Commentary on Theories of Mathematics Education

Complexity Theories and Theories of Learning: Literature Reviews and Syntheses 573
informal notions about the complex and the commonplace into a more formal, stylized
language, one in which intuition and meaning can be more or less faithfully captured in
symbols and syntax. (p. 270)
Next this chapter responds to Casti’s call for formalization and operationalization
of terms and approaches in building complex systems analyses; first by discussing
Camazine et al.’s (2001) work in the field of biology, and then by taking a close look
at John Holland’s (1995) book Hidden Order.
Camazine, et al.: Biological Systems
The approaches to complexity-based theories of biological organization offered by
Camazine et al. (2001) may be very fruitful for extending our thinking about classroom
learning in systems sorts of ways. In Self-organization in Biological Systems,
the authors examine a wide variety of biological systems and assess the development
of those systems in terms of possible “mechanisms of pattern formation” (p. 47) that
may be seen as sources of observable organization. When trying to make sense of
the complex behaviors of such systems, it is often beneficial to move to a higherlevel
vantage point and watch the behavioral patterns evolve from the context of
a larger set of patterns and behaviors. Alternatively, it is sometimes useful to slip
down a level, and observe how the system you are trying to understand offers the
context for a “lower” set of behaviors.
For example, if one wants to think about the rules-set that might govern the familiar
flight patterns of a flock of geese, one can raise his or her vantage point up to
the level of the flock. From that position, you can focus your on an individual agent
and watch the patterns of the flock as a whole, and use this combined perspective
to inform attempts at modeling the rules-system of and individual bird. Similarly,
if one is trying to understand the search behaviors of individual foraging ants, he
or she could slip “down” a level, and focus on the local terrain and distributions of
food sources. It is within this context that the ants’ activity patterns emerge—the
“structure” of the ants’ surroundings exerts a strong influence on observed foraging
patterns.
Although Camazine et al. (2001) fail to offer a single explicit definition for what
they consider a complex system to be, it seems reasonable to suggest that they are
thinking of (agent-based) systems in a fairly standard sense—systems composed of
multiple agents acting according to a hypothesized collection of (internal) rules in
co-operation with local information and other agents. Activity of agents at the local
level generates patterns of behavior at a more global level. These authors define
pattern as a “particular, organized arrangement of objects in space or time” (p. 8),
state that global patterns are seen to emerge from the organization of local activity,
and propose and discuss various ways these patterns emerge in biological systems.
In order to understand the activity of an organizing system, for example, of a
classroom as it learns, careful attention must be paid to the types of things that appear
to drive or encourage the organization. Camazine et al. (2001) categorize ways
of thinking about how activity might be organized in biological systems: strong