Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Plausibility Arguments
Replies: 11   Last Post: Mar 14, 1997 10:04 AM

 Messages: [ Previous | Next ]
 Ted Alper Posts: 118 Registered: 12/6/04
Re: Plausibility Arguments
Posted: Mar 13, 1997 1:35 PM

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

Date Subject Author
3/12/97 Lou Talman
3/12/97 W Gary Martin
3/12/97 Ed Wall
3/12/97 Ted Alper
3/13/97 Lou Talman
3/13/97 Andre TOOM
3/13/97 Michael Paul Goldenberg
3/13/97 Steve Cottrell
3/13/97 Ted Alper
3/13/97 Andre TOOM
3/13/97 Ted Alper
3/14/97 W Gary Martin