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Training Wheels Off

Like virtually all schools, Barringer practices “tracking,” the segregation of students by speed and quality. Tracking is the subject of a lot of educational debate, and arguments can be made for and against it, but nearly every school in the country has an honors track, a standard track, and a remedial track by some name or another.

None of my classes is the fast track. For a high school math teacher, this means fighting a lot of mental blocks: “I’m just no good at math.” Well, it’s true. Lots of people aren’t good at math. But then, you don’t need to be good at math to handle 90% or more of high school mathematics. Almost all of what we do is conceptually simple, the meaning tragically lost in arcane symbols. As a tutor back in college, I repeatedly reached the point where a student who had been struggling with a concept finally heard the penny drop; he or she would invariably sit up straight and say, “That’s all there is to it?” And I would say, “That’s all there is to it.” High school math is easy, if only you can keep a grip on what the symbols mean..

So a high school math teacher often labors to get this point across. “You can do this. You really can. Trust me.”

Case in point: long division of polynomials. Someone who works with math often has need to “find the zeroes” of a polynomial, a process of determining which of a very few values a variable, traditionally X, might take so as to satisfy a given equality, such as 2x^3 +5 x^2 – 14x – 8 = 0. Generally speaking, x can take a number of values equal to power of the polynomial—in this case, three—and to find them, the solver must tease the polynomial apart into separate linear factors, in this case (x-2)(x+4)(2x+1). Since the product of these three factors is zero, one of them must be zero, meaning X must equal 2, -4, or -1/2, according to which of these three factors equals zero. The solver can then test which of his several solutions satisfies the needs of the actual, material problem to which he is applying the algebra.

(See what I mean? The idea is easy. If x=2, then x-2=0, so any product including x-2 must be zero. And conversely, if a product containing x-2 is zero, there’s a good chance that x=2. But talking in jargon like “linear factors of a third-degree polynomial” makes the process seem incomprehensible.)

Often, a solver can spot one “zero,” one possible value of X that satisfies the equation, through a combination of luck and intuition and basic arithmetic. He can then remove one factor from the polynomial and continue working with a simpler problem. Here is where long division comes in. Had we, for example, realized that x=2 is one of the “zeroes,” we could divide 2x^3 +5 x^2 – 14x – 8 by x-2, set aside x=2 as one of our “zeroes,” and find the remaining two “zeroes” for 2x^2 + 9x + 4 = 0, a much simpler prospect. (Really. It’s easier if you can see it done; an online journal entry is a poor place to demonstrate the manipulation of polynomials.)

The process of removing the factor of x-2 from the original polynomial is exactly analogous to long division, the process by which we all learned to divide 155771 by 17. Drawing out that long bracket shape, putting the thing to be divided beneath it, putting the thing by which it is to be divided to its left, and, column by column, constructing the quotient, taking care to keep our columns tidy and watch our signs. The process is exactly the same, except that our columns now represent powers of X instead of powers of ten.

And so I was able to introduce the subject to my students by saying “You can do this,” and follow up with “I know you can do this because you already do it. It’s so easy that you’ve been doing it since you were in grade school. Just nobody told you at the time that you were already doing algebra.”

I think it worked. Most of my kids in that class caught on quickly. Not that the idea was difficult to grasp—again, high school math rarely is, and I was sincere when I told the students they already knew how to do it—but it’s easy to panic when confronted by X’s and Y’s and square roots. To their credit, most of my students didn’t panic.

But the results were not uniformly positive. A few student’s homework was nearly blank; either they were so lazy that they didn’t bother to try, or were so at sea that they didn’t know where to begin. When a math teacher extends his figurative hand and says, “You can do this. You really can. Trust me,” he means it. But of course the student ultimately has to trust, or it all breaks down.

And so begin the exercises in building student-teacher trust.

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