```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."WayneAt 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>
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