Back to factoring, a student who makes a "minor" error in a process and
thereby gets a wrong answer, but thinks that answer is "partly right",
because that is how we grade, has missed the big concept that any
manipulation on a mathematical expression yields an equvalent mathematical
expression.
Martha
>Dear All,
>
>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 te 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 how 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