ABSTRACT
Several years ago I had the rare good fortune to hear George Polya talk about teaching problem solving at a meeting of the American Mathematical Society. I remember quite vividly his closing remarks. He said that mathematics teachers must bear in mind that their primary goal is to get their students to “use their heads.” I must confess that at that time I was a bit disappointed with so “simplistic” a piece of advice. In retrospect I see his advice as both simple and profound because his message is one that seems not to be heeded in most problem-solving instruction. The message is that the ultimate goal of instruction in mathematical problem solving is to enable students to think for themselves. It is my view that most problem-solving instruction not only does not enable students to use their heads, but in fact it does more harm than good. Let me give an example from my own research with elementary schoolchildren to support this position.
