Commentary on Theories of Mathematics Education

574 A. Hurford
leaders, blueprints, recipes, templates, and self-organization. The first four of these
ways organization imposed from outside the system. The fifth, self-organization,
comes from inside the system: “Pattern formation occurs through interactions internal
to the system, without intervention by external directing influences” (p. 7). Beyond
successfully responding to Casti’s call for formalization of systems-theoretical
approaches and language, Camazine et al.’s systems perspective highlights considerations
of control of activity and locates that control internally or externally to the
group and its constituents.
Strong leaders, blueprints, recipes, and templates are all mechanisms for controlling
activity that are viewed as being external to the system, organizing activity by
remote control so to speak. Referring back to Casti (1994), note that systems analyses
are “inherently subjective” (p. 269): what one ends up seeing in part depends
upon which levels of organization one chooses to observe. In high school classrooms
one may view students’ activity as always being driven externally, since, for example,
school attendance (on the level of a “long” time frame) is mandatory for most
students and six or seven periodic location changes per day (on the level of “short”
time frames) powerfully regulate students’ experiences. Mandatory attendance or relocations
notwithstanding, even when students are physically present, they usually
direct their own attention, deciding (more or less consciously) if, when, and how
they will engage in learning opportunities and other activities. This ambiguity—are
students being externally controlled or are they directing their own activity—does
not invalidate using systems perspectives as ways to study classroom learning. What
it does do is demonstrate the need for researchers to be careful about how they delimit,
define, and communicate about the systems they are studying.
For example, although it seems that much of what goes on at the level of classrooms
in a school is actually driven by external controllers such as legislative mandates,
“core curricula”, and bus schedules, it also seems like many of the more
interesting aspects of learning will only come into focus when we (subjectively)
choose to background exterior driving mechanisms and observe classrooms at a different
level of detail. At the level of small groups learning addition facts, whole
groups learning about the Civil War, or individuals learning to read, we can begin to
look for components of complex systems as defined in other fields of research. As
students and groups of students self-organize, selectively negotiating and deciding
which chunks of the curriculum to attend to and composing what they have attended
to into models, it seems that powerful insights into learning (e.g., what is motivating
students, what concepts are salient to the students, or how they use the models they
have created) may become observable.
Camazine et al. (2001) describe self-organization as being a function of the aggregated
activities of self-directed agents inside the system. The primary mechanism
for the self-organization of agents is feedback, which is another of the important
“basic terms” that needs formalization in systems-theoretical perspectives. Positive
and negative feedback are “the two basic modes of interaction among the components
of self-organizing systems” (p. 15). Positive feedback “generally promotes
changes in a system” taking “an initial change in a system and [reinforcing] that
change in the same direction as the initial deviation” (p. 17). Negative feedback