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.