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RE: approaches to content

	HI Peter,
	You asked:

> Again, as with Peg's work, I'm wondering if the students "model" the 
> general
> relation (between amount and rate, etc.) with some sort of special
> diagrams, or formulas, or tables, or special algebraic notation, and 
> whether
> this helps them assimilate the general concept. 
the networked technology we work with (HubNet and Participatory Simulations,
developed by Uri Wilensky and Walter Stroup) includes a couple of simulation
activities that do have students moving from general to particular in the
way you describe. The system involves a network of graphing calculators (the
kind that many schools in the US have; Texas Instruments TI-83) connected to
hubs that communicate with a computer server. Using network technology with
a public display space, 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.  For both activities, graphs are included in the public
display of the emerging system (e.g., traffic flow in the grid, icons
representing individual agents in the population through which disease is
spreading), so that the graphical representations emerge in real time as the
system is evolving. Students' and teachers' discussion and analysis of the
graphs in relation to each other and to the motion in the display has them
exploring relations (e.g., stopped cars over time, average speed, average
wait time in Gridlock; number of sick individuals over time in Disease) in
very qualitative instead of quantitative ways at first. What we see in our
data is that students develop increasingly sophisticated and eventually
quantified notions of relations (e.g., velocity), but those notions are
built on strong conceptual understanding of general relations first.

I will ask my colleagues more about assessing students' understanding...I
have been concentrating on describing and understanding the patterns of
participation we are witnessing. :)


> I'd also be interested 
> if you have
> methods for *assessing* whether students are actually solving problems 
> by moving
> from the general to the particular, or just solving each particular 
> problem as a new
> case.
> That is, how can you assess that:
> > , it is seen as a simpler case rather than a different
> > phenomenon.
> Peter