Math, Dance & the Shoehorn, Together Again

First off, what's the difference between this:

and this?

Which one provides the more meaningful learning experience with multiplication? 

I mean, just look how many ways we can experience and come to understand multiplication! Stunning. Given this reality, why would we want our students to only understand multiplication as a series of facts?

So, now, take a look and tell me the difference between this (done as part of a computer science education project):



And this (start around 0:25 and watch until at least 2:00):



Which one provides you the more meaningful experience with Hungarian folk dance traditions?  

Can you see what happens when the dancing's sole purpose is to be shoehorned into a formal mathematical framework (the sorting algorithms)? 

To me, Video #1 has some interesting footwork but the choreography seems stilted and out of context -- sort of like only ever memorizing multiplication facts.

This may just be my own particular sensitivity but I'm curious what you think. Bonus points for going to the dance/computer science project website, trying out one of the sorts and reporting back how or if the dancing helped you any more than the computer animation they provide.

Learning Math without a Body

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. 

[The Math We See] [The Math We Do]

There are two ways I present Math in Your Feet. The first way is to upper elementary students.  In this version, the five-day residency revolves around the creative problem solving process of making our own percussive dance patterns.  Along the way, this pattern-focused process engages students in a flurry of mathematical activity within the dance making itself.  In addition, relevant math ideas/facts are identified and/or folded in to the experience to help describe, inform and improve our creative work.

The second way I present the program is to teachers. In the current version, our 3-hour workshop revolves around giving educators an experience of making math and dance at the same time; I lead teachers through the same core Math in Your Feet lessons as I do the students. This is a program about learning while doing so it makes sense that if you're interested in teaching the program to your own students, you need to have experienced the work first hand.

I have had the thought for a year or two now that I am dissatisfied with my teacher workshop model in some subtle but important ways. There are few things in particular that have come clear since my most recent teacher workshop:

1. Three hours is enough time for experiencing Math in Your Feet, but not much more. We need another session for processing that learning and what it means for students, making connections between the dance work to other ways we do and learn math (especially on the page) and hashing out all the permutations of classroom implementation.

2. On the whole, people will see the math with which they are already familiar in the work they are doing, often geometry.

3. Math in Your Feet is not a geometry unit.

4. It's time I made this fact more explicit.

5. When I can clarify for others the core of the learning that occurs while making math and dance at the same time I will finally have the conceptual base for creating a meaningful learning experience for teachers as learners first. 

I spent the last two days mulling all this over and came up with this. I think it's a good start:

An in-depth inquiry into patterns is at the core of the program. We spend the entire first day understanding how these patterns are built and structured, and how we might go about making our own.  We begin to understand that within a single beat in a dance pattern we can describe that one moment in at least three different ways (foot position, type of movement or direction).  This is the essence of mathematical abstraction.  This program is not about any one mathematical topic or strand or subject, it is simply about how we think when we do math.

This thinking occurs when we use attributes and variables to make, compare, compose, decompose, sequence, combine and discuss our Patterns A & B. This thinking also occurs when we sort, classify, choose, name and compare the variables/attributes that make up these dance patterns.

This thinking happens in the conversations and creative work within partner teams of two and in the active observation and analysis of patterns during our sit-down group observation and discussion times.

Patterns are our focus, our purpose, and what drives us forward. This important work is informed and supported by geometry concepts as well as the use of mathematical language and spatial reasoning in context. We are also immersed in matters of equivalence:

"What does it mean to dance the same as my partner?"
"How do you know that pattern was reflected?"
"It looked like both the A and the B patterns in their Pattern C were the same, but let's take a closer look and figure out if that's so. If not, how are they different from each other?"

In the end, it all boils down to clarifying the relationship between the math we see and the math we do.

The math we see are the "mathematical objects" that can be identified, named, memorized, tested, and are often the 'things' we hold before others as proof of learning. Many of these things are important, but they in and of themselves are only half the picture of what it means to learn math.

The math we do in Math in Your Feet? Those questions that spark inquiry, noticing, wondering and new questions? This is the process side of math, the action part, the really, truly, super fun part. It doesn't matter whether it's dancing or daily number routines, or working out how many different kinds of hexagons you can make with pattern blocks, or anything else. No matter the vehicle, this is how we think when we do math.

My goal now is to make this process side explicit and to figure out how to design an inquiry for teachers that revolves around the dance making but is as intellectually engaging for the adults as the inquiry I have developed for students. And when I can figure that out, we are going to have SO. MUCH. FUN.

One Huge, Kinetic, Sonic Blur [Residency Notes Day #3]

Note #1:  Day 3 is just one huge, kinetic, sonic blur
The classroom is full of noise, sound, movement and increasing mastery of complex moving patterns, patterns which are created, observed, refined, and analyzed throughout the one-hour workshop.

Having created four-beat Pattern A on Day 2, kids are charged with creating a second dance pattern.  Pattern B needs to be as different as possible from Pattern A. This means returning to the Movement Variables chart as a resource.

Are all your movements in Pattern A jumps and slides? Pattern B should have some other kinds of movements -- steps, turns, touches.

Is your Pattern A full of turns? Try to create something new that feels and looks interesting without any turns.

How about starting and ending position?  If you start and/or end in the center of your square in Pattern A, how about changing that in your second pattern?

You get the picture, yes? What's very clear is that I generally always see much more interesting, creative work in Pattern B, in both kid and teacher workshops.

What is also clear is that giving children agency over an inventory of pattern variables is highly empowering and every class this week has been super focused, productive and engaged.

Note #2: Day 3 is about combinations too 
After creating Pattern B we talk about how to combine the two patterns to make a new 8-beat combination. The options include:

A+B | B+A | A+A | B+B

And, yes, those are simple combinations.  But only on paper.  The challenge is to see where the first pattern in the sequence ends within your dance space and how to execute your second pattern from wherever you ended your first pattern.  Meaning, you can't go back to the starting position of the second pattern (position 0) even if you're in a tricky position or facing a new direction.  These transitions are where the really interesting spatial/math thinking and problem solving occur.

Note #3: Biggest challenge in Day 3?
Says one 4th grader: "When I started combining Pattern A and Pattern B they were separate in my brain. It took a lot of time and effort to make them into AB."

Me: "You mean it took a while before it felt like they were the new 8-count pattern, Pattern C?"

4th grader: "Yeah."

Isn't learning math and dance at the same time fun?  Yup.

[Residency Notes Day #1 | Day #2]

Emerging Voices [Residency Notes Day #2]

This is the day kids really start dancing.

After spending all of day one in various stages of disequilibrium, day two brings an integration of new skills.  If day one was about "Oh my gosh, look what we get to do!" day two is about "Oh my gosh, look what I can do now!" and "Look what we can make!" 

Note #1: Clarifying intent (paying attention to the attributes of moving patterns)

This is also the day we focus on sameness. As in, how can we make our dancing the same as our partner's dancing?  What exactly needs to be the same?

Type of Movement
Me, to different teams of students working on their four-beat Pattern A: "Is that a jump or a slide?"

Choice of Direction
Me, during our periodic active observations of work in progress: "Are they turning the same direction or opposite directions?  How can you tell?"

Foot Position
Me, to dancers: "Are your feet split to the sides or is that a diagonal split?" or "How does it feel to finish with your feet crossed?"

Note #2: Experiment<==>Create

Today I spend a lot of time roving around the room just observing. At the beginning of creating their four-beat Pattern A, their feet and physical intention still emerging. They have to organize and integrate the ideas they see in their heads and communicate it with their bodies.  They also have to sync up with their partner's dancing as well.  This is a fantastic, engrossing challenge. It is also fascinating to watch their dancing/moving/body voices emerging so spectacularly over the course of the hour.

As the class proceeds, they all want to show me what they've come up with. We talk. When I show up near them again, the pattern looks different than I remember. "Oh, you changed it!" I exclaim. "Yeah, we like it better this way," they grin, proud of their agency and resourcefulness. Their dancing is cleaner now too.

Note #3: Thinking bodies (observing the research on gestural thinking, literally in action)

Teammates discuss the similarities, sameness and differences of their (blue, taped, square) dance spaces.  In one class, children noticed: "They both have four corners" and "They have four parallel lines." 

Me: "Are there really four parallel lines?"  Discussion ensues.  At one point, a boy lifts his hands and, without speaking, uses his fingers to trace two parallel lines vertically in the air, and then two parallel lines horizontally.

Me: "Good! So by that I think you mean there are two sets of parallel lines?" He nods. I say, "Okay, let's all trace those lines in the air..."

Note #4: Thinking bodies (hive mind)

Wayne McGregor is a dancer and choreographer who engages in multi-disciplinary collaborative research around how the body thinks and learns, both the individual body, and the larger thinking whole created by a larger social systems. In his 2012 TED GLOBAL talk he provides a helpful primer of what it means to think with one’s body, especially within a dance system:

“So for me, choreography is very much a process of physical thinking. It's very much in mind, as well as in body, and it's a collaborative process. It's something that I have to do with other people. You know, it's a distributed cognitive process in a way …

The work we do in teams of two to choreograph math-informed, math-infused percussive dance patterns is social learning.  Not only do ideas flow verbally and physically between student teams, but also within each class; the energy in the room while kids are making often resembles a beehive. Today, for example, about half way through my most challenging class, something clicked and everyone was working intently; I could literally feel the group working and thinking together on their individual projects. 

[Residency Notes Day #1]

Squaring the Septagon [Residency Notes Day #1]

Note #1: Squaring the Septagon

First, let me just say that I almost met my match in terms of spatial problem solving today.  Almost, but not quite. 

I arrived at the elementary school this morning where I'll be working in the LGI room for the next two weeks with some awesome 4th graders and their teachers.

The LGI room is an extremely irregular septagonal shape, with a long diagonal and without any linear referents on the floor (no patterned carpet or tiles). Add to that our taped dance spaces are square.  It was my job to square the space.

I'll let that sink in. 

Given the difficulty of the task, I think I did a pretty good job.  What do you think?

Note #2: Why do we use so much tape in Math in Your Feet?

Because we do better, more interesting work inside our own dance spaces. Because it helps us focus on the relevant structure and shape of our dancing. Because dancing in limited space is part of many traditional percussive dance styles. That's why!

Note #3: What do I look for on the first day with a new group of students?

Can they keep a steady beat?

Do they know their lefts from their rights?

After a few rounds of our 4-count patterns during warm-ups do they get that we are dancing for four beats, and resting for four beats (signified by claps)?  Do they get the essential structure of the pattern unit and stop their movement on beat 4 or do they keep dancing?

Are their bodies organized? What challenges do they have lifting their feet off the ground? Do they loose track of their personal space and bump into their neighbors or end up far away from the rest of the group? Is it just a few kids, or the whole class?

When I give them words to say while they are dancing (e.g. "Split, cross, split, together") can they talk and dance at the same time? Or does the talking throw them off?

How much new information can I give them before their attention drifts? (Some groups enjoy more words and information, others get overwhelmed with too much input at one time. This tells me how to structure future lessons. The classes that zone out with too much talking need shorter bursts of the dance/talk cycle.)

Despite differences between individual kids and even whole classes, experience has shown that marking this starting spot can help us celebrate success as defined by amount of forward movement and improvement at the end.  Here's more on that: