Commentary on Theories of Mathematics Education

Re-conceptualizing Mathematics Education as a Design Science 135
Fig. 2 Stopping the two systems after 10 seconds
simple functions—where independent variables (A,B,C,...N) go in, and dependent
variables (X, Y , and Z) come out.
The point that we want to emphasize here is NOT that such systems are completely
unpredictable; they are simply not predictable using single-formula models
whose inputs are initial conditions of the system and it’s elements. In fact, web sites
such as http://ccl.northwestern.edu/netlogo/ and http://cognitrn.psych.indiana.edu/
rgoldsto/ give many examples of systems which are far more complex, and in some
ways just as unpredictable, as double pendulums; yet, these same systems also often
involve some highly predictable system-level behaviors. For example:
– In simulations of automobile traffic patterns in large cities, it is relatively easy to
produce wave patterns, or gridlock.
– In simulations of flying geese, groups of geese end up flying in a V pattern in
spite of the fact that there is no “head” goose.
– In simulations of foraging behaviors of a colony of ants, the colony-as-a-whole
may exhibit intelligent foraging behaviors in spite of the fact that there is no “head
ant” who is telling all of the other ants what to do.
For the purposes of this paper, the points that are most noteworthy about the preceding
systems are that: (a) at one level, each system is just as unpredictable as
a double pendulum, (b) at another level, each system has some highly predictable
“emergent properties” which cannot be derived or deduced from properties of elements
themselves—but which results from interactions among elements in the sys-