TAKING THE TRICKS OUT OF MATHEMATICS

Primus: Problems, Resources, and Issues in Mathematics Undergraduate Studies, Mar 2004 by Sprows, David J

ABSTRACT: This note considers some of the mathematical devices or "tricks" that have become a standard part of the presentations of such material as the division of complex numbers or the product rule for derivatives. Alternates to these tricks are discussed as well as the reasons why such tricks should not be used.

KEYWORDS: Complex division, derivative, product rule.

INTRODUCTION

One of the advantages mathematics has over other disciplines is that there is a natural flow to the material. Ideally each new step in the development of a given topic can be viewed as a logical consequence of the previous steps. This is one of the most important aspects of mathematics for students to appreciate. Unfortunately, there are some standard presentations of basic material that sacrifice this natural flow for technical expedience.

A prime example of this occurred recently in a freshman class during a discussion of the division of complex numbers. When the class was asked how to express the quotient of two complex numbers as a complex number, it was clear that everyone in the class had seen this material before. However, the first student who responded stated: "I know there's a trick, but I forget it." It turns out that over a third of the class had forgotten the "trick". Moreover, no one in the class had any idea how to approach this matter without using the trick. This is not surprising since a survey of over a dozen books revealed that each of them introduced the division of complex numbers by multiplying the numerator and denominator by the conjugate of the denominator.

This approach may be the most efficient way of showing how to compute the quotient of two complex numbers, but students tend to view it as involving the introduction of a somewhat arbitrary "trick". Actually it can be shown that this trick is a natural consequence of applying the operation of division to complex numbers. Taking the time to present this background material on complex division does not necessarily contribute to the student's computational skills, but it is definitely worthwhile to add this material to the standard presentation.

DIVISION OF COMPLEX NUMBERS

So in order to express 1/(a bi) without i in the denominator it is necessary to replace 1 by a - bi and a bi by a^sup 2^ b^sup 2^. At this point in the presentation it is not unreasonable to expect that someone in the class will note that this corresponds to multiplying the numerator and denominator by a - bi. At a minimum the class will see that such a step is a natural consequence of applying the operation of division to 1/(a bi).

PRODUCT RULE FOR DERIVATIVES

The trick used in complex division involves multiplying an expression by the number one disguised as a - bi divided by itself. Another trick of the same general type involves adding a "disguised" form of zero to an expression. One of the most common appearances of this trick occurs in the derivation of the product rule for derivatives. A key step in the derivation of the rule for differentiating f(x)g(x) involves taking the limit of a quotient whose numerator is f(x h)g(x h) - g(x)f(x). A fairly standard approach (see for example [1] or [2] ) is to add f(x h)g(x) - f(x h)g(x) to this expression and then proceed.

This approach may result in a mathematically rigorous derivation of the product rule for derivatives, but it adds nothing to the student's understanding of the material. In fact it has the negative effect of conditioning students to accept derivations that involve the introduction of artificial and unmotivated steps. A more natural approach to the product rule can be obtained by using the classical notation and techniques (see [3] for example). Under this approach the change in the product uv is given by (u [Delta]u)(v [Delta]V)-uv which equals u . [Delta]v v . [Delta]u [Delta]u . [Delta]v. If we divide this by [Delta]x and take the limit as [Delta]x tends to zero, we have u times the derivative of v plus v times the derivative of u plus the derivative of u times a term that goes to zero as [Delta]x tends to zero. This classical approach may not be as efficient or as rigorous, but it has the advantage of letting the student see what is involved in producing the product rule.

CONCLUSION

Students and faculty are often impatient to get to the "bottom line". Most students realize that they will not be asked to derive computational rules on tests, but merely to be able to apply them. As a result they tend to have a lack of interest in the details of the material leading up to these rules. This puts pressure on the instructor to use whatever tricks are available to get through the background material as quickly as possible. Using such tricks is a disservice to the students. Most of them will not retain or need many computational skills once they leave school. On the other hand, they should all have a lasting appreciation of mathematics as a unique discipline where each new step in the development of a topic is a natural consequence of the previous steps.