Learn Math: Effective Strategies & Tips for Rudin's Principles

[yoyo100]

Hi Everyone,

I will be beginning an analysis course based on Rudin's "Principles of Mathematics". I was wondering if anyone can share effective strategies to learn higher level mathematics. I realize that different approaches work for different people, but sharing any study strategies that have seemed to work would be great. Specifically, how do you read a textbook to retain material? how do you take notes on the material?, etc. Thanks a lot.

 

[Unknot]

I'm a horrible mathematician but a decent student.

I never take notes.

What I do is reading very carefully, always keeping myself open to possible conjectures or questions that may form in my mind. Axler is quoted as saying that you should read a page for an hour. Now I think that is too much. 5~10 minutes should be enough for a page. Don't read too fast; it's not a novel.

Rudin's book is a classic, and if you skip a sentence it will hurt you a lot. So don't do it. I still do it once in a while and I regret. Make sure you understand every sentence. When reading a theorem don't skip right to the proof. First, try to see why it should be true intuitively. If you cannot do this, you need to go back. Second, formulate some strategies for the proof. Then read the proof.

Once in a while, stop and re-organize all you've learned.

Do exercises. Rudin has a great selection of exercises, which will make your head hurt, which I like very much. I don't do every exercise, but I make sure I read all of them. I only do a few, but probably it is better to do more. What I want to ask others is: how many exercises do you do? How do you find out how many exercises you have to do? Well, easier if you are in a course, isn't it. Homework problems + exams + few interesting questions you pick for yourself should be enough.