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.

"Nutritionism" and Sciencification of Everything

The NYTimes Magazine ran an article, Unhappy Meals, from which this quote comes:

"In the case of nutritionism, the widely shared but unexamined assumption is that the key to understanding food is indeed the nutrient. From this basic premise flow several others. Since nutrients, as compared with foods, are invisible and therefore slightly mysterious, it falls to the scientists (and to the journalists through whom the scientists speak) to explain the hidden reality of foods to us. To enter a world in which you dine on unseen nutrients, you need lots of expert help."

Which makes me think of all the other -isms out there-- all the places we need experts where folk wisdom used to do pretty well. -- nutrition, sleep, gardening, drinking water, teeth cleaning (are teeth cleaner today as a result of all those new toothbrushes?).

I'm down on science education lately. And not the science part of it, but the standards and textbooks and tests and what we think it means to know something and what we think is worth knowing.-- I think a lot of science education works to create this -ism divide between experts and non-experts. It happens when we focus on the things- like the Krebs cycle- that students cannot understand deeply and must rely on experts to tell them about.

I remember David Hammer relating a story about how when he says he's a physics professor people ooh and aah; when he says he's an education professor people start in on their ideas about what's wrong and right in education. But people have far far more experience with the physical world than they do the educational world-- somehow they've learned that their ideas and thoughts about physics are not valid or relevant, while their ideas about education are.

Why teach science.

From Schank's blog:

"Do we really believe that the reason that there so many foreign applicants to US graduate programs is that they teach math and science better in other countries? China and India provide most of the applicants. They also have most of the people. And many of those people will do anything to live in the U.S. So they cram math down their own throats knowing that it is a ticket to America. Very few of these applicants are coming from Germany, Sweden, France or Italy. Is this because they teach math badly there or is it because those people aren’t desperate to move to the U.S.?

In the U.S., students are not desperate to move to the US, so when you suggest to them that they numb themselves with formulas and equations they refuse to do so. The right answer would be to make math and science actually interesting, but with those awful tests as the ultimate arbiter of success this is very difficult to do...

You can live a happy life without ever having taken a physics course or knowing what a logarithm is.

On the other hand, being able to reason on the basis of evidence actually is important. Thinking rationally and logically is important. Knowing how to function in a world that includes new technology and all kinds of health issues is important. Knowing how things work and being able to fix them and perhaps design them is important.

Lets get serious. We don’t need more math and science. We need more people who can think."

-- I would argue "We need more people who can think mathematically and scientifically"-- which is all too often not taught in math and science courses.

Other interesting blogs:

• Susan Ohanian
• Alfie Kohn

Understanding the Math Wars, Implications for Science

In the American Journal of Physics-- a journal for which most of the publications are physics professors sharing interesting problems/labs/ideas, but also some interesting physics education research-- the following editorial was published this month:

School math books, nonsense, and the National Science Foundation


The editorial itself isn't too surprising; it's the kind of editorial that shows up in newspapers and websites when mathematics education reform curriculum is introduced. And it's from a vocal critic of those reforms. What is surprising is that the AJP decided to publish it, with all of its ranting rhetoric.

Some history on the Math Wars is necessary for understanding how to interpret Klein's editorial.

I think science education has different concerns: for one, most parents have less expectations about what students should learn in science than they do for math, are less likely to think they use traditional science content every day, and so are less likely to be vocal opponents to reform curricula. Also, people have an image of scientist as a doer of experiments; a mathematician (perhaps?) they think of as a knower of math. (Actually, I wonder what most people think that mathematics professors do all day?) So science-as-activity is easier to pull off.

But it is concerning that this shows up in the AJP. I composed an editorial in response. (Tried attaching it to no avail. I'll try again later.) We'll see if it gets published.