Agreed! I often emphasize that written proofs are an after-construction--nobody solves a problem the way a proof is set down on the page. The written proof is the result of taking your messy process and setting it down in as concise a way as possible. And when teaching a proof, I do not just go line by line. First I talk about the ideas of the proof, then I go line by line, then I go back to the ideas so that the students can (hopefully) see where the assumptions in the theorem are used and where other tools are used, and whether the proof is straightforward or requires some non-obvious trick. And yeah, of course it's important to motivate why we're doing the theorem at all.
Also useful when teaching theorems and proofs, is to let students try and find examples where one or more of the assumptions in the theorem is not met, and what then happens with the result. (Example theorem: a continuous function on a closed interval has a maximum and a minimum value. What happens if the function is not continuous, does it need to have a maximum and minimum value? Does it necessarily need NOT to have a maximum and minimum value? Where does the proof fail? Etc.)
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Re: Grad school
Date: 2023-01-11 07:37 pm (UTC)Also useful when teaching theorems and proofs, is to let students try and find examples where one or more of the assumptions in the theorem is not met, and what then happens with the result. (Example theorem: a continuous function on a closed interval has a maximum and a minimum value. What happens if the function is not continuous, does it need to have a maximum and minimum value? Does it necessarily need NOT to have a maximum and minimum value? Where does the proof fail? Etc.)