Meaningful Non-Dance Movement in Math Learning

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

Challenging a Literal Approach: Learning Fractions 'through' Music

The last week or so I’ve been thinking quite a bit about what makes real connections between math and dance in a learning setting.  I’ve come to the conclusion that none of those connections can even be attempted without first looking at our assumptions about what math is and what dance is. 

I’ve been sharing my thinking on this blog, to an audience which (I assume) is more familiar with math learning than with arts learning and over at ALT/space, an online writing project I edit and curate for the Teaching Artist Journal.  I’ve not had a huge amount of responses, but the quality of the feedback I have received has been incredibly helpful in moving me forward on this line of inquiry.

Most recently I had a great conversation with friend and colleague Nick Jaffe.  Nick is, among many things, an incredible musician, teacher, thinker, writer and editor.  He’s not a math educator, but he’s definitely not math averse. 

We were talking about how people often try to teach fractions by connecting it to musical notation, which I have always seen as a very literal approach to what people perceive as music (written notation).  Over the course of the conversation I finally understood why I had always viewed that as a very shallow, un-meaningful activity.  Specifically, Nick has a lot of interesting and helpful things to say about discrete and relative quantities when playing music so I thought I’d share our discussion here in case it is helpful to you in any way.
………………………………… 

Me:  I have seen very little evidence that making the correlation between fractions and music have had had any impact. Maybe one study, but they didn't let the kids actually play around with ideas or make their own music -- they just taught them how to play the fractions. And, seriously, do you think like that when you're *playing* music? 

Nick: When people ask me about music and fractions, the first thing we need to establish is that the analogy between musical notation (or the musical reality it describes) is multi-dimensional. Notation denotes both the note as a discrete duration, but also as a division of a larger whole. Which, as I understand it, is also in the dual nature of fractions and an important thing to grasp, and not entirely intuitive.

What seems essential, at least in elementary and middle school math, to understand about fractions is that they have a dual function mathematically--they are a way of signifying two things at once: a discrete quantity, and a relative one. Musical notation functions the same way rhythmically.  In both the abstract, mathematical context, and in the context of music, the same problems arise with regard to this dual nature.  And it is somewhat counterintuitive to students at first to consider this--it's never explained clearly but it's the crux of everything one does with fractions as far as I can see.

I like the question you raise about how one thinks when making music. I think the answer is yes and no. Let's leave aside conscious methods of fractional or theoretically driven music making. Let's consider improvisation which would seem to be the most spontaneous, least theoretical approach to music making. And let's just stick to fractions as a rhythmic concept for this particular argument.

When I improvise I do not consciously think about the duality of a note as a time duration (discreet and part of a larger repeating whole). However I absolutely have to manipulate time, consciously and/or unconsciously as both things.

I think that duality is at the heart of choices about phrasing, for instance. You have to feel a note as a discreet duration, an almost word-like gesture. But you also have to feel it in relation to a pulse, even if there is no pulse, or you are trying to erase any implied pulse--it's still in reference. One does not have to consciously think in terms of fractions to do any of that. However, learning to read notation, and practicing manipulating that duality (playing behind, on, or in front of the beat for instance) can often increase one's control over it.

Finally, analyzing, generalizing and theorizing about that same duality can open up new options, or allow the translation of one dynamic application (actual playing) from one musical context to another.  In that sense theory is both a stimulus to new ideas and means of translating an idea for different contexts. I think theory plays those roles in most disciplines--a generalization that makes possible the prediction of as yet unknown dynamics. 

Me: The problem with using music notation to teach fractions is precisely what you just explained.  The written notation is the quantity, and the actual musicality/music playing is the quality.  What people don't get about mathematics is that 1) it’s not just about the notation (same for music) and 2) it's really about the quality of the quantities.  If you focus just on "how much" you get a dead language or, at the very least, information devoid of meaning. Which is why so many people hate math – for many reasons, and I’m speaking very generally, math education has essentially been bleached of its meaning.  

Nick: I’m actually not that interested in the idea of using musical notation to teach fractions. That seems boring and highly inefficient. What I am interested in, and I think kids might be as well, is working with music and fractional ideas at the same time to make interesting things. Perhaps those things would be musical, perhaps mathematical, perhaps visual, perhaps both.   Undoubtedly one would learn various things in doing such work, but I think it is the making that is what is appealing to me (and perhaps students) and that any real insights and learning depend on the impulse to explore in order to make. 
…………………
 To sum things up, Nick’s thinking about the nature of a mathematical idea (in this case fractions) in relation to his art form is exactly the kind of thinking process I engaged in to find a meaningful overlap between math and percussive dance in Math in Your Feet. It is also a perfect example of how it is possible to teach parts of math starting with your own experiences inside another discipline.  

Where to go next? One of the best music, math and movement resources I’ve run across is from Ellen Booth Church. The activities described in this article describe perfectly how the core concepts of mathematical activity can be connected to musical and movement activity.  Ellen is an early childhood specialist, so her ideas are based on that age group and applicable up to about grade 2, but I think this is the perfect starting place for anyone interested in making these kinds of connections in their own classrooms. 

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 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.

"An Object to Think With"

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 Seymour Papert termed 'body knowledge'. Specifically, my goal, now and in the future, is clarification of what it looks like to combine movement/dance with math learning in meaningful ways.

This post is the result of conversations inspired by this video from Simon Gregg showing an activity he did on factors and primes on the playground with his 8 and 9 year old students.

After he posted it, there was a Twitter conversation between myself and Christopher Danielson which I Storified here. 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 my response.
..........................................

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 it's 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 in concert would, I think, be better suited for 5, 6 and 7 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 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 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 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, bu it is totally worth the effort, so let's keep thinking and talking.

In the end, you may disagree with me and that's okay -- as Christopher said during our recent Twitter conversation, “I would love to hear you argue back."

Starting the Conversation: Meaningful Movement and Math Learning

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!