I went to Brookline High School, which had a very good math and science program. Still, though, (as I expect it is almost everywhere) the focus of the math was on calculations. We saw proofs in geometry, and calculus, but these are often technical, and only steps along the way to deriving some theorem that can help solve some poorly motivated practical problem (Why do I want to calculate the rate that water rises while filling a solid created by a swept polygon??). Also, ideas from computer science were completely missing.
Its not that the more beautiful ideas in mathematics are out of reach of high schoolers -- indeed, my favorite proofs are short, simple, and profound.
If I were designing a highschool senior level introduction to mathematics, I would include:
-- The infinitude of primes
-- The uncountability of reals
-- The notion of polynomial time computation, NP completeness, and the power of randomness
-- The idea of undecidability, through the halting problem
Each of these topics can be presented with elementary mathematics, and each seems to say something profound about the universe. I think that with a course like this, rather than another course focussing on calculation, mathematics and computer science would seem like sexier topics.
Maybe the course could be called "The Nature of Infinity and the Possible"
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