How to Solve It - A New Aspect of Mathematical Method
This book is much more than a simple, or complex, math book --it is be a must read for teachers in general. The most important lesson of the book, in my opinion, is that teachers (besides repeating, repeating, repeating) need not 'give it all away.' The student should never be robbed of working out a solution for themselves; slogging thought the difficulties, feeling the thrill of the hunt, and finally experiencing the joy of victory will instill much more than any rote lesson could ever achieve. Other than that broad advice, the book lays out, in mathematical terms (though it could be applied most generally), how to problem solve by planning around what you know and what you want to know; catalog your knows and unknowns, draw a picture, and work from the end towards the beginning (reverse planning). Other ideas concerning how you might tackle a problem are modifying the problem with the addition or subtraction of data to make it similar to a more tractable problem you are familiar with and pointing out that even failing is progress as long as you learn something in your failure.
The Act of Creation
While likely a less-than-reputable person given his influences and acquaintances, it can not be denied that this, like many of his other works, are at a minimum thought provoking. One of the main concepts explored in the book is bisociation; connecting two unrelated frames into a novel concept. These frames can be learned in any traditional manner (rote, etc) or conditioned into a subject a la Pavlov, Skinner, or Watson's experiments. Either way, to move from one frame to another, solving a problem usually, one needs creativity and originality more than genius (though having both is obviously best). Genius, Koestler and other argue, is a mind with general power toward in a particular direction and the ability to make bisociations.