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Topic: MATHEDU: Mathematics as process
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William M. Farmer

Posts: 1
Registered: 12/6/04
MATHEDU: Mathematics as process
Posted: Jan 9, 1997 11:28 AM
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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.

Bill Farmer


William M. Farmer
202 Burlington Road, A156
The MITRE Corporation
Bedford, MA 01730-1420

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