Commentary on Theories of Mathematics Education

144 R. Lesh and B. Sriraman
Fig. 5 A conceptual schematic of MMP’s
. . . Says Lakoff:
People make decisions . . . based on their value systems, and the language and frames that
invoke those values. (p. xii)
Frames are mental structures that shape the way we see the world. (p. xv)
They are . . . structures in our brains that we cannot consciously access, but though we
know by their consequences . . . People think in frames. . . . To be accepted, the truth must
fit people’s frames. If the facts do not fit a frame, the frame stays and the facts bounce
off. . . . Neuroscience tells us that each of the concepts we have—the long-term concepts
that structure how we think—is instantiated in the synapses of our brains. Concepts are not
things that can be changed just by someone telling us a fact. We may be presented with
facts, but for us to make sense of them, they have to fit what is already in the synapses of
the brain. (p. 18)
The experiential world of the 21st century student is characterized by complex
systems such as the internet, multi-medias, sophisticated computing tools, global
markets, virtual realities, access to online educational environments etc. In spite
of the rapidly changing experiential world of today’s student, our approaches to
studying learning are still archaic. As discussed in this paper, setting up contrived
experiments to understand how students’ think/process mathematical content is interesting
but conveys a uni-dimensional picture of learning with very limited implications
for pedagogy and for future research. Today’s students are more likely to
be engaged in professions that calls for competencies related to understanding complex
real world phenomena, team work, communication and technological skills.
So, in essence there are three kinds of complex systems: (a) “real life” systems
that occur (or are created) in everyday situations, (b) conceptual systems that humans
develop in order to design, model, or make sense of the preceding “real life”
systems, and (c) models that researchers develop to describe and explain students’
modeling abilities. These three types of systems correspond to three reasons why