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Session 3365

Mathematical Forms and Strategies

Andrew Grossfield College of Aeronautics

Abstract

One of the most important mathematical concepts at every level is the concept of form. Starting in elementary school arithmetic, continuing through high school algebra and trigonometry, throughout calculus and in advanced graduate mathematics, students are taught and graded on how to change forms. Many students who master the techniques of such form changes as completing the square, factoring and expanding polynomials do not know why or when to perform these operations. Little is said about the concept of forms or about the important aspects of forms. There is more to know about forms than how to change forms. Form changes should be treated as tactics in the strategies of analytical problem solving. This paper describes the principles concerning forms, which are usually omitted from the appropriate texts, and the important place of forms in the strategies of mathematics.

1. Principles of Mathematical Forms

• It is natural for mathematical objects (integers, rational numbers, other real numbers, functions, vectors, linear transformations, complex numbers, tensors, etc.) to have more than one form. • There is no best form for all purposes. • Useful special canonical forms are found sometimes. • We should be aware of the advantages and disadvantages of the canonical forms that embody the objects being studied. • The principles and techniques involved in changing forms should be made apparent.

2. Elementary Examples of the Principles

First, consider fractions. Many forms for writing fractions exist. A canonical form is the fraction expressed in lowest terms. This form is suitable for multiplying and dividing fractions, but this form is not suitable for the addition or subtraction of unlike fractions. Then we need to know that it is always possible to find numbers that will serve as common denominators (and even a best common denominator) so that we can make the fractions alike. We need to know that the multiplication or division of any fraction by any form of the number one will not change the value of the fraction. With this understanding of form changing for fractions, it is possible to accomplish the four arithmetic operations on any pair of fractions.

Second, consider complex numbers. Many forms exist for writing complex numbers. Rectangular form is suitable for addition and subtraction. While it can be cumbersome to

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Grossfield, A. (1999, June), Mathematical Forms And Strategies Paper presented at 1999 Annual Conference, Charlotte, North Carolina. 10.18260/1-2--7826