Learning Math: Best Practices for New Grad Students

[E01]

I'm a new first year graduate student and I've decided that my way of learning mathematics may not be the most efficient. I often memorize proofs in the book in the hopes that I'll be able to see where to apply the techniques used in the problems(I often understand the logic of a proof but won't remember the techniques at all if I don't try to rote memorize them). I seem to have to go through this process to get started on the exercises otherwise I won't know what to try and do.

I'll try to refine this question more later but I guess I'm asking:
For those who have done fairly well in Mathematics, what processes do you go through while learning new mathematics? Do you work a lot of simple examples? Do you mainly work through exercises? Do you play with the theorems and proofs in the book to see what happens given different hypotheses? Do you have to memorize things actively to be able to remember them for use later? If you could give me concrete "best practice" tips I would greatly appreciate it.

 

[Ackbach]

The best advice I ever got in grad school was this: don't work on the homework that's due next, until it's finished, and then move on to the next one. In grad school, the homework is 5000 times harder than undergraduate. Sheer quantity of time is not going to help you. Instead, do this:

1. Take one class's homework, and give it a good go. If you get into a groove, and you're knocking the problems out, stick with it! But if, as is most likely, you eventually hit a brick wall (no progress at all for, say, 10-20 minutes), then stop.

2. Move on to the next class's homework. Repeat Step 1.

3. Continue cycling through your courses, constantly.

This does a number of things for you. First of all, you might get cross-pollination among your courses, where an idea you get in one course helps you in another. You don't want to cut yourself off from that. Second, it's not the sheer amount of time you spend on problems that's going to solve them, it's the number of fresh starts.

Sleeping can sometimes solve problems! Some people solve problems in their sleep. Sleep, indeed, helps your brain to make connections it wouldn't otherwise make. Yes, I pulled a few all-nighters in grad school, but I always felt absolutely terrible after one. Keep yourself healthy - there's a connection between the body and the mind.

As for memorization, I would recommend memorizing the basic definitions. Memorizing proofs is much less valuable, because the fact is, different theorems, even closely related theorems, sometimes require drastically different proof methods. Understand the constructions used, yes. But really push yourself to imagine, and come up with proofs yourself. Imagination is key to all this. Speaking of which, make sure you give your imagination a workout by not watching too much TV or playing too many video games. Read good books by authors like Homer, Virgil, Dante, Milton, Jane Austen, Charles Dickens and Leo Tolstoy. That'll help as well.

I wish you well in your studies!