Commentary on Theories of Mathematics Education

580 A. Hurford
Diversity
Another property of Holland’s complex adaptive systems, diversity plays a complicated
role in their makeup. To understand diversity, one could think about the
“niches” that agents may fill in a biological ecology and then think about how those
niches might evolve over time. “Each kind of agent fills a niche that is defined by the
interactions centering on that agent. If we remove one kind of agent from the system,
creating a ‘hole’, the system typically responds with a cascade of adaptations
resulting in a new agent that ‘fills the hole”’ (Holland 1995, p. 27). The property of
diversity is a major factor in the evolution of an ecology when, for example, agents
move into totally new territories or when an a single agent is successful in adapting
to a niche. Diversity is a “dynamic pattern, often persistent and coherent like a
standing wave” (p. 29), but it is actually more dynamic than a standing wave, because
diversity itself evolves as a function of adaptations, opening the “possibility
for further interactions and new niches” (p. 29).
Internal Models
The mechanism of internal models plays a vital role in the activities of CAS. Internal
models are the mechanism by which CAS anticipate, and it is through anticipation
and prediction that agents adapt to and thrive in their environments. Although
the mechanism of internal model building seems much more applicable to sentient
systems than to all CAS, Holland (1995) makes the case that even bacteria may implicitly
predict the presence of food (i.e., build an internal model) when they follow
chemical gradients (p. 32). It is important for Holland’s work in developing a universal
model of CAS that he be able to identify a way that prediction and anticipation
work at (essentially) all levels of CAS analyses (thus including lower life forms), but
educators and educational researchers do not share that constraint. In human learning
in general, and in classroom learning in particular, the notions of anticipation
and prediction based on internal models are not at all difficult to conceive of.
According to Holland (1995), the “critical characteristic” of a model is that it enables
the agent to “infer something about the thing being modeled” (p. 33). Internal
models are created by an agent’s selectively attending to building blocks in its environment
and then using this information for the purposes of creating and refining
that agent’s internal structure, its models. The models are then employed as predictors,
elements internal to the agent that enable it to respond to and benefit from the
local environment. Models “actively determine the agent’s behavior” (p. 34). They
are “subject to selection and progressive adaptation” (p. 34) based on new information,
and one may begin to see the possibility of iterative adaptational loops, based
solely on individual agents and local conditions, that can provide powerful insights
into the learning and adaptation patterns of higher-level CAS (meta-agents). In a
classroom example, learners may create and refine their internal knowledge structures
in relation to interactions with their environment, and at the same time, the
meta-agent, the classroom of learners, may change its nature in ways that are not
predictable through careful study of the individuals.