Have you ever stared blankly, with eyes glazed over, at a particular problem in your calculus textbook and then walked away frustrated and bored because you just can't make heads or tails out of it? Well, then George Polya may be your savior.
When faced with a problem, scholarly or otherwise, we are often faced with impediments that prevent us from arriving at a solution. Professor Fletcher had lectured on the differences of how an expert versus a layperson approaches a problem. Consider this silly question from famed mathematical educator George Polya's book "How to Solve It:"
A bear, starting from the point P, walked one mile due south. Then he changed direction and walked one mile due east. Then he turned again to the left and walked one mile due north, and arrived exactly at the point P he started from. What was the color of the bear?
If you happen to be an expert on bears or navigation perhaps the answer is obvious and a proof of your answer is trivial. As evidenced in Fletcher's research, an expert is already familiar with the narrative and can easily progress down a solution path.
The layperson may have a rough time. A naïve problem solver may go through a lengthy parameter search over a potentially large solution space. George Polya had developed an algorithm for applying a set of heuristics that has proven to be useful Roughly the steps are:
1.) Understand the Problem
2.) Devise a plan
3.) Execute the plan
4.) Review and reflect on what worked and didn't work
In each of these steps we may employ any number of strategies or heuristics from devising simpler analogous problems, drawing diagrams, backward chaining, restating the problem in your own words, making a list, and etc. What he has assembled is a method for a layperson to construct a narrative; that an expert may do intuitively.
I have found these steps useful in avoiding some of the pitfalls I encounter in solving certain kinds of problems. I become more aware of when and where I may be fall prey to focusing too much on surface similarities, functional fixedness, or mental sets. Polya's algorithm fails where all algorithms fail: if we can't satisfy a particular step then our solution path falls apart. If I just don't know anything about bears how can I possibly hope to figure out what color a bear is?
All in all, Polya's book is a great read and offers insight into the nature of reasoning from one of the great mathematicians and educators of the 20th century. So, what color is the bear?