How do you study mathematics?

Question

Which is (or was) your method to study pure maths in the university?

I'm interested in knowing your routine when you study maths in a professional way. I think it could help some people to improve their routines and methods and it could be a interesting question to discuss.

Answer 1

Here are some suggested guidelines . . .

• Assuming there's a textbook, take it seriously. If there's no textbook for the course, find one that fits the syllabus and the level of the course. Have some alternate textbooks for additional problems and examples, and to see alternate presentations of a given topic.

• When reading the text, memorize all definitions, and test yourself mentally when you meet those terms again. In your work, try hard to use the notation and terminology of the subject correctly.

• Similarly, memorize the statements of the key theorems.

• If the hypothesis of a theorem specifies some restrictions, make note of examples showing why those restrictions are needed.

• Additionally, maintain a mental repertoire of key examples of objects with a specific set of properties.

• For selected proofs, memorize an outline of the proof, sufficient so that from the outline, you could complete the proof, if necessary.

• Optionally, as you read the text, when you reach a theorem that the author is about to prove, try to prove it yourself, peeking if you get stuck. The same for examples that the author is about to work out.

• As you go, write down questions that occur to you. Try to answer them yourself, but if a question remains unresolved, ask your teacher, or perhaps post it here on MSE.

• Do a wide variety of problems, not just the ones assigned for HW.