Commentary on Theories of Mathematics Education

572 A. Hurford
Table 1 Comparison of Casti’s (1994) “Intuitions” and “Surprises,” with classroom learning examples
Intuition Surprise Classroom example
#1. Small, gradual changes
in causes give small, gradual
changes in effects. (p. 43)
#2. Deterministic rules of
behavior give rise to
completely predictable
events. (p. 85)
#3. All real-world truths are
the logical outcome of
following a set of rules.
(p. 115)
# 4. Complicated systems
can always be understood by
breaking them down into
simpler parts.
# 5. Surprising behavior
results only from
complicated,
hard-to-understand
interactions among a
system’s component parts.
Catastrophe theory—small
changes in parameters can
lead to large discontinuous
shifts in related values. This
is literally the effect of
falling off a cliff. (Chap. 2)
Chaos theory—the “Lorenz
Butterfly Effect” where
minute differences in initial
conditions evolve quickly
into vastly different states.
(Chap. 3)
Incomputability—Gödel’s
Incompleteness Theorem.
The essence of this property
is that “there’s always
something out there in the
real world that resists being
fenced in by a deductive
argument” (Chap. 4, p. 150).
Irreducibility—in complex
systems, due to the nature of
the connectivity between
elements, altering the system
by breaking the connections
irrevocably changes the
nature of the system.
(Chap. 5)
Emergence and selforganization—“surprising
behavior can occur as a
consequence of the
interaction among simple
parts”. (Chap. 6, p. 230)
Small changes in locus of
control from teacher to
students may lead to
significant changes in
classroom learning.
Regardless of how concrete,
straightforward, and
simplistic direct instruction
may be, learners often
emerge with radically
different understandings.
Learning outcomes occur in
classrooms that no theories
of learning are able to
predict.
This is essentially Aristotle’s
truism that the whole is more
than the sum of the parts.
Classroom learning is a
function of the relations and
interrelationships of all the
members of the learning
community.
Exciting learning outcomes
can be achieved by
encouraging learners to
“self-organize,” that is, to
direct and make sense of
their own learning.
systems rather than linear combinations of individual learners—new kinds of theoretical
tools are called for. In Complexification Casti (1994) has paved the way for
the application of complex systems analyses by pointing out that common-sense
attempts at understanding systems are usually inadequate, and he has issued a challenge
to theory-builders to embrace complexity as a science on their way to the
“more ambitious. . . program of creating a theory of models” (p. 278):
...common usage of the term complex is informal. The word is typically employed as a
name for something that seems counterintuitive, unpredictable, or just plain hard to pin
down. So if it is a genuine science of complex systems we are after and not just anecdotal
accounts based on vague personal opinions, we’re going to have to translate some of these