I'm about to embark on a reading of Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, by George Lakoff and Rafael Nunez (thanks to a tip by Michael Paul Goldenberg.) This is a book co-written by a linguist and a psychologist hoping to launch the discipline they would call "the cognitive science of mathematics."
In a similar line of inquiry, the researchers who recently published the article Children's Gestures and the Embodied Knowledge of Geometry concluded that "children think and learn through their bodies."
Here's another recent article on a similar topic: Thinking with your hands: A link between gesturing and intelligence:
"My colleagues at Germany’s Humboldt-Universität zu Berlin , Potsdam University and I have discovered differences between people who gesture frequently and those who gesture rarely. Our study shows that gesturing may be a function of, and may even contribute to, brain development... [Emphasis mine]
...and thoughts about this article, from the Eide Neurolearning Blog:
"We do not know yet whether gesturing facilitates the development of fluid intelligence or whether it is a by-product. But we do know that children who are asked to gesture in certain ways while learning new tasks learn better than children who are asked not to gesture. Considering that gesturing benefits children while learning, it is possible that gesturing plays a role in the development of fluid intelligence, perhaps by simulating action. If this proves to be true, children might be able to literally give themselves a hand in their own development by gesturing more."
"It's interesting to think that teaching children to problem solve certain types of problems should involve strategies that take into account that fact that one is trying to train the imagery of the students. Just verbally saying back the steps of a problem or even watching an explanation won't internalize the imagery. To really 'get' certain problems, we have to enter into the simulation and perceive the question and solution in a bodily way." [Emphasis mine.]
From the Edutopia blog of neuroscientist and teacher Dr. Judy Wills:
My only question is why, with research and thinking like this, many/most schools still insist that keeping kids at their desks, and lengthening the school day behind said desks, is the best way to insure student learning. I know all schools don't think this. I know a lot of really smart folks out there are connecting the dots, but our policies don't reflect the very real evidence that our bodies must be engaged and involved in the learning process.Students need to be explicitly taught and given opportunities to practice using executive functions to organize, prioritize, compare, contrast, connect to prior knowledge, give new examples of a concept, participate in open-ended discussions, synthesize new learning into concise summaries, and symbolize new learning into new mental constructs, such as through the arts or writing across the curriculum. [Emphasis mine -- one of the reasons 'the arts' are helpful in this context is that most artistic activities employ the body during the art making process.]