Date: May 29, 2013 1:10 AM
Author: Wayne Bishop
Subject: Re: When math makes sense - w/ cooking, consruction
Details of the algebra notwithstanding, what in the world did you
think I meant? "Better would be to have them express - and then graph
- - - volume of resulting cylinders as a function of side length of
rectangles of fixed area and find the maximum. Nice algebra and it
greases the skids for eventual calculus. What a concept."
At 01:43 PM 5/28/2013, Louis Talman wrote:
>On Mon, 27 May 2013 19:54:08 -0600, Wayne Bishop <email@example.com>
>>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
>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.
> Department of Mathematical & Computer Sciences
> Metropolitan State University of Denver