It is characteristic of an expert to have a systematic approach to problem solving.
A very good example of this is George Polya's How To Solve It, subtitled "A new aspect of the mathematical method". In it he prepares the math student for a problem solving approach, not so much by the memorization of zillions of dry facts, but an approach and a set of heuristics (read "methods", "rules", or even "hacks") for dealing with a broad range of mathematical questions, discovery and invention.
The core of How To Solve It is a structured approach to problem solving, and a dictionary of techniques which can be applied to this structure.
First. You have to understand the problem.
Second. Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.
Third. Carry out your plan.
Fourth. Examine the solution obtained.
The dictionary of heuristic follows with short pithy entries on technique, questions like "What is the unknown?" and strategies like "Decomposing and recombining".
This book is a math book, but more than that it's a way of thinking about things - and as such, I've found it useful to refer to whenever I embark on a new project to help me understand what I'm after and the sorts of things I need to learn along the way to get to a satisfactory answer. It is as well a brilliant example of how to distill expertise into a handbook which can carry on teaching long after the expert is gone.