The controversy about "reform" or "traditional" math education seems to be
fairly well exemplified by the "factoring trinomials" example so I'll use
it to espouse a viewpoint that is easily generalized.
It is my opinion that students today have absolutely NO need to learn
factoring but a VERY significant need to learn factors.
Before I explain in more detail what I mean by the previous statement let
me say that the students I refer to include both students who are planning
to progress to more advanced math classes and those who are taking their
last math class.
A couple of years ago I was helping a student solve a quadratic equation.
We worked until the equation was in standard zero form and then I told the
student that the next step was to factor the trinomial. He had no idea how
to do that so I suggested he read the previous section (Factoring
Trinomials) and do a few problems before coming back to solve the equation.
He angrilly (sp?) refused, stating that he had already done all those
problems and gotten them right. And so he had. After relearning how to do
those problems by looking at the example he was able to factor the
trinomial in the equation.
I view this as the result of teaching factoring rather than factors. Of
course, what I mean by teaching factoring is teaching the PROCESS of
finding factors. So what would be meant by teaching factors? Clearly
teaching factors would mean teaching what factors ARE rather than some
algorithm for finding factors. The poor student I used as an example above
had learned the algorithm quite well and could factor trinomials easily but
the trouble was that he didn't have even a vague notion of what he was
finding.
As long as the overwhelming majority of American mathematics education
consists of teaching "how to" instead of understanding, it is not going to
matter at all whether we teach students ow to factor trinomials on paper
or on a calculator or computer. It also won't matter whether we teach them
how to factor trinomials or how to gather data and put it into a graphing
calculator or how to approximate the real roots of a polynomial by looking
at the graph.
In an attempt to end(hooray) on a slightly more positive note, it has been
my experience that students are, in general, perfectly willing and able to
learn to understand, communicate, and apply the concepts of mathematics if
it is made clear to them that in order to pass the course they must do
those things. This does of course imply that the teacher can no longer get
by with just working examples and homework problems in class. It is my
fervent hope that that won't bother many of the math teachers. Sorry this
was so long.
Wayne F. Mackey
SCEN #301
University of Arkansas
(501) 575-7661
wmackey@comp.uark.edu