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?
Best Answer
This is a well-known problem. I don't know if it's an option for you but sometimes math departments offer a course which is called something along the lines of "problem solving" (in my home university it was called "Workshop in Mathematics"). In such a course, you are given each week a list of problems to work out on without any context, present your solutions and learn from the solution of other students. This isn't like regular homework where you need to solve all the problems - you need to try and solve some of the problems that interest you and hopefully succeed in solving at least one problem. Some problems might be easy, some will become easy once you'll identify which techniques should be used to attack the problems (so the difficult thing will be to put the problem in the appropriate context) and some will be hard no matter what you try. To toughen you up, sometimes an open problem might "slip in".
The point is that none of the problems are "routine application" of the concepts learned two days ago (for one, because you don't learn any systematic theory which builds up as weeks pass by). There is no time pressure (it is definitely possible to spend the whole week thinking on one problem) nor too much pressure to solve everything (you discover quickly that it is almost impossible). If the problems are chosen well and the atmosphere in the class is good, this kind of class really develops the skills that are often neglected in regular courses. You get to "play with the problem", identify various possible approaches, try and fail a lot, look for unexpected connections, etc. And naturally, those kind of skills are much more relevant later (whether you go into research or apply your knowledge to real world problems).