Commentary on Theories of Mathematics Education

584 A. Hurford
solely on their local environment and the individual’s own internal models. In contrast
to individualized theories of learning, systems-theoretical points of view are
fundamentally concerned with seeing learners and groups as mutually constitutive
agents whose behaviors influence and are influenced by the contexts of their larger
patterns of activity.
Systems-theoretical points of view tend to be very dynamic—characterizing activity
in terms of evolving patterns rather than studying behavior patterns captured
in static “snapshots.” A clear-cut and visual example of this can be seen in Conrad
Parker and Craig Reynolds’ model 7 of the flocking behavior of birds. In this simulation
applet avatars (“boids” to its authors) exhibit amazingly life-like flocking
behaviors. An observer is compelled to build an understanding of the “boids” that
is fundamentally dynamic. The back-and-forth, up-and-down, landing-and-takingflight
behavior of real birds is quintessentially captured by the dynamical nature of
the modeling of this illustrative applet. I believe that the same will be true of dynamical
representations of classroom learning—it will be the patterns of activity
that are the focus of understandings that develop. Rather than static “snapshots” of
individual students’ learning, such as quiz grades or end-of-year tests, assessments
of learning will be made often, in real-time, and with significant regard for the context
and adaptations. The dynamic nature of systems-theoretic views of learning
provides grounding, tools, and a framework for thinking about important patterns of
evolution, development, and adaptation of learners and learning.
Complex situations and complex interactions can also serve to characterize classrooms
and classroom learning. Students’ goals and teachers’ goals are frequently
different and often at odds. In our schools, participants’ social, economic, cultural,
and historical backgrounds are becoming increasingly diverse and potentially at
odds (Fordham and Ogbu 1986; Ogbu1990). Classroom structures may require behavior
that is antithetical to expected behaviors in students’ non-school lives, setting
up dynamic tensions that radically affect learning. Classroom learning is situated
and directed by the contexts of the school, the district, and local, state, and federal
mandates, norms, and expectations. All of these factors and many others combine to
create a dazzlingly complex context for situating learning, and systems-theoretical
approaches provide unique advantages for dealing with such complex contexts. For
example, a systems perspective could cause an observer to move “up” several organizational
levels from the classroom to the surrounding community in order to
inform understandings of students’ resistance to teachers’ learning goals.
Systems theoretical approaches can “see” complex relations and accommodate
their effects. The possibility of multiple “attractors” (Casti 1994, pp. 28–29), that is,
multiple and relatively stable patterns within a limited region is taken as a given in
CAS approaches. As an example from the classroom, consider the conflicting student
goals of wanting to perform well on a test and not wanting to upstage one’s
peers, or for a school, wanting all students to pass first-year algebra but not being
capable of handling large class sizes in second-year algebra courses. Systems approaches
attend to the existence of multiple driving forces and have mechanisms for
7 http://www.vergenet.net/~conrad/boids, Retrieved December 2, 2008.