I have recently started solving some problems from some math problem solving books, and I've noticed a difficulty. When I have to solve problems on the math homework/exam, it becomes a game of "find the way to apply the techniques taught in the section", because that is simply the best way by far to complete the homework and timed exams in a reasonable amount of time.
For example, if in class we are studying the Cauchy Riemann equations and I get a question about the differentiability of a complex function, I immediately try to incorporate the C.R. equations; it would be wildly impractical to try out all other ways, such as writing out the limit definition of differentiability, trying some complicated algebraic manipulations, thinking about the problem geometrically, etc . . .
When I move on to problems that are in a more general setting (outside of class), I find that this mindset is a little hard to shake off; I think only in the context of what was covered in the book, and I find it difficult to let my mind truly run free. If I can't solve a problem and read the solution, sometimes I have the gut reaction "that's not fair, there was nothing mentioned about X so far in the book, how was I supposed to know we were allowed to use that?" It's difficult to truly let my mind run free.
Has anybody else experienced this difficulty? What steps would you take to maintain an open mindset, while still solving classroom problems under time constraints?