i did a lot of math while i was gone so i accumulated some math hot takes which i archived in roam and here they are for your convenience
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1. proofs in textbooks should be commented. the author should be commenting on which steps are routine, which steps are important to understand carefully, which steps are creative and surprising, etc. etc. sipser is the only textbook i've seen do anything like this and it rules
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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