Any way to solve math problems faster?

Question

Is there a faster way to solve math problems? I'm talking about proving theorems, proving certain properties of a function, etc. The way I do it is I write out the problem and all relevant definitions and theorems, and consider simple examples or sketch the function if I'm given one. But it involves a lot of trial and error. Sometimes I feel like I will never be able to solve the math problem. Do you have any ideas?

Answer 1

"When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork." - Paul Halmos

"Painful and creatively frustrating as it may be, students and their teachers should at all times be engaged in the process - having ideas, not having ideas, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other's work." - Paul Lockhart

"If you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem; it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that the theorem is true, you can start proving it." - George Polya