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Re: education, technology & chat (The Mathematics of it)

Thanks, Bill, for such a prompt answer.
I'm afraid I don't know the "TERC Geometry 1 Topic" to unpack " learning
about polygons -- describing and making shapes."
(Just to set some background: I was a co-author of "The construction zone"
with Denis Newman and Mike Cole and I worked with mathematics software in
the early fifth dimensions and in work on mathematics genetically primary
examples [germ cells] written about with Belyaeva and Soldatova.  I have
recently been
looking at very early mathematics education content.  One of the recent
things that has intrigued me is Deborah Ball's thesis that teaching
mathematics is one among other branches of mathematics.  And in that line is
the work of  Ma, Liping.  1999.  Knowing and Teaching Elementary
Mathematics: Teachers' Understanding of Fundamental Mathematics in China and
the United States [Studies in Mathematical Thinking and Learning].  Mahwah,
NJ: Erlbaum.)

We undoubtedly agree that what to tally and whether more or less of
something tallied can be interpreted as helpful for development, depends on
the underlying mathematical concepts.  I say this because you wrote "The
buddies are talking math shapes to each other more often than the ones who
are working individually, although when one child noisily discovers
something new, he draws the attention of many others in his vicinity, "  and
"When she picks partners she is thinking in an integrated way of the
children's social and cognitive development and what sort of mutual zoped
will emerge between the two partners.  The zoped is highly multidimensional
as well as bidrectional."  To me, those two statements suggest you (and the
teacher) rely on analyses of mathematics concepts, the ways they are
represented in talk and act, and the ways that representations are  involved
in children's planning, guiding, monitoring, and checking moves in the
sometimes long and winding road of development of the concepts and of
mathematics proficiency in a more global sense.

So, I want to know more about the mathematics.  The click and drag part
sounds like a kind of tangram activity, so children might be getting at
compositionality and analyzable units within apparent units as well as
stability under transformations (rotate/flip).  So, I wonder how the work is
capitalized on -- for topology, projective geometry, Euclidian concepts,
other domains of mathematics?  How are the acts and talk mathematized?  In
the hands-on blocks and computer versions is the chance taken to get at the
similar (enclosed forms) and the different (3D faces, edges, and corners but
just sides and angles in the 2D)?  (Tangrams mask the 3D-2D contrast; does
the activity you are describing do that in the opposite way?)  Does
discussion of a match in the eyeball activity bring in estimates and
precision about mathematical attributes of mathematical entities (like
number and size of sides and angles) compared to other nice but not
mathematical ones?

As a separate note, have you seen the work that Deb Leong and Elena Bodorova
have done in Colorado and New Jersey with pre-K and K children using
external mediators to promote powerful "buddy" work?  They have big ears and
lips, for example that help self- and other-regulation early on in a pair's
work together (and
eventually get lost).  I mention it to think about times when the teacher
sees a pairing as desirable on cognitive grounds and wants to
engineer/scaffold the social aspects of their development so they can profit
from the pairing.


----- Original Message ----- 
From: "Bill Barowy" <xmcageek@comcast.net>
To: <xmca@weber.ucsd.edu>
Sent: Tuesday, November 09, 2004 5:27 PM
Subject: Re: education, technology & chat

> On Tuesday 09 November 2004 3:27 pm, Peg Griffin wrote:
> > What is the mathematics learning goal for the kids?
> They were working on learning about polygons -- describing and making
> -- following the TERC Geometry 1 topic.  One part of the math software
> the children were using allows the children to click and drag polygons
from a
> tool palette to fill in a line-drawing outline.   There is a similar
> "hands-on" activity ("activity" with a little "a", not the big "A" of
> with plastic polygons to fill in an outline line drawing on paper
> -- copied from the TERC curriculum folder.  One thing I've observed, on
> same day, is that the children are more facile with the hands-on building
> than with the computer, even though the computer constrains the possible
> that a shape can be rotated or flipped.  Hands-on there are endless
> possibilities, but the computer transformations require clicking on a
> transformation icon in the tool palette and then clicking on the shape to
> transform it.  If one gets it wrong, (s)he must select another
> and reapply, whereas manually making transformations with the plastic
> are done in split seconds.
> Another part of the math software shows a shape made of polygons when an
> resembling a set of eyeballs is clicked.  Then the children try to make
> shape that they saw. Jane does a similar activity with the whole class
> an overhead projector (it's one of the TERC lessons) showing a shape made
> polygons for a few seconds, then hiding it and asking the children to draw
> what they see.  I've observed that the children are often tempted to draw
> while the shape is being shown, against the rules of the activity.  Jane
> them to put their pencils back down on the table until she hides the shape
> and then they can draw it.
>  An "affordance" of minor interest in the software is that the children
> be tempted as they can when sitting at tables looking at the overhead
> projection.  Since they are using the mouse, and the shape only appears
> they click on the eyeballs, they cannot simultaneously see the shape they
> trying to remember, while building their copy.  They can stop and peek,
> however, and then resume building.  An affordance more widely understood
> that individuals working at computers can choose their own pace.  The
> computer activity does not require the pulsing out of rhythm by the
> which, with the overhead projector, often proceeds when the last child is
> ready.  I'm left with the impression that, over all, more student work
> done on this kind of activity at the computer.  I'd need to do some close
> tallying to support this claim, but it's not a claim that has any real
> significance, except perhaps for Jane's practice.  The flip side is that
> teacher-directed overhead activity often results in minor but collective
> ebullitions across the tables as the teacher reveals the shape a second
> so children can check their drawings.  There is a more salient emotional
> element involved with the teacher-directed activity than with the computer
> activity.
> The TERC curriculum has been and continues to be hotly debated.  Here's a
> local article that came out today concerning a nearby school system,
> different from the one in which I'm observing,