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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,

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Just spend time meditating on what you've read. –  Matt Gregory Apr 5 '11 at 1:19
    
@Matt Gregory: Thank you. But this is too general, I want more details though. –  Chan Apr 5 '11 at 1:54
3  
Well, simply holding an idea in your consciousness is going to create a deeper impression of it in your memory. The deeper a mental impression is, the more easily it can be recalled back into consciousness, the more easily connections with other ideas can be made, etc. Deep thought is primarily about spending time with ideas. –  Matt Gregory Apr 5 '11 at 2:25
    
@Matt Gregory: "Deep thought is primarily about spending time with ideas", very interesting. I really like this idea. Thank you. –  Chan Apr 5 '11 at 2:33
    
@Chan I think deep thinking means to just spend time philosophizing on one idea. That means to see the idea from different perspectives, wonder what would happen if we were on an other planet, other space etc. How would that effect the idea. Just spend time with it until you feel confidence about that idea.. that you know it upside down. –  Pithikos Oct 26 '11 at 16:05

2 Answers 2

up vote 13 down vote accepted

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.

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Thank you for your suggestion. So I guess, your "deeply thinking" way is to work it out in details. I really like your way. –  Chan Apr 5 '11 at 1:55

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

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Thanks for those references. However, I think my question is not how to solve the problem, but what does "deeply thinking" actually mean? It's easy to understand the context meaning, but the "what" is confusing. Don't you think? –  Chan Apr 5 '11 at 1:57
    
@Chan: It sounds like you have an intuitive feeling for "deep thinking" but don't have an actual definition. So I would recommend finding examples of authors or books which clearly manifest "deep thinking" and study those. Imitate them. Try writing out their proofs in your own words. Absorb them osmotically. Let's use Polya's favorite, most powerful tool, an analogy: consider what happens when you give an elementary piece to a great musician on the one hand, and a child on the other. Both can hit all the notes....(out of space, continued...) –  ShyPerson Apr 5 '11 at 17:28
    
@Chan: ...But the great musician adds so much more depth than just the notes, on so many more dimensions. All great books have this quality. Polya's books on this topic also have this quality: hiding very modestly under simple words is the mathematical power of one of the greatest mathematicians of the Twentieth century. Notice that. Absorb that. –  ShyPerson Apr 5 '11 at 17:29

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