“An Object to Think With” | Malke Rosenfeld

Some thinking about meaningful movement and math learning, in three parts.

First: Prelude
The point of this post is to push back a little on the idea that simply being out of one’s desk and on your feet will create what mathematician and a founder of the MIT Media Lab Seymour Papert termed “body knowledge”. Specifically, my goal is clarification of what it looks like to combine dance with math learning in meaningful ways.  These ideas were originally presented on my personal blog to a very broad audience comprised of people interested in math education. But, merging math with dance, or dance with math, is a two way street, which is why I decided to share it here.

This post is the result of conversations inspired by this video from Simon Gregg, a teacher in France, showing an activity he did on factors and primes on the playground with his eight and nine year old students.

After he posted it, there was a Twitter conversation between myself and math education professor Christopher Danielson which I Storified; this led to my post Starting the Conversation: Meaningful Movement and Math Learning (including a video using modern dance to illustrate statistics concepts). A day later there was a blog response from Simon, and, finally, here is how I responded.

Second: “An Object to Think With”
First of all, I really don’t buy the idea that just by being on one’s feet means there is opportunity for body knowledge. Simon said in his blog post:

“Another thing – one Malke and Christopher didn’t mention – to focus attention, it might work well if some kind of game was involved at some point.”

This sort of makes my point all on its own — Simon’s Year 4 students were on their feet but there were a lot of distractions and, as I’m sure he knows, no opportunity for focused movement. Meaningful movement is created when the focus of the lesson is on a body task and the mathematical concept at the same time – the two together create the focus for each other.  In addition, both need to be developmentally appropriate. What Christopher described doing with the school yard hundreds chart (noticing pathways and the distances between numbers by standing and moving on the chart) highlight, among other things, important numeracy skills but I am looking at it as a movement activity as well.  The kind of body activity Simon and Christopher proposed would, I think, be better suited for five, six and seven year olds and probably provide very little physical challenge for the upper elementary students in question.

So, if you’re interested in creating more focus during ‘stand up’ math learning, what kind of movement skills can you embed into a hundreds chart activity like the one shown in Simon’s video? What is the most important concept you would want your students to understand about primes? When you think of prime numbers, does your mind go to the intervals between them? If so, is the hundreds chart itself really the best visualization for that particular idea? What exactly is the big idea connected to prime numbers? And, if you were going to create a movement game what would be the point of the game physically and mathematically?

If you are having trouble thinking of answers to these questions I think I know why. There is a perception I think many people hold about using movement or dance in concert with math learning — that they are primarily thinking of the body as a drawing tool.

My standpoint on all this is deeply tied to what I see as a huge qualitative difference between mathematical representation and modeling and more literal illustrations of mathematical procedures or definitions (watch this video where the dance is little more than animated illustration for basic ideas of statistics). For one thing, definitions and procedural concerns are not math and dance is much more than literal interpretation. I know I need to develop my argument a little more but, for now, my point is that you are not creating body knowledge if you can just as easily illustrate (draw or identify) a math idea (like a right angle) using your eyes or a pencil and paper. If you can, there is probably no need to do a similar activity with your body.

Your body is an expressive vehicle and, as Seymour Papert intuited, an “object-to-think-with” capable of deep knowing that is rarely called upon or valued in academic learning. The body can be the perfect math manipulative, the perfect tool for children to use as they explore and experience the big ideas underlying mathematical activity – if used properly. Simon mentioned Zoltan Dienes’ principle of multiple embodiment. Here’s what I took away from an article about Dienes’ work I found on the Rational Number Project site regarding the use of math manipulatives (bolding emphasis and brackets mine):

“When individual activities [in this case with the body] cease to be treated as isolated actions and start to be treated as part of a systematic pattern of activities, the student begins to shift from playing with blocks to playing with mathematical structures [that’s what we want] Yet, when concrete materials have been used in instruction, more concern is often given to the “concreteness” of the materials than to the “activeness” of the activity – as though the abstraction were from the materials rather than from the structure that must be imposed on the materials.”

This takes me back to my original point. If using math manipulatives does not automatically connect you to the underlying mathematical meaning then being on your feet is not a guarantee of building ‘body knowledge’ in mathematics or any other domain. It goes the other way, too. Just because you’re a skilled mover does not necessarily guarantee you will be good at math. The meaning is created through thoughtful construction of activities or even whole sequences of investigations where both modes, mathematical and movement, come to influence the understanding of the other. Easier said than done, but it is totally worth the effort, so let’s keep thinking and talking.


Third: IF
I’ve been thinking intently for the past week. I’ve finally come to the conclusion that the challenge we face when bringing dance or movement into the picture during math time is not necessarily related to creating meaningful and effective learning experiences for our students, although these are certainly important concerns.

No, the issues we collectively need to address, before we can even start that process, are our deeply held beliefs about what math is and what dance is.

If math learning means number facts, right or wrong answers, learning algorithms, memorizing procedures, and experiencing math topics in isolation from one another then this video makes perfect sense to me. (I love the energy here, but question the assumptions.)

Or, this — a very strong example of non-dance movement but, again, with the ultimate goal being memorization of math facts.

If, on the other hand, we can come to not only accept but truly understand the following vision of math making and math learning:

Mathematics is a highly creative activity.  Mathematicians solve problems, but they also pose problems. They inquire. They explore relations. Investigate interesting patterns and craft proofs.  They present their ideas to the mathematics community and those ideas hold up only when the logic of those arguments are accepted. They don’t have a wise one who they line up for to check their answers with a red pen.” – Cathy Twomey Fosnot(excerpted from this Context for Learning video)

…and if we can at least consider, as I argued, that the body is more than a drawing tool

…maybe then we could come to accept (and eventually understand) how body knowledge is different from but not inferior to what we see as ‘real learning’: verbal and written discourse and reasoning abstractly through the medium of notated language.  If we could do this then perhaps eventually we could create some clarity on how the body can be more than simply the handmaiden to the goals of other disciplines, specifically math, in educational settings.

I’m still thinking on all of this, and it’s for sure a good kind of think, but I do wonder sometimes if I’m setting the bar too high. I’ll leave you with what I know:

  • Kids love to move.
  • Kids love to move, but there are different kinds of moving and different kinds of learning-while-moving.
  • In her book, Smart Moves: Why Learning is Not All in Your Head, Carla Hannaford said, “Learning, thought, creativity, and intelligence are not processes of the brain alone, but of the whole body.”
  • There are ways to bring dance and movement into math learning and still maintain the integrity of both disciplines. My recent article goes into further detail about how this can come to be.


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