How to effectively and efficiently learn mathematics [closed]

How do you effectively study mathematics? How does one read a maths book instead or just staring at it for hours?

(Apologies in advance if the question is ill-posed or too subjective in its current form to meet the requirements of the FAQ; I’d certainly appreciate any suggestions for its modification if need be.)

Answer

The only way is to put in a lot of time, to not give up, and to keep studying. As Agusti Roig mentions, the importance of doing exercises cannot be downplayed – working things out for yourself is absolutely necessary when trying understand things more deeply.

For me one of the most important things has been asking myself questions. Lots of questions Why does the theorem have theses hypothesis? Why this definition? What was the key idea in the proof? Can I apply this idea, this method of proof to other questions? For what kinds of questions will this method fail, and why does it fail? Is there another proof? Terence Tao has a good blog post about this titled “Ask yourself dumb questions – and answer them.

The more time you spend thinking about a subject, the better you will understand it.

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