How to memorize math proofs?

Question

I know that most of you have dealt with math proofs over your career, so this could be a good place to ask for advice. I´m an Economics student, this year I took some tough courses like Intermediate stadistics and advanced microeconomics. The proofs are not very hard to understand, but I forget them and I´m not able to write them alone. I need to memorize them in like 25 days, and they are almost 50. How do you deal with this kind of exams?

Answer 1

It is one thing to understand a line in a proof, and another to come up with the line by yourself, just as it is one thing to understand a speech and another to write one.

When reading proofs, you need to ask yourself, "why did somebody decide to do this step?" Unfortunately, for some proofs it is exorbitantly difficult to answer this question - much more difficult to answer it than to understand the step or the course material. But for many proofs, there is an answer that you can discover by thinking:

• What premises haven't I used so far?
• What relevant information was introduced earlier in the course and hasn't appeared yet?
• What fact, if true, would lead me to my desired conclusion?
• How did we proceed in a similar proof?
• If I were to find a counterexample to disprove the theorem, what would it look like?
• If I were to write my own proof, what would I do?

If you can remember all the reasons for the non-obvious steps, then you can write the proof independently. You should not be trying to just understand each step (under-ambitious) or to memorise each step (over-ambitious).

This is just like speech-writing, where you should not be reading notecards, but should have the important steps in your argument very firm in your mind, and the exact phrasing of each sentence or wording choice can be loose enough to sound natural but strict enough that you do not lose your way.