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Subject:   RE: More on What is a Trapezoid
Author: lanius
Date: Dec 14 2004
Great who-wants-to-be-a-millionaire story. :-)

Yes, I suspect that someone decided that the hierarchies inherent in the
inclusive definition were too hard for younger students to understand. I don't
think kindergarten materials tell kids that a square is also a rectangle,
unfortunately, and I think that the poodle=>dog=>animal argument should show us
that they can understand if put into the right context. I have to say here that
the Van Hiele model would indicate that maybe they *couldn't* understand it. I
like this site for a van Hiele at a glance.

(And by the way, I'm NOT advocating that if something is too hard for children
to understand that we should distort it to make it easy enough for them. PLEASE
don't anyone conclude that from the last paragraph.)

I would like to talk to early childhood teachers  though, and hear from them
about how children use similarities and differences to both distinguish and
group things. Is it beyond the comprehension of most five year olds to
understand that all squares are rectangles and some rectangles are squares?

So does anyone know of any research in this area?

Much thanks to all who have contributed to this discussion.

On Dec 14 2004, Alan Cooper wrote:
> 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:

> Re: Is a rectangle a square?
Author:       Kit <>
> 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

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
> 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
> 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"

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