The Map is Not the Territory

Last week was was a strange week of firsts for Math in Your Feet. I've been a teaching artist for about sixteen years and started exploring the connections between math and percussive dance in late 2003. Between 2004 and 2006 the program was piloted twice at all nine elementary schools in a large urban school district in Indianapolis, IN; prior to that I spent five years teaching clogging at many, many small, often isolated, rural schools across North Carolina, South Carolina and Kentucky. 

I've seen many schools and many students over the past sixteen years. But never anything like this.

In the first half of April I spent two weeks working at a public elementary school near Indianapolis, IN. This year they have six large classes of fourth graders, averaging 30 kids per class. I taught three of the classes the first week (one hour a day for five days) and three classes the second week. As is typical in my residencies, most of the kids were happy to be with me, worked hard, were proud of their work, and made progress but... During the second week I saw some startling things I've never seen before in all my years of teaching 4th and 5th graders.

This is the first time that... 

• kids made 3-beat patterns without noticing (they actually needed 4 beats)
• kids mistook their starting position as the first beat of their pattern 
• whole classes were still struggling to clarify footwork (directions, movement, foot position) by the time they created their second of two 4-beat dance patterns
•  on the fifth (and last) day of the program most kids were still not dancing fluently at a steady tempo...
• ...and, even more worrisome, a good amount of students in each class were still unable to reproduce their original dance patterns the same way every time.  Not surprisingly, they were also still working on dancing in unison (congruence) with their partners on the last day.  

I have honestly never seen this before and I have been puzzling over these observations for days. Here's what I'm thinking and wondering right now about all this:

1. It is definitely not about whether kids are 'good at dance' or not.
Some people might think that maybe part of these troubles are due to the fact that some kids are just not 'good dancers' but I do not agree.  My entire career has been focused on crafting meaningful learning experiences with my art form for students, no matter their dance backgrounds. This is the reason I developed the Jump Patterns tool in the first place. Jump Patterns provides a framework and basic feel of percussive dance for new dancers. (Interestingly, it provides an awesome challenge for more skilled movers as well.)  Also, I am a very flexible teacher of new dancers; I'm not looking for "good" dancing, just clarity of thought through the body whether dancing fast or slow.

2. I wonder if some of what I observed is about how much movement children are getting or not getting? 
Children think and learn through their bodies. Children develop spatial reasoning by moving their bodies. If their movement is severely limited due to a sedentary lifestyle, or a primarily sit-down education focused on test results, or school policies that use recess as a reward and/or punishment, then children are not getting the movement they need for developing their brains and bodies as a whole system.

The last time I saw difficulty like this was when I was working in very poor, rural parts of South Carolina in the late 1990s. I think the reason that I am writing this post is that only 20% of the children at the school last week qualify for free or reduced lunch. What's going on??

3. I also wonder if this is partly about how math is (generally) taught. 
I am teaching dance and math at the same time by facilitating a robust choreographic inquiry into the creation of multi-layered, three dimensional, moving patterns. Math in itself is inherently action-oriented which is why the body has so much potential in partnership with math learning. 

For example, in Math in Your feet we focus on the action side of math when we make, compare, compose/decompose, sequence, combine and discuss the patterns we are creating. This is mathematical activity. In addition, activities such as sorting, classifying, choosing, naming and comparing the attributes and variables that we use to build our patterns in the process of creating those patterns is mathematical activity.  This is what we do and how we think when we make percussive dance patterns AND when we do other kinds of math.

Because I've watched children think with their bodies for many years, most of that time in relation to mathematics, I think what I observed last week might be, quite literally, a visible deficit in experience with the process side of math, the part that builds conceptual understanding so that we know why and how we got an answer.

I think what I'm seeing is possibly a byproduct of math being taught as answer getting* rather than helping children build pattern-finding skills with numbers and in other mathematical situations. What I saw this week shows me that kids may know how to get answers, follow directions and learn procedures, but it is likely that many of the kids I saw in front of me have not had the chance to develop a strong conceptual understanding of mathematics, including:

- unitizing (the ability to compose and decompose shapes and numbers into smaller parts or larger new wholes)
- spatial language and concepts (built through the body and connected to math through language)
- pattern recognition beyond the (very basic) "red, blue, red, blue..." class of visual linear patterns

And I'm not the only one whose radar is pinging on this one. This blog post includes some of what I wrote on the Math in Your Feet Facebook page mid-week. A teacher who was part of the original pilot year with her students commented:

"We have noted that students need more concrete and visual / spatial experiences than they used to before they can move to abstract reasoning at the fifth grade level. We've wondered if our observations are correct and if they are, why?"

Abstract reasoning means we can take a math idea and use, apply and represent that idea in a number of different contexts. This cannot happen until the learner has built her/his own relationship with and understanding of that math idea. Abstraction itself is a process of coming to understand through conversations, observation, wondering, playing around with ideas, and noticing patterns and relationships. This is answer making.  Without this process an answer is essentially meaningless.

Ideally we should not have to remediate any of this. As a society we should provide our children with developmentally appropriate learning experiences at the time in their development that their brains and bodies need those experiences. In the case of spatial reasoning, unitizing, and pattern making/observing/identifying, this should start in preschool and increase in sophistication through elementary school. And, among many other tools, we should make a point of including the whole body in the math learning tool kit.

The reality right now is sadly quite short of this ideal. This post is simply intended to provide one educator's perspective on what seems to be happening as more and more children learn without their bodies.
_________________

*Thanks to Tracy Zager for giving me the term 'answer getting' which ultimately helped me clarify my thoughts in the post. 

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

The piece in question

Just yesterday I was playing around with a piece of paper, folding it, watching it, seeing how it could move.  I showed it to a friend and said, "I guess what I'm doing is improvisational origami."

Just today I mentioned the same thing to a different friend.  She said, "Have you seen Between the Folds?  You have to..."

Here's the trailer for the movie -- it's on hold at the library.  Hopefully I'll get to watch it soon.  Can't wait, can't wait, can't wait...

Every day I am finding more and more of the pieces that connect math, art, dance, rhythm, science, expression and creative practice.  I think the reason I'm so excited is that I recently found my way back from a wrong turn I took with this inquiry. But, that's part of the process, too, no matter what the medium. After determining what I don't want or need to do right now, the missing pieces seem to be falling into place.

[Edit: This post seems to be getting a lot of traffic, for some reason. If you're interested, here is the follow up post I wrote after viewing the documentary.]