Test Prep Authority, a test prep company, will now be sending “Math God” problems to those students preparing for the SAT and ACT that want them. The idea behind this, according to their press release, is that using very hard problems for training purposes is useful if you want to be able to solve easier problems quickly.
So far they’ve put out three of them: one, two, three. They don’t strike me as being particularly difficult so much as annoying. But they’re annoying in the same way that SAT problems are annoying, only more so, so from the point of view of training for the narrow slice of mathematics that is on these tests they’re pretty reasonable.
I’m actually of two minds about this. On the one hand, I instinctively shudder when I see test prep resources, because I’m afraid that they’ll be devoted to “how to take tests” and not actual learning. (Test Prep Authority seems to be pretty good in this regard.) On the other hand, the philosophy of doing harder things than you “need” to be able to do is certainly one I can get behind. When it really matters, you don’t want to spend all your time working at the limits of your capabilities; that’s exhausting. You may never need to drive faster than, say, seventy miles an hour, but you wouldn’t want a car that only goes that fast.1 Just because you never need to run to actually get somewhere doesn’t mean that running won’t make walking less tiring.2 And of course they say that you never really learn algebra until you take calculus, or that you never really learn a subject until you have to teach it.
I’m rather curious about that last statement. How would you test that?
1. This describes my first car, a 1987 Subaru DL which I drove for much of 2000-2001. Around seventy it started shaking pretty violently. But that’s not a horrible trait in a car for a new driver, who might otherwise go too fast.
2. I should take this advice, but I don’t, because I really can’t stand running.
God plays dice 
If you were to create a testing regimen which would inculcate more learning, what would you do? I remember taking American Math Competition tests pretty fondly – should that be the goal?
That last statement rings true at least for me. I’ve always loved math, was the geek in school who did all 50 assigned exercises from the book each night and wished for more. But that was the shame of it, I did a lot of exercises and very little problem solving. All through school I went through the motions without the heart and frustration and perseverance that a good problem should elicit every time.
Exactly, then came calculus, and I thought, Oh Shit. Grasping calculus was the eye opener to everything I’d learned about math that came before it.
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