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 pedagogically

useful examples. But even in a geometry class there are lots of

good false "proofs"... most of the ones I can think of off

the top of my head involve diagrams that get the order of

points wrong (and perhaps Geometer's Sketchpad can catch

a lot of these)... there's a great one in which one proves

that all triangles are isoscceles.

It's hard to do these in ASCII without even the carefully drawn

misleading diagram, but here's a simpler one that constructs a

triangle with two right angles -- don't use geometer's sketchpad,

or a ruler and compass on this, just draw the picture freehand at

first:

Draw two intersecting circles, of somewhat different radius -- call

their centers A and B (call the intersection points C and D). From C

draw the diameters for both circles (CE and CF, where CE goes through

A and CF goes through B). Now draw line EF, and label the points where

it intersects circle A X and label the point where it intersects

circle 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 F

are colinear; therefore angles CXY and CYX are right angles... so

triangle CXY has two right angles!

OK, maybe you see right through this -- "properly" presented on the

blackboard 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 innocuous

before.

Ted Alper