Commentary on Theories of Mathematics Education

Complexity Theories and Theories of Learning: Literature Reviews and Syntheses 581
Building Blocks
The last of Holland’s seven basic ingredients of complex adaptive systems is directly
related to the mechanism of internal models. This mechanism provides a
means for generating useful internal models of a “perpetually novel environment”
(Holland 1995, p. 34) by the distillation from experience of reusable “building
blocks.” Agents acquire building blocks through a process of selective attention—
decomposing information from their environment into constituent elements that can
be combined and re-combined into novel internal models. Through iterative use and
testing agents accumulate building blocks that enable the construction of useful internal
models, models that enable those agents to anticipate the probabilities and
consequences of potential future actions. Iterated and mutually influential development
of building blocks and internal models is a key source of the adaptation of
complex adaptive systems.
Summary
John Holland’s (1995) treatment of the fundamentals of complex adaptive systems
is a detailed and comprehensive systems-theoretical framework. It provides a welldefined
“litmus test” for deciding whether or not a system is complex and adaptive
and for defending such a decision. Beyond that, it provides powerful conceptual
tools—the properties and mechanisms—for analyzing the activity and pattern developments
of CAS.
This discussion of Holland’s model of generalized complex adaptive systems is
intended to provide a subsequent framework for building persuasive arguments that
classroom learning, and human learning in general, can be profitably studied from
systems-theoretical points of view. That project might proceed by first relating observed
patterns of classroom behaviors, to Holland’s more general mechanisms and
properties and then using those parallels to try and generate a model of learning
based on his systems approach. Although a thorough treatment of classroom learning
as CAS is well beyond the intended scope of this article, a plan for how research
might proceed in doing that will be sketched out in a later section.
Each of these three perspectives on dynamical systems, Camazine et al.’s (2001),
Casti’s (1994), and Holland’s (1995), provides an ontological pathway for making
sense of complex systems and of the collective behavior of aggregates of agents.
These perspectives also serve to define and clarify the types of considerations necessary
in undertaking a complex systems analysis. Next I turn to a discussion of the
utility of these systems perspectives for making sense of the dynamic and complicated
activities inherent in classroom learning.
Individualized and Systems Approaches to Learning
The great majority of theory building in constructivist and cognitivist learning
theories has been focused on the learning of an individual. Behaviorist perspec-