The thread concerns artifacts (and their materiality), activity (in the
sense of providing a horizon of experience that includes artifacts), and
affordances (as a description of a relation between the person and aspects
of the environment they are in/make).
Here goes...
THREAD 1
I finished a dissertation in 1990 (citation below) in which problem-solving
outcomes were related to the kinds of representational artifacts being made
and used by students, teachers, and advanced undergraduates in computer
science as they solved "algebra story problems." Costall and Still's piece
was very useful for me, and I ended up doing three things that I think
might be relevant to this discussion:
(1) I did a brief historical analysis of "algebra story problems" as
cultural artifacts that have a varied and LONG history, both in mathematics
curricula (e.g., dating back to the Treviso Arithmetic in 1478), in
information processing studies of algebraic reasoning and transfer of
training, and in research on mathematics education. The gist of the
analysis is that these little texts are VERY durable historical artifacts
that circulate between different communities and mediate claims about
mathematical competence and the role of "realism" or "authenticity" in
instructional settings.
(2) I took seriously a distinction between making and using representations
(see Howard Becker's paper, "Doing things together," for a striking
analysis of pedestrian navigation using automobile association maps in San
Francisco). Seeing "mathematical" representation as something made and used
on the spot, I asked how different types of "notational structure" (e.g.,
arithmetic/algebraic expressions, drawing, arrays) were combined with
different views of quantity (e.g., enactive or prospective "states" in a
situation versus composite retrospective "roles") at different points
during people's attempts to solve the problems they were given. I called
these local constructions "material designs," both to underscore their
materiality and the fact that they were designed for some purpose within a
solution attempt. These ideas (materiality and design) were, I thought,
closely related to Eric Livingston's sense of "lived work" in following a
mathematical proof. A major difference, of course, is that solving a
problem is more like MAKING a proof, something we know relatively little
about.
(3) I borrowed Gibson's notion of an "affordance" to ask if different types
of material designs, as made an used, created different affordances for
inference and calculation, activities that are (I claim) at the heart of
attempting to "solve problems." The major result of the dissertation, in my
view, was being able to show that inference and calculation play out very
differently across types of material designs. It turned out that the
standard or sanctioned material designs of algebra instruction (e.g.,
algebraic expressions, formula specific tables) were the loss leaders for
making correct inferences about problem structure. On the other hand,
nonstandard or less clearly sanctioned material designs in algebra
instruction (e.g., a drawing that incrementally unpacks motion, a narrative
describing relative motion) were terrific for introducing correct
inferences and even for repairing prior conceptual errors in a solution
attempt.
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. I've followed these ideas into thinking about
represenation as an activity rather than an object, but that could be
another longish set of threads.
THREAD 2
Luciano Miera did a similar analysis in his 1991 dissertation (citation
below), arguing that the "design of material displays" was a productive way
of following how goals emerge in ongoing activity as kids "solve problems."
Again there is the contrast between problems as given and found (i.e.,
emergent goals, to use Saxe's term), a distinction between representations
that are standard and nonstandard (e.g., artifacts that have a value laden
history), and a careful focus on materiality in relation to mathematical
activity. Luciano (now at Universidade Federal de Pernambuco, Recife) has
recently published parts of his dissertation in Cognition and Instruction
(don't have this citation handy, but its 1995).
THREAD 3
Jim Greeno working variously with Joyce Moore, Randy Engle and others at
Stanford, has proposed a fairly elaborate combination of situation theory
(ala Barwise and Perry), contribution analysis (ala Clark), and what they
call "activity nests" (borrowing some aspects of activity theory). Again,
the basis of the analysis is kids solving given math problems, and I
believe they will be presenting a paper on this at the upcoming AERA
meetings in New York.
This has gotten long, so I'll close with two citations. I'd be keen to hear
any reactions, and maybe I should get on xmca to find them. The citations
follow. Rogers
Hall R.P. (1990) Making mathematics on paper: constructing representations
of stories about related linear functions. Doctoral dissertation, Technical
Report 90-17, Department of Information and Computer Science, University of
California, Irvine. Also appears as Monograph 90-0002, Institute for
Research on Learning.
Meira, L. (1991) Explorations of mathematical sense-making: an
activity-oriented view of children's use and design of material displays.
Dissertation, University of California, Berkeley.
____________________________________________________________________
Rogers Hall
Division of Education in Mathematics, Science
and Technology
University of California
Berkeley, CA 94720
rhall who-is-at garnet.berkeley.edu
(510) 642 3489 (office, 4641 Tolman Hall)
(510) 642 4206 (division office)
(510) 642 3769 (fax)