I very rarely find that people are able to articulate most of the heuristics and intuitions they use constantly unless you walk them through it - it requires much more metacognition than most people routinely deploy.
I think it's pretty clear that -- so far, at least -- the greatest mathematician to devote serious thought to pedagogy is Israil Moiseivich Gelfand. Compared to the attention he gave to mathematics itself, of course, his pedagogical efforts pale into insignificance. But still.
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I'm not familiar with Gelfand's work on pedagogy (or any of his work, honestly) - do you have a good starting point?
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It's hard to exaggerate Gelfand's significance to 20th century mathematics; he was one of the titans. His work in pedagogy of mathematics was relatively modest, but still far more impressive than that of any other mathematician of similar stature. See: http://bit.ly/2qen58Q
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