FW: Inclusive and exclusive definitions... again!
Allan Turton
mean that they extend the acute angles even further than what they had drawn
originally. I guess they might have been exposed to the idea that a diamond
must be "pointy", rather than a diamond must have equal sides.
The holly leaf difficulty is something I've not seen. Perhaps it's related
to how some people, some of the time, do things as reflections. For example,
I've noticed that some young children will suddenly write their name
backwards, sometimes with the letters reflected too, sometimes not. I've
also noticed that in groups I do paper-folding with, maybe 5-10% of children
and adults will do all the steps as a mirror image even when the
instructions are right in front of them. For example, they fold the top-left
corner instead of the top-right corner.
As for when not including squares as rectangles becomes a problem, there are
a couple of instances I can think of, but mostly confined to the classroom
environment. One is in testing situations where knowledge/ignorance of
square-as-rectangle can trip up students on badly chosen test items. The
second is when students are trying to establish relationships between shapes
to see the value of class inclusion. Outside of school, knowing a square is
a type of rectangle seems to have almost no value except at trivia nights.
Perhaps this is why curriculum writers can't be bothered getting it right -
people function extremely well without knowing any different.
Allan
On 10/3/10 1:02 AM, "Mary" <maryk...@att.net> wrote:
> If these remarks are inappropriate please forgive me. I'm diving into the
> middle of a discussion without knowing what's been said except for four
> posts in my email.
>
> Certainly I have been irritated by the lessons, etc., that do not include a
> square as a rectangle. I do not know whether or when this original, taught
> misconception becomes a handicap.
>
> My concern is whether you have considered the tendency of young children
> when copying the typical diamond with long axis vertical to exaggerate the
> two sharp points by putting "ears" on them. Also I can remember by own
> frustration by being unable to copy the shape of a holly leaf. Time and
> again I drew scalloped edges, lobes out, which I could see was wrong but
> could not correct. I do not know how old I was, nor do I remember noticing
> when that changed, nor do I know whether this is as typical as the diamond's
> ears.
>
> These considerations do seem to be relevant to the problem of appropriate
> shape names for young children.
>
> Mary Krimmel
------ End of Forwarded Message
Mary
In my recollection, the ears really do look like animal ears, rather like
donkey ears, and that is what child psychologists called them. Have you seen
them? I think they showed when a child (perhaps about two?) was too young to
have been exposed to any shape discussions. Still, one asked the child to
copy a drawing; I doubt that it was ever spontaneous. I do not know whether
the tendency was much investigated, whether the orientation of the figure
could be changed for a similar result, how sharp the angle had to be, what
other developmental signposts it correlated to, etc. Worth looking into?
Maybe.
The holly is something I've neither heard of nor seen, except in myself.
Like you, though, I'm well aware of mirror imaging, have seen many examples
of entire names and of indivdual letters and numerals. Lower-case b-d
confusion is ubiquitous and notorious. Old and childish, I notice that I am
increasingly unsure of left-right, especially when something is upside-down.
An example is table setting. At least experience has taught me to be wary,
think carefully. I've had no experience with observance of paper-folding,
believe I'm in the 90-95% group.
You're certainly right about square-rectangle distinction being a non-issue
for most people. I find the spatial approach fascinating, surely productive.
I've just made models of a triangular dipyramid and a rhombohedron (hope I
have the names right), intend to make more and play with them.
A possibly illustrative aside: I attended a workshop where students were
developing and testing alternative, green life-styles. A couple were working
on a complicated mechanism for powering a sewing machine with a stationary
bicycle. They were unaware of the original treadle Singer I learned to use
as a girl. My ingenious electrical-engineer father eventually rigged up my
mother's machine to run on electricity! Even interest in Dutch-style
windmills is returning, along with the most advanced new designs.
And thank you for posting to the Math Forum, too.
Mary
- ----- Original Message -----
From: "Allan Turton" <a_tu...@origo.com.au>
To: "Mary" <maryk...@att.net>; <app...@support1.mathforum.org>;
<geometry-p...@support1.mathforum.org>
Sent: Wednesday, March 10, 2010 3:03 PM
Subject: Re: Inclusive and exclusive definitions... again!
> P.S. If this email doesn't get through to the email list (as happened with
> the ones to and from Walter) I'll post yours and mine at the Math Forum if
> you don't mind.
Michael de Villiers
You might be interested in visiting this link on my website where I suggest ideas to develop understanding of inclusive classification informally BEFORE introducing definitions by using dynamic transformations:
http://math.kennesaw.edu/~mdevilli/quadclassify.html
You might also be interested in this extended classification of quadrilaterals
http://mysite.mweb.co.za/residents/profmd/quadclassify.pdf
which is an excerpt from my book "Some Adventures in Euclidean Geometry", which discusses quite extensively why inclusive definitions of quadrilaterals are more useful than exclusive ones. For more info go to:
http://www.lulu.com/product/paperback/some-adventures-in-euclidean-geometry/5414956
Hope that helps.
Regards
Michael