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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: Mathman Date: Jan 2 2005
On Jan  1 2005, Alice wrote:
> These are the kinds of questions we should encourage the students to
> generate. They should be shown two or more conflicting definitions
> in textbooks and decide, as this discussion has, what would change
> if we altered the definition. Then they could go on to answer
> Cynthias initial question. What a great geometry course that would
> be.
Then they will really understand what math is and what
> mathematicians do..and not just gifted kids either....but any
> students who can pass a geometry course. Great discussion!!!

I agree.  It all comes under the heading of "necessary and sufficient" perhaps;
something that I observed in another discussion.  The thing is that it takes
some practice before insight is gained to the extent that most [or just some?]
can readily see what is required by a problem.  At the very least, the
definitions allow the student to look for information that is both necessary and
sufficient, ...and so no more, or less than that... to resolve a problem.  That
is; the student learns to ask himself/herself, at each point, "What do I need to
show that this is a rhombus, and what information do I have shown directly?  Now
what further information do I need to complete the proof?"   This is still an
arduous task at best, and I wonder if the school curriculum these days has the
time to do this with this study and with other areas at the same time?  That's
part of the problem as I see it ... there are now too many other topics,
elements of which are slipped down from above, that occupy time that should be
spent in more detail with topics at hand.  We can't have our cake and eat
it.

David.