More Than Red, Blue, Red, Blue

First, a little context.  I've just finished four days of hands-on workshops for a summer camp of kids ages five to twelve.  Our hour-long, mixed age classes have been full of rhythm and patterns in the feet as well as exploration of other kinds of patterns including ones we can find in nature (Fibonacci numbers, hexagons in bee hives, etc.).   I've also told some stories about squares who are completely bored with their straight edge/sharp corner existence who want and need a change. (See a version of these stories in a recent post called Scissor Stories: Tales of Transformation.)

This kind of summer programming up in the city finds me working with kids whose backgrounds I know nothing about.  I don't know where they came from or where they're going, so I mostly just try and go with the flow, try to meet them every day where they're at instead of where I think we should be. 

Although grant sources require some 'evidence' of learning or growth, my job is to do the best I can in four days.  To the kids I describe our work as using and making and understanding patterns of all kinds, and we spend most of our time doing just that; instead of explicitly talking about patterns, we're just trying to make and use them.  As a result, kids may or may not have absorbed new vocabulary to use when it's time to assess their learning on the final day.  I think my goal is simply that they have a reserve of experience to call on when they're back in school learning math with their paper and pencils.

But, like I said, I still need to ask them questions at the beginning and at the end.  Here are the questions I asked this week.  I'm still not sure that the first question is a helpful one for assessing learning but, as you'll see, it did provide me with some very interesting information:

Question One: "What is a pattern?"

First of four days, summer camp children ages five to twelve: 
Something that is ABB and it keeps going...
Two or more things on paper...putting it on paper over and over.
Square then rectangle, keep going.
Repeats.
Something that gets put together.
Different colors.
Numbers -- 1, 2, 1, 2...

Last day:
Numbers, shapes
Rhythm
Nature
Sequence, repeating.
Heel, heel, toe, toe... (a clogging step)
Circle, circle, square, hexagon (We studied a hexagon as one of our nature's patterns inquiries.)
Take things and put them in order.
You know it's a pattern when it's the same, something that repeats, like red, yellow, red, yellow...

Question Two: How can you make patterns with your feet?

First day:
Different kinds of shoes and socks.
Switch shoes.
1,2,3 on one foot, 1,2,3 on the other foot.
Bang them.
Mix with other people's feet.
Big foot, small foot...
(Alternate) movement.
Switch shoes.
Number of taps.

Last day:
Steps, slides, jump, turn.
(Use directions) left right front back diagonal.
(Put your feet) together, split, crossed.
Big, small, big, small (movements)
Heel, heel, toe, toe (a clogging step)
Sounds
Use toes, heels, kicks.

It's heartening to see that the answers to the second question showed much more understanding after four days of dancing.   As for the first question, it's clear to me that, even after four days of playing around with patterns, the idea of pattern these children (and pretty much every other child I've seen this summer) have internalized is a very narrow conceptualization limited to colors and shapes that repeat in a linear way one after another, almost always "on paper." 

Driving home today (it's over an hour each way) I had a lot of time to think.  I was thinking of all the ways we can harness kids' love of doing and making in the elementary years to the goal of engaging in an deep and meaningful exploration of patterns of all kinds.  And that, in the the process of this kind of exploration, kids would get a chance to represent and experiment with this oft perceived 'simple' concept in a multitude of ways: 2D on the page (so many ways to do this), 3D with their bodies as well as all the great math building materials out there, and even 4D using time and rhythm. 

While I drove I also had a lot of questions in my head: How can kids learn to see patterns in numbers if all they know is 'red, blue, red, blue'?  How can they understand what patterns are if they don't have personal experience with constructing them, taking them apart again to explore the pieces, and transforming them into something completely different?

I am certain it is possible to provide deep, meaningful, artistic, open ended explorations of patterns at the elementary level.  We do it all the time in Math in Your Feet with percussive dance; I am starting to understand how I might move this approach toward other mediums.  (To see a really nice curated collection of math art, go to the Math Munch blog.)

I have a lot more thinking and learning to do on this topic, but for now I'm clear on one thing:

Almost every one of the 180 kids I encountered this summer, no matter their age or their dancing ability, were unable to identify or describe patterns outside the standard textbook context.  I think they can handle more.  Not only that, I think they want more. 

A Big Discovery

The kid and I were playing around with tangrams the other day and -- all of a sudden -- I had an Aha! moment.  It was a chain of connected thoughts and realizations which provided clarity around an answer I've been hunting down for the last fifteen months.  Yeah, I know, this big!  Here's how played out:

I recently read a post by Shelisa at Think Magnet which I found at the Love2Learn2Day Math Monday Blog Hop.  The post was about a brilliant game Shelisa plays with her kids every time the family sits down to a meal. 

I was completely impressed with the game because not only did it grow out of a spontaneous observation by one of her kids (my favorite kind of learning) but she was then inspired and able to turn the moment into a wonderfully complex and endlessly interesting activity.

I can't really do justice describing the game here; you'll have to go see for yourself and you'll be glad you did!  But, the upshot is that the game is all about attributes and learning how to discern differences and similarities.  It is also about just how inspiring a playful approach to learning can really be. 

For the most part, I've been following my own kid's interest and questioning related to math over the last six months, an inquiry which has focused on measurement, maps, number patterns, and geometry discoveries.  Reading Shelisa's post I started wondering how I might incorporate it more directly into our own math explorations.

So there I was, later that day, mulling all this over at a semi-conscious level in the back of my brain while sitting on the kitchen floor with my daughter.  We had just re-discovered a whole box of tangrams, with four or five puzzles all mixed together. 

"Look Mama!" the kid exclaimed, "I made a parallelogram out of triangles!  And I can use two more triangles to make it into a rectangle!" 

At the time I was playing around with building squares and I decided to start building onto her rectangle.

"Oh, look!" I said, "I can make a big square out of two triangles.  And I can make a second square out of four smaller squares.  If we make a third square and add it on, we'll have made your rectangle even larger!  How can we make the third square differently from the first two?"

And, in my mind, I said: We can play around with position, size and color.  Adding attributes like these increases complexity and interest...

And that's when it hit me.  How many ways can you make...?  That's the question I had been asking the kid about the number twenty or the number ten.  And here I was asking a similar question (How can you make the third square different from the first two?) about combinations of shapes, and colors, and sizes. 

Attributes and combinations. 

I ask this type of question in Math in Your Feet too: Which of these movement variables will you choose as you create your four beat pattern?  How can you make your second pattern different from your first?  To help them find their 'answer', students use a tool I created called Jump Patterns to choreograph their own foot-based percussive dance patterns.  Jump Patterns are created by working with the elements of a percussive dance step (the related math term being 'pattern unit').  Take a look at the chart I use in the classroom:

Students use this chart to experiment with what I have always called Movement Variables, except now I realize that, in the math sense, they're a collection of....attributes!

For the classic set of attribute blocks it's five shapes, three colors, two thicknesses and two sizes.

In Math in Your Feet kids choose from five foot positions, six types of movement, and six directions to create each of the four beats in their Pattern A, and then they make choices again when they choreograph Pattern B.  There are two other categories to work with as well, including tempo and starting position. 

Here's why it's a big deal:

The math topics in Math in Your Feet have always been clear to me: concrete, kinesthetic experience with spatial reasoning, combinations, congruence, transformation, reflection, and rotation.  These are the things kids and their teachers know are being taught.  They've always been the 'selling points' of the program because they are easily recognized and understood as 'math'.  

But, after years of  working out connections between these topics and percussive dance I knew there was more going on than simply learning 'about' math.  My hunch was that there was some real mathematical thinking going on, I just needed to figure out where it was occurring and how to explain it.

This hunch was one reason I embarked on my current math quest (chronicled in this blog over the last fifteen months).  I've been working hard to figure out what it means to 'do' math.  And now, my new understanding about the nature and purpose of identifying attributes in elementary math education has provided at least part of the answer:

When children are able to identify the elements (attributes) that they have used in their creative dance work they are thinking mathematically.  When they use this understanding to analyze and critique others' creative work they are thinking mathematically.  Attributes are 'charactaristics of an object or a shape' and I'm confident that this definition is broad enough to include moving pattern units as well.

I am struggling at the moment to articulate my thoughts about the differences between identifying attributes in a traditional math education setting vs. choosing attributes in a creative or design process.  But, no worries!  After more than a year of asking, surfing, conversing, reading and learning, I feel like I have won some kind of prize!  And, thanks so much to everyone who is sharing this journey with me.

p.s. If you want to learn more about Jump Patterns, here's a link to an article I wrote that details the whole story of what they are, how I developed the tool, and how they are used in the classroom.

Teaching Math in Your Feet...Without Me!

The program Math in Your Feet was developed as a five-day artist-based residency, led by me, the percussive dancer.  I created the program in collaboration with Jane Cooney, an elementary math specialist in Indianapolis, IN.  I hammered out the dance/math integration and dance class activities through sheer repetition, trial, and error in classroom after classroom.  In 2006 I was lucky enough to also have the opportunity to develop a teacher workshop through association with the Kennedy Center and Clowes Memorial Hall at Butler University.  I've been doing both student residencies and teacher workshops ever since. 

Meg Mahoney, an elementary dance specialist in Seattle, read the article I wrote about the development of Jump Patterns as a teaching tool in my program Math in Your Feet (published in the Teaching Artist Journal, April 2011).  As a big fan of her work I was thrilled when she told me she was going to try it out with her own fourth and fifth grade students!  Even more amazing, she has never seen the program in person or on video.  True, she is a fabulous dance teacher, having worked for fifteen years in an academic setting, but still, it take a lot of guts to commit six weeks of your school year to something brand new like this. 

In a recent post on her blog, Meg says:

The article unwraps the dance/math residencies Malke Rosenfeld teaches in public schools. The fact that she shares her methodologies with classroom teachers for use in the classroom lit a spark for me. Even without being a step dancer myself, maybe I could lead my dance students through the jump pattern curriculum!

Apparently she was quite successful!  Read on...

I’m about to begin Week 5 of 6, finishing the jump patterns with my second set of 4th & 5th graders (six lessons per group), and we’re all enjoying it. Malke’s outline provided lots of material to work with, and I’ve worked the pacing & focus of instruction for each lesson to fit my ELL learners & my circumstance. The movement variables are broken into malleable chunks, and we’ve explored the math-related concepts of precision, congruency, reflection, and turn symmetry, with students choreographing patterns in teams of 2 and 3. In addition to integrating dance & math, there’s a problem-solving (choreography) component that parallels the "workshop/conferencing” structure that my students are familiar with through Writers Workshop, allowing me time to confer with & jump-start individual students. In addition, there’s a spatial arrangement that supports classroom management (personal dance spaces for each team — wow, what a concept!). Add in some dance videos to “mentor” the kids in their choreographic process & journaling questions to provide feedback on what students are learning, and it’s no wonder we’re all engaged!

A perfect Math in Your Feet moment, courtesy of Meg's blog.  These boys are in the middle of a 270 degree turn.  Nice!

My first groups of 4th & 5th graders finished the unit before the holiday break, with some of them performing their patterns, both congruently & in mirror symmetry. They nailed the precision steps they’d created, even without the support of their personal dance spaces, and their peer audience was able to talk about what they were seeing with insight and new vocabulary. What a pleasure to watch…

In this description of how how she used the Math in Your Feet curriculum, Meg shows us all that the program is flexible enough to be adapted by teachers in a number of key ways:

- Easily reproducible outside the artist-residency setting 
- Engages and inspires both learners and teachers to learn and grow in exciting new ways
- Creates opportunity for new insights into topics and practices in both math and dance
- Encourages fluency with new math and dance vocabulary, in context
- Allows exploration of, and play with, math concepts in a dynamic, physical, choreographic process
- Adaptable to ELL learners
- Adaptable to the particular circumstances/expectations of a school and/or district

I'm continuing to work away at producing the Math in Your Feet curriculum guide and instructional DVD so that any elementary classroom teacher, or PE, dance or music specialist can reproduce the same energy and engagement that an artist-based residency provides.  In a way, the Math in Your Feet program is poised to be more effective in this new form because, as a teaching artist, I am just a short-term visitor who really knows very little about the individual learning needs and goals of each student, let alone the specific circumstances and culture of individual schools and districts.

Based on Meg's success, and the successes of teachers who have taken my professional development workshops to learn more in-depth about the methods and content of this program, it is not hard to imagine hundreds, maybe thousands, more children jumping (and sliding, stepping, turning, hopping...) their way through math class led by their very own teachers!

Marveling at Moving Patterns (Video)

I took my daughter to the Nutcracker Ballet this afternoon at Indiana University.  We had a great vantage point from up in the balcony, perfect for a six year old actually.  We could see the entire audience and into the orchestra pit, usually hidden from view.  We had a great view of the stage.  And, it was the perfect place to take in the big picture of moving bodies in space.

Personally, I enjoyed the corps de ballet pieces the best.  I'm usually very sensitive to timing and phrasing, but every group piece was so well performed that I was just able to relax and take in the moment. 

I had a thought, while watching, about how the lines and movement through space (both in the bodies and around the stage) were made more understandable, and beautiful, because of the amplification of the patterns.  By this I mean, sixteen dancers on stage dancing the same choreography highlighted the patterns and rhythms in a way that a solos or a pas de deux does not. 

I also thought about how the real meaning to be found in patterns is in the change and movement between one moment and the next.  We often think of patterns as fixed moments in time, but even visual artists know that without a sense of movement on the paper, the patterns lose meaning.  At this point in my inquiry into such topics I know enough to say confidently that math, science, social sciences, history, literature, and arts of all kinds ALL assign some value to what happens between Point A and Point B.

That is what we do in Math in Your Feet. We move from Point A to Point B.  We figure out how we're going to get there, and which way we're going to turn.  We connect the four individual pieces of time to make a larger whole and once we've got the flow of that, we then find a way to connect our patterns together -- where does one end and the other begin?  These are the questions of mathematicians and scientists and artists and philosophers as expressed through the mind and body of a typical fourth grader.

As we sloshed home through dark late afternoon rain I suddenly remembered seeing a video almost a year ago of micro-origami unfolding in water; they have a very fractal-like quality.  I recently watched Between the Folds, a documentary about origami, and was moved to tears at the depth of meaning inherent in the process of folding.  Since we think of origami as a fixed and finished object we often don't observe or think about what happens between a flat, uncut square and the final 3D object. 

This video of micro-origami, below, will show you, in reverse, the movement, order, folds and structure used to create each piece. 

The original silent video of Etienne Cliquet's Flottille (2011) is here but I chose this (shorter) video that was presented with music, to share with you.  Enjoy!

So Many Levels, So Little Time

I've been asked to do an interview about Math in Your Feet for what seems to me to be a big deal blog.  I'm excited!  But in the planning of it, the writer asked to see my program in action.  Not having any video to share at the moment (I know, I know...I'm working on it!) he suggested rounding up some kids and video taping me working with them as a supplement to the written post.  Here's my response:

"With a few kids you would get to see what the dance looks like, but it would be important to keep in mind some other things that can't be observed unless it's a larger group.  This program is multi-layered in its approach to learning, working on social/emotional challenges (collaborating with a partner, independent learning and creating, no one 'right' answer), physical challenges (coordination and beat competency), intellectual challenges (pattern recognition, permutations, combinations, transformations), etc.  The program is also sequential, and the math and dance build on each other as the program progresses; the real mathematical thinking is not as obvious in the beginning stages when the kids are still learning how to create their Jump Patterns.  So, we might need to talk more about the different layers of math within the program after watching the basics."

It's so worth it to go deep, but these days, it's easy for schools to say, "That's nice, but we don't have time for that." or, "We'll let them have some fun, but then it's time to go back to math class." 

I say, if you can go this deep and cover so much ground in just five days (although you know I'd love more time with the kiddos) why not jump in with both feet and a smile? 

Give This a Try!

A student's illustration of his Jump Pattern A, at his own volition, no prompting.

Start in center.  Feet together.  Assume all four beats move on a Jump.

The first two beats work just fine as written.

In Beat 3 you are still facing forward, right and left foot are not labled, but the arrows give you a clue.

How far do you turn on Beat 4?

Can you do this pattern backwards (a reflection of the beat order, e.g. 4, 3, 2, 1)?

How else can you rearrange these four beats to make a different pattern?  Did it work?

Which combination of beats/moves did you like the best?

Jump Patterns: The Full Story!

I'm am thrilled to announce that my article, Jump Patterns: Percussive Dance and the Path to Math, has just been published in the Teaching Artist Journal, the only peer reviewed publication for my (still very new) profession.

The article focuses on how my approach to teaching percussive dance led me to search for, and find, relevant math connections.  I outline how I developed and formalized 'Jump Patterns,' a teaching tool which illustrates the elements of percussive dance (similar to how creative movement teaches the elements of other formal dance styles like ballet or modern dance).  I also describe how this tool has helped children become creative in my art form and make many meaningful connections to math topics at the same time.

Although I have shared much about my work in this blog over the last five months, the newly-published article is the first time I have provided a comprehensive, detailed description of Jump Patterns and how they are used in the classroom.  The article can be found here. 

I am very interested to hear your thoughts, observations, and questions about my approach as well as any similarities to other approaches out there.