The Cognitive Science of Math v. the Mathematics of Math

One of my favorite books about links between math and mind is "Where Mathematics Comes From" by Lakoff and Nunez. A mathematician who reviewed the book critiqued it for its mathematical claims, rather than the cognitive science claims-- something I see in physics (and science) education quite frequently.

Lakoff and Nunez, in their reply to her criticisms, note:

"We have to keep in mind, however, that our goal is to characterize mathematics in terms of cognitive mechanisms, not in terms of mathematics itself, e.g., formal definitions, axioms, and so on. Indeed, part of our job is to characterize how such formal definitions and axioms are themselves understood in embodied cognitive terms.

We simply have a different job than professional mathematicians have. We have to answer such questions as: How can a number express a concept? How can mathematical formulas and equations express general ideas that occur outside of mathematics, ideas like recurrence, change, proportions, self-regulating processes, and so on? How do ideas within mathematics differ from similar (but not identical) ideas outside mathematics (e.g., the idea of "space" or "continuity")? How can "abstract" mathematics be understood? What cognitive mechanisms are used in mathematical understanding?"

This is a great way of understanding some of the difficulties in math or science education-- on the one hand you have the logic of math (and science)- not contextualized, impersonal, precise; on the other hand you have the language the mind uses - embodied, contextual, analogical. Understanding each is side is crucial to understanding how to teach.