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Ask: How do I memorize mathematical proofs for my calculus exam?

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Re: Ask: How do I memorize mathematical proofs for my calculus exam?

#11
Memorisation sucks - I remember having to memorise a similar number of proofs, years ago. (And then again a year or so later, in electromagnetism.)

Here are some things that helped me, personally:

Proofs usually have two things that you can focus on if you're having trouble getting them internalised: intro formalism and some key twists.

The intro formalism is just the stuff that will be used in the proof - you've seen this: new definitions, maybe new symbols, and maybe some assumptions and 'wordy stuff'. Focus on understanding and recalling these first. Even if you can find it easier to recall other, more mathematical facts about the proof, it's possible you'll end up with a muddled argument, because the intro stuff usually makes things precise, and eliminates edge cases.

Once you've got the beginning formalism sorted out in your mind, the rest is the hard stuff: the key twists, the actual logic being performed. What I found useful is to avoid memorising every logical step in a derivation as a separate thing. Instead, memorise a few in-between steps and construct exercises for yourself by trying to go from the intro stuff and the in-between steps to the actual final proof.

I used to have a little notebook with my own exercises in them, all constructed from proofs that I had to memorise. I didn't do this for each and every proof, just for the ones that were conceptually troublesome.

The key is to make the memorisation of proofs more like regular mathematics with exercises - with the full proofs being the solution to the exercises.

Another point - and this is somewhat unorthodox: for very hard proofs, the key mathematical machinery being used can be set at a level that's actually too high for your current course. I found this quite frustrating, so (in these cases only) I decided to focus on (almost) rote memorisation first, and then understanding. I know that goes against the usual advice of understanding always trumping mere recall, but exams that force you to memorise lots of proofs are an artificial condition anyway; real-world maths isn't like this. So weigh this for yourself - maybe start with memorising hard stuff, then gradually chip away at it and try to understand it if you can.

All this obviously takes time, and for ~50 proofs this implies a substantial amount of work. I'm guessing you still have a fair amount of time left until your exam, so it's best to do this stuff early and consistently.

I've got a chapter on memorisation in my book about maths-heavy exams [1] - but it's not a free ebook. However, if you want it, send me an email and I'll send you the .mobi if you wish (or we can share via dropbox, rapidshare - we're spoilt for choice :)) - maybe some parts of it could help.

[1]: www.amazon.com/Exam-Mastery-excel-maths-heavy-ebook/dp/B00AMU20HE/ref=sr_1_4?ie=UTF8&qid=1355929591&sr=8-4&keywords=exam+mastery

Re: Ask: How do I memorize mathematical proofs for my calculus exam?

#12
It's basically impossible (or inadvisable) to learn proofs without understanding. Break the long proofs into sections like you might with a long function. Try to write summaries of the higher order steps. Try to uncover the intuition behind the proof or theorem.

My personal method for studying maths was to have latex versions of my notes. This is great because it makes lookup very easy (just ctrl f). I have a program that gets all the theorems and quizzes me on them and records how much I can recall. When it asks a question I rewrite the proof in full and only check back when absolutely necessary, then compare to the notes. Your quizzing program can focus on theorems that you did badly on.

It's important to focus your efforts. Pay special attention to the proofs that seem important or that a lot of other theorems depend on.

Re: Ask: How do I memorize mathematical proofs for my calculus exam?

#13
post #12

It's basically impossible (or inadvisable) to learn proofs without understanding. Break the long proofs into sections like you might with a long function. Try to write summaries of the higher order steps. Try to uncover the intuition behind the proof or theorem. My personal method for studying maths was to have latex versions of my notes. This is great because it makes lookup very easy (just ctrl f). I have a program…

what's the name of the program?

Re: Ask: How do I memorize mathematical proofs for my calculus exam?

#14
post #13
post #12

It's basically impossible (or inadvisable) to learn proofs without understanding. Break the long proofs into sections like you might with a long function. Try to write summaries of the higher order steps. Try to uncover the intuition behind the proof or theorem. My personal method for studying maths was to have latex versions of my notes. This is great because it makes lookup very easy (just ctrl f). I have a program…

what's the name of the program?

It's just something I wrote myself. Fairly adhoc - it just checks through the latex notes for theorems and keeps them in a list with a current score. The picking is done using softmax selection which picks randomly with a bias towards very low scores. When I complete a theorem I enter how well I did at recalling it and then it updates the score using temporal difference learning. I'm going to clean up the code and put something like it on github sometime.
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