My name is Bill Farmer and I am a research mathematician and computer scientist at The MITRE Corporation in Bedford, MA. MITRE is an independent, not-for-profit company that provides technical support to the government.
I very much agree with the point Alan Schoenfeld put forward. I believe that mathematics is a discipline for exploring our world and for creating new ways to see our world. The heart and soul of mathematics is the creative/explorative process in which one creates mathematical models and then explores them by stating and proving conjectures. Of course, "problem solving" is a major element of this process. In a nutshell, the problem with mathematics education in the U.S. is that the average American gains very little experience in creating and exploring mathematics. Consequently, the average American cannot use and does not understand the mathematics process. Instead, he usually learns a collection of mathematical facts that he finds little use for and eventually forgets.
As a researcher in industry, I have learned firsthand that understanding how to do mathematics is far more important than knowing particular definitions and theorems. Many of the problems that researchers are asked to solve in industry involve mathematics. Although the mathematics may not be particularly novel or interesting, it is often outside what is traditionally taught in college and graduate school. This means that to solve the problem the researcher must learn or create mathematics that is new to him. Knowing something about group theory or topology may be of no help, but "mathematical maturity" and experience in doing mathematics is essential. Unfortunately, many engineers and technicians do not have the level of mathematical maturity that they need, and their managers do not understand mathematics well enough to realize the deficiency.
Just as learning to write well requires writing, writing, and more writing, learning to effectively do mathematics requires lots and lots of practice in applying, exploring, creating, and communicating mathematics. How can we, as mathematicians and educators, help students to be participants rather than spectators?
I am convinced that an important part of the answer lies in the use of "interactive mathematics laboratories". An interactive mathematics laboratory (IML) is a computer environment with a set of integrated tools that is intended to facilitate a wide range of mathematical activity. It should provide support for both calculation and deduction and include a large library of mathematics. An IML can assist one in doing mathematics in the way that a text editing system assists one in writing. It is a laboratory in which mathematics can be practiced in the way a foreign language is practiced in a language laboratory. In a future posting, I will explain further what I mean by an IML, and I will argue that, with an IML in hand, a student will be able to travel with greater ease in the world of mathematics, and as a consequence, will learn more mathematics and have more fun.
William M. Farmer 202 Burlington Road, A156 The MITRE Corporation Bedford, MA 01730-1420