```Date: Mar 13, 1997 1:35 PM
Author: Ted Alper
Subject: Re: Plausibility Arguments

Lou Talman wrote:>Purposeful construction of misleading arguments strikes me, in most>instances, as misguided showmanship.  There is a place for such>constructions at post-calculus levels, where proof begins to become>central to instruction.  (I maintain that proof is important, but not >*central* until then.)  I do agree that one wants to separate showmanship from pedagogicallyuseful examples. But even in a geometry class there are lots ofgood false "proofs"... most of the ones I can think of offthe top of my head involve diagrams that get the order ofpoints wrong (and perhaps Geometer's Sketchpad can catcha lot of these)... there's a great one in which one provesthat all triangles are isoscceles.It's hard to do these in ASCII without even the carefully drawnmisleading diagram, but here's a simpler one that constructs atriangle with two right angles -- don't use geometer's sketchpad,or a ruler and compass on this, just draw the picture freehand atfirst:Draw two intersecting circles, of somewhat different radius -- calltheir centers A and B (call the intersection points C and D). From Cdraw the diameters for both circles (CE and CF, where CE goes throughA and CF goes through B). Now draw line EF, and label the points whereit intersects circle A X and label the point where it intersectscircle B Y.  CXE is a right angle, since CE is a diameter of circle A;CYF is a right angle, since CF is a dimater of circle B; X,Y,E, and Fare colinear; therefore angles CXY and CYX are right angles... sotriangle CXY has two right angles!OK, maybe you see right through this -- "properly" presented on theblackboard it can stump a lot of students! >I maintain that the most effective way to deal with the situation is to>present the "prover" with an example that clearly contradicts what has>allegedly been proved.  And then step aside.Absolutely. You want students hunting for the gap in the argument,investigating carefully the chains of reasoning that seemed innocuousbefore.Ted Alper
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