We don't need no (constructivist) education!

In the world of mathematical research, the word "constructivism" refers to a debate addressing the existence of objects that cannot be explicitly constructed. If such assumptions are disallowed, some aspects of the infinite become very difficult to deal with. However, in the field of mathematics education (which is simultaneously close to and distant from mathematical research), constructivism instead applies to an approach to learning based on individual experience.

For a little more than a decade, so-called Math Wars have raged across the country. Textbooks and curricula advocating an exclusively constructivist approach to mathematics have been adopted by many boards of education both on the state and local levels. A very insightful comparison between a constructivist and a deductionist approach to the Pythagorean Theorem was written by G. D. Chakerian and K. Kreith of the UC Davis Department of Mathematics when the Math Wars reached the secondary schools of Davis.

What I find saddest about this situation is that Mathematics education does benefit from taking constructivism into account. The best teachers from whom I've ever had the pleasure to learn made excellent use of examples to be worked out by the student, often when only a partial understanding of the mathematical theory had been presented. However, these instructors made absolutely certain to declare clearly to the students the general principles governing the behavior of these examples. Examples present students with data; theorems provide a framework for making sense of these data. Without teaching students the theorems that control the behavior of the numbers that surround us, we tell them either that a few examples suffice to determine general behavior, or that every situation is a special case and general principles cannot inform our understanding. I'm not sure which is a more dangerous lesson to learn.