>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.