Friday, December 30, 2011

Teaching Math in Your Feet...Without Me!

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

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

In a recent post on her blog, Meg says:
The article unwraps the dance/math residencies Malke Rosenfeld teaches in public schools. The fact that she shares her methodologies with classroom teachers for use in the classroom lit a spark for me. Even without being a step dancer myself, maybe I could lead my dance students through the jump pattern curriculum!
Apparently she was quite successful!  Read on...
I’m about to begin Week 5 of 6, finishing the jump patterns with my second set of 4th & 5th graders (six lessons per group), and we’re all enjoying it. Malke’s outline provided lots of material to work with, and I’ve worked the pacing & focus of instruction for each lesson to fit my ELL learners & my circumstance. The movement variables are broken into malleable chunks, and we’ve explored the math-related concepts of precision, congruency, reflection, and turn symmetry, with students choreographing patterns in teams of 2 and 3. In addition to integrating dance & math, there’s a problem-solving (choreography) component that parallels the "workshop/conferencing” structure that my students are familiar with through Writers Workshop, allowing me time to confer with & jump-start individual students. In addition, there’s a spatial arrangement that supports classroom management (personal dance spaces for each team — wow, what a concept!). Add in some dance videos to “mentor” the kids in their choreographic process & journaling questions to provide feedback on what students are learning, and it’s no wonder we’re all engaged!
A perfect Math in Your Feet moment, courtesy of Meg's blog.  These boys are in the middle of a 270 degree turn.  Nice!
My first groups of 4th & 5th graders finished the unit before the holiday break, with some of them performing their patterns, both congruently & in mirror symmetry. They nailed the precision steps they’d created, even without the support of their personal dance spaces, and their peer audience was able to talk about what they were seeing with insight and new vocabulary. What a pleasure to watch…
In this description of how how she used the Math in Your Feet curriculum, Meg shows us all that the program is flexible enough to be adapted by teachers in a number of key ways:

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

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

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

Thursday, December 22, 2011

Sneaky Math: You Know, UNO!

My six-year-old daughter is hip to my game.  You know, that I'm interested in stretching and deepening my personal understanding of what math is and how we make it.  She has become highly sensitive to moments where I might be trying to teach or show her something math related.  She's on to me.  She calls me, accusingly, "Math Mommy!" 

This is an attitude shift, actually.  Back in September, when she had plans to run away with her best friend, she very clearly relied on me to help her learn the math she would need for when they finally headed out into the wilderness. (Or, the elementary school playground, which has a huge field and lots of trees around the edges.  Whichever.)

Now, however, she wants nothing to do with me and my math.  And, after being quite self-motivated and curious about measuring, comparing amounts, comparing sizes, spontaneous chant counting by tens and twos, relationships between numbers, and relationships between shapes the whole summer and fall, her inquiry into all these things has slowed somewhat.

Except, she has become a W-H-I-Z at two person UNO.  We are UNO addicts; we play UNO every day, sometimes twice a day.  UNO perks us up: Having a bad day?  Let's play UNO!  Been in the same argument loop all morning?  Let's play UNO!  And, yes, I know there is math in this game but DON'T TELL HER!!!  Honestly, I have been giddy with glee that I now have an outlet to influence her mathematical thinking and move her math skills forward without her knowing!

Well, I mean, she knows she's adding up points, for example, but she wouldn't do it at all if I just asked her to outright.  She's already shot down my suggestion that maybe, perhaps, we could, let's say put down a 3 and 4 to match that yellow 7 that's on top of the pile?  "Math Mommy!  I just want to play the game!"  So, I've come up with my new 'sneaky math' approach.

What she doesn't know is that, faced with such perceptive resistance, I'm loosing on purpose (sometimes) and I'm quite gleeful about the way that it's working out.  I hand her my handful of lost points and say, "How big a win was it for you?!" giving her an opportunity to be very specific about the magnitude of her victory.  I'll casually say, "How many tens can you find?"  or "What's fifty plus twenty [Wild Draw Four + Skip cards]?"  As a result she's naturally skip counting by tens and sometimes fives, easily finding different combinations of numbers to make ten, adding numbers to sums way past twenty, using her fingers less and less and, in the process, improving her capacity for mental arithmetic.

Examples of different ways to make 'ten'.  I've somehow found my way to guiding her to find as many tens as possible and add those up first, which capitalizes on her love of tens, hundreds, and thousands.  We did get stuck today, though.  She found two 10's but then we were stuck with three 6's.  In the end I got out the paper to show her the 'easy way' to finish up the adding, thus sneaking in a little two digit addition on paper.  She balked a bit, but I said, "How are you going to know if you beat me or not??"  That was motivation enough!  (I'm still not sure, though, when to bring in a calculator.  I'm pretty sure being able to mentally calculate numbers is an important life skill in many ways, so I'll stick with fingers, skip counting and the occasional paper and pencil for now.)
She is quite publicly gleeful herself as she gloats about the magnitude of her victories.  Me? I am secretly thrilled that I've found a way for my incredibly enthusiastic and fiercely independent learner, enigma that she is, to enjoy her math without knowing her mama is enjoying (and influencing) that math along with her. 

For Christmas, since the kid is so into games these days, she's getting Junior Monopoly and the cube version of Quirkle.  Oh, and Mancala, plus some really cool fractal fridge magnets to go with our fridge tangrams.  I've got my sneak on, big time!

Sunday, December 4, 2011

Marveling at Moving Patterns (Video)

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

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

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

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

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

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

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

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

Wednesday, November 30, 2011

Between the Folds

The piece in question
Just yesterday I was playing around with a piece of paper, folding it, watching it, seeing how it could move.  I showed it to a friend and said, "I guess what I'm doing is improvisational origami."

Just today I mentioned the same thing to a different friend.  She said, "Have you seen Between the Folds?  You have to..."

Here's the trailer for the movie -- it's on hold at the library.  Hopefully I'll get to watch it soon.  Can't wait, can't wait, can't wait...

Every day I am finding more and more of the pieces that connect math, art, dance, rhythm, science, expression and creative practice.  I think the reason I'm so excited is that I recently found my way back from a wrong turn I took with this inquiry. But, that's part of the process, too, no matter what the medium. After determining what I don't want or need to do right now, the missing pieces seem to be falling into place.

[Edit: This post seems to be getting a lot of traffic, for some reason. If you're interested, here is the follow up post I wrote after viewing the documentary.]

Tuesday, November 29, 2011

Playing Math Every Day from Moebius Noodles

If you haven't heard of Moebius Noodles, I highly recommend you check it out!

From the Moebius Noodles blog:
We are creating an advanced and accessible math book for young kids and their parents, called “Moebius Noodles” and an online knowledge exchange hub to support it. It’s an off-the-beaten-path travel guide to the Math Universe for adventurous families. A snowflake is an invitation to explore symmetry. Cookies offer combinatorics and calculus games. Floor tiles form tessellations. The games in “Moebius Noodles” draw on these rich properties of everyday objects in ways accessible to parents and kids, even babies. As the world turns into a mathematical playground, it transforms, one family at a time.

The most recent post has a fabulous menu for 'playing math every day' for the week of November 28 through December 4.  Activities include playing ball outside and then exploring a type of fractal called Apollonian gasket, fun subitizing activities, and starting a math journal. 

Try it out and let me know what your favorites are!

Monday, November 28, 2011

Bal-a-vis-x: Rhythm is the Key Element!

"Bal-a-vis-x: Rhythmic Balance/Auditory/Visual/Exercises...requires focused attention, demands cooperation, promotes self-challenge, and fosters peer teaching."  

Wow, a program that uses rhythm, cooperation, crossing the midline, and visual tracking all in the pursuit of an integrated nervous system that is ready learn.  There are many similarities here to aspects of Math in Your Feet including rhythmic movement in concert with others, self-challenge, peer cooperation, and cross-lateral movements.  Sounds like a great idea whose time has come! 

Wednesday, November 23, 2011

A Moving Classroom: "Find Your Center"

"Nothing exists without a center around which it revolves, whether the nucleus of an atom, the heart of our body, hearth of the home, capital of a nation, sun in the solar system, or black hole at the core of a galaxy.  When our center does not hold, the entire affair collapses.  An idea or conversation is considered 'pointless' not because it leads nowhere but because it has no center holding it together." 
-- Michael S. Schneider, A Beginner's Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science

I just picked up A Beginner's Guide to Constructing the Universe.  I still have unanswered questions about what kinds of math are interwoven with a creative inquiry into percussive patterns.  That's not to say that I am confused.  I know exactly what kind of math the kids are learning, but I am sure there is more and I am searching for evidence.  Some days, like this one, I find I understand math more than I think.  So do you, by the way. 

Here's is a reposting of one of my very first pieces on this blog, The Power of Not Moving (October 13, 2010), which attends to the value of a single point of stillness and focus, the center, in the moving classroom.

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.

Thursday, November 17, 2011

Number Discovery

In retrospect, this amazing discovery started with parental neglect and erroneous math.  I freely admit I was at fault, but it all worked out amazingly well in the end.

It was all precipitated by an innocent question, yelled from another room: "Mama, what is half of 38?"

I did a little mental math: "Sixteen!" I yelled back.  In my defense I was in the middle of something very important.  That's why I didn't insist she figure it out herself, you see. 

When we were finally in the same room it took me a minute.  I was actually fairly impressed with her thinking.  Even though it was all wrong, she still had logic and structure to her reasoning.  Can you figure out what her 'rule' was?

[And, yes, at six she still writes many things backward; sometimes, as is the case here, she even writes from left to right.  Ah, the growing brain!]

Luckily, I had my wits about me and simply said, "Cool, look at that!  Hey, let's check your work with the Cuisenaire rods!"  This was a brave move since, prior to this moment, we have done absolutely nothing with 'taking away' or 'difference' in any formal way let alone using the rods.  Fortunately, the taking away part was so wonderfully obvious in this visual/tactile realm that I had no problem explaining it and the girl got it right away.  During this process I also noticed that in the intervening couple months between our first major experience with Cuisenaire rods and today, her ability to visualize and attribute amounts to the rods has become second nature.

Here the total number was 31, take away nine.  It was so very satisfying to physically take away a number and literally see the difference. 

Sorry this is so blurry, but hopefully you can see that we started with 37 and took away 15.  I'm not completely sure why we didn't start with the original number in question, 38, but like I said, it all worked out in the end.  Just look at it!  It's beautiful, don't you think?  By the fourth equation I asked her if she could figure out what would come next.  She guessed right for the last five equations, but wanted to check her work with the rods each time anyhow.

Today I am basking in the joy of unexpected discoveries and a growing mind. 

Monday, November 14, 2011

Geometry Discoveries

One of my most enjoyable creative endeavors these days is accompanying my daughter (age six) on her math learning journey.  Sometimes we play games, or work with pattern blocks or Cuisenaire rods.  Sometimes she wants to earn and save money so she can eventually run away with her best friend.  Whatever we do, she learns best through conversation and narrative. 

She also learns math without me.  There are plenty of times when she happens upon something 'by accident' by which I mean: things (books, puzzles, making supplies, marbles, tape...) left around the house, acted upon, forgotten for a while, ultimately to be rediscovered a month or three later and acted upon again, but this time in new ways.

This happens a lot 'round here. 

For instance, we have tangrams in almost every room of our house.  There's a magnetic puzzle book set in the car and approximately six plastic sets on the game shelf, of various colors and all mixed together. I have four sets of magnet tangrams that I got at the NCTM Annual Meeting in Indianapolis this past April.  They have been on the fridge in the kitchen since then with minimal interaction...until recently, and, last I looked, they were all in use in some kind of design or another.

Here's what I found my dear child doing with the set of car tangrams (inside for some reason) over the last few days.

By way of setting the scene, I should tell you that the kid generally initiates and works independently on any project she pleases during her afternoon quiet time, usually with the company of an audio book.  A few days ago during this time she informed me she was "making a math book."  I had observed her tracing tangrams, but was surprised to find that she was doing it in a deliberate way, and that she was also able to clearly describe her discoveries and observations to me. Here is the book, so far.  I am just a scribe here -- these are her words as she dictated them to me, except where noted:

"Four of these triangles that you see here can make a square.  If you pull these triangles apart you can see that they're little triangles.  But you can see on this page that they make a square."  [I see that she has drawn them as individual shapes, which, over the course of her illustration, merge together into her intended shape.]

"This rectangle you see is made up of a parallelogram and two triangles.  Really they're just shapes, but when you put them together they make a rectangle." [It looks like she's numbered the inside angles of the individual shapes.  It also looks like she is again showing the process individual shapes merging into the intended new shape.]

"You see the wheels of this bike as rhombuses but really they're squares turned so their points are facing up and down, and to the side." [She was gesturing this first, and at first she used the word 'flipped' to describe the orientation of the square wheels.  I focused her on the orientation of the corners to describe how the square was turned.]

"The square and the rhombus that you see here, their edges are both the same length.  The difference is a rhombus is a squished square, squished to its side.  The rhombus has two larger angles and two smaller angles than the square.  But the square has the same angles on each corner."  [These are actually tracings of shapes from the pattern blocks set we have.  The ruler markings was her idea for comparing the two shapes.  I supplied some new vocabulary in the form of 'angles' and made some observations about the difference by overlapping the two shapes, in an effort to help her observe and further articulate the difference between the two.]

Writing about all of this made me think to look up more about the van Hiele levels for geometric reasoning.  I need to spend more time with that to analyze how I'm interacting with future discoveries of hers.  I also think I'd like to focus more specifically on flips, turns and slides.  Maybe I'll be super ambitious and have us play around with flips and turns using both 2D and 3D shapes!  

Wednesday, November 2, 2011

What Do You Make of This? (Build What I Have, Redux)

In my recent post Conversational Math: Part Two I described a game that my daughter and I like to play with Cuisenaire rods called "Build What I Have".  It's really fun.  So fun, in fact, that my daughter asks for it every other day or so.
Here's a funny thing that happened today:

It was my turn to give directions.  To start, take a look at how my design looked when I was finished giving directions.  I should point out that there is only one layer of blocks laid out on the board we were using.  I should also point out that we hid our work from each other.  She is trying to replicate my design using only my spoken words as a guide.  Ironically, no gestures are allowed in this game!

Here's how I started: "Take three light green rods and put them together so they are all touching and all perpendicular to the bottom of the board.  It should look like a square that is three units across.  Now add a fourth light green rod on top of the square.  It should be placed perpendicular to the other green rods and parallel to the bottom of the board.  It should look like a rectangle now."

So far so good!

Me: "Take an orange rod and place it on top of the green rectangle."
Kid: "Like a teter totter."
Me: "Yeah, I guess so! Now, take two white blocks and put them on top of each end of the orange rod" (etc.)

Then it was time for the big reveal...tah dah!

Geez Louise, what happened?!?  Her orange rod IS on top of the light green rectangle, but it appears the kid built up while I built...up?  I suspect we were both right, but I am at a loss to explain why or how.  I think it is accurate to call the kid's work three dimensional, but so is mine because it is built with polyhedra and not 2D lines on a flat surface.

Hmmm....any ideas?

Sunday, October 30, 2011

Marx Brothers Math: Transformation & Reflection

I'll wager that each one of us looks into a mirror at least once a day. Surprisingly, what we see in the mirror is up for some debate; at least that's been my experience when talking with fourth graders about the subject of reflection.  The result of these conversations is that I now firmly believe that when we use movement to explore the concepts of transformation and reflection, we gain a truly three-dimensional understanding of the subject.  Here's a little peek into how it all goes down:

Me: "What do you see when you look in the mirror?"
4th Grader: "Myself."
Me: "But is it really you? There's only one of you!  There's no one else like you in all the world.  You are an original!"
Different 4th grader: "It's your reflection!"

Later, after my 'magic wand of transformation' has turned the entire class into multiple reflections of me and they've had a chance to experience what it's like to exist on the other side of the mirror, I ask:

Me: "How many of you think it would it be fair to say that your reflection is doing the same thing as you?"
Half the class raises their hands.
Me: "Or, is your reflection doing the opposite of you?"
One third of the class raises their hands.
Me: "Or, how many of you think it might be both, the same and the opposite?"
One or two hands shoot up, other hands raise and lower tentatively.

We work through answering this question in class using our creative dance work.  In lieu of this experience here is a video clip for you from the Marx Brothers movie 'Duck Soup' (below).  I find this video to be simultaneously fun, highly entertaining, and instructive about the process of reflection.  Remember that transformation is essentially about change, and I assert that movement is a particularly effective way to make the process of change visible.

A few things to consider before watching the video, below:

Most of the time we are looking into a mirror straight-on. We brush our teeth, wash our faces, or comb our hair, all while looking at our faces and the fronts of our bodies. In this orientation is easy to think that the reflection is doing the same thing as us.

But remember, the mirror can reflect all sides of our bodies.  As you watch this video you will see Groucho and Harpo directly facing the "mirror" but also walking along the length of the mirror (shoulders to the mirror line) and turning toward and away from the mirror. There's even a fun bit where their bottoms are closer to the mirror than their heads!

In Math in Your Feet, children reflect their dance patterns by deciding who will dance the original pattern and who will reflect that pattern; the reflection changes the original pattern in small but very important ways.  Based on the narrative arc in this particular video, Groucho is the homeowner (original) and Harpo an interloper (reflection).  As you watch, ask yourself:

When is the reflection doing the same thing as the original?

When is the reflection doing the opposite of the original?

I'll give you a couple examples to get you started. When Groucho first sees his 'reflection' in the 'mirror' they both move in toward the mirror and then away from the mirror. In this case they are doing the same thing. Then, still facing each other, Groucho's right hand goes to his chin, but it is his reflection's left hand that goes up. Both hands go up to the chins, but they are using opposite hands.

One more example: At 0:35 Groucho turns away from the mirror over his right shoulder, for a total distance of 180°. Harpo also turns 180°, but over his left shoulder.

How many examples of same and opposite can you find? Can you find any mistakes? I had a hard time tracking if they were using opposite rights and lefts in their footwork, for example. Have fun and don't forget to try out some of the activities listed below when you're done watching!

How'd you do? Ready for a little application of the concepts?

Try this at home:
Put a line of tape on the floor. This is your mirror, otherwise known as a line of reflection.
Decide who will be the original and who will be the reflection.
To start, the reflection has to be the same distance from the mirror line as the original.
Move slowly at first so the reflection has a better chance of accuracy.
Most important: don't forget to experiment with having different sides of your body be 'reflected' in the mirror.

Extra challenge:
Make up a short piece of choreography with a variety of moves and levels (high, medium and low).  In Math in Your Feet, the foot based patterns are units of four steady beats.  See if you can make a four- or eight-beat combination of moves using your whole body. 
Both people practice doing this choreography congruently (everything the same).
Then, do the choreography with the line between you. The original needs to move slowly while the reflection figures out what parts of the choreography needs to change (hint: everything is the same except the reflection uses opposite rights and lefts).
When you're well-practiced and have it a tempo that both people can do comfortably, show off your work!

Extra, extra challenge:
Perform your choreography with your partner first congruently (everything the same) and then reflected (opposite rights and lefts).
If you want a triple challenge, change roles and have the other person become the reflection.

Friday, October 28, 2011

Friday Fun! Video of Some Awesome Dancing

Melody Cameron is one of my favorite dancers, and a former teacher of mine. She is from Cape Breton Island, Nova Scotia, Canada and has her own particular, fluid, musical style of Cape Breton step dance.

Here she is improvising to a piece of 'mouth music' sung in Scots Gaelic by Joy Dunlop.


Tuesday, October 18, 2011

"Children Think and Learn Through Their Bodies"

Needless to say, as someone who harnesses movement as a teaching and learning tool, I am fascinated every time cognitive  science reveals another link between the body and learning, what is often called 'embodied cognition'.

I'm about to embark on a reading of Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, by George Lakoff and Rafael Nunez (thanks to a tip by Michael Paul Goldenberg.)  This is a book co-written by a linguist and a psychologist hoping to launch the discipline they would call "the cognitive science of mathematics." 

In a similar line of inquiry, the researchers who recently  published the article Children's Gestures and the Embodied Knowledge of Geometry concluded that "children think and learn through their bodies."

Here's another recent article on a similar topic: Thinking with your hands: A link between gesturing and intelligence:
"My colleagues at Germany’s Humboldt-Universit├Ąt zu Berlin , Potsdam University and I have discovered differences between people who gesture frequently and those who gesture rarely. Our study shows that gesturing may be a function of, and may even contribute to, brain development... [Emphasis mine]
"We do not know yet whether gesturing facilitates the development of fluid intelligence or whether it is a by-product. But we do know that children who are asked to gesture in certain ways while learning new tasks learn better than children who are asked not to gesture. Considering that gesturing benefits children while learning, it is possible that gesturing plays a role in the development of fluid intelligence, perhaps by simulating action. If this proves to be true, children might be able to literally give themselves a hand in their own development by gesturing more."
...and thoughts about this article, from the Eide Neurolearning Blog:
"It's interesting to think that teaching children to problem solve certain types of problems should involve strategies that take into account that fact that one is trying to train the imagery of the students. Just verbally saying back the steps of a problem or even watching an explanation won't internalize the imagery. To really 'get' certain problems, we have to enter into the simulation and perceive the question and solution in a bodily way." [Emphasis mine.]
From the Edutopia blog of neuroscientist and teacher Dr. Judy Wills:
Students need to be explicitly taught and given opportunities to practice using executive functions to organize, prioritize, compare, contrast, connect to prior knowledge, give new examples of a concept, participate in open-ended discussions, synthesize new learning into concise summaries, and symbolize new learning into new mental constructs, such as through the arts or writing across the curriculum. [Emphasis mine -- one of the reasons 'the arts' are helpful in this context is that most artistic activities employ the body during the art making process.]
My only question is why, with research and thinking like this, many/most schools still insist that keeping kids at their desks, and lengthening the school day behind said desks, is the best way to insure student learning.  I know all schools don't think this.  I know a lot of really smart folks out there are connecting the dots, but our policies don't reflect the very real evidence that our bodies must be engaged and involved in the learning process. 

Monday, October 10, 2011

Kitty Census: Vet Edition

We have a LOT of kitties in our house.  And they're all real, too.  Most are stuffed, some are ceramic or plastic.  Others include a rare bookmark breed and one in a frame.  We also have an amazing example of the exotic puzzle breed, one who is an imaginary friend, an actual breathing, chipmunk-catching cat and, finally, one who is a special kind of human cat with white fur and a brown spot on her head.

When faced with multitudes like this, a mama's mind turns to math: Just how many kitties do we have and can we remember all of their names?  How many different types of kitties are living here?  How old are they?  Who is related to whom?

This morning the girl wanted to play vet.  I wanted to revisit the Kitty Census we started back in the summer.  As with many things, we compromised. 

ALL the cats went to the vet. 

I was the intake coordinator.  Each cat was registered by name, age, type/description, and malady.  Some of the issues included: "a little sad", "lying around, droopy whiskers", "smelly", "bedraggled", "music slows down at end" (that was the music box cat), and "hurt paw".  In the end, twenty eight showed up for well visits and eleven had some serious issues.  After a triage that took about an hour, we attended to the sickest first. 

This is the bunch that were there for their well-checkups.

We will revisit this data later.  For today we were concerned with 'sick' and 'well'.

Aventurine, of the rare 'puzzle breed', was assessed at intake as "needs fresh air and run around time without breaking."

First, the sick kitties, pictured above, all had a thorough examination, including temperature taking and whisker inspection.  Of the group, the Webkins family all had strep, two were afflicted by a 'fever' of undetermined origin, and two needed immediate whisker surgery.  Two others, originally thought to be sick ('lying around'), were determined to simply be acting like cats.

In the end, I think we simply set the stage for more questions about our brood.  I had her chart out the number of sick kitties compared to the well.  She was curious about how many kitties there were all together, and initiated a count starting at 28 (well) and counting up the sick column to find the sum total.

We'll revisit the intake records and mine it for more data in the near future.  How many family groups do we have?  How many different breeds and how many in each category?  What else do we need to find out about our household of kitties?

Until then, one thing's for certain: We've got a whole lotta' kitty loving going on 'round here.

"Mama!  Lucy's purring!"

One Awesome List

The participants of the Math in Your Feet teacher workshop at Clowes Memorial Hall (Butler University) last Thursday night created this list at the start of our three hours together:

Smiley faces indicate that point is no longer an issue.  A check mark means that this
will always be an issue, but not insurmountable. 

This was the most comprehensive list thus far compiled in one of my teacher workshops.  By the end of the workshop, it was collectively agreed that most of the items on the list were no longer viewed as challenges.  Here's why:
  • Large class size:  Taping down fifteen pairs of boxes onto the floor takes some spatial problem solving in limited space, but it is doable.  Students do all their work in the little boxes, there is no need to move around.  So, if you have the space for the boxes, you've got space for this program.
  • Room size/obstacles: This one got a check mark because there are always going to be issues with room size and obstacles, but they are not insurmountable.  One idea that teachers have had is to have kids push their chairs under their desks and tape the squares directly behind their chairs.
  • Behavior management/self control:  The rules are clearly laid out -- stay in your square and do whatever you like, except for 360 degree turns on one beat.  Because kids are focused on self-initiated work and collaborate with partners, there is always something for them to do, including simply sitting down and taking a break.  If things do start going off track, then it's 'group practice time,' time for teams to demo their work, time to add some new math, or even just time to stop.
  • The teachers were initially worried about how to keep order and whether adding movement would lead to chaos.  They collectively agreed this would not be a problem using the room layout and my suggested formula of 3-5 minutes of moving, creative work and/or practice, followed by a few minutes of new information or observation, followed by a little more movement, etc.
  • Overloaded kids: This is a really active, often noisy program, but there are regular times where we stop and quiet our bodies.  During the workshop I modeled a number of ways for kids to move and maintain self-regulation at the same time; some of this I already knew I was doing, others strategies were pointed out by the participants.  I purposefully use my 'count down to silence' to bring us back to a group focus.  It ends with smooth quiet movements and voices, by which time we are all on the same page and ready to focus on new information.  There are also times I communicate without talking (hand gestures, pointing to words on a poster, etc.)  Someone pointed out that everything in my class is really predictable, including the flow of events -- stand/sit, move/quiet, etc. as well as my expectations for staying focused on creative work.  This is an on-task program with a creative process broken down into manageable bits, one bit at a time. 
  • They were initially worried that not every child would want to participate -- shy kids, kids who are normally disinclined to participate, kids who don't like to move, etc.  There are always slow starters, ones who need to observe before jumping in.  As long as they don't get in others' way, that's totally okay.  There are also always a few teams of kids who never volunteer to show their work.  I tell those kids privately that it's okay if they don't want to come up now, but by the end of the week they will be expected to show what they've been working on.  Final presentations are never an issue on the last day because of this.  A participant noticed that 'everything is a positive' in this program meaning no matter a person's skill level we can always find success.  This aids participation because as long as they've moved forward in some way from the first day, mathematically, choreographically or socially, it is time to celebrate!
  • Another big worry was about the attitudes of school administrators: "Why are the students out of their seats?!?"  By the end of this workshop the teachers said that all you need to do is show the list of vocabulary and concepts used in this program to those who might doubt the value of such an approach to illustrate the relevant learning happening while the kids are moving. 
The teachers in this particular workshop had between eight and 37 years of full-time teaching experience.  What I love is that, based on evaluations, they all found this three-hour workshop to be a positive learning experience and that the ideas presented would be useful in their teaching.

Saturday, October 8, 2011

Happy Anniversary!

...and I nearly missed it.  Just this week I was thinking, "Gee, I think I've been blogging for about a year now.  I should go back and check the date."

I remembered just now, and good thing, too.  It's been exactly a year today since my first post.  I had just submitted my article for peer review to the Teaching Artist Journal and felt I had more to write.  And write I did.  When I started I didn't really know if anyone would read this blog; I haven't had a ton of readers, compared to other blogs, but am so grateful for the folks who have subscribed, followed, checked in, and commented. 

This blog has been a chance for me to illustrate and explain my work integrating percussive dance with elementary math topics, describe my work as a teaching artist more fully, and make connections between math, dance, and other similarly creative pursuits.  It's not really all over the 'map' but I do recognize that this might be categorized as a 'multi-topic' blog. That's fine with me -- I enjoy having multiple interests that intersect in sometimes fascinating ways over time.

This space has also been a way for me to connect with really interesting and smart folks in the mathed world, folks who have been really patient with me as I ask questions, share my ideas and generally expand my understanding of math thinking, topics and practices.  Sue VanHattum at Math Mama Writes, Maria Droujkova of Natural Math, Julie at Living Math, and Bon Crowder at MathFour have all provided wonderful support, forums, and conversations as I explore the world of math education.

It's been a whole year, but I feel like I'm just getting started.

Friday, October 7, 2011

Supporting 'Math Values in a Rich Context' / Origami as Math

I see a lot of geometry in origami but have always wondered what other math you can find in this kind of paper folding. 

Here is what Maria Drujkova had to say about origami during a recent blog post about the activities in her Natural Math Club:

Origami has the same values as mathematics, such as precision, modular reasoning (“bird base” as a group of folds), attention to detail, modeling, algorithmic thinking… Thus origami can be used to support math values in a rich context. The same goes for music.

I love the idea of supporting math values 'in a rich context.'  I love even just the idea of a 'rich context.'  I think all learning should happen in such a place.  Onward.

Book Giveaway: Square Cat

Over at love2learn2day there's a fun giveaway of the book Square Cat, by Elizabeth Shoonmaker.  You can find a review and lesson plan of the book there as well.

I just had to repost this because we are all into cat-based education here at our house.  There's more to this story but, for now, I will just say:

Check back soon for the results from the KITTY CENSUS.

Tuesday, October 4, 2011

Conversational Math: Part Two

In trying to capitalize on the kid's penchant for 'talking math' I recently decided to try a game with her that I found in the booklet that came with our set of Cuisenaire Rods. 

The game is called Build What I Have.  One person describes a design they are making with their rods and others try and reproduce that design by listening closely.  One of the main points in this game is to introduce and/or reinforce math vocabulary.

The suggested age range for this activity is 2nd-8th grade; even though the kid is a young six I knew we could still get something out of it.  I decided that, to start, I would capitalize on concepts she already knew (parallel, points, edges, top, bottom, sides, etc.) and introduce some new ideas (perpendicular, horizontal, vertical). 

The rest we'd muddle through somehow, I figured, but she did surprise me by knowing her lefts and rights.  "We've been doing that in ballet class, Mama," she stated mater-of-factly.  Fabulous.

To start, we hid our designs from each other.

This is the first design.  I led and she followed, trying to make her design match mine by following my instructions. I started by saying: "Lay your blue rod parallel to the bottom of our work surface."  She already knows the concept of parallel really well, often times finding and noting examples of parallel lines when we are out and about.  "Then," I continued, "take your green rod and place in perpendicular [holding the rod in the air] up and down like this, and place the end in the middle of the blue rod."  Success!  Our designs matched!
This is the game she led.  To start she told me to put my orange rod parallel to the bottom of the workspace, but about an inch up.  The second orange rod was to be 'a couple' inches above the first one, but when she told me to put the blue rods on the sides to 'make a rectangle' I clarified the distance.  "Looks more like three or four inches, to me," I said.  I asked her to clarify the placement of the blue rods -- do they go on the outside ends of the orange rods, or inside?  Notice that this design is mostly made up of parallel lines, a concept she is most familiar with.
This is the second design I led.  I said, "Take your three light green rods and put them so they are together and vertical, up and down, in your workspace...Oh look!  They make a nice little cube!"  At first she thought she needed a fourth one to make it a square, but I clarified and said we're not making the outline of a square, but a solid shape.  When we revealed our designs to each other we saw some differences! 

This is how she recreated my instructions.  The white cubes are essentially in the right areas, but I had actually challenged her to put each white block 'point to point' with each corner of the light green square.  The dark green rods are essentially in the correct place; I knew that was somewhat complicated to execute.  And, I just noticed, the light green rods are horizontal, not vertical.

This is the last design in our session, which she led.  Perfect!  She wanted to use a bunch of different rods, but everything is still parallel here.
Here is what I find fascinating:  

My daughter's designs were much simpler today than normal and I think it might be because she had to describe what she was doing as she built them.  There is an equivalent experience that I find to be true in my work with 4th and 5th graders as well.  Often times I tell those kids that they are doing complex mathematics in their bodies and grade-level math on the page; they understand more math in their bodies than they can communicate through words or symbols.  Sometimes it is impossible for them to notate their Jump Patterns because they are just too complex for their current stage of symbolic mastery.

Often kids can do, know, and understand way more than they can communicate symbolically.  If we only judge a kid by her output on paper, we're not really seeing the whole child.  There are many ways represent comprehension: we need to listen and watch carefully for other indications of understanding as well. 

It wasn't too long ago when I brought the word 'parallel' into my daughter's universe.  It will be exciting to observe her body and conversations show me she's 'got' the concepts of perpendicular, horizontal and vertical. 

Conversational Math: Part One

Being able to use math terminology freely and appropriately is a big goal in elementary math education.  In my opinion, there's no way you can really use math vocabulary intelligently unless you've first had a chance to experience the concepts in a concrete manner.  This is a huge part of what happens during a Math in Your Feet residency -- I am never surprised when an astonished teacher tells me: "I can't believe how easily they're talking about math!"  It would not be an exaggeration to say that most of us need to do math with our bodies in some way before we can talk about it.

Much of the math my daughter and I do together is hands-on and verbal; as a learner, she is best served by conversation.  We'll be in the car on the way to somewhere and she'll spurt out some new computation she's been working on in her head and with her fingers.  She may only be working with numbers one through twenty but, based on the frequency and variety of these kinds interchanges, she seems to be going pretty deep into her questioning. 

The other day I wrote Maria Drujkova from Natural Math with a (very) basic math question: 

"My daughter is telling me all the different ways she can 'make ten'," I wrote, "what's that called in math language? You know, 2+8, 6+4, 9+1...?"

Maria, being the awesome person she is, got right back to me:

"These are called 'number friends' by elementary teachers. 10 is important as the base of our number system. She is playing with a newly-discovered idea of an 'equivalence class.' You can try it with other things. For example, numbers whose difference is her favorite number (10, 2, whatever) - like 15-5 or 12-2 or 99-89. It's a road to ratio and proportion, too - which come up a lot in music signatures and dance!"

Thanks to Maria, it appears my daughter and I will have a lot more to talk about, well in to the future! There are other ways we are talking math these days, as well.  You can find out more in Conversational Math: Part Two.

Tuesday, September 13, 2011

Survival Math -or- How Cuisenaire Rods Contribute to Independent Living

From South to North: Our House, Bloomington, Park,
Nashville, Columbus (IN), Ohio (see the archway?)
Darling girl is an early riser and a morning learner.  Every day she rises, recites her list of questions and plans, and sets it all in motion.  This morning the dear child drew herself a map.

Soon after, she was really, really frightfully intent on writing number sentences.  All before my first cup of caffeine.  Normally I'd just let her chatter away and let it all wash over me (it was only 7am after all) but today the tide was just too strong.  I didn't really understand why there was such a need to do all this before breakfast but sometimes you just need to act first and ask questions later. 

She had already started her 'math lesson' by writing down 1+1=2.  Then 2+2, 3+3, 5+5.  At some point she said, "Every time I add, the numbers get bigger!"  The answer to 5+5 was doubled, which was doubled again.  80+80 gave her some pause, but by then I had drunk my tea and decided we might as well get out those Cuisenaire rods again. 

I wasn't completely sure if she was ready to move on to labeling each rod with a unit number or, for that matter, if I really understood how to use the rods to represent addition, or if we were going in the 'right' sequence of learning with these things.  But why let any of that stop me?

I said, "First, can you just put the rods in order from smallest to biggest?"  Done.  "Okay, let's look at your first number sentence, one plus one.  Here are two of the white blocks.  Which block is the same length as two white blocks?"

She picks up the red rod.  I say, "Okay, how many whites makes a red?  So when you write 'two plus two' this is what it looks like.  Which rod equals four whites or two reds?"  She pulls the purple rod to match.
 Then I had a bright idea:  "Can you figure out how many white blocks make an orange?  What other ways can you combine the rods to make ten?"  Girl interrupts my line of thought, "I want to make a line of a hundred!" She knows that ten tens make one hundred so she starts stacking them together to make sure she has ten of them before she lines them up.  I persist, "How many more ways can you make a ten?"
She wanted to see how long a line the ten tens would make when lined up end to end. 

This is how long it was and there are apparently many different ways to combine the rods to make ten.  We didn't get to all of them, I'm sure.  Then she wanted to see what a line 200 units long would be...

"Look Mama!  I'm 110 units long!"
Overall, my biggest takeaway from this morning is that, although she has enough questions to keep us busy for now, I need to do some more research on the best way to use Cuisenaire rods -- quick!

Later in the day I did find out why she was so intent on doing math this morning.  It's actually a secret so if you see her don't let on that you know.  "Mama," she said, "You know why I've been wanting a longer math lesson and have been so interested in maps and money?  Because T. [a good friend] and I were wondering what would happen if you [the mommies] stopped loving us and those things would be important to know."

Apparently, she and her great friend T. have hatched a sort of secret survival plan for an undefined point in the future when "our mommies don't love us any more."  They've decided they need to know how to do math, figure out how to work with money, and know how to survive in the outdoors.  After I assured her repeatedly that I will ALWAYS love her, we went to the library to get some books about how to identify and cook with edible plants. 

She said, "Don't tell T. that I told you.  And even if you never stop loving me, it's still helpful information to know."

"I won't tell," I replied, "But it's probably good that I do know so I can help you find the information you need. Learning all this will be really helpful when you grow up and live on your own."

"But I'm going to live with you forever."

Thursday, September 8, 2011

White...Red...Light Green...Purple...Cuisenaire Rods!

I pulled out the cuisenaire rods yesterday.  The set had been sitting on a shelf for almost a year.  If you don't know what they are the best I can say right now is that, essentially, they are unit blocks of specific colors designed to (at least at first) help kids develop number sense as well as strengthen conceptual understanding of addition, subtraction, multiplication and division. 

The girl immediately remembered the long ago time when we played a 'trick' of putting the first four (white, red, light green, purple) behind our backs.  One of us would tell the other which color to bring forward, and every time we'd be right!   I can tell I'm going to do a little more research on how to use these things, but I know from what I've seen so far is that the first thing you do is focus on the attributes of each color in relationship to each other before assigning a unit number.  Here's how things developed today:

Girl: "I'm going to put them biggest to smallest..."  Mama: "Say it with me!  Orange, blue, brown, black...  Okay now, with our eyes closed!  Orange, blue, brown...Yay we did it!  Now smallest to biggest, white, red, light green, purple..."

Mama: "Hey look!  A blue with a white makes an orange.  Oh, and a red with a black makes an orange, too!"  Girl: "Let me do it."

Girl: "It's divided here."  Mama: "Oh, I see it, on the diagonal.  Let's put a block down to show where it is."

Girl: "I want to pull them apart."  Mama: "Look, they're in the same order!"

Girl: "Now I want to make a design."

What she figured out is that to make this design work you had to skip colors: orange + 2, blue, (skip brown), black, (skip green), yellow, etc.

So many more things to do with these -- I can't wait!!


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