Commentary on Theories of Mathematics Education

136 R. Lesh and B. Sriraman
Fig. 3 Stopping the two systems after 20 seconds
tem. Consequently, if the goal is to control such systems, then what needs to be
controlled are the interactions—not just the initial conditions. . . . Consider the Paper
Tearing Experiment described in Fig. 4. Now consider the kind of systems that
mathematics educators need to understand and explain—such as: complex programs
of instruction, plus complex learning activities in which the complex conceptual systems
of both students and teachers or students will be functioning, and interacting,
and adapting. . . . Within such a systems, it is clear that the system will involve feedback
loops (where A impacts B which impacts C which returns to impact A) and
where the systems-as-a-whole develop patterns and properties which result from
interactions among elements of the systems, and which cannot be derived or deduced
from properties of elements themselves plus properties of any “treatment”
that might be used.
In spite of the obvious complexities in educational systems, a prototypical study
in education tends to be thought of as one that shows what works—even in situations
where (a) nobody was clear about what “it” really was that worked, nor what “working”
really should have meant, and (b) the assessments themselves were among the
most powerful un-tested parts of the “treatment” that presumably were being tested.
. . . In fact, as we observed earlier, most tests are chosen precisely because they
were intended to influence outcomes. So, in cases where the things they assess are
not consistent with the goals of curriculum innovations that they are being used to
assess, then they become important parts of the treatments themselves. Furthermore,
if they are only used as pre-tests and post-tests, then they neglect to measure the sin-