## Thursday, November 29, 2012

### Chicken or the Egg? A Math Integration Tale

I recently had the opportunity to have my teaching work critiqued by a group of colleagues.  They viewed a ten-minute video I produced which illustrated what success looks like in my classroom. The feedback I received was all at once enthusiastic, thought provoking and puzzling.

I teach elementary students the elements of percussive dance and then, within a structured framework, give them the freedom to create their own percussive patterns.  Along the way we use and talk about a lot of math which both describes their patterns and informs their creative choices.  It seems straightforward to me, so I think that is why I was flummoxed by a question they all had:

“When your students are choreographing their percussive dance patterns, how much of that activity is about their math understanding?”

I can answer that question.  The answer is, “All of it.”  I don’t see a separation between the two.   In fact, I think the dance and the math are essentially the same activity.

Here is an example:  a video of some traditional Irish figure dancing with accompanying percussive footwork.  You only have to watch a minute of the dancing to notice it is full of geometry and symmetry and all sorts of other wonderful kinds of math:

The shifting, curving patterns move through space and time while undergoing symmetrical transformations.  The dance choreography explores permutations and combinations of moves and steps by arranging and rearranging dancers at a dizzying rate in time to the music.  The footwork traces invisible maps on the floor.  The math in the percussive footwork is a reiteration of the figure dancing, but on a smaller scale with more specific and precise patterns.  Unsurprisingly, precision is a hallmark of mathematics which has, by a popular meme, been called ‘the science of patterns’.

All this is well and good, but what my colleagues really wanted were more specifics about my evaluative criteria.  How exactly do I gauge my students --within the medium of percussive dance or with regard to the math?  Again my, possibly controversial, answer:  Both.

And a question back: Why do we think of them as separate activities?  I think part of the issue might have a lot to do with how we, on the whole, perceive mathematical activity.

Generally conceived, dance is a three-dimensional, kinesthetic endeavor.  Math is rote memorization of algorithms and concepts and inhabits a two-dimensional symbolic realm.  Everything we’ve learned in school bears this out, except that it’s really not true!  When I started to really investigate what it means to do math I found that it’s completely different than what I did in school when I was a kid (and you too, probably).

And, as I dug deeper, I also realized I had been thinking mathematically all my life – I just never recognized it as a mathematical activity.

One of the things I’ve come to realize is that, really, people who do math don’t spend a lot of time plugging numbers into memorized algorithms.  Instead,they formulate and/or approach questions that don’t have immediate solutions.  They spend time thinking, talk to others, sketch out ideas on napkins (or whatever), and build models.  And then, when they think they’ve got something that resembles a solution, that’s when they start writing it down.  The notation is the end result of a process of questions, trial and error, and conversations.  Sounds a lot like what we do in Math in Your Feet, actually.  Take a look:

When I first started wondering about whether or not there was math in the dancing I did with students, I knew I needed an interpreter, someone who really understood math and how it was taught to children. I was lucky to be connected with Jane Cooney, a classroom teacher with deep experience and love for teaching math.  Our collaboration in creating Math in Your Feet included long discussions about the best ways to retain the integrity of both content areas.  We weren’t going to make up the dance to fit the math and I wasn’t going to make up the math to fit the dance.

We didn’t and I haven’t.  There was no need. There is enough overlap between the two that, if you hit it right, you often can’t tell where one starts and the other ends.  However, I have consciously created specific lessons to identify and learn the math that we’re going to use in our dancing.  Not only is math a tool we need to understand in order to use it properly, but I think it’s also important to know exactly how math is involved in our physical and creative work.

Like the old chicken/egg conundrum, it really doesn’t matter which one comes first because they’re both part of the same process.   And that is why, when I watch my students share their work throughout the week, I can see clearly if they have both the dance and the math and to what degree.  But that’s another story!

[This story originally posted in the Teaching Artist Journal's ALT/space, 11/28/12]

1. Without knowing the folks who were interrogating your work with those particular questions, I don't want to dance to any conclusions about their pre-judices. But it seems at first blush that they're thinking in a very one-track-only rut: math is math, dance is dance, art is art, and never with they meet. And how do you assign the grades??? (These days, assessment being everything. Will there be any clog dancing multiple choice questions on Indiana's high-stakes tests? No? Then what you're doing is just fuzzy and a huge waste of time).

For the traditionalists with whom I've been warring about mathematics education for 20+ years, the dance would be fluff, a distraction, an excuse for "math avoidance," as one of them likes to say. In other words, your work has to be crap because if you really knew mathematics, you'd be doing mathematics with the students without all this hippie dance crap. It looks way too fun and hence must be without value in drilling students into the little holes that all kids must be forced into (and not just about math).

Do I sound cynical and bitter? I'm not, really, because I don't allow ANY of the above rhetorical baloney to stop me from my work. I'm having a blast working with two groups of 4th - 6th graders at a local private progressive K-8 school. This is probably the most I've enjoyed mathematics teaching in my career, because I'm actually getting students to engage in honest mathematics. They have no choice but to think, and I'm not restricted to a book or a set curriculum. It's challenging for ALL of us. If I could dance, I'd happily add something like that to my classroom, but I know my limitations. There are more ways through the woods than one, of course, and I have so many fine options. Thus, so do the students.

Keep up the work, as I know you will, and remember, "Illegitimi non carborundum." :^)

2. The colleagues mentioned are well versed in arts integration and arts education but not all of them were familiar with math education content or practices. Their questions were actually quite helpful in clarifying some things about my work that I've wanted to give voice to for years but never could until now.

My main point, besides the finer points of what it means to seamlessly integrate seemingly disparate content areas,is that our conception of 'mathematical activity' has to shift -- I've had some great conversations with Paul Salomon about this topic that have been quite enlightening. One thing he said today is fabulous, and I quote:

"If we think of math as a collection of concepts and techniques, then you have to work a little bit to point out the symmetries and patterns of the dance. But if we open ourselves to the idea that we learn mathematics for mathematical habits of mind, and that doing math is merely thinking mathematically, then this is CERTAINLY math. In order to perform those patterns and rhythms one has to maintain and manipulate so much information. The knowledge of the pattern isn't written down like sheet music, its built up out of smaller chunks, each of which may themselves be made of smaller chunks. This is EXACTLY the structure of mathematical knowledge. The smaller pieces eventually become so familiar and manageable, that we no longer trouble ourselves with them. We move our thinking to higher levels of abstraction and structure in order to perform more complex actions."

I am completely confident in my approach to teaching math and dance as the same activity, especially since my approach to teaching percussive dance echoes Paul's descriptions of building mathematical knowledge. I can also justify my work based on the number of more recognizable math 'concepts' that are used AND the fact that much research points to the benefit of representing mathematical understanding in multiple and diverse ways.

And, as I tell teachers in my PD workshops -- the teacher doesn't have to dance -- just talk your best mover through the movement variables ahead of time and you're all set! ;-)

3. Wow!

Malke, it is such a pleasure watching your evolution through your posts here. I love this post, and I also love how it comes out of what you've written in the past - your struggles to understand math more deeply, and the clarity you've gained as you've worked with your daughter.