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John_Maxwell_IV comments on Is Scott Alexander bad at math? - Less Wrong
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Interesting. I think I've read research suggesting that answering questions is significantly better for learning than just reading material (similar to how Anki asks you questions instead of just telling you things).
Val at CFAR likes to make the point that if you look at what students in a typical math class are actually practicing during class, they are practicing copying off the blackboard. In the same way maybe what most people are "actually practicing" when they do math homework is flipping though the textbook until they find an example problem that looks analogous to the one they're working on and imitating the structure of the example problem solution in order to do their homework.
Consider that you're given a magic formula (the derivative) to determine the vertex of a quadratic equation when learning how to graph equations. That nonsense is how mathematics is -taught-. It shouldn't surprise us when students adopt the "magic pattern" approach to problem-solving. (And my own experience is that most of the teachers are following magic patterns they don't understand themselves, anyways.)
That would explain why story problems seem to be perceived as hard by average students at the high school level. I remember being confused by that, since mathematically they were usually the easiest problems in a set -- but they wouldn't be trivially pattern-matched to sample problems.