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ARE FUNCTIONS REAL?

Andrew Grossfield

Vaughn College of Aeronautics and Technology

Abstract

Some think that the calculus reform movement, consisting of computers, calculators, hands-on, interactive and applications, is going to make calculus and other mathematical analyses understandable and accessible to high school and college students. I remain skeptical and hope that many others share my skepticism. The reason students have problems learning math is that the explanations they receive and the organization of the their texts and other teaching materials are not clear. Many students do not see that functions are real and that functions are central to the study of calculus. This paper will be a discussion of scientific reality as well as mathematical reality. Of course, functions are real, and it is our responsibility as math teachers to guide every student to the awareness that the study of functions is worthwhile and to provide a framework, which will enable students to manage their study. This paper is intended to serve students who may find themselves adrift in their studies of algebra and calculus.

Scientific Reality

What is meant by real? Science is concerned with that which is observable by the community or is repeatable or demonstrable and non-contradictory. Of course, occasionally theories fail under new experiments and perspectives. The phlogiston theory of combustion, while never disproven, had to yield to Lavoisier’s more extensive theory of oxidation. Unicorns, werewolves and Superman obviously can be dismissed as unreal. There is no end to the list of unrealities that people believe in. In our modern world, many students study the sciences, and yet say they believe in astrology. As a scientist, I am mystified.

What are Functions?

A study of a simple quantitative system might be started by identifying the variables that describe the system. It is possible that there may be no relationships between these variables. However, in special cases the values of some variables determine the values of other variables. When one or more variables determine the value or values of another variable, the relationship between the variables is called a function. The relationship might describe variables that happen to move

“Proceedings of the 2005 American Society of Engineering Education Conference &Exposition Copyright © 2005 American Society of Engineering Education”

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Grossfield, A. (2005, June), Are Functions Real? Paper presented at 2005 Annual Conference, Portland, Oregon. 10.18260/1-2--15093