Hi,
Below is a short passage taken from the book "Letters to a Young Mathematician" by Ian Stewart, which provides us with a good insight on how mathematicians think, from which we could model after them to improve our mathematical thinking.
According to Jacques Hadamard, author of The Psychology of Invention in the Mathematical Field, ideas in mathematics seem to arise in three stages.
First, it is necessary to carry out quite a lot of conscious work on a problem, trying to understand it, exploring ways to approach it, working through examples in the hope of finding some useful general features. Typically, this stage bogs down in a state of hopeless confusion, as the real difficulty of the problem emerges.
At this point it helps to stop thinking about the problem and do something else not related to the problem. This gives the subconscious mind a chance to mull over the original problem and try to sort out the confused mess that your conscious efforts have turned it into. If your subconscious is successful, even if all it manages is to get part way, it will "tap you on the shoulder" and alert you to its conclusions. This is the big "aha!" moment, when the little lightbulb over your head suddenly switches on.
Finally, there is another conscious stage of writing everything down formally, checking the details, and organizing it so that you can publish it and other mathematicians can read it.
The great mathematician Henri Poincare called the first stage "preparation,", the second "incubation followed by illumination," and the third "verification." He laid particular emphasis on the role of the subconscious in one famous section of his essay Mathematical Creation:
For fifteen days I strove to prove a difficult result. I was then very ignorant; every day I seated myself at my table, stayed an hour or two, tried a great number of alternatives and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until they converged to a solution. By the next morning I had established the approach for the proof, which took but a few hours to write.
To end, I'd like to quote an interesting phrase from Albert Einstein which reads as follows: Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.
Thank you!
Cheers,
Wen Shih