Introducing the Tape Chronicles Project

It's really no secret I love tape. I should have bought stock in it long ago.  In my mind, tape is the ultimate open-ended, the world is your oyster, creative, hands-on learning and making supply.  I mean, just look at how versatile it is:

You can change an environment in an instant!  Need kids to be able to visualize diagonals while creating foot-based percussive patterns? Voila!

Are you six and want to visualize a trajectory to the moon?  Tape's for you!

Even clear packing tape can be harnessed in the pursuit of art making and invention:

Check out the endless ways tape can be employed in the interest of math, art, kinesthetic exploration, invention and education -- and then consider submitting some examples of your own!  The Tape Chronicles are housed over at the Math in Your Feet website.  Hope to hear from you!

Sidewalk Math: Functions!

After a mild winter we had a lovely and quick blooming spring which allowed us to get out and about earlier than we might have otherwise.  In March I posted about an outside adventure where we discovered a veritable treasure trove of circles in juxtaposition with other shapes.  I think this might have been the origins of what I've started calling 'sidewalk math'.



Sidewalk math is fun because, generally, all you have to do is keep your eyes open.  If you've got a camera to record your observations, all the better.  This is not necessarily an original idea; the photographer Tana Hoban has a whole series of books with photos of the math all around us.  Her camera is the eye through which we can notice math in the physical world.  There are also the engaging Math Treks developed by Maria Droujkova of Natural Math.

For us, sidewalk math is a combination of these two approaches and has turned into a large percentage of our first grade math classroom.  It capitalizes on my daughter's propensity to notice everything, fulfills her need for movement while she learns, and bypasses her resistance to formal lessons.  It's also an opportunity for us to make observations and pose questions in a collaborative way, which is an approach that works for both of us.  For example, on a recent walk my daughter notice a crack in the sidewalk that initiated an hour-long in-depth conversation and exploration into the nature of triangles as we traveled to the hardware store and back.

(And it's apparently it's sticking with her: As I'm writing this my daughter calls down to me to report that she and her dad saw "seventeen triangles on their way home from the park this morning....did you know that part of an arrow is a triangle?!" )



But, in this story, sidewalk math plays another role, that of salvaging my initial attempt to introduce functions to my young daughter.  You can read about my first attempt here where she was wholly and unequivocally unimpressed with my presentation of the subject and took matters into her own hands.  I ended the post wondering what to do next.

I was understandably thrilled when I came across the book A Game of Functions by Robert Froman.  It's part of the Young Math Series from the 1970's and is out of print.  A quick Google search found copies available for purchase between $17.00 and $115.00!!  Luckily, my husband works at a university with a very comprehensive library and I got my hands on a copy.  I read it to my daughter one morning.  She wasn't having a great day, but she didn't protest, and we got through most of it.  I let the idea sit, waiting patiently for an opportunity to put the ideas into action as the book suggests. 

The book starts out with an introduction to the idea of 'function', as in 'whether we go the park this afternoon is a function of the weather -- if it rains this afternoon we will go shopping, if it is nice we will go to the park' (I'm paraphrasing here).  Or, as in this example below, how long it takes you to run around the outside of your house depends on on whether you crawl, walk or run. How quickly you go is a function of your mode of movement.

At that point, the book introduces the functions 'game'. 

You find a nice big area and draw a line across and a line up.  Lucky for us I had sidewalk chalk on me and we were at a park with a parking lot that looked almost like graph paper!

When I asked the kid what her 'rule' was, she said she wanted to take ten steps over and ten steps up.  We quickly realized that we needed a way to make sure her steps were the same length so we landed on her personal foot length, heel to toe.  She made a little white chalk X at ten steps.

And then, ten steps up from the X, and marked with chalk.

Although I helpfully informed her that she didn't need to go back to the beginning each time to 'add ten' to the last result, it was interesting watch her ignore me completely and then figure it out for herself.  And it didn't take her long -- by the time she was working on 30 steps, she realized she should just add ten to the second X (twenty steps) instead of count 30 from zero.  It wasn't an 'I told you so' kind of moment, just a little bit more proof that if the kid wants to figure something out on her own I should just let her do it. Lesson learned and internalized!  (For her and me.)

When she had used as much space as she could I asked her to stand in the corner and look at all the Xs she had marked up into the space.  "They go in a diagonal!" she observed.  And then she ran from (50,50) all the way to (0,0).  

We also worked on another rule for a little while (nine out and eight up) but she was running out of steam.  That was a lot of thinking for one morning.  It was perfect timing, too.  As we were packing up, four cars drove into the parking lot and covered her work!

At the very least, I feel like I've redeemed this concept for her (or, more likely, myself).  I haven't labeled what we did 'functions' but I did use the word 'rule' a lot, for example "The rule is to 'add ten' so your next move is ten more steps than the last time...let's see what happens when you do that a bunch of times!"

As you can see, above, the book goes on to show how you can do the same work on graph paper.  I'm thinking about how to make it a game...maybe two rolls of the dice determine the rule?  I could do my line and she could do hers and then we could compare?  Steepest line?  Line with the most graphed points?   Which one gets to the top of the paper with the least number of graphed points?  And, maybe include the question: "How steep your line is depends on (is a function of)....?"

Who knows?  This journey is full of adventure and surprises.  It's not always smooth sailing, but we're learning a lot, her and I.  And, one thing's for certain, there is more sidewalk math in our future.

Power

You are nine (or ten, or eleven). You have power.

You think mathematically while you move.  You make your own choices about how far to turn and in which direction, what kind of movement to use, where to put your feet, how to combine your patterns and whether to transform your pattern with symmetry.

Your choices are good, especially when they take time to perfect. 

You can enjoy the process...

...or not.

You can have fun and work hard at the same time.

You can be the one who usually gets sent around the building 'for laps' because you can't focus in class, and yet, in this case, you are the one who makes up the most fantastical, awesome pattern ever.

You marvel at how fast an hour goes.  Where does that time go?

It goes into your feet.

You have agency over your body, your brain, your ideas, and your learning.

Now that is power.

[It's also images of Math in Your Feet in action!  Photo credit: Ryan Richardson.]

Mapping the Familiar

There was a bright, blue sky up above, and at ground level there was frozen fog.  The result was a morning delicately dressed in sparkles.  Perfect for a walk.

Down the driveway.  Pick a piece of lavender, covered with millions of tiny ice crystals.  Which way do you want to turn, left or right?  Next intersection, right or left? 

We walked through the fog, already burning off.  We could feel the sparkles entering our lungs with each breath.

Busy road and cars zooming.  Three choices, right, left, or straight ahead.  "Mama, I used to be afraid of those trees, but I'm not any more."  We pick tiny pine cones off the tips of the branches. 

Next crossroads, "Mama, that's University Street down there, let's go there."  Thinly frozen puddles alongside thinly disguised reading practice: "What does that P stand for, do you think?  The one with the red circle and the cross over it?  What does the rest of the sign say?" I ask.

Right turn down University and an old dog barking at us across the lawn.  "Mama, I used to be afraid of dogs, but now I just don't like their licking."

Find two sticks, clap them together and then we're marching.  Left turn.  Right turn. Rose street!  Find the stone pig, march on home.  "I wanna make a map!" the girl says.

She wanted red and blue like the roads in the atlas.  We recalled which way we turned, what we saw, remembered the crossroads, one block at a time.  All of a sudden, she notices corners and the geometry of the street layout.  "I always thought the blocks went in a circle.  I didn't know they were squares."  The things she loves go on the map (friends' house, stone pig) as well as the things that 'used to' frighten her.  Her long battle with anxiety has its landmarks as well.

It may not be to scale, but when we go out again this afternoon to follow our map, the issue of scale might come up.  I'm also pretty sure we didn't get all the streets on there especially at the end of our route, but next time we'll bring paper and make some notes as we walk.

Mapping the familiar twists and turns and landmarks of our neighborhood -- bodies first, memories second, paper and pencils third.  I marvel at the human brain inside my six year old daughter's head that is so driven toward representation of her experiences and activities; driven toward it even though she is still just learning to decode print and write using 'the rules' .

She makes maps of other kinds, too.  Sewing "patterns", also not to scale, but clearly a sequence of steps mapped out:

I recently read a fascinating article in the New York Times about teachers taking their young students on walking field trips as a way to develop literacy.  This kind of activity is literally a step in the right direction.  Without concrete, kinesthetic, physical experiences like these, no child can fathom the meaning behind the marks on the page or develop full mastery of the human brain's greatest gifts. The order needs to be sensory experience / memory / symbols, not the other way around.

To Each its Own: Targeting My Professional Development Workshops

In the last couple months I've had whole bunches of fun presenting professional development workshops in a variety of settings, to a variety of people.  Let's see...math teachers from all over the U.S., PE teachers from across Indiana, classroom teachers from Indianapolis, fellow Teaching Artists from a variety of disciplines, and arts education administrators from Young Audiences affiliates from around the country. 

Each session bore the title 'Math in Your Feet' and was a combination of big picture information and hands-on experience, but that is where the similarity ended and my job got really interesting!

For the classroom teachers I started by focusing on the challenges of using movement in a classroom setting.  As they started to move and experiment with foot-based percussive patterns they became more comfortable and sure in their own movement.  This approach usually leads to a greater willingness to embrace, sometimes for the first time, the possibility of leading their own students in movement-based learning.  To some extent I am also encouraging them to have fun with math, many for the first time.  I consider the 'doing and making' of percussive dance patterns in this program the same as the 'doing and making' of math so, in every teacher workshop I do, I walk them step by step through the intersection where math and dance meet.  We're so used to focusing on the symbolic, static realm of mathematics that we don't always recognize when we see math happening in front of our eyes.  It helps to have a guide.

For the self-identified math teachers at the NCTM annual meeting I also started with a message of 'anyone can lead movement in the classroom and here are some tools' but then quickly moved toward 'here is an opportunity for your students to represent their math understanding in a new way within the kinesthetic realm'.  I also drew their attention to the fact that the processes of solving a problem in both math and dance (choreography) are often similar -- question, understand what tools it might take to answer the question, experiment with ideas, use your resources, find an answer that seems to work, evaluate and then ask more questions. 

The group of 80 or so PE teachers was a new one for me simply because there was not one bit of trepidation or reluctance to get up and move!  Not all of them were comfortable with the idea of dance, at least initially, but they were definitely game.  I was only with them for about an hour, and I couldn't go very deep, so I stuck with active modeling of the bridge between my particular brand of movement with an academic content area.  If I had had more time with them, I would have focused on the  process for moving the dance to the page -- speaking the words that describe aspects of our movement as we move, writing those words down, turning these words into symbols, and graphing foot positions on a coordinate grid.  I did the point that Math in Your Feet can be a collaboration between classroom and specials teachers, just like it is when I lead my residency.  The concrete movement and math activities can be done in PE or music class which then build the bridge to the formal, written, symbolic realm of math back in the regular classroom.

At their conference the arts education administrators were focusing on how to add the A in arts to STEM topics (STEM to STEAM).  I gave a general overview of the program and laid out my process for building the program and integrating the dance with the math.  The most important issue for me is that when you are thinking about integrating any art form with another content area you really need to be honest with yourself and ask 'is it a good fit?'  If the answer is no then it is not worth forcing the issue.  If you think 'maybe' then do a little more work to explore the connections.  In the end, though, the connections need to be more than skin deep.  Just because we count our beats in this program doesn't mean I consider that a good example of what math and dance have in common.  I also gave a similar account of how I combined math and dance to my fellow Teaching Artists.

My favorite moments while teaching teachers are when they ask me questions that show me they are imagining how they will do this work with their own students.  It's similar to house hunting, I suppose.  The minute you start imagining where you're going to put your furniture the realtor knows you might really be serious!  I love hearing all the different ways engaged and caring education professionals imagine tailoring my ideas for their own particular learning environments. 

The Importance of 'Doing' Math: A Conversation with Maria

Maria Droujkova, from Natural Math, read and commented on my recently published article about the development and use of Jump Patterns in the classroom.  Here are some excerpts from an interesting exchange we had today on the Natural Math forum based on her questions after reading the article:

Maria: Great article, Malke - thanks for sharing! I loved the photos, and especially the cool graphic organizers and visuals you use. Do kids like to use the charts? Does it depend on the person?

Malke: What plays out again and again in this program is that teachers are really surprised when they see how enthusiastic their students are when it comes to writing about their experiences in Math in Your Feet.  Recording their patterns using the one best word to describe each category of each beat *is* challenging, but they are motivated toward accuracy because it is *their* pattern.  Also, it usually plays out that within each team of two, one person is more comfortable in the 2D realm of the page than the other, and one is more comfortable moving than the other -- it's a team effort, which makes it more comfortable for everyone. Once the kids do the tough work to record their pattern using the descriptive words, it's actually quite easy for them to plot their feet on the simple grid.  I still think there is a better, maybe more mathematically accurate way to do this, I just don't know what it is yet! 

There are, however, whole groups of kids who still just need the physical portion of the program (more and more, sadly).  These are kids who never had a chance to develop spatial reasoning in preschool, for instance. They don't have enough math, even in 4th or 5th grade, to use the program to take them further -- I find that they begin to understand the math concepts as if it's the *first* time they've ever seen or heard about them.  In these cases, I require just the minimum in their workbooks, and I purposefully stay in the physical realm.  It may be the only time they will ever have to just 'play' with math.

Maria:  Mathematics is "embodied" in that its grounding, basic metaphors come from bodily experiences and observations. You can't skip over that and go into formal math. Even working with adults, I find that you need to go through folding, building, mirroring, measuring and other physical activities and/or stories if math does not make sense to them.

Malke:  This is great to hear, and I believe it wholeheartedly based on what I see kids do in my program and in my personal math (re)learning...I just gave a very well attended 90 minute hands-on presentation at the NCTM [National Council of Teachers of Mathematics] annual meeting and it was surprising how many of these adults were really quite challenged.  It has nothing to do with being 'good' at dancing and everything to do with not having enough experience working with and within a physical realm.  I attended a session on the van Hiele Levels [for developing geometric thought] and realized that this probably applies to adults as well -- experience is key to understanding.

What is interesting to me is that my 'hunch' eight years ago, that there might be math in what I did as a percussive dancer, is now more true than I initially imagined.  At that point in my life I believed, as most of us probably do, that math is primarily symbolic.  I realize now that the math I bring to children in the form of rhythm and dance is some of the experiential math they may not have ever had, and that they need this kind of experience to move forward.  I've heard that only 10% of us will understand the symbolic realm of mathematics without needing to first have, as Maria says, "...bodily experiences and observations".  Just this fact alone makes DOING hands-on, experiential math that much more of an imperative.

Here's the link to our full conversation on the Natural Math forum.

Yep

From Spark: The Revolutionary New Science of Exercise and the Brain, by John J. Ratey, MD, a wonderful gift last week from Templeton ES PE teacher Monica Chapin:

In the Introduction:

"In today's technology-driven, plasma-screened-in world, it's easy to forget that we are born movers - animals, in fact - because we've engineered movement right out of our lives.  Ironically, the human capacity to dream and plan and create the very society that shields us from our biological imperative to move is rooted in the areas of the brain that govern movement.  As we adapted to an ever-changing environment over the past half million years, our thinking brain evolved from the need to hone motor skills.  We envision our hunter-gatherer ancestors as brutes who relied primarily on physical prowess, but to survive over the long haul they had to use their smarts to find and store food.  The relationship between food, physical activity, and learning is hardwired into the brain's circuitry." Page 3

In the chapter Learning: Grow Your Brain Cells:

"The body was designed to be pushed, and in pushing our bodies we push our brains too.  Learning and memory evolved in concert with the motor functions that allowed our ancestors to track down food [the reason we learned how to learn in the first place], so as far as our brains are concerned, if we're not moving, there's no need to learn anything." Page 53

So...if you are not moving you are not learning?

I think that's it in a nutshell.

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!

The Mind/Body Split: A Guest Post from My Former Self

I found my former self in some files recently, but I wasn't sure what to do with her, until now.  Here's what happened:

A friend sent me a link to an article titled Children's Gestures and the Embodied Knowledge of Geometry.  This is an incredibly important finding.  Here's the abstract:

There is mounting research evidence that contests the metaphysical perspective of knowing as mental process detached from the physical world. Yet education, especially in its teaching and learning practices, continues to treat knowledge as something that is necessarily and solely expressed in ideal verbal form. [Emphasis mine.]  This study is part of a funded project that investigates the role of the body in knowing and learning mathematics. Based on a 3-week (15 1-h lessons) video study of 1-s grade mathematics classroom (N = 24), we identify 4 claims: (a) gestures support children’s thinking and knowing, (b) gestures co-emerge with peers’ gestures in interactive situations, (c) gestures cope with the abstractness of concepts, and (d) children’s bodies exhibit geometrical knowledge. We conclude that children think and learn through their bodies. [Emphasis mine.]  Our study suggests to educators that conventional images of knowledge as being static and abstract in nature need to be rethought so that it not only takes into account verbal and written languages and text but also recognizes the necessary ways in which children’s knowledge is embodied in and expressed through their bodies.

Reading this abstract I am reminded that when I was in college I was concerned with, among other things, the continuation of a dichotomous world view; you know, seeing only us/them, self/other, intellect/emotion, good/evil and not any of the gray area in between.  In particular, one of my main concerns at the time was of the mind/body split.

That's where my former self comes in; it turns out she actually has a lot to offer this discussion.  Here is an excerpt from the introduction of a paper I wrote at the end of a quarter of self-directed study in 1989.  The paper is grandly titled "Connections: Healing the Split within Western Thought" and although I'm pretty sure it wasn't my best work, I find this portion to be particularly interesting. 

(Please excuse my youthful idealism.  And, please keep in mind the first line of the above abstract as you read: "There is mounting research evidence that contests the metaphysical perspective of knowing as mental process detached from the physical world.")

"In a dream I had a couple years ago, I cut off my head and walked around with it under my arm.  But the funny thing was, I could still hear and think, and feel.  I sat it down on a bed and we looked at each other for a while.  Then it was time for me to put the head back on my neck; I tried to keep it there but it flopped around like a little baby's head.  Finally, I got it settled, but then realized that I had to sew it to my neck -- but I hate to sew!  So I just let it sit there and walked around with a bright red line of blood around my neck.  I never did connect my head to my body, until now."

"The dream of severing my head from my body and then being reluctant to reattach it in any permanent way was, I know now, an unconscious image of the dichotomous schema that pervades Western thought.  Polarities such as self/other, us/them, male/female, black/white, mind/body, human/animal, and life/death permeate our mentality and prevent us from seeing the necessary relationships and connections that run between them. [...]

"In the following pages, I touch on the issue of how we have become so disconnected with the world.  And, more importantly, I look at the steps we can take to heal the split to become whole again.  In Part One, I trace briefly the major steps in history that have contributed to the dichotomy of thought that exists in our society today.  In Part Two, I present the foundation of holistic thought, which includes a brief discussion of Gregory Bateson's epistomology -- how we can learn to know the world as it really is.  Part Three investigates the characteristics of creativity and how it can help everyone reach her or his potential in life.  And, finally, Part Four shows how, when combined, Systems Theory, Batesonian epistemology, and creativity create an understanding of how we can, at least in theory, heal the split."

Um.  Okay, I know that was a big bite of thinking I took back then but let's but fast forward to now.  I suppose you could say that in creating a curriculum that integrates math and dance, using concrete movements and patterns executed by the body to illustrate and illuminate the often abstract world of mathematics, I am attempting to 'heal the split.'  I should also say that, at the time, I was acutely aware that I was short on life experience with which to illustrate my ideas.  My twenties were spent searching for a solid tether; when I found step dancing five years after this paper was written I felt a huge relief that I was finally doing something meaningful and real. 

And now?  Now it seems I've come full circle because from where I sit it is clear to me that finding the connection between mind and body has been my life's work all along.  I also like reading about the seeds of my interests in understanding the uses and benefits of creativity outside the realm of artistic process.  But that's for another day. 

(A side note from me as the spouse of a digital archivist -- if I hadn't printed out my papers back in 1989 and only kept my now inaccessible 'floppy discs' there would be no 'former self' to contribute to this blog.  How will you be accessing your digital files twenty years from now?  Just sayin'!)

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.

Math & Movement Lesson: Basketball Court Pathways

Ooooh, I just had an idea!  Last year I developed a preschool math/movement program by creating simple pathways through an empty space with different colored tape on the floor, and adding locomotor movements down the different paths.  Within a couple months the kids could follow more complex pathways using combinations of locomotor movements.
 

Look at all those lines!

I've been thinking recently that what I did with four year olds could be adapted for the K-2 set.  The problem is that putting down tape can take a long time.  However...

Most public schools and sports centers have a gym.  A gym with lines already on the floor.  Basketball court kinds of lines.  Lines that are straight and curved.  (It's hard to make curved lines with tape. Really hard.)  To take advantage of all these lines and all the open space, here's an idea to use with five-to-eight year olds that I came up with that merges an exploration of space with locomotor movements. 

The lesson, below, is definitely in the 'map is not the territory' category.  I've had the idea, I've based it on previous experience, I've mapped it out for you to try, but we won't know how it works until someone tries it out.  That's one of the reasons I am sharing this lesson, because I probably won't have a chance in the near future to try it out with kids and it seems like such an exciting idea!

The lesson is more like a whole unit of activities and you'll need to decide how to break it up into manageable chunks.  Also, since repetition is the key to learning, I encourage you to repeat a lesson until it's clear that everyone understands it physically and cognitively.  In even smaller chunks it could also serve as a movement break when needed in the course of a learning day. 

Finally, this lesson starts out looking a lot like dance, and it will build dance skills.  As you build those skills, the more math you'll be able to explore.  If you try out any aspect of this lesson I'd really love to hear how it went and if you have any suggestions or questions.  And, I'd love to hear from you because I'll have questions for you, too!  Leave a comment here or e-mail me at: malke (dot) rosenfeld (at) earthlink (dot) net! 

Basketball Court Pathways
©2010 Malke Rosenfeld, http://www.mathinyourfeet.blogspot.com/ and http://www.mathinyourfeet.com/
Users of this lesson have permission to share it with others with proper acknowledgement, copyright notice, and website links (as above).  If you want to share this lesson with forums, educational groups, wiki sites, etc. please consider sending me a message to let me know where you put it.  You can e-mail me at:  malke (dot) rosenfeld (at) earthlink (dot) com

LESSON OVERVIEW:
After exploring a variety of paths around a basketball court by following the lines, 5 to 8 year olds will:

• decide on a pathway that has a clear beginning, middle and end;
• create a pathway that includes both straight and curved lines as well as directional interest and some repetition;
• decide on two to three locomotor movements  (skip, hop, run, walk, slide, gallop, hop, leap, jump) to use while moving down the pathway and which part of the pathway gets what movement;
• map out the pathway on paper, including color coding and notating when and where to do their movements.

LEARNING GOALS:
Children will:

• Use intentional, meaningful movement to gain experience and competency with spatial relationships, a foundation for mathematics understanding; 
• Make creative choices about the length, shape, direction and design of the pathway;
• Express creative choices with appropriate math and dance terminology;
• Bring their kinesthetic experience to the symbolic realm on the page by creating a simple map of their dance; and, 
• When appropriate, integrate the concept of scale and coordinate systems when mapping the pathway. 

VOCABULARY:
Locomotor Movements: skip, hop, run, walk, slide, gallop, hop, leap, jump
Other Movements & Attributes: turn, smooth, sharp, slow, quick, big, small, long, short, high, low
Spatial/Directional Terms: left, right, on, around, curve, straight, forward, backward, corner, on, off, double, single, length (time and distance), intersecting lines


MOVEMENT ACTIVITIES:

1. Start by playing follow-the-leader around the gym to introduce kids to the different combinations of straight and curved pathway choices.  Start by walking on the lines -- as long as you stay on a line you're playing the 'game' right.  Model the idea of a starting and ending location by saying "We'll start at this corner, where should we finish our path?" and "Now we have finished this path, where should we start the next one?"
2. As you play this introductory game, start giving kids choices about which locomotor movements to use (skip, hop, run, walk, slide, gallop, hop, leap, jump).  You can stay in the lead or give kids turns taking the lead which will help keep the game fresh.  Spend as long as you like on this, and perhaps even repeat the activity a few times a week for a couple weeks.  You can vary this 'game' by giving different challenges such as:  'How slowly can we move this time?' or 'How smoothly can we move?' or 'When we turn a corner, let's make it a sharp turn!' or 'Let's make our movements big on the straight lines and small on the curved lines.'  Keep it playful!
3. After you're sure they have the 'follow the lines' concept, put on some music (examples below) and let the kids experiment with the lines to find their own pathways.  At this point it should just be about the path, not the movements. The goal is that eventually every person should have their own unique pathway.  After they've experimented for a couple minutes, have them 'freeze' and reinforce this goal as well as the... 
4. Rules of the Road: If they cross paths or eventually share part of a pathway with another child, challenge them to be 'good drivers' and share the road.  Also, remind them that they need: a starting point and an ending point, to use at least 1/4 of the gym, and to include repetition (for example, two trips around a circle, or double back down a line).
5. Let them work for two minutes then gather them in a group and see who wants to share their work.  Ideally, pick a kid who looks like s/he already has a pathway and is able to repeat it.  Get a couple kids to show first then send everyone back out to finalize a pathway they can repeat the same way every time. 
6. Make sure every kid gets to show his or her pathway before moving on.  Use this time to give feedback; you'll want to make evaluative comments like "The lines you've chosen are all straight lines.  I wonder what it would look like if you added a curved line to your path?" or anything else you've noticed about their work.  Because this is a creative activity, there is no completely wrong answer/path, just decisions to make.  So, try to pose questions that will help the child become conscious of the decisions s/he is making.  When everyone has shared their work, this may be a good time to stop the lesson for the day.  Or, it may be a good time to go directly to the Mapping Activities section and complete Activity #1.
7. Once everyone has a pathway, take some time away from the paths to review basic locomotor movements by saying, "Who knows what a gallop looks like?  Who would like to show me what a gallop looks like?  That's right, one foot in front of the other!"  Have one child at a time illustrate the different locomotor movements, naming each one as you go.  This is essentially a mini-lesson focusing on locomotor movements where kids get a chance to practice their locomotor movement skills by follow one of the lines on the court instead of a more complicated pathway.  That's a good way to assess where their skills are at.  You can never do too much of this kind of cross-lateral movement, which is why this is good for a movement break as well as a dance/math lesson.
8. By now, you should have decided on your own pathway too.  The next step is to add locomotor movements to the pathways, so model for them what you are going to do with your path.  The best bet is to have one choice of movement per line and then change to a different movement when the line changes (straight to curved, or after you turn a corner.)
9. Some final reminders for the pathways: turn all corners sharply, and find smoother movements for moving on the curved lines, which will enhance the attributes of a curve. 

MAPPING ACTIVITIES:

1. Have kids review their pathways.  Using black marker, pen or pencil, have them draw their pathway as best they can on a piece of unlined paper.
2. Make a little key of the movements used while traveling the pathway.  Write the moves down (i.e. skip, run, hop) and assign a color to each move.
3. Redraw the pathway on a second piece of paper, this time using the assigned colors to create each section of the pathway.  An alternative would be to color the existing black-lined map using the assigned colors.   
4. If you think it would work (7 or 8 year olds) have the kids trade maps and see if they can recreate the other person's pathway.

EXTENDING THE ACTIVITY:
Remember, the movement itself is furthering spatial understanding and this experience (up through the mapping, above) may be enough for five to eight year olds.  However, if you think your kids are ready, here are some additional suggestions to further the exploration of math concepts:

1. Have the kids assign a certain number of skips, hops, gallops, etc. to each section of their pathway.  Make sure it can be danced first, and then transfer to the page.
2. Measure the space and the length of the lines and then create a scale drawing/map of the pathway.
3. Using the measurements of the space (above), create a scaled-down version of the pathway using an x and y coordinate grid.  The intersection of x and y would be oriented to the center of the space your path runs through.

To further the movement/rhythm concepts (some of which turn out to be math related!) try these suggestions:

1. Develop beat competency.  Using one of the music selections below, work on moving 'to the beat' while moving on the pathway.
2. Basic phrasing.  Each line segment in the pathway will have a certain length which can accommodate a certain number of steps.  Kids can figure out how many hops they can do on the line before getting to the corner or the start of the curve and then mark that on their map.  Which brings up another point...
3. How many small hops on the line?  How many if you do your hops bigger?  An issue of scale, I suppose.  Lots of experimentation and questions (from you and the kids) along with a 'let's try it' kind of attitude can bring out some amazing math connections that none of us know are there yet!  Let me know what you find out!

MUSIC SUGGESTIONS:
This music is essentially for background color during the creative work.  Dancing to the beat is a whole other ball of wax, so please just start out using the music as inspiration for the creative work time.  That being said, you can spend some parts of your dancing time just on locomotor movements, and that would be a good time to work on dancing to/with the beat.

Artist/Album/Song -- all on iTunes
Chiwoniso/Rebel Woman/Listen to the Breeze (Modern African)
Vishten/Live/Figeac (Traditional Canadian)
Solas/Sunny Spells/paddy taylor's (Traditional Irish)

More Than The Sum of It's Parts

I am always thinking about better ways to describe what exactly is happening in Math in Your Feet.  It's actually been quite difficult for me to explain because, in the end, the total experience is more than the sum of it's parts.  Think about it -- this program brings together two subjects which communicate, in their own mystifying language, about space, time, and movement.  Teachers who have been through it once often advise first time teachers that they'll "understand it after they're done," which is not ideal.  Luckily, I'm meeting with some teachers tomorrow to plan for an upcoming residency.  While preparing for the meeting I took the opportunity to update my thinking about what is really going on while a bunch of kids jump around in small boxes taped on the floor.  Here's what I came up with:

Specific Learning Areas in Math in Your Feet (Upper Elementary)

INTEGRATION
Both the dance and the math content are focused on equally; finding connections between the two creates a stronger understanding of both content areas.

KINESTHETIC LEARNING
Engaging the vestibular system through intentional cross lateral and patterned movements improve learning.  Math concepts are experienced first through the body.  Words are connected to the movements and then used in reflection journal entries, word studies, and in the process of recording patterns on the page.  This everyday language is then converted to a more abstract symbolic language in the mapping activities.

REFINE/STRENGTHEN/REMEDIATE UNDERSTANDING OF SPATIAL RELATIONSHIPS
Firm grounding in spatial relationships (best learned through the body) is vital to a strong understanding of math concepts. 

INTENSIVE STUDY OF PATTERNS
Higher order thinking and problem solving skills are strengthened during the process of creating, manipulating, combining, observing, transforming and analyzing foot-based dance patterns.

MATH VOCABULARY LEARNED IN CONTEXT
Teachers consistently report that their students use new math terminology and vocabulary appropriately and with ease in conversations about their work in the program.

CONCRETE GRADE-LEVEL MATH TOPICS
This program is not about numbers, formulas, or procedures, but there are discrete math topics learned within the experience.  Angles, degrees of turns, directions, basic fractions, symmetries, reflections and rotations are all covered in the dance class.  Extension activities in the Student Workbook also touch on combinations, tangrams, lines of symmetry, lines of reflection, scale drawings, and perimeter and area.

IMPROVED ATTITUDES TOWARDS PROBLEM SOLVING AND MATH
At the center of the students’ experience is their role as creator, using just the elements of percussive dance and a few guidelines.  There is nothing quite so empowering as being able to create something by yourself out of (almost) nothing.

What do you think?  Does this answer any questions you may have had about the how's and why's of this program?

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!

More Than Counting

Every once in a while I do a Google search for my name or my program's name, Math in Your Feet.  It's instructive and sometimes surprising to find out just how far out in cyber space I am.

Case in point, I recently found an undated conference paper by an associate professor from the Institute of Mathematical Sciences and Physics at the University of the Phillippines Los Baños.  Among other things, the paper makes a case for "the cognitive and aesthetic similarities between mathematics and dance."  Math in Your Feet is cited and described in one of the opening paragraphs.  Never mind the fact that it's just a rehashing of promotional language direct from my website, the rest of the paper makes a pretty good argument for the importance and relevancy of integrating math and dance.

My favorite section of the paper is on pattern recognition (a major connecting theme in my program) and includes a quote from a book called The Math Gene by Keith Devlin, mathematician, NPR's Math Guy, and (among other things) a senior researcher at Stanford University's Center for the Study of Language and Information, who argues:

"mathematics is not about numbers, but about life.  It is about the world in which we live.  It is about ideas, and far from being dull and sterile, as it is often portrayed, is full of creativity.  It is the science of patterns."

Within this one quote are ideas I've been trying to communicate for years to the folks who hear about my program and inevitably reply, "Oh, you mean like counting your steps?"  I always try to find a nice way to say, no, not really, but there's never enough time to explain my reasoning fully.  However, I do have time and a little space here to say mathematics is more than numbers and counting and balancing your checkbook and that a program which teaches math through dance (or dance through math) is about more than counting too.

Mathematics is essentially a language invented to describe, manage and understand the physical world. (And if there are any mathematicians checking in on this blog, feel free to correct me!) That's why connecting math to the real-world helps kids better understand it, because the real world is math.  And, no matter what your opinion is about dance being lumped in with 'things in the real world', please remember that we all move and without a firm grasp on spatial relationships, one of the foundations of a good mathematics education, you will most literally be lost.

That's why even the preschool version of Math in Your Feet has less to do with numbers and more to do with exploring and understanding where you are, where you want to be, and how you're going to get there.  Add to that a chance to develop patterns of movement embedded in time as well as make creative choices while moving and you now have a powerful mathematical experience in real time and real space.  The critical piece (for upper elementary kids, anyhow -- the younger ones just need as much kinesthetic experience with the concepts as their teachers are willing to fit into a school day) is that kids get a chance to bring the math back to the page.  After all, math is a symbolic language, but it is really much more than counting.

I'm sure I'll be revisiting this theme in the future.  But first I have to run to the library to pick up a couple of Keith Devlin's books.

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!

What You Can Learn from A Square

What can you learn from a square?  Well, before you can learn from a square you first have to understand it.  And, to do that, it's best for you to have a chance experience 'squareness' using use all your SENSES.  As Albert Einstein said:

"Learning is experience.  Everything else is just information."

To start, you need to SEE it.  You've probably been looking at squares all your life.  You can correctly identify a square and find examples of it in your environment.  But do you really understand it? 

Possibly.  You might be able to give some facts about a square (number of sides, number of vertices, measurment of angles, etc.) You may even be able to draw one or all four of the lines of symmetry through square.  But if you're a fourth grader being asked to make your body execute a move on the diagonal (opposite corners), it's actually pretty hard for you to do that without a visual reference.  That's where tape comes in. 

So, LOOK AT your taped 2'x2' square on the floor and sit inside it.  Use your fingers to TOUCH and TRACE the parallel sides, the vertices, and the equilateral sides.  TOUCH the directions forward, back, left, right and all the diagonals, and while you do this SAY the words that identify where you're touching. 

By now, it's probably safe to say that you are oriented to your two-dimensional dance space.  It's also likely that you have become inordinately fond of this little piece of property you call "home" and are unwilling to let anyone usurp your territory.  Ah, to be nine again.  No matter, it's all good.

It's also time to count up how many SENSES you've used so far to further your understanding of  'squareness.'  Let's see -- seeing, hearing, and touching.  There's actually one more sense you need to use and it's not smelling or tasting.   It's your VESTIBULAR SYSTEM.  Carla Hannaford in Smart Moves: Why Learning is Not All in Your Head puts it this way:

"...just as important to our development and our lives is the integration of sensory input, which gives us information about gravity and motion, and about our body's muscular movements and position in space -- the vestibular system and proprioception.  These play a surprisingly significant role in our awareness of the world and also...in our ability to understand and learn."

Okay, so now it's time to STAND UP and utilize your body for some high powered learning.  Put your FEET TOGETHER and STAND in the center of your space.  JUMP forward, JUMP center, JUMP back, JUMP center.  SPLIT your feet to the sides of the square.  SPLIT your feet to the diagonals.  TURN your body right, towards the first side.  TURN toward the next side.  TURN toward the third side.  TURN toward the fourth side.  How many times did you have to TURN to get all the way around?  How far, in fractions, did you JUMP on each turn?  How far was each turn in degrees? 

If you JUMP and TURN half way around your square, how far have you gone?  How many jumps will you need to get all the way around?  If you start your movements facing forward, JUMP and TURN 90 degress to the right and then JUMP and TURN 180 degrees to the left, where will you end up?

Are you learning anything new from a square yet?