How to think deeply in mathematics way? [closed]

Most of the time, I solve a problem by doing the following steps:
1. Look for a related theorem.
2. Look for a related formula.
3. Find a similar problem or sub-problem.
4. Try to adapt the one that I had done.
5. Asking at Math.SE is an alternative solution that I've just found out.

When I read the article named "Eight Laws Of Memory", they claimed "People poorly remember what they read because they do too little thinking", and this is exactly my problem. I read the answer of the question that I asked. Then I try to solve it in my own way, couple days later, I forgot what I did unless I see it again.
So, by "deeply thinking" what do we have to do, especially in solving Maths problems. My guess was "deeply thinking" means breaking the problem into all possible cases, and observe each outcome, comparing one with others. Can anyone share some experiences about how to think deeply? Any feedback or suggestion would be greatly appreciated.

Thank you,

Solution 1:

This is just a comment on how I try to learn things. Of course, solving problems requires that you understand the material. From personal experience, I feel I absorb information better if I'm actively writing down thoughts, questions, observations, etc. as I read through a text.

So my basic advice is this, don't just sit and read a textbook and try to understand it in your head. When you finish reading a proposition, lemma, theorem, whatever, try to reprove it by yourself in your own words and write it down. Keep doing this until it sticks. Change the way you present the proof, so that the ideas are fluid to you, and thus the exercise is not one of memorizing the sentences of the proof, but getting a feel for general approaches to problems as well. This will force you to engage the material and think about what you're reading. Read the author's words like a skeptic, and rigorously prove little details that the author may find obvious, but a newcomer wouldn't.

I think there is some quote somewhere that when learning mathematics, you should produce 10 pages of writing for every one page you read.

Solution 2:

Fortunately a fine mathematician, George Polya, has already worked on this problem and has written a remarkable book called How to Solve It. Although his techniques are heuristic rather than algorithmic in nature, I've found numerous times that his techniques can at least get me "unstuck", a much nicer place to be than "stuck".

There's no substitute for reading the book. But if you're in a hurry, at least look at the Wikipedia entry, especially the table of heuristics:

http://en.wikipedia.org/wiki/How_to_solve_it