17 September 2009

The Calculus

In this mini TED talk, mathematician Arthur Benjamin endorses an idea that's near and dear to SB7's heart: calculus does not need to be the pinnacle of secondary math education:



Here's what I said about calc last winter:
I think the best thing about calculus for high school students is that it's the most rigorous class they're likely to face. You spend an entire year confronting only two ideas (differentiation and integration) and if you don't grok them you can't really fake it. Contrast this to, say, American Government where you can have some fleeting and incomplete notions of separation of powers and the Bills of Rights and the New Deal and hand-wave your way to passing marks. Of course calculus' rigorous and monolithic nature is also it's weakness: if you don't grok those two principles the entire year goes over your head. Students are left feeling like all of mathematics is beyond them.

What I'd like to see taught is probability theory, hopefully with some statistics as well, discrete math, and perhaps some linear algebra. I like probability and statistics because I think it helps people understand the world in a quantitative way better than any other math class. Even if the only benefit was that no American with a diploma ever played the slots again I'd consider it worthwhile. I think discrete math has the advantage of geometry in the you can do pretty advanced reasoning based on very simple components and relationships. Discrete math also has a lot of subfields that can be taught, which overcomes that weakness of calculus by allowing a student that didn't understand set theory to move on after a few weeks to graph theory where they might fair better.
Benjamin seems to hold mostly the same idea, but keeps his endorsement limited to stats. He mentions the divide between continuous and discrete mathematics, but frustratingly doesn't actually endorse discrete math as a discipline. This isn't too surprising, because in my experience no one without a background in math, CS or EE has taken a discrete math course or typically knows what it is. Here's a wikipedia breakdown of topics covered by the umbrella of "discrete math":
  1. Logic
  2. Set theory
  3. Information theory
  4. Number theory
  5. Combinatorics
  6. Theoretical computer science
  7. Operations research
  8. Discretization

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