Commentary on Theories of Mathematics Education

578 A. Hurford
and category construction is a fundamental requirement for building models. It is
what we do as model builders. We chose which aspects of a system to aggregate
in order to simplify the system’s complexity—this is one of those “artful” activities
that helps us to make sense of the world. In studying a system, or thinking about
our worlds, we very naturally create inclusive and exclusive categories such as cars
or trucks or gifted learners. A subtlety that figures later into the conversation about
“building blocks” (below) is the idea that the categories we create are “reusable [emphasis
added]; we almost always decompose novel scenes into familiar categories”
(Holland 1995, pp. 10–11). Aggregation in this (simpler) sense speaks to the categorization
of components of CAS that are selected for the purposes of highlighting
certain features and backgrounding others in service of a particular investigation or
model.
The second sense of the term aggregate is the one mentioned in the foregoing
introduction to Holland’s approach. This sense of the term refers to the coalescence
of individual agents at one level of complexity into “meta-agents” at the next higher
organizational level, and is a fundamental property of all CAS. Careful study of this
type of aggregation is one of the primary means by which we can make sense of
these systems and this complex systems approach. Holland poses several questions
germane to this type of aggregation:
What kind of “boundaries” demarcate these adaptive aggregates? How are the agent interactions
within these boundaries directed and coordinated? How do the contained interactions
generate behaviors that transcend the behaviors of the component elements? We must be
able to answer such questions if we are to resolve the mysteries. . . (p. 12)
These questions point at means by which we can begin to use elements of CAS
analysis for the purposes of furthering understanding of particular systems. In the
case of classroom learning for instance, what will be the composition of spontaneously
forming “small groups” in a given classroom, and how will those aggregates
evolve over different time scales? On what basis, under which sets of rules,
will these groups form? What are the effects of these group formations on learning
at the individual, small group and whole class levels? One possible means for
investigating these questions is the first item on Holland’s list of seven basics—
tagging—and it is to this mechanism that our attention will now turn.
Tagging
This is one of the mechanisms of CAS that enables adaptation and promotes the formation
of aggregates. Tagging is a process of identifying features in the environment
of a CAS that become salient and useful in determining its future activity. A CAS
selects salient features (building blocks) from all the possible inputs in its environment
as a function of a currently active set of tagging rules and these rules structure
agents’ parsing their environments by motivating and driving selective attention.
When a CAS first encounters a situation, a preexisting set of tagging rules relevant
to the particular situation becomes active, and the rules specify particular things for
the CAS to expect and to look for. “Well-established tag-based interactions provide
a sound basis for filtering, specialization, and cooperation” (pp. 14–15), and these