Memorising lots of maths theorems/lemmas

Question

In a few weeks I'll have my summer exams in a Senior Freshman Mathematics course. Two of my modules have a huge number of theorems, lemmas and definitions - over 200 definitions and 250 proofs by my last count - and I'm struggling to find a way to memorise them all. I know the key to memorisation is to understand the topic (which I do) however I still need to be able to perfectly recall the exact proof/definition in question. (It doesn't help that a lot of the material is similar and can be confused with a different proof/definition.) So my question is, how do I memorise all this information? Is there something similar to the memory palace/method of loci, but for maths stuff?

Answer 1

Formal proof of theorems is a sequential process.

[0] Start by absolutely knowing the definitions used in the theorem. You need to know what you're talking about.

Try to understand the theorem. If you don't understand it all at the beginning, then identify the parts of it that you do understand.

[1] Begin by making a list of the hypotheses.

If you are making up your own theorem, to construct a proof you can keep adding to the list of your hypotheses. Keep it up until there are enough hypotheses to imply the desired conclusion that you want to arrive at (the QED - look it up).

If you want to understand someone else's proof, you need to take ownership of the theorem as if you were creating it yourself. It pays to own the theorem and your proof of it.

[2] Then you start adding a list of intermediate conclusions.

Each intermediate conclusion should follow from

• the hypotheses. together with

• the previous intermediate conclusions.

Keep adding to the list of intermediates until you finally are ready to conclude.

[3] the conclusion is the statement of what needed to be proved.

I have to memorize by repeated engagement with the process of writing the steps down. Generally I can do it by memory somewhere around the fourth run through the steps.

In a less formal setting, say where you want to explain something to someone unfamiliar with the topic, you paint a background picture of what's going on in the context of the theorem.

Then you keep building the picture, making it more and more detailed, until there is no escaping the conclusion. Repeat four times.

Memorization follows from the engaged repetition, but NOT from rote repetition. See if you can generate a sort of emotional involvement. You have an intention, and you are emotionally committed to fulfilling that intention. Then I get the (emotional) payoff that goes with accomplishing the goal.

You might as well. It's kind of like solving a puzzle or winning a game. Ideally you'll be having a lot of fun and having a heck of a good time. That makes the repetition struggle-free.

Ideally, since the definitions are declared hypotheses, they will be absorbed by your memory as you repeat the proof, engaging committedly with the accomplishment of your goal.

Answer 2

Then you keep building the picture, making it more and more detailed, until there is no escaping the conclusion. Repeat four times.

Memorization follows from the engaged repetition, but NOT from rote repetition. See if you can generate a sort of emotional involvement. You have an intention, and you are emotionally committed to fulfilling that intention. Then I get the (emotional) payoff that goes with accomplishing the goal. You may as well. It's kind of like solving a puzzle or winning a game.

Ideally, since the definitions are declared hypotheses, they will be drawn into your memory as you repeat the proof, engaging committedly with accomplishing your goal.