cahn | Historical Characters, Including Frederick the Great, Discussion Post 40

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I'm trying to use my other account at least occasionally so I posted about my Yuletide gifts there, including the salon-relevant 12k fic that features Fritz, Heinrich, Voltaire, Fredersdorf, Saint Germain, Caroline Daum (Fredersdorf's wife), and Groundhog Day tropes! (Don't need to know canon.)

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From:

luzula

The whole problem was that I was taught to study math one page at a time, one sentence at a time, and never move on to the next thing until you understood the current thing.
That's definitely not how I read a math text, anyway! I start by identifying what the main theorem is, and reading whatever definitions and preliminaries I need in order to understand the theorem, and also the bits about why the theorem is important. Then I identify what theorems/lemmas are needed to prove the main theorem, often by drawing a diagram showing how the different theorems/lemmas hang together and lead to each other. And then after that I dive into the proofs (if I actually need to--sometimes you just need to use the theorem).

I don't know that I necessarily needed to have harder problems in high school, or early in my university education. It might have been good, but actually I think we got a fairly okay progression. The master thesis was a sort of mini-graduate project, after all, and before that was various smaller projects. I think I just hit...well, it was partly about my abilities (I don't think I have it in me to be a brilliant mathematician), but also about my interests. I'm often serially geeky, and I had moved on to other geeky interests.

From:

mildred_of_midgard

That's definitely not how I read a math text, anyway!

That's now how anyone should read a math text, in my opinion. What I needed was someone to tell me I could look at later chapters at a high level to just grasp what they were about and what we were building up toward, instead of going, "No, Mildred, you're getting ahead! We must go line by line or it's not rigorous."

...Yes, if you're writing a proof. There is a difference between a rigorous proof and a rigorous pedagogy. The needs are different.

(My math and physics profs seemed to think that if they walked you through proofs line by line that was the same thing as teaching you, and you would then be able to 1) grasp the concepts, 2) apply the concepts to concrete problems without further effort. Maybe that works for the

cahns of the world, but the Mildreds need "This pointless-looking lemma is building up to Galois theory, which is covered three chapters from now, and the point of Galois theory is blah blah," before learning the proof for each pointless-seeming lemma in isolation.)

actually I think we got a fairly okay progression

More evidence you went to school in Sweden!

From: (Anonymous)

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.)