"So the conclusion (massively hedged) was that material designs, as
constructed on paper, produced quite different affordances for knowing and
doing mathematics among the people who made and used them. At the same time,
forms of representation that were generally considered nonstandard or even in
some cases childish in relation to algebra instruction, played a strong
generative role."
This raises two sets of questions for me:
(1) Could differences in uses of material artifacts in joint activity have
to do more with the development of "perspective" (as articulated by Holland &
Reeves (1994) than with the affordances of the tools. The "same" activities
with the "same" tools may embed similar tensions within them, to which
participants construct different perspectives on their attempted resolution.
Which leads to
(2) Your claim about the generative role of certain forms of representation.
Is the generative dimension of representation as activity (or, as I would
argue, mediated activity) due to the the affordances of tools, or can it be
interpreted in terms of a necessary "slippage" and evolution of tools as they
migrate from particular activities to other particular activities or, as
Bakhtin would say, a tension between the "centripetal" and "centrifugal"
aspects of utterances (a special species of action)?
I have put these perspectives at odds in an either/or way, but that is not to
suggest that I think the answers lay in choosing one side--the encounter of
voices, though, I think might lead to some generative representations on its
own :-).
Bill Penuel
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PreventionInventions
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