2. students should be taught how to "debug" proofs. here is a basic technique that people don't get taught: if you proved something you know is wrong, you can figure out which step is wrong by stepping through the proof *with a counterexample*
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3. it's weird how we taught students to encode their understanding of math in their understanding of social permission (what they're "allowed" to do). permission has nothing to do with it. the question is what procedures turns true statements into true statements
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can you recommend good books on HOW to develop this thinking process?
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i have never read a book about it! people say good things about polya's "how to solve it":
amazon.com/How-Solve-Math
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short of downloading your knowledge into my brain this is the next best bet
thanks, I'll check it out!
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yeah but it could happen faster if people talked about it more explicitly i think. we sort of expect math students to at some point just osmose this stuff implicitly which is a little unfair
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it's like... there's a few distinct transitions you go through on the way to building up "mathematical maturity," a few really key "developmental milestones" for absorbing how mathematicians think about math that i think could be organized into a rough stage model
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i've def met smart reasonable adults who did well in math classes who later realized they did not understand this and were just manipulating expressions syntactically the whole time
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not at that level but like people who thought of themselves as being good at math in school