Commentary on Theories of Mathematics Education

Complexity Theories and Theories of Learning: Literature Reviews and Syntheses 577
environment, and that environment includes other agents doing the same thing. The
net effect being the evolution of complex patterns of activity when viewed from “one
level up” as Holland might put it. At this point the notion of adaptation is probably
sufficiently defined for the present purposes: adaptation can be viewed as learning
based upon emergent interactions that an agent has with its environment, a major
component of that environment is other adaptive agents, and the patterns of activity
generated by agents’ individual and mutual adaptation provide the observable
organizational characteristics of CAS.
An important correlation with Piaget’s work is that Holland relies very heavily on
“genetic algorithms” (p. 69) and molecular biology for developing his theory of the
mechanisms of adaptation in CAS. At the same time, lest the readers think that Holland’s
work is solely biological, it should be noted that he also applies his adaptation
schemes to the (iterated) Prisoner’s Dilemma, economics (pp. 80–87), and many
other diverse systems. It is also important to note that as agents and their behaviors
evolve, so does their environment. Although Holland does not discuss this explicitly,
the idea that agents and their environs can be seen as mutually adaptive provides an
additional and potentially useful piece to the puzzle of CAS. The patterns of activity
and adaptation of the agents in a CAS are influenced by and influence the environment,
and so it seems that watching the evolution of the surroundings as well as the
evolution of the agents should further enhance understandings about CAS.
The Seven Basics: Mechanisms and Properties
These “seven basics” that Holland (1995) considers common characteristics of all
CAS “are not the only basics that could be selected” (p. 10) as ways of understanding
the activity of complex adaptive systems. He reminds us that, even as
researchers, we still need to be a bit artful in choosing which characteristics will
provide useful foci for our particular investigations: “This is not so much a matter
of correct or incorrect. . . as it is a matter of what questions are being investigated”
(p. 8). At the same time, Holland’s work is intended to generate a model that can be
used for studying all CAS and he claims that “all the other candidates” for mechanisms
and properties that he has encountered can be “derived” from “combinations
of these seven” (p. 10). This is a strong claim, but for now, let us accept these
seven basics as sufficiently characterizing CAS and take a closer look at the basics
in hopes of developing a general understanding of their applicability for building
useful understandings of complex adaptive systems. 4
Aggregation
This attribute has two interpretations that are applicable to CAS. First, the simpler
sense of this term has to do with the natural human process of building categories,
4 Holland discusses the seven basics not according to whether they are mechanisms or properties
of CAS, but rather in a manner that emphasizes their interrelationships, and that order will be
maintained here, for the same reason.