews and Dick Askey want to completely banish even calculators from the curriculum, fearing that students who rely too heavily on calculators and computers will lose the basic feel for numbers.

Bruno Buchberger agrees that they do have a point. Hence Buchberger introduced the White-Box Black-Box Principle, asserting, like Solomon, that there is time for everything under Heaven. There is time to see the details, and work out, with full details, using pencil and paper, a simple non-trivial example. But there is also time to not-see-the-details. Having mastered the algorithm or concept or whatever, the student (and researcher!) should be allowed to use the computer for other examples. I would insert another stage, that could be called "Gray Box", that consists of working out a more complicated example, interactively, using Maple or Mathematica as a calculator, but working out all the steps. Finally, I ask the students to program themselves the algorithm, which is a much better way of having them internalize it than have them do, by rote, many complicated examples by hand. Finally, once they programmed it, they are allowed to use the built-in implementation, if it exists.

I applied this principle a few weeks ago when I taught the Buchberger algorithm to my Algorithmic Discrete Math class. First, they had to, during class, compute the Grobner basis of {x+y,x^2+y^2}, all by hand! Then, as homework, also by hand (no cheating please!) they were asked to find the Grobner basis of {x+y+z,x^2+y^2+z^2,x^3+y^3+z^3}. Then, they were asked, using Maple, to perform the same calculations step by step, computing S-polynomials and reducing. Then they were encouraged to try and program their own version, even though it is unlikely to be as efficient as the built-in Maple command. Finally, for ever after, they were given permission to use the built-in command gbasis, and use it as a complete Black Box, without worrying about the details, enabling them to do new research.

Seeing all the details, (that nowadays can (and should!) be easily relegated to the computer), even if they are extremely hairy, is a hang-up that traditional mathematicians should learn to wean themselves from. A case in point is the excellent but unnecessarily long-winded recent article (Adv. Appl. Math. 34 (2005) 709-739), by George Andrews, Peter Paule, and Carsten Schneider. It is a new, computer-assisted proof, of John Stembridge's celebrated TSPP theorem. It is so long because they insisted on showing explicitly all the hairy details, and easily-reproducible-by-the-reader "proof certificates". It would have been much better if they would have first applied their method to a much simpler case, that the reader can easily follow, that would take one page, and then state that the same method was applied to the complicated case of Stembridge's theorem and the result was TRUE. For those poor people who are unable or unwilling to run the program themselves, they could have posted the computer output on their websites, but please, have mercy on the rain forest! You don't need 30 pages, and frankly all this EXPLICIT LANGUAGE of hairy computer output is almost pornographic.

Hence, both the Gelfand and the Buchberger principles are as useful when we teach our students (at all levels, K-grad_school), as when we try to teach our colleagues, in other words, write understandable research papers. But, last but not least, when we teach ourselves, in other words, do research.

Added June 21, 2005: Read intriguing feedback by Tim Gowers and Volodia Retakh

Added May 26, 2006: Read Gustavo Lacerda's fascinating feedback

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