Commentary on Theories of Mathematics Education

564 R. Lesh
increasingly impacted by systems that range in size from global economic systems
and ecological systems to small-scale systems of the type that are embodied in the
gadgets that are ubiquitous in modern societies. Some of these systems are designed
by humans; others are only strongly influenced by humans. In either case, many
of the most important “things” that need to be understood, explained, or designed
today are systems in which: (a) catastrophic changes may result from interactions
and resonances among simple, continuous, and incrementally changing factors; (b)
feedback loops occur in which global second-order effects overwhelm local firstorder
effects; or (c) systems-as-a-whole often develop properties of their own which
are quite different than those “agents” or “objects” constituting these systems.
In systems theories, these latter types of properties are referred to as emergent
properties of the system-as-a-whole. Examples include wave-like patterns associated
with traffic moving through a city or along the freeway, the letters and words
spelled out by a marching band on a football field, jazz bands, or the collective
“personalities” that are developed by students in a classroom. Such emergent properties
can certainly be influenced by properties of their elements, but patterns often
emerge that go well beyond the scope of activities of the individual elements.
My main point here is that, unless citizens of the world develop powerful ways
of thinking about these types of systems, unforeseen catastrophes will arise much
more frequently. Global climatic change, world-wide economic recessions, or the
fragmentation and radicalization of political systems will emerge—the question is
whether these things happen as abrupt surprises or are anticipated in proactive ways
that can leverage non-linearities in the system dynamics and soften the blows. In
modern graduate schools responsible for developing leaders capable of dealing with
such issues, “case studies” are often used to help students understand how, in complex
systems, decisions that seem to be wise locally often lead to results in which
everybody loses at the global level.
A hallmark of the preceding kinds of systems is that the whole is more than the
sum of its parts—and that the system is changed in significant ways if it is partitioned
into disconnected parts! The systems-as-a-whole often are alive in the sense
that they are not inert entities lying around dormant until the time that they are stimulated
into action; and, when you act on them, they act back. Mathematicians often
refer to such systems as dynamical system and, another of their characteristics is
that they often function even in situations where they don’t quite fit—such as when
the coordinated actions that are involved in hitting baseballs are applied to the hitting
golf balls, tennis balls, or ping pong balls. In such situations, the system may be
referred to as assimilating the new situation; or, when adaptations are needed in the
systems themselves, systems may be described as accommodating to the situations
in which they are. In fact, these systems can be profitably thought of as complex in
the extreme: complex, dynamic, self-regulating, mutually constitutive and continually
adapting systems.
Whatever led us to think that most of the most important human experiences can
be described and understood adequately using only a single, one-way, solveable,
and differentiable relation—where actions are not followed by reactions or interactions,
and where multiple actors never have conflicting goals? The essence of complex
adaptive systems is that their most important (and emergent) properties cannot