Four ways
Our tutor is faced with too many alternatives.
Many of our tutor’s mathematics students are now dealing with polynomials, and specifically finding their roots. For general polynomials of degree higher than about three, certainly five or more, this is a mathematically hard problem. That means it’s a challenge to the best minds, and computer algorithms need to be chosen carefully. Students, however, are more or less guaranteed problems that will work out evenly and without too much trouble. (This gives a somewhat deceptive impression of math, but is necessary for teaching.) One set of problems that receives much attention is that of factoring quadratics. For those of you who haven’t done it recently (which covers almost all adults), that means something like turning x² – 2x – 35 into (x – 7)(x + 5). You need to find numbers that add up to one of your coefficients and multiply out to the other.
Our tutor learned to do this in a sort of logical-guess way, and with some practice was able to spot useful patterns quickly. Some of his students are taught a similar method. But there are at least three other techniques: grouping (which is also useful for higher-order polynomials, but takes a certain cleverness), the diagonal-X method and (one he saw only within the last year or so) the box method. The difficulty lies in the fact that a student will only learn one technique, and explaining another just causes confusion. And indeed, as our tutor has found, keeping all four in your head at once is very difficult; he tends to use just one of them and to forget how to operate the others. Similarly, he forgets how to do Synthetic Division, since he can get the same results through Long Division as quickly, and can apply the latter to a wider variety of problems.
Why worry about any of this? Given that almost all students will never factor anything in their later lives, and the task is the mental equivalent of a weight-room exercise, maybe there’s no point in spending too much time on the alternatives. It’s fairly straightforward to check an answer (and to teach the student to do so), and getting the student to find his or her own mistake is a worthwhile exercise.
But students need various levels of help, and our tutor is tasked with supporting each one. He is now facing the necessity of relearning each of the four techniques of factoring polynomials. In the same way, he is faced with the task of relearning other topics he sees rarely and tends to forget in the meantime: things like buffered solutions and reaction mechanisms in Chemistry, and 3D equations of lines and planes in Multivariable Calculus (he is much happier with the more general techniques of four-space). It comes with being a teacher. But he’d really rather no one came up with another method of factoring quadratics.
