Conservapedia talk:Critical Thinking in Math

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I have a number of comments on the proposed curriculum. Much of this relates to the difference between something that might be titled "mathematical topics in popular culture" and "critical thinking in mathematics". The latter requires very careful presentation, with adherence to the axiomatic method and the understanding of correct proof procedures. This concern applies particularly to the material on the Axiom of Choice and the Gödel incompleteness theorem.

I believe that there is at least a 10-year gap in educational achievement between 9th graders (even future Riemann's!) and some of the things mentioned. We couldn't say anything about the Axiom of Choice or Wiles' proof that wouldn't be dumbed-down to the point of worthlessness, and, IMHO, not what we want for this class.

I am not aware of any a controversy about the method of proof by contradiction. It is so totally ingrained into the subject matter of mathematical proof that nothing further needs to be said. I still remember being taught about it, in 9th grade, before we did any "real" math.

While we are discussing the 17 proofs of the infinitude of primes, we might discuss the fact that there are over 300 proofs of Pythagoras' theorem. Including one by President James Garfield!

The "transcendental numbers - how do we know they exist?" is not the way I would put it. I assume you are referring to the Cantor "diagonal" construction, layered on top of a Dedekind or Cauchy construction of the reals. But there's actually a lot of tedium in the proof that the reals can be represented by decimal expansions. Do you want to gloss over that and just use the "intuitive fact" that the reals correspond to the decimals? But I have another problem: "How do we know they exist?" has a sort of mystical ring to it, particularly because of the connotations of the word "transcendental". The same thing construction could be used to prove the existence of irrationals, without the mystical baggage. They're uncountable too. And there is a very clever and approachable proof from the ancient Greeks about the existence of irrationals, that does not rely on countability arguments or diagonalization. Or do you genuinely want to use the transcendentals because the ancient proof can't be used?

"Problems solved only with proof-by-contradiction". As above, I consider the method of contradiction to be so fundamental that I'd be reluctant ever to say that somethng can be proved only that way. Better: In the introductory explanation of proof by contradiction, using the "always a prime between N and N+1" theorem, or the "infinite number of primes" theorem, as an example to motivate it.

Definition of the integers. Do you really want to do a Peano-postulate program here? As in Landau's "Foundations of Analysis"? I don't think people in this age group will actually appreciate that the tediousness of this is worth it. (Of course, we know that it's worth it, but they donn't.) Also, a Dedekind-cut construction of the reals would actually contain material that the students don't already "know", in that it shows, for example, that the square root of 2 really exists. But of course, skipping over the integers and going straight to the reals is somewhat unesthetic! The students already "know" the result of the Peano program, they just don't realize that they don't really know it.

"Interesting problems in number theory and Euclidean geometry". Yes! There's a lot of really cool, engaging, and surprising material in geometry, above and beyond its use to show the axiomatic method. Geometrical inversion, for example.

"Axiom of Choice ... Wiles' proof". I have a problem with this. There's really very little we can say about AC, and even less about Wiles' proof, that would be accessible. The simplest places that I know of in which AC appears are the Tychonoff Theorem and the construction of sets that are not Lebesgue measurable. Not what I would want to do in a course like this.

The Gödel incompleteness theorem, and the Church or Turing analysis, are extremely tedious to go through if they are to be understood properly, and take a great deal of background that high school age people won't be able to get through. Once again, we would have to dumb it down to the point of worthlessness. Though there are presentations of what is effectively Russell's paradox, which is related, and that are accessible and compelling.

There are other things that can be covered that are very good for high school students: projective geometry, conic sections, vectors, matrices, operators, eigenvectors/values, function spaces, introduction to differential equations using eigenfunctions, complex numbers in electrical engineering, Fourier series, inner products, geometrical inversion, ....