Background for Algebra

Algebra is built upon a foundation of arithmetic. This much is obvious. But how much of a background in arithmetic is needed before it makes sense to begin the study of algebra?

At one time little if anything was done to introduce students to algebra prior to high school, and in high school algebra was something that was only taught to the more academically inclined students. In an effort to upgrade mathematics education in America around the time of Sputnik, we began teaching algebra more widely, and we began introducing students to algebraic concepts at an earlier age.

Fifty years later we have students who are starting to study algebra earlier. They are spending more years in high school studying algebra. And they are taking dumbed down SATs to make the comparison with students who took the test before Sputnik look less embarrassing than it would otherwise. It is true that more students are now going to college, and more students are now taking the SATs--but that only explains the decline if we assume there has been no real improvement in mathematics education despite massive time and effort devoted to that goal.

Before Sputnik we did drag our feet, so to speak, delaying the teaching of algebra far more than we needed. But in many ways the reforms in math education have only made things worse. Students need to have a math background that will help them to make sense of algebraic concepts. These concepts can be confusing to many students. And like people, algebra has only one chance to make a first impression. When we teach math, it is better to wait a little longer before introducing students to algebra, than rush to introduce algebra, and make the first impression that students get of algebra a confusing one.

What if algebra has already made its first impression, and that first impression has not been a good one? In that case we will want to go back and reintroduce you to algebra. Just as in human relationships we can overcome a bad first impression--but it does take some effort, and it does take a recognition that something of this sort needs to be done.

EZ Algebra rests upon a foundation consisting of three pillars. The first pillar is basic arithmetic: counting, addition, subtraction, multiplication, division, and straightforward problems that require the use of these basic arithmetic operations.

A second pillar is algebraic notation. The best introduction to algebraic notation is one where we use formulas such as p = 4s as a shorthand notation for a verbal rule such as the perimeter is 4 times the length of the side of a square.

The third pillar is non-straightforward arithmetic or what I call verbal algebra. The more challenging percentage problems are an example. If adding a 20% tip to a bill makes the total $60, what was the cost of the meal without the tip? A common mistake would be to subtract 20% of $60 and get $60 - $12 = $48. But this would be wrong. The tip isn't 20% of the total including the tip. It is only 20% of the total before the tip--and that total is something we don't know. But if we think of the problem in a different way, we can solve the problem. Adding 20% to our basic 100% gives us 120% of the basic bill or 1.2 times the basic bill. Dividing $60 by 1.2 will give us our answer of $50. The reasoning we have used here is quite similar to what we do in algebra, and having a background that includes working with problems like this one will give us a leg up when we begin our study of algebra.