Commentary on Theories of Mathematics Education

586 A. Hurford
The systems perspective developed by Camazine et al. (2001) offers another possibility
for research directed at classroom learning. These authors offer five possible
mechanisms of control of the activities of organized systems: strong leaders,
blueprints, recipes, templates, and self-organization. The idea is that the activity
of aggregates of agents (systems) can be directed externally (through the first four
mechanisms listed) or internally through the process of self-organization, in which
the activities of individual agents, acting on their own accord based on local information,
combine to form complex patterns and structures (e.g., termite mounds and
beehives). In relation to classroom learning, one can view any of these five “organizers”
as the driving force behind particular classroom episodes. The line of possible
research might be descriptive, much like the above proposal based on Casti’s work,
trying to identify and capture episodes of activity where each of the five organizers
is the dominant source of activity patterns. Beyond this type of descriptive analysis,
and once substantial patterns of organization and organizers have been established,
research into the relative value of each for given learning outcomes could be pursued.
For example, it may be that direction from a “strong leader” would prove to
be most productive for acquisition of declarative-knowledge learning goals such as
rote memorization, while self-organized learning opportunities might be shown to
be more effective for developing higher-level thinking and problem-solving capabilities.
The perspective that seems to have the most potential for development of a
systems-theoretical model of learning is John Holland’s (1995). Holland has tried
to develop a model of a “universal” complex adaptive system. Holland’s careful
analysis and informative examples of complex systems provide potentially powerful
grounding for the projects of: (1) describing classroom learning and individual
learning as CAS and, much like the proposal above based on Casti’s work, ultimately
defining such learning as complex; and possibly, (2) extending and using
such an analysis in order to model, explain, predict, orchestrate, and assess learning
in classroom situations.
Certainly each of these suggestions for research is tentative. The point here has
been to continue the complexity in learning discussion and to lay out some ways
that systems perspectives might be used for investigations into learning at multiple
levels of classroom organization. I believe that is safe to say that human learning
is not well described by simple models, linear relations, and static snapshots, and I
hope that this review of selected systems perspectives will promote extensive and
fruitful investigations into learning based on complexity, systems theories, and, as
John Casti (1994) puts it, a “science of surprise.”
Conclusion
The time has arrived and the tools are at hand for educational researchers to build
models of learning that embrace its inherent complexity in ways that have not been
possible before. We are now prepared to move beyond models that reduce learning
to the simple pairing of stimulus with response or to static collections of “data