Commentary on Theories of Mathematics Education

Complexity Theories and Theories of Learning: Literature Reviews and Syntheses 571
Saying what complex systems are not can make a first step in Casti’s rendering
process. What they are not are simple systems: systems that have predictable behaviors,
that involve “a small number of components” and “few interactions and
feedback/feedforward loops” (Casti 1994, p. 271). Simple systems generally have
limited numbers of components and are decomposable—if connections between
components are broken the system still functions pretty much as it did before. By
counter-example we can begin to see what complex systems are: they involve many
components (elements, agents), and they are highly interconnected and interactive,
exhibiting multiple negative and positive feedback/feedforward loops. These systems
are further characterized by internal, agent-level behaviors-producing rules and
they are irreducible—“neglecting any part of the process or severing any of the connections.
. . usually destroys essential aspects of the system’s behavior” (p. 272).
This kind of complexity forms the foundation of a “science of surprise” and surprise
occurs literally when our expectations and observations are at odds with each
other. Casti (1994) prefaces five chapters with what he calls “intuitions,” actually,
misconceptions, about how the world behaves, that have their genesis in linear and
simplistic models, and that routinely land their followers on the “tarmac of surprise.”
Before proceeding, we should make it clear that we intend to use the ideas from
Casti’s Complexification to inform an emergent understanding of classroom learning.
In this case we are calling the classroom of students a system, and that system
is being viewed by us as an inherently complex system. The previous paragraph
describes well why classrooms should be viewed as such—they are not simple, as
defined above, and they are complex. Classrooms are composed of many agents—
learners and teachers—and each is a decision-maker whose deciding may affect the
behaviors of any fraction of the whole. The elements are highly interconnected and
irreducible—changing elements changes the dynamics—and activity in the classroom
is can be characterized as having by multiple feedforward and feedback loops.
Having set the context for view the classroom, let us continue with a brief treatment
(Table 1) of Casti’s “causes” of surprise and try and point to a way in which each
might be related to classroom learning.
In Casti’s “roots of surprise” we have the beginnings of a general understanding
of complexity as well as the beginnings of a rationale for thinking of classroom
learning as a complex system.
To summarize, John Casti (1994) does an excellent job of identifying the challenges
in trying to understand experience from a systems-theoretical point of view.
Systems that are complex have instabilities and non-linearities that can result in
catastrophic reorganizations, and they often exhibit “deterministic chaos,” eventually
settling in on a few “strange attractors” (p. 29; see Circle-10 System example,
pp. 33–37) via chaotic and apparently random paths. Complex systems are those in
which logical rules do not necessarily lead to predictable behaviors. They cannot
be studied by being broken into constituent components because local and global
connectivity are critical to the system’s activity. Finally, complex systems are ones
in which unexpected patterns at one level can emerge from relatively simple interactions
between agents at a lower level.
When we undertake the business of trying to make sense of learning in classrooms
and to look at the “big picture”—modeling classrooms as connected, whole