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

Very interesting discussion, Mike, Peg, Bill, and all!

Following up Mike's comment, I would be interested to hear your opinions of how, when technology (say, software), is used, one incorporates the curriculum *content*
in a chat-like way.

In particular, in a given content area, might there not be software that enables the children's assimilation of the "germ cell" of the subject? If so, what would this look like
in different school subjects?

In our use of Davydov's math curriculum, we have played with creating software within which the children assimilate the action of measuring/constructing a quantity (area, length, etc.) using a unit, where the action of measuring/constructing is
mediated by one of Davydov's "models" (a diagram or formula).

In the case of written language acquisition, I suppose one might create software based on Elkonin's boxes and other diagrammatic supports for the concepts of
word, phoneme, syllable, etc.

What should the content of the technology/software be like, in a given subject area, to help the children "get" the germ cell so they can ascend from the abstract to the concrete?


I hope people will pay attention to Peg's emphasis on conceptual
content of the domain in question when talking about obuchenia
(teaching/learning). I and many others all too easily overlook the
centrality of content so key to Devydov's
thinking, in my case, in part, because 5th Dimensions are
chock a block with all different kinds of content, partly because my
grad training was 100% process.

With respect to another comment: yes, I believe technology is
a very broad term, the sub-parts of which are very interesting
to consider, but a term which is badly underextended in discussions of
technology and human affairs.

Work is progressing on fortifying and improving xmca, glad to see that
there is life despite the spam.

On Wed, 10 Nov 2004 10:47:40 -0600, Peg Griffin
<peg.griffin@worldnet.att.net> wrote:
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

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/