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Discussion: Dynamic Geometry Exploration: Properties of the Midsegment of a Trapezoid tool
Topic: Midsegment of a rectangle?
Related Item: http://mathforum.org/mathtools/tool/15621/


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Subject:   RE: More on What is a Trapezoid
Author: Alan Cooper
Date: Dec 14 2004
Getting back to Cynthia's question re source of the popularity in textbooks of
what many of us consider a wrongheaded definition, I don't have any historical
insight but I do have some speculations re the motivation.

One is that it has less to do with mathematics than with some other context in
which the shape occurs. For example a table is stable against folding sideways
only if its legs are not parallel, and the perspective view of a rectangle can
be a trapezoid but not a non-rectangular parallelogram.

Aside from possible roots in such non-mathematical usage, it seems to me that
the popularity of the exclusive definition in textbooks may have two sources -
one "good", and one "bad".

The "bad" would be a pedantic attachment to some notion of "precision" that
mistakenly identifies it with having lots of detail and being somehow
inconsistent with generalization (which is in fact the higher value from a
mathematical perspective)

The "good" source is a genuine belief that exclusive definitions are easier for
children to grasp and so are more appropriate for elementary textbooks.  I don't
know if there is any evidence for such a belief, though, and am inclined to
doubt it as children learn from an early age to handle "nested" concepts (eg
poodle is dog is animal)

On Dec 14 2004, Alan Cooper wrote:
quoting Floor van Lamoen

> I am afraid that if one teaches pupils to be precise
> on these exclusive definitions, one teaches them to focus
> on the wrong things, and perhaps forget the important concept
> of generalization.



Oh. and here's another item from the MathForum geometry-precollege archives
which shows that the divergence between encyclopaedists and mathematicians on
this issue is not a purely American phenomenon:

Subject:      Re: Is a rectangle a square?
Author:       Kit <steinadler@gmx.de>
Date:         28 Sep 04 05:21:38 -0400 (EDT)

Pamela Paramour wrote:
>Is a square a rectangle?  ...
> If you refer to Webster, ...

Here is a nice story realy happend in german tv, sorry for my bad
english.

In the german quiz-show "Wer wird Millionär" (Who becomes a
millionaire) from January, 31 2003 the 8000-Euro question was:
Every rectangle is:
(a) a rhombus
(b) a square
(c) a trapezoid
(d) a parallelogram.

In this show _allways_ exactly one answer is (has to be) correct.
The candidate was so confused, she didn't know if c or d is thw right
answer, so she skipped the question and went home (with "just" 4000
Euro). In the following days the broadcast station got tons of mails,
letters and phone calls. Nearly all "mathematicians" regarded c _and_
d as correct. The broadcast station told, that they looked up in three
different encyclopaedias, all three saying that trapezoids have only
one pair of parallel sides. Taking this definition only d is correct.

That's the problem. Who is right: More than 90 percent of the
mathematicians saying a parallelogram is also an trapezoid or three
encyclopeadias saying the opposite?

The Solomonian solution. In the next week the candidate got a "new"
8000-Euro-question.


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