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.
Peg
----- 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
shapes
-- following the TERC Geometry 1 topic. One part of the math
software
that
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
CHAT)
with plastic polygons to fill in an outline line drawing on paper
worksheets
-- copied from the TERC curriculum folder. One thing I've observed,
on
the
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
ways
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
transformation
and reapply, whereas manually making transformations with the plastic
blocks
are done in split seconds.
Another part of the math software shows a shape made of polygons
when an
icon
resembling a set of eyeballs is clicked. Then the children try to
make
the
shape that they saw. Jane does a similar activity with the whole
class
using
an overhead projector (it's one of the TERC lessons) showing a shape
made
of
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
asks
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
cannot
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
when
they click on the eyeballs, they cannot simultaneously see the shape
they
are
trying to remember, while building their copy. They can stop and
peek,
however, and then resume building. An affordance more widely
understood
is
that individuals working at computers can choose their own pace. The
computer activity does not require the pulsing out of rhythm by the
teacher,
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
gets
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
the
teacher-directed overhead activity often results in minor but
collective
ebullitions across the tables as the teacher reveals the shape a
second
time
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,
http://www.boston.com/news/education/k_12/articles/2004/11/08/
mathematical_unknowns/