How does one go about reading a math book?

This is Joe Blitzstein's answer to a question on Quora about how to read a math book.

Paul Halmos, who was famous as a mathematical expositor, had some great advice about this:

Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?

That is, try to deconstruct and reconstruct each result. What is the motivation behind it? What is an example where it applies? What is an example where it does not apply? What were the key strategies used in the proof? Why is each assumption needed, and can you find explicit counterexamples if you delete an assumption? How does the theorem connect with other results?

Reading as actively as possible, by thinking about these kinds of questions and doing a bunch of exercises, is very important. Textbooks often create the impression that problem-solving is a separate activity done after reading the material (if at all), since they often relegate the exercises to the end of each chapter. Instead, I recommend interweaving reading with problem-solving (the problems can both be those provided in the book and questions that you come up with while trying to deconstruct and reconstruct the results).

Also, math is a connected web of ideas but books are inherently linear, so it is especially helpful (with a well-written book) to reread chapters to reinforce your understanding and look for connections between ideas.

Clearly this is a very time-consuming process, requiring dedicated effort. But studying in this way should help give a much deeper understanding of the material compared with passive reading.

Reprinted from：www.quora.com/How-does-on…