Commentary on Theories of Mathematics Education

Re-conceptualizing Mathematics Education as a Design Science 143
made more-and-more elaborate—rather than considering the possibility that they
might be fundamentally flawed.
Theories are cleaned up bodies of knowledge that are shared by a community.
They are the kind of knowledge that gets embodied in textbooks. They emphasize
formal/deductive logic, and they usually try to express ideas elegantly using a single
language and notation system.
The development of theory is absolutely essential in order for significant advances
to be made in the thinking of communities (or individuals within them). But,
theories have several shortcomings.
Not everything we know can be collapsed into a single theory. For example,
models of realistically complex situations typically draw on a variety of theories.
Pragmatists (such as Dewey, James, Pierce, Meade, Holmes) argued that it is
arrogant to assume that a single “grand theory” will provide an adequate basis for
decision-making for most important issues that arise in life (Lesh and Sriraman
2005).
• Models are purposeful/situated/easily-modifiable/sharable/re-useable/multi-disciplinary/multi-media
chunks of knowledge.
• Models are both bigger than and much smaller than theories.
• Here are some ways that models are bigger than theories.
– They often (usually) integrate ideas from a variety of theories.
– They often (usually) need to be expressed using a variety of representational
media.
– They are directed toward solving problems (or making decisions) which lie
outside the theories themselves—so the criteria for success lie outside the relevant
theories.
• Here are some ways that models are much smaller than theories.
– They are situated. That is, they are created for a specific purpose in a specific
situation. On the other hand, they not only need to be powerful for the this one
specific situations. Models are seldom worth developing unless they also are
intended to be:
– Sharable (with other people)
– Re-useable (in other situations)
• So, one of the most important characteristics of an excellent model is that it should
be easy to modify and adapt.
Concluding Points
The powerful pull of ideology is becoming apparent even in the popular press—and
even with respect to domains of knowledge that have nothing to do with emerging
fields of scientific inquiry. For example, consider George Lakoff’s best selling book,
Don’t Think of an Elephant, which attempts to explain why, in the last presidential
election in the USA, so many citizens clearly voted against their own best interests.