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Re: Plausibility Arguments
Posted:
Mar 13, 1997 1:35 PM
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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
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