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."

Wayne

At 01:43 PM 5/28/2013, Louis Talman wrote:

>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>