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Re: Plausibility Arguments
Posted:
Mar 12, 1997 8:43 AM
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Lou,
This is, in my opinion, a strong analysis of the role of proof in the
mathematics curriculum, especially the final sentence. I hope you are
writing up some version of this for the MT Focus Issue on Proof.
A personal experience on the misguided zeal of "EIT"'s. One of my biggest
surprises as a curriculum developer for high school geometry came when my
advisory board discussed a number of issues related to proof. I wanted to
take a softer, more common-sense (I thought) approach to proof -- for
example, avoiding tiresome issues like congruence of segments vs. equality
of their lengths. My reasons were purely pedagogical: These things trip up
students who are perfectly capable of deductive reasoning but don't see the
importance of all the picky detail. I was more concerned with their being
able to see the broad picture of proof, and why proof is important. (Note
that the Van Hiele levels are also applicable to this argument.)
My strongest allies on the advisory board ended up being the research
mathematicians. My sharpest critics were classroom teachers. I found this
fascinating. The "real experts" cared about whether a plausible argument
was made. The teachers were more concerned about the particular details of
a proof.
The danger that I see is that many teachers read the Standards as saying
rigorous proof is not important and that it should not be the sole focus of
geometry, missing what I think is the real message -- proof should always
be done at an appropriate level of rigor, and it should be a part of all
areas (including geometry). And should always be aimed at deepening
students' levels of thinking about mathematics, "teasing [them] deeper into
more arcane issues".
Gary
At 1:29 -0500 3/12/97, Lou Talman wrote:
...
>The crux of the matter is this: A good teacher can be precise without being
>pedantic. Issues that an expert-in-training (EIT) will think important will
>seem pedantic to both the novice and the real expert. Precision demands those
>issues be dealt with; the EIT is learning this and must find them and deal
>with them publicly--not only to convince herself that she can but to convince
>her teachers as well. The expert knows about the issues and deals with them
>privately. (Dealing with issues privately: One of my favorite teachers
>taught me that it is impolite to compute in public.) The novice doesn't yet
>know about the issues--let alone understand them or know how to deal with
>them.
>
>In point of fact, almost all "proofs" are really plausibility arguments. Good
>teaching is the art of casting our plausibility arguments at a level that
>convinces the EIT but keeps him stretching for that conviction, teasing him
>deeper into more arcane issues, forcing him gently but inexorably toward the
>expert level.
>
>--Lou Talman
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