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Session 2465

Mathematical Definitions: What is this thing?

Andrew Grossfield College of Aeronautics

Abstract

One sure way to turn students from the study of mathematics is to give them a mathematical definition. These one liners which jump out of nowhere are intended to be the conceptual starting point for five and six hundred page texts. An intelligent student is provided no basis to contest these mandated definitions, even if he does not understand or is confused by them.

New mathematical ideas have always been viewed with suspicion and historically have only been accessible to insiders. Mathematicians have always been under the gun of other mathematicians to justify everything asserted. As a result, a format of mathematics presentation arose that would provide for this justification. Nothing had to be easy to understand, as long as it made justifying, proving easy. Accept as a principle that it makes no sense to prove anything that is not understood. Math teachers conventionally violate the principle. They define something and then state and prove a theorem in the next breath. That is the model to which all traditional math teachers aspire.

Any student who objected too strongly to the meaninglessness of the mathematics regimen was banished to study Social Sciences or Arts. Few texts departed from the program. Only teachers who were schooled in the texts and proud of their mastery of the texts were considered competent.

Almost everyone teaching math today is a survivor of that dictatorial system, a master of the mathematical party line. Now the vogue is reform. Reform of what? Will computers or applications provide new, understandable definitions? Will computers or applications say what we are studying and why?

This paper considers mathematical definitions: π, variables, trig functions, functions, limits and derivatives. It provides examples of good and bad definitions in search of principles by which all students, even those who teach, can glimpse whether they should accept or reject what is being put on their plate.

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Grossfield, A. (2000, June), Mathematical Definitions: What Is This Thing? Paper presented at 2000 Annual Conference, St. Louis, Missouri. 10.18260/1-2--8554