I expect this can get one past some limitations of raw ability that wouldn’t otherwise be lifted using just problem-solving practice, but I’m not sure how far.
A lot farther than most people realize. Few people actually try going meta, because social structures don’t encourage it. (At least not in a near sense.)
Not going meta for developing reliability of problem-solving took a lot of points from me. I just relied on the magical intuition, which was good enough to solve some hard problems (to figure out solution method, without knowing how it was being figured out), but not good enough to reliably solve those problems without errors.
As a result, when I was applying to college, I was afraid of the regular admission exams which I couldn’t reliably ace (because of technical errors I wouldn’t notice, even though solution methods were obvious), and instead used the perfect score given to winners of Moscow math and physics olympiads, which required solving some hard problems but not solving all problems without errors. Which is a pretty stupid predicament. It just never occurred to me that production of perfect scores can be seen as an engineering problem, and I don’t recall any high school teachers mentioning that (even smart college professor teachers administering cram school sessions).
I feel like the advice in your earlier comment is good for obtaining insight, but I can’t see how it would be useful on a test. I haven’t taken many tests where I have had enough time to solve each problem in several ways!
I’m eager to learn more if I haven’t understood correctly, though.
For harder tests, the benefit is in not ignoring low-hanging fruit, and training to look for any opportunity to get better reliability, performing cheap checks and selecting more reliable of any alternative sub-steps. On the other hand, ordinary exams are often such that a well-prepared applicant can solve all problems in half the time or less, and then the failure would be not taking advantage of the remaining time to turn “probably about 90% of solutions are correct” into “95% chance the score is perfect”.
A lot farther than most people realize. Few people actually try going meta, because social structures don’t encourage it. (At least not in a near sense.)
Not going meta for developing reliability of problem-solving took a lot of points from me. I just relied on the magical intuition, which was good enough to solve some hard problems (to figure out solution method, without knowing how it was being figured out), but not good enough to reliably solve those problems without errors.
As a result, when I was applying to college, I was afraid of the regular admission exams which I couldn’t reliably ace (because of technical errors I wouldn’t notice, even though solution methods were obvious), and instead used the perfect score given to winners of Moscow math and physics olympiads, which required solving some hard problems but not solving all problems without errors. Which is a pretty stupid predicament. It just never occurred to me that production of perfect scores can be seen as an engineering problem, and I don’t recall any high school teachers mentioning that (even smart college professor teachers administering cram school sessions).
What do you mean by “going meta”?
I feel like the advice in your earlier comment is good for obtaining insight, but I can’t see how it would be useful on a test. I haven’t taken many tests where I have had enough time to solve each problem in several ways!
I’m eager to learn more if I haven’t understood correctly, though.
For harder tests, the benefit is in not ignoring low-hanging fruit, and training to look for any opportunity to get better reliability, performing cheap checks and selecting more reliable of any alternative sub-steps. On the other hand, ordinary exams are often such that a well-prepared applicant can solve all problems in half the time or less, and then the failure would be not taking advantage of the remaining time to turn “probably about 90% of solutions are correct” into “95% chance the score is perfect”.
Gotcha. That’s much more clear to me—thanks.