>From the NY Times article posted by Chris Green:
>"My whole experience in math the last few years has been a struggle
against the program," Jim said recently. "Whatever I've achieved, I've
achieved in spite of it. Kids do not do better learning math themselves.
There's a reason we go to school, which is that there's someone smarter
than us with something to teach us."<
It goes deeper than that. Vanishingly few of those *teaching* the subject
matter could have come up with the ideas themselves. Even 'elementary'
parts of mathematics were derived and developed originally by people of
outstanding intellectual abilities. To expect more than a small minority
of children to arrive at the concepts themselves, even with the guidance I
presume they must get in "constructivist math", is absurd -- and becomes
more so with even modest increases in the complexity of the mathematical
ideas.
But it's not a matter of "either/or" ("rote learning" or "understanding").
A good schoolteacher will not simply present new notions to a class and
say "there it is now use it". He/she will set the mathematical notions
about to be presented in practical context, continually invite suggestions
how to proceed, give clues about what might be done at each stage, and so
on. In other words, it is not a matter of what is often derided as "chalk
and talk". It should be a constant two-way process in which students are
drawn into the process of arriving at the final result, even though in
most cases they couldn't have derived it themselves.
So there's no necessary contradiction between "understanding" and knowing
"by rote". Ideally the student should understand the derivation of the
formula/procedure, become familiar with it by practice so that it comes to
be known automatically, i.e., by rote. (Sometimes this will mean the
deliberate memorising of a formula.) The formula/procedure is then
immediately available for use in more advanced work. Only in that way can
a student develop a reasonable mastery of the subject at each stage.
So, to recapitulate, there should be definite steps in the process of
teaching students how to handle problems of the kind cited by Michael
Scoles:
Solve 15 - 7 =3D
First explain the concepts by means of which the problem is solved
(usually, in this case, by analogy with balancing scales). Then get the
students to solve a few similar equations, writing out the steps in full.
Move quickly on to getting them to solve similar problems, "saying" the
logical steps *in their minds* as they do it. Finally, the procedures
should become automatic, so that the student doesn't have to think out the
logic of the argument each time, and doesn't write down more than the
minimal number of steps. The procedure has then become an integral part of
the student's mathematical thinking, to be applied in any appropriate
situation as required (including on psychology courses!).
This "traditional" approach to the teaching of mathematics in schools is
not only, I believe, more effective than the constructivist approach, it
is also far less time-consuming, enabling considerably more material to be
covered.
Allen Esterson
Former lecturer, Science Department
Southwark College, London
[EMAIL PROTECTED]
---------------------------------
Wed, 9 Nov 2005 21:31:34 -0500
Author: "Mike Palij" <[EMAIL PROTECTED]>
Subject: Re: "constructivist" math
> On Wed, 09 Nov 2005 17:34:07 -0800, Christopher D. Green:
> >
> >Here's a NYT article about a basic educational dispute that may
> >well outstrip the evolution "debate" in terms of its long-term
> >implications. I notice that most of my stats students know next
> >to nothing about trigonometry or probability -- topics I first
> >learned about in high school. About half have never seen a
> >factorial and a few every year do not know what exponents are.
>
> Meanwhile, over on the Math Psych mailing list, people are
> discussing what can be done to get more people interested and
> involved in mathematical psychology -- especially as fewer
> and fewer U.S. students are interested in math in general.
>
> Well, now we see why. RIP Math Psych?
>
> -Mike Palij
> New York University
> [EMAIL PROTECTED]
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