How To Prepare Students for Algebra

teaching of algebra in the national spotlight. The present national goal is not only “Algebra For All,” but also “Algebra in the Eighth Grade.” Because algebra has come to be regarded as a gatekeeper course—those who successfully pass through will keep going while those who don’t will be permanently left behind—the high failure rate in algebra, especially among minority students, has rightfully become an issue of general social concern. Many solutions of a pedagogical nature have been proposed, including the teaching of “algebraic thinking” starting in kindergarten or first grade. I will argue in this paper that no matter how much “algebraic thinking” is introduced in the early grades and no matter how worthwhile such exercises might be, the failure rate in algebra will continue to be high unless we radically revamp the teaching of fractions and decimals. The proper study of fractions provides a ramp that leads students gently from arithmetic up to algebra. But when the approach to fractions is defective, that ramp collapses, and students are required to scale the wall of algebra not at a gentle slope but at a ninety degree angle. Not surprisingly, many can’t. To understand why fractions hold the potential for being the best kind of “pre-algebra,” we must first consider the nature of algebra and what makes it different from whole number arithmetic. Algebra is generalized arithmetic. It is a more abstract and more general version of the arithmetic operations with whole numbers, fractions, and decimals. Generality means algebra goes beyond the computation of concrete numbers and focuses instead on properties that are common to all the numbers under discussion, be it positive fractions, whole numbers, etc. In whole number arithmetic, 5 + 4 = 9, for example, means just that, nothing more, nothing less. But algebra goes beyond the specific case to statements or equations that are true for all numbers at all times. Abstraction, the other characteristic of algebra, goes hand-in-hand with generality. One cannot define abstraction any more than one can define poetry, but very roughly, it is the quality that focuses at each instant on a particular property to the exclusion of others. In algebra, generality and abstraction are expressed in symbolic notation. Just as there is no poetry without language, there is no generality or abstraction without symbolic notation. Fluency with symbolic manipulation is therefore an integral part of proficiency in algebra. I will give an illustration of the concepts of generality and abstraction and how they are served by the use of symbolic notation by considering the problem of when the area of a rectangle with a fixed perimeter is largest. This of course would not be appropriate as an entry-level algebra problem, but we choose it because it is an interesting phenomenon and because it illustrates the nature of algebra well. As preparation, let us begin with some well-known algebraic identities. If x, y and z are any three numbers, then