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emr comments on Innate Mathematical Ability - Less Wrong
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... but a key point of my post is that context-free abstract pattern recognition ability is innate and can't be learned :-). You can learn how to answer standard Raven's matrices type questions, by learning patterns used to construct the items, but the skills built aren't transferable – if given a different kind of test of context-free abstract pattern recognition ability, you would do no better than you would now. It is possible to improve a great deal as a mathematical thinker, but trying to build this sort of skill is not the way to do it.
"Context-free abstract pattern recognition" can be partially resolved into more legible subcomponents, some of which can be learned, and some of which can't.
So working memory is one such component, and is often theorized as a big pathway for (intuitively defined) general human intelligence. It doesn't look you can train working memory in a way that generalizes to increased performance on all tasks that involve working memory (although there's some controversy about this). And as with other traits, increased performance on formal measurements of working memory might not translate to the real-world outcomes associated with higher untrained working memory.
At the same time, it seems that the universe must come packaged with a distribution over patterns, and so learning a few common patterns might transfer fairly well. The Raven pattern is XOR, a basic boolean function. The continued fraction is self-similarity, which is an interesting pattern (meta-pattern?), because while people already recognize trivial self-similarity (invariance, repetition), it look like people can be successfully taught to look for more complicated recurrences in math and CS classes.