Xref: utzoo sci.math:17257 comp.edu:4312
Path: utzoo!utgpu!news-server.csri.toronto.edu!rpi!usc!cs.utexas.edu!turpin
From: turpin@cs.utexas.edu (Russell Turpin)
Newsgroups: sci.math,comp.edu
Subject: Re: Preparation of mathematics teachers
Summary: Knowing math *is* important.
Message-ID: <19722@cs.utexas.edu>
Date: 5 May 91 22:04:12 GMT
References: <1991May1.192513.11714@watmath.waterloo.edu> <1991May2.195751.22316@psych.toronto.edu> <11890@mentor.cc.purdue.edu>
Followup-To: sci.math
Organization: U Texas Dept of Computer Sciences, Austin TX
Lines: 43

-----
In article <11890@mentor.cc.purdue.edu> hrubin@pop.stat.purdue.edu (Herman Rubin) writes:
> ... Not knowing how to do differential equations, or even
> calculus is irrelevant. ...

Somewhere along the way, we learned that the volume of a
parallelepiped is the surface area of the bases times the
perpendicular distance between them.  An argument was made for
this in terms of stacked plates shaped like the base, which could
be made quite thin.  (I think the teacher used a couple of decks
of cards and pushed them askew against a rule to make the point
that for the volume, it did not matter whether the stack was
straight or slanted.)  Later, we learned that the volume of a
pyramid was one-third of the base times the perpendicular
distance between it and the peak.  Again, this has an intuitive
argument.  Later, we learned the volume of a sphere.  But why is
it *that*, I asked? 

The teacher had no easy argument.  But, it was promissed, there
is a later course, called calculus, where I could learn this, and
how to calculate the volume inside many curved surfaces.  Thus,
long before I took calculus, I had some intuition about what it
was partly about, and the study of limits, which is to many
students largely unmotivated, I knew had ties to surfaces and
volumes.  Perhaps it would have been better had the teacher been
able to lead me through a simple kind of argument for the volume
of the sphere.  But overall, the answer was not a bad one. 

What struck me was how different it was from the answers I had
had from previous teachers.  In elementary school, as soon as we
learned squares, we were taught the Pythagorean theorem.  But why
is it so?  To this question, the teacher at the time responded:
things are just that way.  I was left with no idea that it could
be proven, or that some of the stuff we had learned about
triangles and squares was the way to do it, nor even that its
demonstration (as opposed to just memorizing it) was part of
mathematics.  (I suspect the teacher did not know enough about
these things to pass on any useful information.)

It may not be important for secondary math teachers to know how
to do calculus or differential equations.  But they should have a
good idea of what these and other subjects are, because they pass
on to students (or fail to do so) what mathematics is all about.