Multiplication Models Poster

Even though I don't do anything with multiplication in my program, I still thought this multiplication models poster from Natural Math was too good (not to mention too beautiful) to not share!

Update August 2013: This poster is available for purchase here. :-)

Looking Forward...to the NCTM Annual Meeting!

The 2011 Annual Meeting of the NCTM is coming up in about three weeks, and it happens to be in Indianapolis, IN this year, about an hour from my home. 

I'm starting to get excited!  There are many things to look forward to including (but not limited to):

• the possibility of meeting Keith Devlin, NPR's Math Guy
• the possibility of meeting Scott Kim (one of the MathDance guys)

Both Keith and Scott are presenting.  Also:

• the possibility of meeting other really interesting people, still to be discovered but no less exciting
• the sheer volume of workshops on interesting arts connections to teaching math

And, last but not least:

• a chance to share my own work, and to give teachers some concrete experience with integrating percussive dance and elementary mathematics as well as tools and lesson plans that they can take right back to their classrooms after the conference is over!  And all in 90 minutes.  

You gonna' be there?

Learning to Listen

Are kids smarter than they were five years ago, or is it just that I've stopped looking for the 'right' answer?

Are my students more perceptive than they were, or is it just that I am more receptive to what they have to say, even if it's not 'time' for it?

Are fourth graders smarter than they used to be, or is it just that I am less interested in 'moving the lesson forward' and more interested in listening to their reasoning?

Are kids understanding the program better now or is it just that I'm finally really hearing what they have to say?

Have they always had insightful observations, or is it just that I am more open to multiple interpretations of the same event?

I think we know where this is going.  Kids have always been smart and I haven't always known what to listen for.  It's been really exciting this week to hear my students' observations, thoughts and insights.  I am, for the first time, really enjoying the process of teaching this program. 

Divided Loyalties?

In the process of exploring the intersections between math and dance I find that I seem to camp out in one subject for a while, and then pack up and move on back to the other camp.  Frankly, it leaves me feeling a bit fractured, which is interesting since I have declared in the past that in Math in Your Feet the two subjects come together 'seamlessly.' 

Maybe for the kids, yes, but not for me.  So let's focus on me for a moment.

This issue shows up in my teaching.  When I am thinking too closely about the math, my feet don't move in class.  They don't talk.  I forget to use the one great attention getter I have that nobody else does -- my tap shoes!  I forget to start our sessions with the art and instead hurry forward to the math. 

My most recently completed residency at Christel House Academy has made me think of this.  The residency was part of a pilot project by Young Audiences to evaluate the structure of their new Signature Core Services and the elements that interact to create a quality arts learning experience.  They wanted to know if and how these elements were brought out in this particular arts-based programming.  But me? I was worried about how I was teaching the math and was surprised that they didn't even care about that (for the purposes of the pilot).  That shows you where my mind is these days, doesn't it? 

I've also spent a lot of time overly focused on the dance (mostly in the first few years of developing the program) but these days?  These days, as I teach, I have mostly been taking the dance for granted.  I am quite surprised and somewhat alarmed to realize that I've been seeing dance as a means to an end.  I'm horrified, actually! This is not really supposed to happen in an integration and I'm really supposed to be on the side of the dancing right?  Yes, we're dancing and moving our bodies at least two thirds of each workshop, but I am realizing now just how much my mind has been over at the math camp.

I think this might be because I've been spending much of my creative work time this fall focused on learning about math.  I've been trying to figure out what math means:  What is math really?  What other kinds of math are we doing in this program besides the ones I know about already?  What kinds of conversations are these kids having about the math while they're dancing?  The learning I've engaged in, I believe, has been a full out creative process, one full of questions, dead ends, frustration, problem solving (What don't I know?  Where do I need to go to find out?  What sorts of questions do I need to ask?  Am I asking the right questions?) and 'aha!' moments. 

My intellectual explorations have affected my teaching in the past as well.  Earlier in the development of Math in Your Feet, when I was still trying to figure out how the dance and math fit together I was deeply engaged in exploring the integration aspect of the program and  my teaching at the time reflected this.  Although I did have questions about the value of teaching the integration (and still do; mostly I think it's a good tactic) I was working it out in the classroom none-the-less.

So here's a question: If the basic flow of the residency content stays the same, does it matter if my interests bias our focus, even if slightly?  Does it matter if I'm over in the math camp while teaching a dance-based program?

Maybe my loyalties to both camps are like siblings vying for their mother's attention.  I am essentially acting as the content specialist for both camps simultaneously and it's hard to pay attention to both equally at the same time. 

Maybe I embody the reality of integration and the inherent tensions therein?  It's interesting to reflect on this question because, since it's just me responsible for both content areas, we can't blame the tension on personality conflicts between teaching artist and classroom teacher.

Maybe none of this matters because one's best teaching comes when you're enthusiastic, engaged and truly interested in the subject matter. 

Maybe it doesn't matter which camp I'm in on any given day or year because the truth is the kids are moving, they're dancing, they're thinking intensely with their bodies and they're learning a whole lot of math, all at the same time.

I've been told on numerous occasions that I think too much, but who was it that said an 'unexamined life is not worth living'?  (I just Googled it, it was Socrates.)  I think it is important for me to ask questions about where my loyalties lie even if, in the end, it doesn't matter.  I'm a process person, I live in the moments between.  And, I know I'm a better teacher when I'm inspired, whether it be by math or dance or reading Carla Hannaford's book Smart Moves like I did in the Fall of 2009.  Hopefully my dance friends will put up with my math obsessions for a while.  I'll be back soon!

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!

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.

Making Things

Curried red lentil soup with sweet potatoes and greens.

I know a lot of people who make things -- dinner, for example.  I know people who grow their own food, make their own music, build their own rock walls, design and build their own houses, decorate their homes with their own artwork, sew their own quilts, and bake their own bread, all as a matter of course.  I know people who knit their own socks!

Then again, when I go into schools, I meet legions of children who never even considered the possibility that they could make something, let alone their own percussive dance patterns.  Here the thing, though -- most kids, even if they don't regularly engage in making things, get excited when it's their turn to make their own dance patterns in class.  It's amazing and energizing to watch, honestly.  It's like I'm giving them permission to take their rightful place in the human race.

I have been reading back issues of the Teaching Artist Journal and finding common themes emerging in what I read.  One of the things that comes forward is that many Teaching Artists (like myself) teach with less of a focus on the technical aspects of their art and put more emphasis on helping their students make things within the structure of an artistic practice and process.  When a Teaching Artist takes their art into the classroom (or community center, or after-school program, or a nursing home, or wherever people are)  they work with people who may have never taken a dance class or made music or used paint in a meaningful way in their entire lives.  They usually have limited amounts of time to work with a group of students and so they become masters at helping people make personally relevant art with a limited vocabulary of skills.  

This is what I do in Math in Your Feet.  In an upcoming article to be published in April 2011 in the Teaching Artist Journal, I go into detail about a teaching tool I created which I call 'Jump Patterns.'  Essentially, Jump Patterns are to percussive dance what creative movement is to ballet or modern dance.  Jump Patterns reflect the elements of percussive dance and allow children to be creative and successful in this particular dance style, but without great need for technical development. 



Don't get me wrong, skill development is important, but sometimes a focus on technical issues creates a huge obstacle to experiencing the incredible joy and importance of just making...things.  Art.  Ideas.  Music.  Shoes out of paper plates.  Even having this particular discussion is part of the cultural baggage we carry, a particular mindset that gets activated and reactivated whenever we think of art solely as something to be 'good' at  instead of an opportunity to make things.  And that's a pity, because I believe that making things is one of the most meaningful activities in which a person can engage. 

An 'old fashioned dress' the creation
of which was facilitated by safety pins and
some adult modeling.

Right about now you might be asking yourself, 'Just how can someone make something without skills?'  The answer is facilitated making.  Kids are like everyone else, you do have to give them something to work with before they can begin to create.  But, in the end, all you really have to do is help kids explore the medium, find a way for them to  investigate the processes of the art form, help them formulate some questions or just wait until one is asked, and, when it's time, add instruction on specific skills when the student is ready for them.

Here are some examples of what I mean by facilitated making:

Example I:  In Math in Your Feet, Jump Patterns are the tool that facilitate the making of foot-based dance patterns.  Once a kid learns three of the four elements of percussive dance (leaving out the most technical aspect, parts of the feet) they are then ready to experiment and create their work.  They become participants in the traditional aesthetic -- everyone has something to offer, everyone has their own style or take on the dance, everyone is welcome.  Not everyone is good but, more importantly, everyone participates.  Without participation there is no form.

Example II: A Teaching Artist friend of mine works with kids who, more often that not, have no formal music training.  When he sets up digital recording studios in their schools they are able to get immediate feedback on their own ideas and find ways to evaluate and adjust their ideas to make real, and often interesting, music and lyrics in a collaborative way with their friends.  He spends parts of each class talking about the science of sound or new techniques for using the equipment but mostly the kids experiment on their own and come to him if they have questions.  In this case, the facilitation is the TA's approach to using the equipment with his students. 

Mermaid's Tail

Example III: My daughter, who is five, has always had many ideas and many of them get expressed through her art supplies.  When she was two, she wanted paper cut into shapes so she could make something with them.  She had an idea but still did not yet have the skill to cut the paper the way she wanted.  Did I say, "Sorry darlin'.  You'll have to wait until your hand muscles are stronger and more coordinated before you can get that idea out of your head"?  Nope.  I cut the shapes for her and she did the rest.  Did it change the fact that it was her artwork?  Nope.  A few years later, when she was a few months shy of her fourth birthday, she had an idea for a mermaid's tail.  By then, facilitating her making process simply consisted of making sure the tape was secure so all those little pieces stayed together.

Making things is not rocket science...well, unless you're trying to build a rocket.  But what comes before that rocket?   All the times when kids get a chance to ask questions, experiment with materials, find their own answers and get some help and guidance from adults along the way.  We can sometimes get wrapped up in giving kids what we think they need to learn, but it's in the doing where the meaning is made. 

And Now? A Standards Antidote

Okay, here's my special challenge (and I take comfort in knowing that out there are other people who understand exactly what I am about to say).  I am an artist.  I am a teacher.  I am a teaching artist, curriculum developer, website designer, market research specialist, performer, agent, writer, editor, math student, calf wrangler...I am going insane!  Well, not really, but I do feel like I exist in multiple worlds simultaneously, sometimes to the point where I can completely contradict myself and agree with myself in the same moment!

Today I was all over those Common Core State Standards I wrote about yesterday.  I just spent the better part of the afternoon working with the new language.  I'm not making anything new up, just translating what I do, what I've always done, into the language of the most current stated education goals.  I've also been working on a grant project for the second pilot stage of Young Audiences, Inc. Signature Core Services program, which has meant working with new forms and terminology.

Along parallel lines, I have been spending the better part of my after-the-kid's-in-bed time systematically reading all my back issues of the Teaching Artist Journal (I'm into year five at this point).  I started this particular project in December because I figured that, as a recently appointed Associate Editor for this publication, I should probably have a working knowledge of what kinds of articles, topics, research etc. have been published in the past before I move forward with my new position.

Re-reading TAJ's has actually been really interesting and inspiring, and much less an exercise than my husband thought it might be.  It's been interesting because the process of reading them one after another is bringing me the big picture I never had getting each new TAJ one at a time, three months apart.   I'm learning a lot!  And, it's been inspiring, because I've been running into really great thoughts like this one (below), which balance out all my standards-based wonkyness of late.  So, without further ado, here's my standards antidote for today:

"Although making art does encourage curiosity, critical thinking, and empathy -- necessary tools in today's world -- these very qualities may also demand that one speak truth to power or insist on beauty.  Nourishing imagination may inspire one to declare along with Blake: Everything that lives is holy.  Such a vision might make one less suited for production lines, prison cells, or political speech.

"So how do we, as human beings and as artists, speak honestly about art?  I'm glad that we talk about the connection between art-integration and academic progress, art's value in passing down community wisdom, and the transferable skills that making art develops.  But I hope we also remember to say that making art is a voyage into the unknown and, therefore, not a process inherently practical, polite, predictable, or proper."

From Judith Tannebaum's article"Oh How We Sparkled: One Vision, Two Themes."  Teaching Artist Journal 4.4 (2006): 248-9.

Residency Reflection

I started this post a few weeks ago.  I guess I needed some time, space and perspective to finally finish it.  Here you go: 

I just finished a big week with four classes of fourth graders.  Their teachers told me they were a tough group on the whole, but of course it's only through experience that I figured out what that meant, exactly. 

I've had tough groups before, but these kids confounded me, actually.  There were bright spots in each class, and there was also some good thinking and creating, when it happened.  But, on the whole, they could barely stay in their squares.  They would forget any verbal instructions I gave.  I had to stay within two feet proximity to get any attention at all.  Most of them could not consistently find the center of their dance spaces.  They took every gentle reminder as a power struggle.  Transitions such as moving from sitting to standing or dancing to not dancing with control was a herculean effort for them.  

Don't get me wrong.  These were not 'bad' kids, just kids with a lot of challenges.  I've spent a lot of time over the years figuring out how to help kids control their bodies while moving and then again when the moving is done.  I've also spent many years helping kids become more aware of where they are in space and where in space they are intending to go, with good results.  But this particular week, none of my strategies seemed to work.

I often compare my week-long residency to the first week of school.  By the end of our time together I've figured out exactly what each particular class needs to be successful, but by then it's time for me to go on to the next school, or the next group of kids.  Not really understanding who I'm teaching and what they need as individuals and as a class is my least favorite part of the job.  If I had the week to do over, this is what I'd do:

• 30 minute class length, or less.  I am contracted to implement my program within certain parameters including the amount of time each class has with me (usually 60 minutes at a time).  Classroom teachers who implement Math in Your Feet in their own classrooms have the flexibility to work on it in ten minute chunks or whatever length they choose.  My 4th graders this week had to hold it together for a whole hour with me each day, which was a loosing proposition. 
• Take it out of the gym and find a more enclosed space.  The gym for this particular week was not a bad gym in terms of sound issues, but it was a gym none-the-less.  I never use the whole gym and always define the limits of our dance space (with tape!), but if we could have put walls around us to enclose the space somehow, it might have helped the kids feel safer and may have helped the focus issues. 
• Reduce transitions.  It's a hard reality, but we have to get up and move, and then we have to sit down and listen.  We do both in short bursts, which these kids needed, so I'm not sure exactly how I would have done this.
• Dance for them more to keep them focused on the reason for why we were together. 
• Teach one two-person team at a time. The only times they were really quiet, focused and, well, not argumentative and actually helpful, was when a team was up in front of the class showing work in progress or demonstrating a math concept through their dancing.  If I could do this over I'd take the plunge and just teach one team at a time while the others watched.  I think we would have gotten a lot more acomplished this way.
• Acknowledge the issue of 'first reaction' sooner and with all the classes.  I think I got this idea from a parenting book I read in the the last year.  Essentially, each one of us reacts in our own way to new situations.  We have either a negative first reaction or a positive first reaction.  Knowing how we operate in this regard can help us overcome the challenges each present.  For those with a negative first reaction (hold back, regard with suspicion, decline to participate, or exhibit unhelpful behavior) it just means making sure that they know that I understand they're not into it at the moment and that's okay with me.  At that point I usually outline what my minimum expectations are for participation-- stand when the rest of the class is standing and sit when the rest of the class is sitting, and that's it.  I've had success in the past acknowledging the reality that kids may not love what we're doing as much as me, but I still expect them to give it a shot.  By the time I realized what I was working with, it was too late in the week for this strategy to have much effect.   That usually gets us over the hump, but not this week. 

So, all in all, we did get through the week.  I suppose the bright side of all this is that I'm a little more ready the next time I encounter a group like this.  And, I'm still a big supporter of using dance and movement in integrated ways for all children.  Just because it didn't work for them the way I wanted this time doesn't mean that it can't.  It just needed to be different.  Too bad my time machine's in the shop, or I'd try again.

Like I said at the beginning, there were some bright spots.  This is what one girl wrote in response to a reflection journal prompt:

"Ten years from now I will remember how hard it was to learn and how much fun it was.  The challenges I had were putting pattern A and pattern B together.  The thing that helped me work through them were because I had D. as my partner and he was always hard working and an on task friendly boy. :)  I loved this program."

Featured NCTM Workshop!

Hey, check this out!  My workshop at the NCTM 2011 Annual Meeting and Exposition is featured in their promotion of hands-on workshops at the conference this coming April.  I see a few there I'd like to attend myself.  Maybe next year I can do one on tape as the ultimate math manipulative?  Big talk, I know, but I'm on a mission to prove my theory.

Will you be there?

Link: Resources for Integrating Dance with Curriculum

Hey y'all!  Look what I found this morning -- a beautiful and thoughtful blog from a dance specialist in the Seattle Public Schools. 

I really wanted to share because this is the real deal.  I don't know for sure, but I think many of my readers are curious about how dance and/or movement fit together with math, and maybe other subjects as well.  There's a lot to sort through out there in web-world, so I was excited to find this. 

Here's a great link from her blog with a list of resources for integrating dance with curriculum.  You might also like the big picture of her work here.

On Being A Teaching Artist

My dad reads this blog; he's been following it from the start.  It's something you'd expect from a dad, I suppose, but my father is also an artist.  He grew up with his parents and two older brothers in the back of little grocery store in Chicago in the 1930's.  This little room, where they all lived, was the subject of some of his first paintings.  In the 1950's he went to the Art Institute of Chicago to focus on oil painting, and later he moved to graphic design when that still meant lots of brown paper, dial-a-type, darkrooms, and wax rollers.  Ah, the toys of my childhood!  Now he's put his away his brushes and instead works with fabric, making quilts that are his paintings. 

He recently e-mailed. "I read your blog," he said "and really don't pretend to understand half of it since I base my art on intuition and instinct, allowing the sub-conscious to take over when faced with a visual problem." 

I love his description of his creative process because it is exactly what art is about!  Giving way and letting the mysterious parts of your brain mull something over until one day, the solution seems to come to you out of the blue.  Art and creativity (no matter the form) are about deep thinking with parts of yourself that have no words, just images, feelings, urges, emotions.  Even some writers, I bet, are not thinking entirely in words but possibly visually as well when they imagine the stories and ideas they are trying to put on paper. 

My father probably knows more about his own creative process than he thinks, but he brings up a good point.  He's an artistic person so why doesn't what I have to say here in this blog make sense to him? 

Maybe it's because I am engaging in a second creative pursuit called 'being a teaching artist'.  Not only do I pursue my own artistic and creative visions, but I also focus on how to teach my art in all it's complexities, in a way that I hope makes sense to young learners in academic settings.  I am an interpreter and a guide, using words to make my process clear to others. 

Being a guide to my art form requires me to have perspective on percussive dance as a whole (different styles and traditions, technique, and history) and at the same time be able to articulate to others about the specifics of my creative process as well as my intent as an artist.  For example, clogging is often taught in schools with the focus of teaching a traditional American art form.  Often, traditional artists will tell stories about someones grandfather who passed down a step to his granddaughter, or tales of which step came from whom, or how so-and-so took that step and turned it into something else, or an old square dance that is from a certain part of Kentucky that is still done today.  I did not grow up within a traditional dance and music community. I came to these wonderful traditions as an outsider; those stories are not my stories and  I don't feel like my teaching would be authentic if I brought clogging or step dance to kids that way.  So, I've decided to teach clogging a little differently.  The stories I tell kids through my teaching are more about how I hear the music, how I make choices about what sounds to make with my feet or how I go about making up my steps, and the thrill of taking part in music made by people who are in the same room, playing together.

That is my voice.  The beauty of any and all kinds of art is that there are an infinite number of ways to find a space for yourself and your individual voice.  But if you're a teaching artist you really must step back and reflect about how you use your voice, you must reflect on your own process, because not only are you are teaching about your art you are teaching about yourself as well.  The most important aspect of my job may very well be to ask the questions: Who am I?  What do I believe?  How do I see my art form?  What is my approach?  What is important about my work?  And then, take those answers into the classroom.

Not everyone wants to do this kind of work to make their process visible to others.  I'll admit, it is difficult to switch back and forth between multiple mind-spaces.  The artist part of me is the non-verbal, sensing part that experiences, creates, questions, experiments, listens, responds -- all of this in the moment.  The teaching artist part of me is the interpreter, picking just the right words to frame my work and create lessons that build skill, understanding and connections.  I also have a third role to fill -- the teacher as an artist.  In the end, I have young souls in front of me, many of them unsure of what we're about to do.  I know there is value and relevance in my work in relation their lives, but they don't know it...yet.  Every group of kids I work with is different in their temperament, interest, and school culture.  To really reach them I have to use, as my dad said, "...intuition and instinct, allowing the sub-conscious to take over" when faced with the challenge of real kids in real time.  It seems that what I do in the process of teaching dance is similar to my father's efforts to solve a 'visual problem' and that means he and I have more in common than he thinks. 

As always, I'd love to hear your thoughts!

The Power of Limits 4: A Jazz Metaphor in Mathematics Education

Stephen Nachmanovitch, in his book Free Play: The Power of Improvisation in Life and the Arts, includes a chapter titled 'The Power of Limits.'  This is the fourth in a series of posts inspired by this chapter, exploring how limits not only enhance creative problem solving but are actually a requirement of such a process.  

Today, however, my focus is on the idea about how 'playing outside' an established structure (e.g. a limit) is a way of exploring the limitations themselves, a process which can often lead to new innovations, strategies and ideas.  The article Playing outside: An introduction to the jazz metaphor in mathematics education is the inspiration for this post.

Stay with me here...

After debuting my blog in mid-October, an arts friend sent me a copy of an article, as referenced above, from the Australian Senior Mathematics Journal 18[2] authored by Jim Neyland, Victoria University, Wellington, New Zealand.

I read it and promptly penned (typed, actually) the post Building a Bridge which starts out "I am not a math teacher."  Since then I've been revising that particular view, but that's really the subject of a future post.

Today I picked up the article again because I needed something to read while the kid was jumping over alligators in dance class and found some very interesting ideas!  This excerpt from the article succinctly pulls together two of this blog's main topics -- connections between the arts (specifically music and dance) and mathematics education:

"It is becoming evident that experienced [math] teachers operate in a way that is better described by what is called a 'complexity' model, a radical alternative to the linear model...The teacher's role in the complexity model is that of an artist; but not any kind of artist.  The teacher is not the sort of artist that turns lumps of clay into pottery, or a blank canvas into a painting.  He or she is an improvisational artist who participates in the process of emergence, but in a special way.  The improvisational teacher uses an 'attractor' -- that is a technical term used by mathematicians when referring to the way some chaotic systems eventually settle to an emergent order, and in teaching can be taken to mean what is called a 'rich mathematical activity'...and watches what happens when the students engage with it." 

I read "rich mathematical activity" as something similar to a rich, productive creative process, one where ideas are experimented with, dropped, added, and revised in the process of becoming a fluent thinker within a discipline. I think that happens in my classes when children, after learning the basic vocabulary of percussive dance, are given the freedom to experiment and reach a level of comfort and fluency within a relatively short time frame of five days.

The author goes on to say,

"Much of what I have been referring to about the complexity model is evident in the way jazz improvisation occurs [...] In jazz, 'playing outside' refers to a radical form of improvisation that deliberately transcends the established [musical] structure [...] Why is playing outside done at all?  It creates a high degree of tension.  It is a way of exploring the limitations of the established structure.  It is a way of keeping the structure secondary to creative improvisation...Playing outside, to put it differently, is playing with [emphasis mine] the structure, not within it as happens in normal improvisation.  As such, playing outside is essential in the study of mathematics."

In reference to my own creative process of developing the program Math in Your Feet, I'm pretty sure I was 'playing outside' the traditional view of my dance form as I asked questions and, over the course of a couple years, experimented with the best fit between a rhythm-based dance form and elementary math topics.  There was actually quite a bit of tension as I did this -- a push and pull around what my role was as the artist in a math context, what kind of ratio of dance to math within my lessons, and whether or not what I was doing was 'real math' or not, etc.  In the end, I think I remained true to the spirit and the structure of the traditional dance forms I know and love, but the context for exploring the dance changed.

Also, because of the limited amount of time I have with kids, I'm not sure if I really 'play outside' the structure of what I am trying to teach; at the very least I am 'playing inside' (improvising) the structure of my lesson plans as I respond to the varied skills, experiences, and motivations of my students.  In terms of math education, however, the students' time with me may very well be the first time they have had the opportunity to 'play' with math ideas, free from procedural concerns, for a little while at least. 

It is possible that I will pick up this article again in another month and see something completely different, but equally as thought provoking.  For now, though, I'll leave it here as road marker on the path of this discussion about limits and creativity.  As always, your thoughts and feedback are welcome!

More Than One Right Answer

There's a video showing up lately in the different places I'm visiting on the Internet.  It's an RSA talk by Sir Ken Robinson, world-renowned education and creativity expert, called Changing Educational Paradigms.  This is surely a subject that's outside the scope of my experience and this blog, but there are certain things he said during the talk that speak to what I think about when I'm working with or creating programming for children.

"Creativity is the process of having original ideas that have value.  Divergent thinking isn’t a synonym, but it’s an essential capacity for creativity.  It’s the ability to see lots of possible answers to a question, lots of possible ways of interpreting a question, to think laterally, to think not just in linear or convergent way.  To see multiple answers not just one."

There is also another interesting RSA talk I recently watched called The Surprising Truth About What Motivates Us.  It's a talk given by Dan Pink and although it is focused on workplace motivation, I think there are some parallels to a school setting. 

"[This study on what motivates people] has been replicated over and over and over again by psychologists, by sociologists, and economists.  For simple, straightforward tasks, those kinds of incentives, 'if you do this then you get that,' they are great.  For tasks that are an algorithmic set of rules, where you have to get a right answer [emphasis mine] if/then rewards, carrots and sticks – outstanding.  But when the task gets more complicated, when it requires some conceptual, creative thinking, those kinds of motivators demonstrably don’t work. […] There are three factors that the science shows that lead to better performance, not to mention personal satisfaction.  Autonomy, mastery, purpose."

My purpose in Math in Your Feet is to create opportunities for children to develop a level of mastery using the language of percussive dance to solve problems in a creative context. This context is naturally open ended and a place where there is more than one right answer, indeed an infinite number of right answers.  My philosophy is to give children limits (defined work space, four and/or eight beats only) and some tools (elements of percussive dance, a clearly defined process) and then let them work it out from there. 

The students work with a partner.  As early as the second day of our residency, students are taking control of their ideas, making choices, collaborating, and creating.  During their creative work time, I say over and over, "There are no right or wrong answers, only choices that have to be made.  What works, what doesn't work? Decide that and go from there..." 

What can you create within the limits that I set?  The 'answer' for each pair/team of students is two four-beat dance patterns sequenced into an eight-beat pattern and transformed with reflection or rotation symmetry or sometimes both.  In the many years I've been doing this, I've never seen the same pattern twice but they are all 'right' answers.  In fact, by the end of their time with me, many classes understand the potential of this structure so well that they still have ideas they want to try, directions in which they want to go. 

By the end of our week, children begin to understand and see that their ideas are ones "that have value."  I ask them if they are proud of the work they have done in their week with me (in both dance and math) and the answer is always a resounding

YES!

Teachers: Be There and Be (in a) Square!

Are you going to the NCTM 2011 Annual Meeting in Indianapolis, IN in April? 

I'll be there, so let me know your plans!  My 90 minute hands-on workshop Math in Your Feet: Teaching Geometry through Rhythm and Movement is one of the 650 presentations offered.  I was excited to see that it's been included in their presentation sampling for the 3-5 grade band. 

Come even if you're just curious, but know that teachers can learn to do this too and many before you have successfully implemented Math in Your Feet programming in their own classrooms!  A few years ago I developed a two-part professional development series in association with Clowes Memorial Hall of Butler University and Young Audiences of Indiana.  As part of this process I participated in the Kennedy Center's Seminar, "Artists as Educators: Planning Effective Workshops for Teachers."  This six-hour professional development series presents Math in Your Feet as a sequential program of activities which anyone can teach, even if they don't wish to do much moving themselves. 

My hands-on NCTM workshop will present a portion of the Math in Your Feet program.  You will be engaged in an in-depth investigation of transformations using simple foot-based patterns.  In particular, we'll harness the power of kinesthetic learning for understanding reflection and rotation symmetries in 3-D, a process which I'm sure will bring you some exciting new insights on the subject.  In addition to the math content you will experience learning with and through the arts. 

Hope to see you there...in a square!

The Space Between / Transitions

I think about a lot of things, and something I have thought a lot about are transitions.  When I teach clogging or step dance I think about how to explain to my students the variations and subtleties of transitioning weight or how to finish one step successfully and move on to the next.  My child, who has a regulation disorder, has needed me to pay close attention to the daily transitions/routines in her life since birth. I watch my own reactions to unexpected changes in my concept of what 'should' be happening in my day.  I am aware of, mostly after it's already started to happen, the subtle shift of life as it moves from now to past. 

(This seems to be a somewhat philosophical post, but even that is starting to shift...)

So it probably won't surprise you to hear that I think about transitions every time I walk into a classroom.  When one works with moving kids a transition can be a moment ripe with opportunity or a potential stumbling block, a great teaching moment or a train wreck.  I find these moving-while-learning kinds of transitions the ones that require a particular kind of attention and respect.

I suppose my approach to classroom transitions stems from my experience as an artist whose medium is space and time.  From where I stand, a transition is a moment when you have to simultaneously distinguish between something as it was, as it is, and as it will be.  Dancers (and musicians, too) encounter moments when they need to talk with themselves, the choreographer or other dancers about how to move from one moment or step to the next.  This attention to detail is just part of the process but its also a process that can be fraught with opposing viewpoints, confusion, and tension over all aspects of interpretation of the moment.  And, because dance (and music) exist in such temporal space (meaning here and then not-here in a split second) one usually spends 1000 times more time working out the transitional moments than actually performing them.  When resolved, the agreed upon transitional moment can be a thing of beauty, a seamless moment of artistry.

In my role as a dancer and a musician, dealing with transitions are part of the job.  In that particular context, if transitions don't get a certain amount of attention or respect all you'll end up with are unconnected pieces of movement or sound devoid of form or meaning or, at the very least, lacking the quality you would desire.  It's practically the same experience for me when I bring my art into an academic setting. 

I consider teaching (whatever the subject) the art of working in time and space to organize, focus and energize multiple bodies and minds.  If I walk into our dance space with my lesson plan but have paid no attention to the spaces between the bullet points, it is highly likely that my day will be quite unsatisfactory, perhaps even, as I said above, fraught with opposing viewpoints, confusion, and tension over all aspects of interpretation of the moment.

That sounds heavy, but just how do you stop bodies and ideas in motion?  The child's choice to pay attention is just that -- a choice.  It is, of course, expected that in school you pay attention and follow directions, but what if you are paying attention, just that all of your focus and intent is on what your feet are doing?  My own daughter gets so caught up in her projects that we continually have to strategize on ways to disengage her from her work when it's time for school, or dinnertime, or bedtime.  This is all to point out that transitions can be HARD, especially if you're emotionally invested in what you're doing now and are still in the now even when it's time to go to the next thing.

How do you stop bodies and ideas in motion, when needed?  My strategy has become to make the transition part of what we're learning and to make sure that as we go from here to there we are still on-topic.  Here's an example of a classic moment in my program, one that has often been "fraught with opposing viewpoints, confusion, and tension" for me, at least until I paid attention to it and found myself a solution.  This classic moment is called:
 

Me working the "Countdown to Silence"

Kids Are Standing and Need to Sit Down

Just telling a moving kid to sit down and expecting them be instantly ready to focus on verbal or visual directions is a hit or miss strategy, if you don't mind me saying.  A moving body needs a few focused seconds to calm itself down and turn on the eyes and ears.  Because I value the power of a group making rhythm together, I created a simple rhythmic countdown (which you can learn for yourself if you ever happen to see my work) which seems to do the trick.  It brings the whole room back to focus.  Kids can find their center even while sitting.  This countdown is a simple verbal and physical rhythmic reminder every time we transition from moving to sitting and is usually enough to help kids get ready to bring in new information.  This kind of transition also helps to maintain consistent composure of the group over the course of the lesson. 

Sometimes, if you just pay attention to and clarify the intent of a moment, especially an in between moment, the solution will be waiting.  A philosophical yet potentially useful mindset.

The Power of Not Moving

So much of what we know about how the brain learns points to using all the senses -- moving, touching, smelling, looking, leaping, running, talking, writing, tracing, solving, thinking, responding, producing, revising, doing.

But, what about stillness? 

What about a moment of doing nothing except making your body balanced and quiet, ready for learning?

In Math in Your Feet we call it 'finding your center.'   This means, quite literally, to stop what you're doing and put your two feet in the middle of your square dance space.  Arms by your sides. Eyes on the teacher.  Mouth quiet.  In control and in charge of your body.  Ready and waiting for the next thing to happen.

One of the biggest concerns teachers have about bringing kinesthetic/movement/dance learning into their classrooms is that it's going to be chaotic and uncontrollable.  I hear it every time I lead a professional development session for teachers.  By the end of our workshop, however, they realize that just because it's movement doesn't mean that self-control is absent.  This is something I bring up with kids, as well.  Continuing to return back to your center is one of the ways that you can remind your body what it feels like to be in control.  And then, you are ready to move again.

Movement is crucial to helping children learn, even if it just means there is time in the day to get up and move around the classroom.  However, when you are using a lot of movement, children need stillness to counterbalance all the activity.  This idea is worked into the flow of each class I teach in an academic setting.  My formula has always been 3-5 minutes of moving, followed by 3-5 minutes of sitting and focusing on other things -- watching and responding to others' creative work, receiving information, clarifying content.  This kind of non-moving time, coupled with 'finding your center,' becomes a powerful counterpart to all the jumping, sliding, stepping, turning, talking, collaborating, resolving, and creating the students do the rest of the time.

Whether you're a teacher, a teaching artist, a dancer or a parent (or a combination!) I'd love to hear what you think about the topics in this blog.  Please consider sharing your thoughts and ideas in the comments sections or by sending me an e-mail.  I hope to hear from you!