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JRMayne comments on How valuable is it to learn math deeply? - LessWrong
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Comments (79)
Random thoughts:
The decision that smart high school students should take calculus rather than statistics (in the U.S.) strikes me as pretty seriously misguided. Statistics has broader uses.
I got through four semesters of engineering calculus; that was the clear limit of my abilities without engaging in the troublesome activity of "trying." I use virtually no calculus now, and would be fine if I forgot it all (and I'm nearly there). I think it gave me no or almost no advantages. One readthrough of Scarne on Gambling (as a 12-year-old) gave me more benefit than the entirety of my calculus education.
I ended up as the mathiest guy around in a non-math job. But it's really my facility with numbers that makes it; my wife (who has a master's degree in math) says what I am doing is arithmetic and not math, but very fast and accurate arithmetic skills strike me as very handy. (As a prosecutor, my facility with numbers comes as a surprise to expert witnesses. Sometimes, they are sad afterward.)
Anecdotally, math education may make people crazy or attract crazy people disproportionately. I think that pursuit of any topic aligns your brain to think in a way conducive to that topic.
My tentative conclusions are that advanced statistics has uses in understanding the world; other serious math is fun but probably not optimal use of time, unless it's really fun. "Really fun," has value. This conclusion is based on general observation, and is hardly scientific; I may well be wrong.
I agree that basic probability and statistics is more practically useful than basic calculus, and should be taught at the high-school level or even earlier. Probability is fun and could usefully be introduced to elementary-school children, IMO.
However, more advanced probability and stats stuff often requires calculus. I have a BS in math and many years of experience in software development (IOW, not much math since college). I am in a graduate program in computational biology, which involves more advanced statistical methods than I'd been exposed to before, including practical Bayesian techniques. Calculus is used quite a lot, even in the definition of basic probabilistic concepts such as expectation of a random variable. Anything involving continuous probability distributions is going to be a lot more straightforward if approached from a calculus perspective. I, too, had four semesters of calculus as an undergrad and had forgotten most of it, but I found it necessary to refresh intensely in order to do well.
"Computational biology," sounds really cool. Or made up. But I'm betting heavily on "really cool." (Reads Wikipedia entry.) Outstanding!
Anyway, I concede that you are right that calculus has uses in advanced statistics. Calculus does make some problems easier; I'd like calculus to be used as a fuel for statistics rather than almost pure signaling. I actually know people who ended up having real uses for some calculus, and I've tried to stay fluent in high school calculus partly for its rare use and partly for the small satisfaction of not losing the skill. And probably partly for reasons my brain has declined to inform me of.
I nonetheless generally stand by my statement that we're wasting one hell of a lot of time teaching way too much calculus. So we basically agree on all of this; I appreciate your points.
Calculus has value for signalling intelligence to colleges. I'm told that for professions (e.g. economists) that do use calculus, real analysis plays more-or-less the same role- a rarely used signal of intelligence.
It seems to me that making it mandatory for everyone to learn math beyond percents and simple fractions is even less useful than the old approach of making ancient Greek and Latin mandatory.
When I first read your comment, I thought, "that's not obvious to me". Then a few seconds later I realized: less useful given the opportunity cost of not learning the best possible alternatives. And while math is useful (so are Greek and Latin), there are much better alternatives for mandatory high-school education, basic programming for one.
Exactly. Not sure about programming being any better, though.