The Map is Not the Territory

This post is an addendum to my last post on meaningful non-dance movement in math learning.  After some reflection, I realized that for any of my thoughts to make any sense, I need give some concrete examples of what I personally see as a math learning through the body outside of a dance context.  

I homeschooled my daughter for first and second grades but I did not explicitly employ any kind of kinesthetic approach to learning math or anything else, for that matter. She wouldn't accept anything formal for the first year so we spent a lot of time out of  the house -- on walks (with lots of opportunities to talk math), math games, thrifting (always lots of history lessons there), reading books, listening to audio books, library visits, making stuff. 

For a while I wasn't completely confident in my approach, but over time I realized she was showing me what she was learning in many different ways: through conversation, through her art work and other creations, and, very often, through her physical movement.  

Here are some summaries of and links to blog posts from the past couple years that documented this phenomenon of "the mind needing a body to think with".  At the very least this will give you a peek into what I see when as I watch a child physically interact with her world. 

I'll start with a potent example in full, and give excerpts for the rest.  My daughter was six and seven in these examples.
 
Think Like a Straight Line (June 14, 2012)
It's been a loooong time since the kid has ridden her bike.  So long it seemed like the first time again today.

She felt wobbly.  Steering was a challenge.  So, she gave herself a pep talk as she worked to reacquaint herself with the activity.

"Okay, all I have to do is think like a straight line in geometry..."

She rode back and forth across the basketball courts chanting her new her mantra.

"Think like a straight line, think like a straight line, think like a straight line in geometry."

When she'd get to the end of the court, she'd get off the bike and turn it around.  

Then she figured she could make the turn without getting off.

"All I have to do when I get to the end is think like a circle...."

I'm sure she'll be back in the swing of things in no time.  Plus, I love the thought that pathways have specific intentions.  She's in the math, man.  Totally in it.

"Look Mama!  I can do Origami with my body!" | Origami Twirling Bird: Points, Edges, Turns, Poetry and Poses | August 25, 2011



"We've read Sir Circumference the first Round Table a number of times.  Now she has a game she made up where she leaps towards her blow-up wading pool in what she calls the "diameter jump' -- I hold my breath every time as she leaps, finger tips to toes stretched out in one long line to touch the front and back of the pool at the same time, literally flying, flopping almost on the other side of the pool."  Spontaneous Math / Math All Around | August 19, 2011



This next post relates to body knowing because it is built around the fact that we went on daily walks all over our little city.  Many times we would set out and I'd let my daughter navigate us downtown. The map of our city and our experience in the real territory in the map made for a very potent game. | Totally Territorial: Cats, Maps, Area and Multiplication (April 3, 2012)



How we came to understand scale: "If an ant weighed fifty pounds (the weight of a human child) how many pounds could it lift?  My girl counted it up on her fingers and immediately sprang up and ran around the living room trying to lift up all the chairs.  I nixed that idea, but it was such an immediate reaction that it sparked the idea that this needed to be an interactive experience." | Ten Times Better, Longer, Faster, Farther: Understanding Scale | January 11, 2013

This final example is from some summer work in the city: "The girls in the room were hanging out with me before class while I set up and helped me tape out the floor.  Any time I have a chance to let kids help me tape, from preschool to upper elementary, my helpers invariably end up spontaneously exploring their newly taped environment without any prompting.  This is actually my favorite time with kids -- manipulating the floor space with tape and then seeing what they do when they first discover it.  Here's a peek at the space and the only part of their exploration I could capture on video." | Floor Tape How Do I Love Thee? (Video Edition) | July 15, 2012

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

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 recently, 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 in the Teaching Artist Journal goes into further detail about how this can come to be.

I had an interesting conversation on Twitter today with Christopher Danielson about body knowledge (Papert) in relation to mathematical knowing and learning. I've Storified it now but here's what showed up in my in box later in the day -- perfect timing, as they say.

Some thoughts before you watch:

1. This is the first dance/math video I've seen that I've not been grumpy after watching it. I think I also finally understood statistics. That being said...

2. Could these concepts be illustrated effectively in some other way, meaning without the moving human bodies?

3. What, if anything, have we learned about dance in the process of watching this video?

4. What, if anything, could be learned by turning this into a dance lesson?  Would we understand anything more or differently after creating a dance based on the ideas in this choreography?

5. Would the dancing make sense without the text on the screen before and after the dancing?  Would the math in the dancing make sense without the text?



If we are serious about using movement, or a specific dance form, in our math classrooms, I think it's worth thinking and talking about these kinds of questions. I'm not sure I have good answers for all of them, and I will weigh in, but I'm curious to hear your perceptions and thoughts first!