Commentary on Theories of Mathematics Education

Complexity Theories and Theories of Learning: Literature Reviews and Syntheses 585
dealing with the concomitant positive and negative feedback effects on the behaviors
of the system.
Systems perspectives encourage a reflexive and reflective shifting of levels, from
considerations of the individual’s perspective to a focus on the patterns of the aggregate
and back again. Individuals make personal decisions based upon their own
“rules sets” and their immediate situations. The synergetic sum of individuals’ decisions
gives rise to the patterning of the activity at the group level. A further shift in
levels, say from the classroom to the district, might facilitate decision making about
funding, curricula, and policies. A systems perspective enables multiple levels of
focus for the purposes of understanding and making sense of learning.
Finally, in addition to all of the benefits outlined above, systems-theoretical approaches
offer a subtler and perhaps much more important benefit. They give researchers,
teachers, and learners a new and self-consistent language with which
to talk about classroom learning. The terminology of complexity theory is finding
its way into educational discourse at nearly every level. From state-of-the-art
researchers to curriculum designers to classroom teachers people are beginning to
employ systems ideas in many arenas. This appropriation of the language and the
common-sense ideas of complexity by inquirers into learning may be the most compelling
argument of all for the utility of the approach. Thinking about groups of
learners and the consequences of nonlinearities in learning systems becomes much
more likely and productive as educators continue to appropriate the language and
ideas of complexity.
Future Research Directions
The implications of systems-theoretical perspectives for research into learning remain
largely unexplored, and what follows represents but a few of the possibilities
suggested by the foregoing discussions of the work of John Casti (1994), Camazine
et al. (2001), and John Holland (1995).
From Casti’s (1994) perspective, there is a research opportunity in terms of arguing
for a definitive and research-based demonstration that classrooms and classroom
learning can indeed considered to be “complex” (p. 269) systems. Although the idea
of classrooms as being complex systems is apparently being taken as a given by the
education research community at large, a concerted effort to identify the characteristics
that Casti describes as the “stuff of complexity,” such as catastrophic changes,
irreducibility, and emergence in real-life classroom activity could be very important.
It seems that one of the first steps toward Casti’s sort of formalization of a systems
approach to classroom learning should be a careful characterization of the range
and types of complexification that occur in classrooms. Identification and exemplification
of Casti’s five sources of surprise (Chaps. 2–6) as may be found in existing
classroom videotape footage, accompanied by careful a careful analysis of the nonlinearities
and their sources would constitute a valuable contribution to research on
learning.