Representing Math Concepts Through Percussive Patterns

Not quite congruent.  One partner has landed while the

other is still up in the air.

This week I'm working at Christel House Academy, a charter school up in Indianapolis.  This is part of a grant-funded pilot project for Young Audiences' Signature Core Programs.  The fifth graders are fantastic!  They are perfectly perfect in all their 11-year-old-ness, and quite observant and thoughtful to boot.  They make connections easily and ask interesting questions that show they are really thinking about how this all works.

This is an interesting situation for me.  I am usually invited to schools where kids are at least a grade level or more behind in math and my role is to assist in catching them up.  At this school, the fifth graders know and understand quite a bit so we are in the position of applying what they know to a new situation instead of learning it for the first time.  But the really fascinating thing for me is that, although they 'know their math' they are still challenged by representing it physically.

In my reading about mathematics education, I've come across an idea called 'the power of three'.  Essentially, the idea is that to really understand a math concept a child needs to represent it in at least three different ways.   This would be through pictures or some other means.  I'm just beginning to realize that one of the strengths of Math in Your Feet is that it provides an opportunity to experience and represent math concepts in the kinesthetic realm.  Part of this challenge lies in the fact that these patterns are not static, but require students to literally be 'in' the pattern.  Just today I had an interesting conversation with some boys about whether to record a turn as being on the third beat or on the fourth.  We eventually came to the agreement that the turn was actually happening between the third and fourth beat, but that since third beat ended in one position and the fourth beat was in the new position, we had to record it as being on the fourth beat.  My system may not be perfect, but it does create a structure to ask these kinds of questions. 

So, here's how it works.  Kids make up a four-beat dance pattern using the elements of percussive dance that I've outlined for them.  They learn to make their dancing congruent by producing (with pre-teen bodies!) the same tempo, foot placement, movement, and direction as their partner.  After that, we start transforming these patterns using different symmetries, starting with reflection.  At that point, all the pathways forged between the body and the brain have to be shuffled around as one partner dances the original pattern and the other (on the opposite side of the line of reflection) has to change the pattern by dancing the opposite lefts and rights.  For example, a turn to the right would be reversed to go left, or a right foot would be switched to a left foot.  This all sounds rather straightforward as I'm writing about it, but after observing the CHA fifth graders this morning, I realize that no matter how well they understand it in their heads, and no matter how 'smart' their bodies might be, it's still a challenge!  There's quite a bit of thinking going on here, in both body and brain, and it takes a lot of practice to remember a sequence of the four moves that make up their pattern.

This is only the third day and we have a couple more to go.  Things do get more interesting and more challenging when we start combining individual patterns into larger ones (i.e. start the second pattern where you ended the first, not at your original starting point and then try the reverse) and also when we transform the patterns using turn symmetry which seems rather straightforward in a static representation on paper, but is absolutely spectacular when you see it in motion. 

I'll keep you posted!

You Can Count on Monsters

You Can Count on Monsters, by Richard Evan Schwartz
Paperback, 244 pages, A K Peters

I happened to be listening to NPR's Weekend Edition recently and I heard a conversation between Scott Simon and NPR's math guy Keith Devlin about this new book You Can Count on Monsters by Richard Even Schwartz.

From Schwartz's website:

"The book starts with a 20 page introduction, written at an elementary school level. After explaining multiplication, prime numbers, and factoring, the introduction lays out the general idea for the rest of the book, as I'll now describe. To each prime number, we associate a pattern of dots and a monster.

"There is something about each monster that has to do with the prime. Part of the fun of the book is figuring out how the monster is related to its prime. For each composite number, we factor the number into primes and then draw a scene that involves those primes. We also show an arrangement of dots and a factoring tree that helps explain the picture. (A factoring tree is a kind of diagram that shows one way to factor the number into primes.)"

Near the end NPR piece, Devlin said about the book:

"The thing that distinguishes mathematicians is that we, at some stage in our development, we develop this understanding that numbers do have personalities, they have structures, they have relationships.  We form that, but most people don't manage to get it.  What Schwartz has managed to do is use his own skill as an artist to bring out some of the personalities, and the point is that what he brings out through his art is actually the structure and the personality that those of us in the business have always seen, we just haven't got the tools and the ability to make it accessible the way Schwartz [has].  It's his skill as an artist that makes this work [emphasis mine]." 
-- Keith Devlin on NPR's Weekend Edition, Saturday, January 23, 2011

As a dancer who integrates percussive dance and elementary math, I am in the business of making math accessible. I work to illustrate math concepts through a thoughtful sequence of activities; the children build original percussive dance patterns and learn and apply the math that arises naturally from this creative process.  I have spent many years learning and building my own understanding of the math content and practices that relate to this work.  And, I have carefully built a learning bridge that makes meaningful connections between the two subjects.

Now that I have built my bridge and my curriculum is where I want it, for now, I have become fascinated with searching for and finding examples of other kinds of bridges to math.  I am also trying to figure out just what it is that mathematicians see that the rest of us can't.  I'm coming at this task from a couple angles (no pun intended). 

First, going on some information I heard recently that it is most effective to learn a new language like a baby does (there's been some research findings about this, but I can't locate them right now), I'm working on (re)learning math myself alongside my five year old daughter by exploring math concepts through hands-on experience.  And, because I'm not five, I'm also looking ahead to where we might go next.  A few years down the line we might both be ready for You Can Count on Monsters. 

I'm also finding articles and online communities that are focused on how to teach math concepts for comprehension (not just for memorization of procedures) and learning from others' descriptions of how they teach and the kinds of questions they ask students.  I'm also on the lookout for quality examples of how art in general can help build a bridge to real comprehension of math concepts.  Schwartz's monster book seems fit perfectly into the bridge category, in a big way!  By the way, not only does Schwartz appear to be a working artist he is also a Chancellor's Professor of Mathematics and Director of Undergraduate Studies, Department of Mathematics, at Brown University.

So, happy reading and happy learning!  I'm off to the library to find myself a copy!

Play

"The artist is accustomed to working in the open spaces of creativity, ambiguity, uncertainty, opinion, and personal story.  Woven into this relationship with the discipline is a sense of play.  Play may be a key to understanding how people learn and how artistry and scientific thinking are linked. [...] The playfulness of artistry can be absorbing, exuberant, intense, and transcendent.  It can provide temporary perfection and sanctuary of mind that is a refuge from the mundane banalities of ordinary existence. Playing often involves rules, but it also includes freedom, imagination, risk unanticipated outcomes, and the possibility for  participants to become deeply immersed in the activity."

From Mark A. Graham's article "How the Teaching Artist Can Change the Dynamics of Teaching and Learning."  Teaching Artist Journal 7.2 (2009): 89.

Family Math Night Fun!

I did a family math night at Christel House Academy in Indianapolis this week and we had a blast!!  This math night extends the dance residency's themes so that anyone can participate from 10 months to 100 years old.  The same principles that run through my residency program show up in the family night as well:

Making is fun.

Patterns are fun.

Choice is fun.

Art is fun.

Dance is fun.

Math is fun.

Challenge is fun.
Learning is fun.

Glue, tape and paper are fun.

The more colorful the better.

The more choice the better.

The following pictures don't represent every activity in the night, but you can probably tell by the focused hands that everyone was engaged in what they were doing.  Fun!

The Math in Your Feet Family Math Night is called "Pizza, Patterns & Problem Solving."  The key ingredients to this night are MAKING and CHOICE.  Sure, everyone is designing a paper pizza that illustrates turn symmetry, but every pizza is different.  



I made up this name pattern activity based on the frequency of vowels and consonants in one's name, but I'm sure it's not an original idea.  It's fun to watch kids discover the larger pattern that emerges from repeating their name pattern over and over. 

This family night is an extension of the Math in Your Feet residency topics.  For example, in the dance residency, kids use flips, slides and turns to transform their dance patterns.  Here they do the same thing with tangram puzzle pieces as they work to make the shadow shapes.  

  

This is a night for the entire family, so there are little babies all the way up to grandparents making and solving next to each other.  It's quite amazing to see the generations working together, all in one room.  I am always heartened to see the adults trying new things.

Mirror symmetry.



Patterns in the feet.







A List

Mathematical aesthetics. 
I had heard about the beauty of mathematics, but I never really understood what that meant.  Now, I think I'm 'getting it'; not with numbers, but definitely on a visual level.  Sue VanHattum from Math Mama Writes posted this video called Doodling in Math Class: Infinity Elephants.  When I watched it something just clicked for me. 

More pictures of tape in action.
I've got this idea in my head to find as many examples of tape being put on the floor (or wherever) to further a child's learning, or to change an environment to promote exploration of space.  Send me yours!

Heading toward the front left diagonal!

Find new music and dance to it. 
Always on the look out for a great tune.

Gestures and embodied cognition in mathematics learning. 
After reading research findings about this, I've been more aware of how people move their hands while talking, especially when they're trying to describe a procedure or a design.  A friend was describing a plaid shirt in her closet; her hands moved across the front of her body horizontally and vertically while she said the word 'plaid'.   

Clear the mind.
Find new music and dance to it.  Better yet, I'm teaching kids next week!  There's nothing like a class of moving fifth graders to keep one in the present moment.

Building an icosahedron by folding paper plates. 

How many paper plates will it take?  I've made a two-frequency tetrahedron so far, which is four plates.  The reason I'm interested is that I've read that Labanotation (a method for notating dance movements and choreography) was created by visualizing the human body inside an icosahedron. 

This is actually an open icosahedron
from http://www.wholemovement.com/

Van Hiele Levels of Geometric Reasoning. 
Is this useful to me as a dance teacher teaching math?  I teach a lot of geometry.  I think the kinds of questioning employed that are intended to help move children from level 0 to level 2 might be helpful.  Need to look into it more.
 
Learn how to teach math with Cuisenaire Rods using daughter as guinea pig.  

I wish my math education had consisted of these, but I can re-learn math as I teach my own daughter.  These unit blocks are great for developing a real sense of what numbers mean, but when when you grow tired of that focus, you can use them in other ways!  I drew a line of symmetry and made up a game where one person puts down a rod, and the other person 'reflects' it on the other side of the line.  In this case my daughter led and I followed, but you could take turns in any number of ways.  Maria Droujkova from Natural Math posted some very interesting, videos of kids using these rods.