My conversations with Christopher Danielson over the last couple months about dance, math, Papert and learning have inspired me no end. He's a great provoker, and I say that with the utmost respect, especially in the area of question asking.
One big question he had for me has gone unanswered for what seems like months, even though it's been just over a week. I've been thinking intently about other related topics but his question has been in the back of my mind the whole time. Christopher asked:
"Do you have examples of meaningful movement in mathematics teaching that are not dance?"
The answer may take many lifetimes of work, but we can still benefit from partial answers and that is what I provide here.
To start, meaningful movement in mathematics learning can be either dance or non-dance. Dance implies a meaningful system in itself -- in my work, for example, percussive dance steps can be created using a variety of movement variables authentic to the art form coupled with a musical aesthetic.
Examples of non-dance movement in mathematics learning has been a little harder for me to nail down. This is what I have so far, please feel free to add to this list.
1. Meaningful non-dance movement in math learning happens in the natural body system of gesture and everyday movements (as shown in the work of Susan Gerofsky and this study that showed that 'children think and learn [math] through their bodies. Also, here's a past blog post of mine with links to more research and thinking on this topic).
This body system of gesturing as both a way of expressing knowledge and a way to think through ideas (mathematical ones, specifically) is at work whether we have noticed it or not. As with anything related to body knowledge, we need to grow our movement/math learning eyes so we know what to look for in our learners. A recent post on Christopher's Talking Math with Your Kids blog shares a story of a child using her own body knowledge to essentially discover one-to-one correspondence. This is not necessarily gesture, but it is a potent example of something we could notice in this realm.
The idea of gestures as non-verbal expression and thinking makes sense to me. On the whole, we tend to consider real knowing/learning as verbal and symbolic output. All I can say is that this hyper-focus on educating ourselves "from the neck up" coupled with the disappearance of hands-on learning-by-making in school (shop class, art, music, etc.) has alienated generations of children and convinced them to think they are really not all that smart when in fact that is not the truth at all. I was one of them.
2. Meaningful non-dance movement happens in a system where the child is using previous experiences in her body, or creating new understanding through her body, during exploration of mathematical ideas and concepts in school or with adults...and has agency over the exploration.
What does this mean? Seymour Papert's LOGO turtle geometry as described in his book Mindstorms is a great example. Papert coined the phrase 'body syntonic' to describe this kind of body knowledge -- "ideas which are compatible with one's own feelings of being in a body." [source]
But, as I've stated recently, just because you have a body does NOT mean you will automatically be able to develop ideas from it or access it in learning. Papert was presenting a way of learning that still, for the most part, has not been fully understood in formal educational settings. Essentially, the work Papert was doing with his turtle was to create a learning environment that provided enough structure for children to learn mathematics on their own. [Take a minute to let that sink in...]
This is agency -- the freedom to lead your own exploration and make mistakes on the path to new understanding. When a child is thinking about the choices she wants to make with the Turtle, it is her own body knowledge she relies on. That's agency. What is not agency is setting up that turtle and giving a class the exact same step-by-step directions on how to make it draw a flower, or a house, or whatever.
Where else can a child build and call on body knowledge in a setting that allows learner agency? I just got home from the FroebelUSA conference where we got to experience most of the ten Froebel Gifts. Friederick Froebel was the guy who invented kindergarten over 200 years ago. If you have blocks and math manipulatives in your classroom or home, then you are experiencing Froebel's legacy. If you know of Waldorf and Montessori, then you know a little bit about Froebel because they are shoots from the root of his system. If you are familiar with the names and work of Buckminster Fuller, Kadinsky, Frank Lloyd Wright or the Bauhaus, then you know of people who attended Froebelian kindergartens (ages 3 through 7 but extendible to any age.)
Froebel's gifts are essentially what Papert might call 'objects to think with' -- starting with a wooden sphere that can fit in the palm of your hand, on to a solid cube, and then various interesting divisions of the cube, and other gifts to explore surface, point and line. Here's a picture of Gifts 3 and 4:
In the Froebel system there are three main ways to experience the gifts: using a narrative context in which to explore the properties and powers of the materials, explore the mathematical properties of the materials, and as a 'form of beauty' including exploration of symmetry and patterning. Sometimes the gifts are presented in a guided way, but it seems that there is plenty of opportunity in the Froebel system to explore these materials freely, with personal agency.
My main point:
As Professor Eugene Galanter (one of the founders of cognitive psychology) said during his keynote at the start of the FroebelUSA conference:
"The mind needs a body to work in."
As I write this I am finding all sorts new questions in these ideas I've set out and it's clear that this topic requires much more than a single blog post. I've written more in depth about the differences between dance/math exercises (low to no agency), lessons (potential for limited agency) and truly exploring and making mathematical meaning through dance making (high agency) in the Math in Your Feet program. You can read and download the newly published article here.
Let's keep this going: Am I missing anything? Any holes in my argument? What other examples of meaningful non-dance movement learning can you think of?
Addendum, October 15, 2013: Here are some more specific examples of a child (my own) learning math through her body. "Thinking Like a Straight Line": Examples of a Body Learning Math