Educating for better problem solving is important for K-12, college and graduate education, business, health, science, the arts, and every discipline. But before getting in to the nitty gritty of 'how', it's also worthwhile to think about two very different types of problem solving - the brain 'easy' and brain 'hard' divisions that have very different timing and characteristics, and should have different preferred educational characteristics.
The brain 'hard' version - is what you might expect from heavy critical thinking: like this picture from the mathematical calculating prodigy: It requires huge areas of cortex, both sides of the brain, and conscious manipulation of lots of facts, relationship and data. It looks like you might expect for a heavy bit of number crunching.
But this is not the only way problems are solved. There is the brain 'easy' way too. With insight-related problem solving, only small areas of brain are activated, and these areas only switch on right before a solution is recognized. The area that lights up is that dreamy area of the right temporal lobe that might access more personal knowledge and experience - or 'autobiographical' memory. It might be that right analogy or metaphor is struck, the pattern is recognized and... Aha! The problem is solved.
The differences in these approaches are important to consider whether we are thinking about fostering the problem solving ability K-12's, software engineers, or corporate executives. Some creativity training approaches are unevenly weighted toward one approach (perhaps reflecting the inventor of the method?), but being facile in both approaches leads to more powerful problem solving.
When People Solve Verbal Problems with Insight
Mental Calculation in a Math Prodigy