First, here is a famous excerpt from Seymour Papert's seminal book Mindstorms. In the preface Papert introduces the idea of body knowledge in a story about his childhood fascination with a set of gears. He says:

*“Piaget’s work gave me a new framework for looking at the gears of my childhood. The gear can be used to illustrate many powerful ‘advanced’ mathematical ideas, such as groups or relative motion. But it does more than this. As well as connecting with the formal knowledge of mathematics, it also connects with the ‘body knowledge,’ the sensormotor schemata of a child. You can*be

*the gear, you can understand how it turns by projecting yourself into its place and turning with it. It is this double relationship—both abstract and sensory—that gives the gear the power to carry powerful mathematics into the mind.”*

My question is around this idea of 'body knowledge.' It seems Papert uses that phrase to mean that body experience

*in the world*can, at a later time, be brought to bear on understanding mathematical ideas. His Logo geometry (which used a programmable object called a turtle) is said to be something you can “walk” through. From my perspective, programming a moving object is not actual walking or actual movement; it is, like Papert mentioned in his gears story, a*projected*walking—projecting body knowledge into the object itself.
Here's my question:

When thinking about the body's role in math learning,

**I wonder if Papert ever considered the inverse to such a process,**specifically, how a body might participate in*actively creating*body knowledge of mathematics. The double relationship he mentions ("both abstract and sensory") seems to relate only to the*sensory memory,*not to the using the senses in the moment.
Any thoughts will be incredibly helpful. If I'm barking up the wrong tree, tell me. If you want me to clarify anything, let me know.

Thanks in advance.

*(Edit 12/9/13 -- check out the comments!!)*
Hi Malke,

ReplyDeleteYou write:

"...that body experience in the world can, at a later time, be brought to bear on UNDERSTANDING mathematical ideas." (I added the emphasis)

I agree with your questioning of the "inverse" to this. Namely, that bodily experiences fundamentally shape the CREATION of mathematical knowing/knowledge/objects. I know I have mentioned Rafael Nunez's work here before, but I believe much of his work in cognitive science supports this view...that mathematics does not exist a priori, that it is a mind-made creation, and that this creation is largely based on conceptual metaphor drawn from bodily experience.

In fact, I think the first quote (about understanding) is only one that an observer could make as a sort of post hoc analysis. What it basically says is "I, the observer, have a particular way of knowing and I now see bodily movement as helping an other to think in a way that I recognize as similar to my own." It only makes sense from the perspective of the observer, because the other (mover? learner?) is in a process of CREATION...they "have" no other way of knowing to compare to their own...no"thing" TO understand.

Thanks for a thought-provoking post!

Bryan

Hi Bryan -- thanks so much for your thoughts.

ReplyDeleteI had this question while drafting a new chapter in my new book. Your comments reminded me of something else that I drafted yesterday, around that idea of post hoc analysis:

"Papert recounts his story of the gears through the lens of a highly trained and schooled adult. What did that experience look like when he was a child? Did an adult observe him playing and engage him in conversation? What did he know of or think of the gears at that moment? At what point did he make the connection between the mathematical ideas illustrated by the gears and the mathematics itself? Is body knowledge something we can create simply by looking at an object and imagining we are part of it? What other ways can we understand how the body knows math?"

In the end, I think that movement/dance and math learning will be better understood by looking more closely at how Papert conceptualized "an object to think with". But, I think there is a fundamental misunderstanding about 'body knowledge' that we have to address first -- namely, what exactly it is and what body experiences build the kind of body knowledge useful for learning math?

I'm hoping other readers will share their thoughts as well so we can get to the bottom of this! ;-)

Oh, and in a different (e-mail based) conversation I had today I started to clarify another question I've had related to what you said in your comment in relation to 'understanding' math:

ReplyDeleteI'm investigating if there's a difference between 'learning about' [UNDERSTANDING] math with the body -and- the body 'learning with' math. For example *using the arms* to discover permutations instead of having arms be part of a more expressive process that *uses/explores math ideas* (like permutations) to create something new (which harnesses the nature & potential of the body to think inventively in new/creative ways). I've got to figure out a way to make the distinction between the two and then adequately describe the continuum on which this kind of work can happen.

Thanks again, Bryan, for the conversation.

It seems as though the inverse of body knowledge would be this question: "Is it the case that the mind can inform the ways we move our bodies?" In that case, I think the answer is obviously yes. But maybe it's the wrong question. All applications of mathematics are founded on this assumption, right? That an analytical approach to a physical problem will result in a solution we can instantiate.

ReplyDeleteI have a feeling that thinking about the relationship this way may provide me some breakthroughs in thinking about body knowledge. That is, if I have the inverse right, then I understand it much better than I do the original function. So clarifying how the inverse works, maybe I'll gain a better understanding of the original mapping you and Seymour have set out for us.

Maybe.

And then again, maybe not.

Christopher, I think my inverse and your inverse are different. It was good to have your thoughts b/c it makes me think I see the body differently from Papert (and you).

ReplyDeleteAs I understand it, he is essentially saying that experiences in our body provide empathy for the object -- I think his phrase was body syntonicity or body syntonic, meaning ideas which are compatible with one’s own feeling of being in a body, which has become a way of conceptualizing programming (I think).

When I look at and read about things like the turtle or the gears, I see how the body is or isn't intentionally engaged in creating mathematical meaning. So, perhaps there are no connections between what I do and Papert's work, like I first thought. Or, perhaps I need to create some kind of corollary to Papert's programming metaphors but related to mathematics specifically. As you'll remember, the Turtle is the focus of the activity *through which* the mathematics is learned. Sort of like the body activity is the focus of MiYF *through which* mathematics is learned. Except it is the body *and* the mind that are learning together when the body is involved in math learning with dance. I doubt very much the same can be said when programming the turtle. So, maybe there is no corollary to be made.

I'm all in tangles, but it seems like a productive struggle. Thanks for your thoughts. Helpful as always. :-)