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RE: approaches to content
I neglected to say that the work we are doing in math/science/technology is
6-12, and that we concentrate on pre-algebra, algebra and courses that are
often thought of as 'remedial'.
It seems, from Peg's description of measurement in preschool math, that we
share the idea that giving students access to rich, complex math concepts
early is important to their conceptual development of powerful ideas. For
example, instead of starting with linear functions, we tend to work with
students on activities that involve varying quantities of amount and rate,
so that when they do encounter the 'degenerate' case of a linear
relationship, it is seen as a simpler case rather than a different
phenomenon. Students don't encounter linear relationships nearly as often as
they do more complex ones in life, so it is much easier to draw on their
prior knowledge in supporting their thinking about and mathematizing
relationships when we start with more complexity. I think the way Peg (and
her co-author) are thinking about measurement has similar notions and power.
> From: Peg Griffin
> Reply To: email@example.com
> Sent: Wednesday, November 10, 2004 4:48 PM
> To: firstname.lastname@example.org
> Subject: Re: approaches to content
> It is very stimulating to be in a forum with people immersed in later
> 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
> (Catherine King and I) make is about the mathematics-science link:
> Science in preschool requires a mathematics that goes beyond countable
> entities. Properties characterized by continuous quantity are of
> 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
> 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
> 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,
> data, recording it, analyzing the patterns within it, and reporting the
> results and conclusions.
> ----- Original Message -----
> From: "Ares, Nancy" <email@example.com>
> To: <firstname.lastname@example.org>
> 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
> > 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
> > we attend to 'content,' while agency and participation
> > are central features of social interactions that structure the
> > An example of math structuring social space:
> > Participatory simulations are networked activities where learners act
> > the roles of individual system elements and observe how the behavior of
> > system as a whole emerges from their individual behaviors. These
> > results then become the focus of discussions and analyses. Using network
> > technology with a public visual display, students can, for example,
> > 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
> > stoplight in a simulated city's traffic grid and together work toward
> > improving the traffic flow. Not only is dynamic-systems modeling the
> > being introduced into the curriculum, the learning itself is organized
> > terms of the classroom becoming the dynamic system. By assuming iconic
> > in a system, mathematical ideas like emergence, feedback, and complexity
> > 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
> > of slope and rate, as well as their representation in graphs. Here,
> > used the mathematics of the network-mediated activity itself to expand
> > 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,
> > representation. The space of mathematical objects was enlarged through
> > 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