# Re: approaches to content

```It is very stimulating to be in a forum with people immersed in later phases
of mathematics -- long winded it wasn't in my view.

It seems to me that your systems might recover the engagement impetus for
learning mathematics that pre-schoolers often exhibit but all too easily
lose during conventional primary grade mathematics curriculum with
conventional instruction.

Thanks for the descriptions.

And, by the way, the other part of the pleas about measurement that we
__________________________________
Science in preschool requires a mathematics that goes beyond countable
entities.  Properties characterized by continuous quantity are of scientific
interest, so measurement is needed.  With an entity like time, it is not
difficult to get around its continuous nature.  Change over time in a
science investigation can be treated as change over a counted number of
days.  The countable aggregate is used loosely and without delving into the
fact that it is a unit of time measurement.  Volume and area can similarly
be given a pass by using countable stand-ins: a cup, a pie wedge, tiles.
Tables, charts, and graphs can be constructed with these tactics in order to
treat continuous quantity as if it were not continuous.  For length and
weight, though, interesting preschool science often calls on real
measurement of the continuous property for formulating questions, collecting
data, recording it, analyzing the patterns within it, and reporting the
results and conclusions.

_____________________________________________
Peg

----- Original Message -----
From: "Ares, Nancy" <nancy.ares@rochester.edu>
To: <xmca@weber.ucsd.edu>
Sent: Wednesday, November 10, 2004 3:27 PM
Subject: approaches to content

> Hello,
>
> I, too, am enjoying the discussion of education, technology and math,
> and appreciate the use of data to foster dialogue.
>
> My colleague Walter Stroup and I have been working on notions of
> generativity
> in technology design and use for math and science classrooms. We ground
> our work in the notion of a dialectic of math and science as both socially
> structured and socially structuring. The idea is that we can design
> technologies and activities in ways that both use 'big ideas' in math or
> science (e.g., dynamic systems, proof, parametric space, statistics) to
> structure the social
> space of learning and highlight social interaction
> as structuring the math or science that emerges from activity.
> Space-creating play (as in
> mathematical space) and dynamic structure are central features of the ways
> we attend to 'content,' while agency and participation
>
> are central features of social interactions that structure the 'content'.
> An example of math structuring social space:
> Participatory simulations are networked activities where learners act out
> the roles of individual system elements and observe how the behavior of
the
> system as a whole emerges from their individual behaviors. These emergent
> results then become the focus of discussions and analyses. Using network
> technology with a public visual display, students can, for example, become
> agents in a population where a disease is introduced and be part of the
> system as the disease spreads. In another simulation they each can control
a
> stoplight in a simulated city's traffic grid and together work toward
> improving the traffic flow. Not only is dynamic-systems modeling the
content
> being introduced into the curriculum, the learning itself is organized in
> terms of the classroom becoming the dynamic system. By assuming iconic
roles
> in a system, mathematical ideas like emergence, feedback, and complexity
are
> literally embodied by the network-supported learning activity.
>
> An example of social interaction structuring the math:
> in one participatory simulation used by a teacher to explore
> positive/negative integers, students also recognized and explored concepts
> of slope and rate, as well as their representation in graphs. Here,
students
> used the mathematics of the network-mediated activity itself to expand the
> content and representations involved. Their agency and participation
> involved making contributions to critical aspects of practice, as in
> students' expansion of the activity to include concepts of rate, slope,
and
> representation. The space of mathematical objects was enlarged through
their
> play-full engagement in the generative activity.
>
> One of the critical things, we think, is that we must attend to the
> structuring relationship
> between content and activity to both understand learning and to develop
> powerful use and design
> of classroom technologies.
>
> I know this is a bit long-winded...
> Nancy Ares
>
>
>

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