Date: May 28, 2013 4:43 PM
Author: Louis Talman
Subject: Re: When math makes sense - w/ cooking, consruction

On Mon, 27 May 2013 19:54:08 -0600, Wayne Bishop <wbishop@calstatela.edu>  
wrote:

> If they really know the formulas, they would start by using the given
> information to compute the volumes based on the evidence presented.


How very interesting.

Wayne prates incessantly about the necessity of avoiding math avoidance by
teaching kids the power of algebra. And then he suggests the weakest
possible mathematical strategy for arriving at a correct solution to this
problem. And it's a solution that uses algebra in name only, substituting
numbers into a memorized formula in order to compare numerical results.
But let's give him some credit: This is consistent with his rote approach
to word problems ("word problems by type"). You get a few points for
consistency, Wayne, but none for mathematics---which looks at the
*context* of a problem, and not just its answer.

If Wayne's approaches aren't algebra avoidance, I don't know what is. And
Robert acquiesces, suggesting that "a student of algebra" would never
avoid algebra---leaving us to guess that he agrees that Wayne's strategy
is Real Algebra.

(In fact, students of algebra generally avoid algebra as much as they can:
Learning new ways of thinking involves work that they'd rather avoid.)

Here's a *real* algebraic approach: The volume of a cylinder is Pi r^2 h,
where r is the radius and h is the height. Let's begin with a sheet of
paper of length L and width W = k L, where k is some positive real number
that's at most one. If we roll the paper up along an axis parallel to the
L side of the paper, the radius of the resulting cylinder is k L/(4 Pi),
so the volume we've formed is

V_L = Pi [k L/(4 Pi)]^2 L = k^2 L^3/(16 Pi).

If, on the other hand, we roll the paper up along an axis parallel to the
W side of the paper, the radius is L/(4 Pi), and the volume of the second
cylinder is

V_W = Pi [L/(4 Pi)]^2 k L = k L^3/(16 Pi).

So V_L = k V_W.

Now we've answered the question---but we've done much more than that.
We've explained the answer in a way that applies in more general
circumstances. Moreover, we're building an understanding of the
relationship between measurements of length, area, and volume.
Substituting the given numbers into the cylinder's volume formula
accomplishes none of these things.

What Wayne's sermons and Robert's acceptance of them give us is something
less than a half-measure.

But the two of them are right in one respect. Without what I've just done
here, the activity isn't complete. That doesn't mean that it's useless,
though. Understanding of a phenomenon is based on repeated examination of
the phenomenon in a variety of circumstances, of which this activity
provides one. It should be part of a progression that leads students to
ask "Why?" And that leads, ultimately, to the analysis I've given above.

- --Lou Talman
Department of Mathematical & Computer Sciences
Metropolitan State University of Denver

<http://rowdy.msudenver.edu/~talmanl>