Wednesday, October 30, 2013

Happy Twittereen: The Power of Tape, Example No. 10,233

It's #twittereen over on Twitter.  The last few days I've been completely confused by people's profile pictures/avatars as they 'dress up' as someone else.  I was pretty sure I would just let this one roll right by -- I don't usually move or think fast enough for social hyjinks -- but then I saw this. I wasn't sure what it was, at first -- I know my taping anywhere, and this looked suspiciously less professional different:

Turns out, Christopher (@trianglemancsd) dressed up like me!! This is the picture that I use on Twitter (notice the precision tape job ;-).  Fun stuff!

Later, it became clear those are his kids jumping in the photo he took! And, an even better surprise was that completely without prompting, his action of putting tape on the floor turned into a story that confirms everything I know about tape on the floor. He wrote:
I laid down some tape on the kitchen floor for the photo last night, then proceeded to do the dishes. Tabitha [who is six] wandered in and proclaimed "There's tape on the floor!" She then spent the next 20 minutes making up games using the tape lines. There was one where you needed to jump from one tape line to the next, always landing on one foot, with an extra challenge of landing as close as possible to the end of the line. There was another where you stand in one square, put your hands in the other then jump your feet over to the one where your hands are, with extra credit for doing it with style.
She was eager for me to finish the dishes so that she could teach me the games, which she did and we spent another lovely 15 minutes on that before I had to cut off for the photo and then bedtime.
It was a beautiful little family moment.
See?! This is what I have been saying for years! Put down some tape, even just one straight line, then wait and watch. Changing the environment, even slightly, will invoke an explore response in kids and, inevitably they will find some way to figure out this 'new' space with their bodies. (And, it's a body thing, but for kids it's also a body knowledge math thing.) Fantastic.

Christopher also mentioned:
Also, I have a square on the floor of my office (more carefully taped, I'll have you know) that I laid down when I was practicing for my #miyfeet lesson. People notice it and comment on it when they come in. But NO ONE plays with it. Common questions from my colleagues include whether it is where they need to stand while they talk to me, or where I put students who don't do homework. Students tend not to comment on it. Tabitha hasn't been in my office since I laid it down, but I predict great fun will be had when arrives someday.
No, it's a KID THING. That's why adults (even teachers unfortunately) don't get it right away and why I am always in need of more proof/examples of the power of tape to inspire movement and explorations of space, especially in younger math settings. Harvard professor Eleanor Duckworth, in her book of essays The Having of Wonderful Ideasspeaks to my reasoning around tape:
"In my view, there are two aspects to teaching. The first is to put students into contact with phenomena [of which tape on the floor is a good example] related to the area to be studied -- the real thing, not books or lectures about it -- and to help them notice what is interesting; to engage them so they will continue to think and wonder about it.  The second is to have the students try to explain the sense they are making and, instead of explaining things to students, to try to understand their sense." 
Children can 'explain' their sense making through their bodies if you have the right glasses on. Just before my dance workshop in St. Louis last week I noticed kids out in the lobby playing (jumping around) on an inset in the tile floor - darker tiles that made up an array and a specific space within the larger floor area. These were not just kids getting out some energy (although that was part of the story) but kids having their own new, wonderful ideas about how to explore the new and interesting floor patterns. Kids will play in new ways (to them) and in new physical environments with their bodies first. That's the truth. 

Saturday, October 26, 2013

Mathematical Weaving in St. Louis

My morning with hundreds of homeschoolers from babies to adults started like this... a quick powerpoint introduction of how to use these paper strips to make beautiful things. Create your warp with number multiples (3,3,3) or (2, 2, 2) or (5,5) or whatever you want and decide on a color pattern. Create your weave pattern rule (number of overs and unders) and then see how it affects your design when you invert that rule or change the weave color.

Result? You be the judge:

Make a rule. Ask questions about it.

Weave from top down.

Think qualitatively about numbers...

...but think visually as well.

No wrong answers.

Experiment with warp numbers and colors.


 Enjoy your results!

A marvelous Monday making math!

Thursday, October 17, 2013

Allegory of the (Math) Cave

Is this math?

No. It's a bike rack that, with the sun's help, created a shadow of the symbol for phi. 

Is the symbol itself math? 

No. Plato says a shadow is not reality.  Here's a description of his cave allegory (bolding and brackets mine):
"Plato has Socrates describe a gathering of people who have lived chained to the wall of a cave all of their lives, facing a blank wall. The people watch shadows projected on the wall [symbols] by things passing in front of a fire behind them, and begin to ascribe names to these shadows. According to Plato's Socrates, the shadows are as close as the prisoners get to viewing reality. He then explains how the philosopher [math teacher?] is like a prisoner who is freed from the cave and comes to understand that the shadows on the wall do not make up reality at all, as he can perceive the true form of reality rather than the mere shadows seen by the prisoners." [Wikipedia]
A recent post over at Christopher Danielson's blog Overthinking My Teaching got me thinking about this image of the shadows at the back of Plato's cave, especially this comment by Sue VanHattum:
"Have I reached Malke’s beautiful standard of embodied math? Nah. But it’s in my head as a goal, whenever I can make sense of it.
"Malke, here’s one body thing I do that probably doesn't count. The graph of y=x^2 is a parabola. I like to think of it as having both arms up. I like to think of the graph of y=x^3 as having one arm (left) down and the other (right) up. We are working with graphing more complicated polynomial functions, and I ask them to show me (what I call) the big picture with their arms. I’m hoping to get them to hold this visual information more firmly in place while they work through the detailed parts, so that their graph will reflect their thinking on both the big picture and the details. I’m using body movement just as a cuing device, really."
My reply:
"Sue, I think making sense of what math is in the body is a lengthy ongoing proposition for most of us, even the dancers.  My goal is to at least raise some questions: Is the graph the math? Are the arms doing the graph the math? My answers: The graph is the representation of the idea and the arms are the representation of the graph.  That is why my blog is called 'the map is not the territory'.  Those teachers Christopher mentioned are part of the legions of people who, by no fault of their own, mistake the textbook (map) for the math (actual experience of thinking and doing math) because they've never had a chance to travel the actual terrain of math land.  Personally I think there's no problem with a 'cuing device' or mnemonic as long as you are aware of where it lies in the continuum of math learning and your teaching goals."
I repeat: "...the philosopher is like a prisoner who is freed from the cave and comes to understand that the shadows on the wall do not make up reality at all..."

Most of us (myself include) have learned shadow math in the caves they call school.  Here's a question from Christopher that might get us out of the cave:
"Math comes from, and lives within, textbooks. I am not OK with this.

"So what can we do in every lesson every day to represent mathematics as a subject that comes from, and lives within, the minds (and bodies) of our students?"
These are the things I am thinking as I try to make a worthwhile case toward using the body in math learning for something other than re-drawing the 2D representations in our textbooks.

Tuesday, October 15, 2013

"Think Like a Straight Line": Examples of a Body Learning Math

This post is an addendum to my last post on meaningful non-dance movement in math learning.  After some reflection, I realized that for any of my thoughts to make any sense, I need give some concrete examples of what I personally see as a math learning through the body outside of a dance context.  

I homeschooled my daughter for first and second grades but I did not explicitly employ any kind of kinesthetic approach to learning math or anything else, for that matter. She wouldn't accept anything formal for the first year so we spent a lot of time out of  the house -- on walks (with lots of opportunities to talk math), math games, thrifting (always lots of history lessons there), reading books, listening to audio books, library visits, making stuff. 

For a while I wasn't completely confident in my approach, but over time I realized she was showing me what she was learning in many different ways: through conversation, through her art work and other creations, and, very often, through her physical movement.  

Here are some summaries of and links to blog posts from the past couple years that documented this phenomenon of "the mind needing a body to think with".  At the very least this will give you a peek into what I see when as I watch a child physically interact with her world. 

I'll start with a potent example in full, and give excerpts for the rest.  My daughter was six and seven in these examples.
Think Like a Straight Line (June 14, 2012)
It's been a loooong time since the kid has ridden her bike.  So long it seemed like the first time again today.

She felt wobbly.  Steering was a challenge.  So, she gave herself a pep talk as she worked to reacquaint herself with the activity.

"Okay, all I have to do is think like a straight line in geometry..."

She rode back and forth across the basketball courts chanting her new her mantra.

"Think like a straight line, think like a straight line, think like a straight line in geometry."

When she'd get to the end of the court, she'd get off the bike and turn it around.  

Then she figured she could make the turn without getting off.

"All I have to do when I get to the end is think like a circle...."

I'm sure she'll be back in the swing of things in no time.  Plus, I love the thought that pathways have specific intentions.  She's in the math, man.  Totally in it.

"Look Mama!  I can do Origami with my body!" | Origami Twirling Bird: Points, Edges, Turns, Poetry and Poses | August 25, 2011

"We've read Sir Circumference the first Round Table a number of times.  Now she has a game she made up where she leaps towards her blow-up wading pool in what she calls the "diameter jump' -- I hold my breath every time as she leaps, finger tips to toes stretched out in one long line to touch the front and back of the pool at the same time, literally flying, flopping almost on the other side of the pool."  Spontaneous Math / Math All Around | August 19, 2011

This next post relates to body knowing because it is built around the fact that we went on daily walks all over our little city.  Many times we would set out and I'd let my daughter navigate us downtown. The map of our city and our experience in the real territory in the map made for a very potent game. | Totally Territorial: Cats, Maps, Area and Multiplication (April 3, 2012)

How we came to understand scale: "If an ant weighed fifty pounds (the weight of a human child) how many pounds could it lift?  My girl counted it up on her fingers and immediately sprang up and ran around the living room trying to lift up all the chairs.  I nixed that idea, but it was such an immediate reaction that it sparked the idea that this needed to be an interactive experience." | Ten Times Better, Longer, Faster, Farther: Understanding Scale | January 11, 2013

This final example is from some summer work in the city: "The girls in the room were hanging out with me before class while I set up and helped me tape out the floor.  Any time I have a chance to let kids help me tape, from preschool to upper elementary, my helpers invariably end up spontaneously exploring their newly taped environment without any prompting.  This is actually my favorite time with kids -- manipulating the floor space with tape and then seeing what they do when they first discover it.  Here's a peek at the space and the only part of their exploration I could capture on video." | Floor Tape How Do I Love Thee? (Video Edition) | July 15, 2012

Monday, October 14, 2013

Meaningful Non-Dance Movement in Math Learning

My conversations with Christopher Danielson over the last couple months about dance, math, Papert and learning have inspired me no end.  He's a great provoker, and I say that with the utmost respect, especially in the area of question asking.

One big question he had for me has gone unanswered for what seems like months, even though it's been just over a week. I've been thinking intently about other related topics but his question has been in the back of my mind the whole time. Christopher asked:

"Do you have examples of meaningful movement in mathematics teaching that are not dance?"

The answer may take many lifetimes of work, but we can still benefit from partial answers and that is what I provide here.

To start, meaningful movement in mathematics learning can be either dance or non-dance.  Dance implies a meaningful system in itself -- in my work, for example, percussive dance steps can be created using a variety of movement variables authentic to the art form coupled with a musical aesthetic.

Examples of non-dance movement in mathematics learning has been a little harder for me to nail down.  This is what I have so far, please feel free to add to this list.

1. Meaningful non-dance movement in math learning happens in the natural body system of gesture and everyday movements (as shown in the work of Susan Gerofsky and this study that showed that 'children think and learn [math] through their bodies. Also, here's a past blog post of mine with links to more research and thinking on this topic).

This body system of gesturing as both a way of expressing knowledge and a way to think through ideas (mathematical ones, specifically) is at work whether we have noticed it or not.  As with anything related to body knowledge, we need to grow our movement/math learning eyes so we know what to look for in our learners.  A recent post on Christopher's Talking Math with Your Kids blog shares a story of a child using her own body knowledge to essentially discover one-to-one correspondence.  This is not necessarily gesture, but it is a potent example of something we could notice in this realm.

The idea of gestures as non-verbal expression and thinking makes sense to me. On the whole, we tend to consider real knowing/learning as verbal and symbolic output.  All I can say is that this hyper-focus on educating ourselves "from the neck up" coupled with the disappearance of hands-on learning-by-making in school (shop class, art, music, etc.) has alienated generations of children and convinced them to think they are really not all that smart when in fact that is not the truth at all. I was one of them.

2.  Meaningful non-dance movement happens in a system where the child is using previous experiences in her body, or creating new understanding through her body, during exploration of mathematical ideas and concepts in school or with adults...and has agency over the exploration.   

What does this mean?  Seymour Papert's LOGO turtle geometry as described in his book Mindstorms is a great example. Papert coined the phrase 'body syntonic' to describe this kind of body knowledge -- "ideas which are compatible with one's own feelings of being in a body." [source]

But, as I've stated recently, just because you have a body does NOT mean you will automatically be able to develop ideas from it or access it in learning.  Papert was presenting a way of learning that still, for the most part, has not been fully understood in formal educational settings.  Essentially, the work Papert was doing with his turtle was to create a learning environment that provided enough structure for children to learn mathematics on their own. [Take a minute to let that sink in...]

This is agency -- the freedom to lead your own exploration and make mistakes on the path to new understanding.  When a child is thinking about the choices she wants to make with the Turtle, it is her own body knowledge she relies on.  That's agency. What is not agency is setting up that turtle and giving a class the exact same step-by-step directions on how to make it draw a flower, or a house, or whatever.

Where else can a child build and call on body knowledge in a setting that allows learner agency?  I just got home from the FroebelUSA conference where we got to experience most of the ten Froebel Gifts. Friederick Froebel was the guy who invented kindergarten over 200 years ago. If you have blocks and math manipulatives in your classroom or home, then you are experiencing Froebel's legacy.  If you know of Waldorf and Montessori, then you know a little bit about Froebel because they are shoots from the root of his system.  If you are familiar with the names and work of Buckminster Fuller, Kadinsky, Frank Lloyd Wright or the Bauhaus, then you know of people who attended Froebelian kindergartens (ages 3 through 7 but extendible to any age.)

Froebel's gifts are essentially what Papert might call 'objects to think with' -- starting with a wooden sphere that can fit in the palm of your hand, on to a solid cube, and then various interesting divisions of the cube, and other gifts to explore surface, point and line.  Here's a picture of Gifts 3 and 4:

In the Froebel system there are three main ways to experience the gifts: using a narrative context in which to explore the properties and powers of the materials, explore the mathematical properties of the materials, and as a 'form of beauty' including exploration of symmetry and patterning.  Sometimes the gifts are presented in a guided way, but it seems that there is plenty of opportunity in the Froebel system to explore these materials freely, with personal agency.

My main point:  

As Professor Eugene Galanter (one of the founders of cognitive psychology) said during his keynote at the start of the FroebelUSA conference:

"The mind needs a body to work in."

As I write this I am finding all sorts new questions in these ideas I've set out and it's clear that this topic requires much more than a single blog post.  I've written more in depth about the differences between dance/math exercises (low to no agency), lessons (potential for limited agency) and truly exploring and making mathematical meaning through dance making (high agency) in the Math in Your Feet program.  You can read and download the newly published article here.

Let's keep this going: Am I missing anything?  Any holes in my argument? What other examples of meaningful non-dance movement learning can you think of?

Addendum, October 15, 2013: Here are some more specific examples of a child (my own) learning math through her body. "Thinking Like a Straight Line": Examples of a Body Learning Math

Thursday, October 10, 2013

Challenging a Literal Approach: Learning Fractions 'through' Music

The last week or so I’ve been thinking quite a bit about what makes real connections between math and dance in a learning setting.  I’ve come to the conclusion that none of those connections can even be attempted without first looking at our assumptions about what math is and what dance is. 

I’ve been sharing my thinking on this blog, to an audience which (I assume) is more familiar with math learning than with arts learning and over at ALT/space, an online writing project I edit and curate for the Teaching Artist Journal.  I’ve not had a huge amount of responses, but the quality of the feedback I have received has been incredibly helpful in moving me forward on this line of inquiry.

Most recently I had a great conversation with friend and colleague Nick Jaffe.  Nick is, among many things, an incredible musician, teacher, thinker, writer and editor.  He’s not a math educator, but he’s definitely not math averse. 

We were talking about how people often try to teach fractions by connecting it to musical notation, which I have always seen as a very literal approach to what people perceive as music (written notation).  Over the course of the conversation I finally understood why I had always viewed that as a very shallow, un-meaningful activity.  Specifically, Nick has a lot of interesting and helpful things to say about discrete and relative quantities when playing music so I thought I’d share our discussion here in case it is helpful to you in any way.

Me:  I have seen very little evidence that making the correlation between fractions and music have had had any impact. Maybe one study, but they didn't let the kids actually play around with ideas or make their own music -- they just taught them how to play the fractions. And, seriously, do you think like that when you're *playing* music? 

Nick: When people ask me about music and fractions, the first thing we need to establish is that the analogy between musical notation (or the musical reality it describes) is multi-dimensional. Notation denotes both the note as a discrete duration, but also as a division of a larger whole. Which, as I understand it, is also in the dual nature of fractions and an important thing to grasp, and not entirely intuitive.

What seems essential, at least in elementary and middle school math, to understand about fractions is that they have a dual function mathematically--they are a way of signifying two things at once: a discrete quantity, and a relative one. Musical notation functions the same way rhythmically.  In both the abstract, mathematical context, and in the context of music, the same problems arise with regard to this dual nature.  And it is somewhat counterintuitive to students at first to consider this--it's never explained clearly but it's the crux of everything one does with fractions as far as I can see.

I like the question you raise about how one thinks when making music. I think the answer is yes and no. Let's leave aside conscious methods of fractional or theoretically driven music making. Let's consider improvisation which would seem to be the most spontaneous, least theoretical approach to music making. And let's just stick to fractions as a rhythmic concept for this particular argument.

When I improvise I do not consciously think about the duality of a note as a time duration (discreet and part of a larger repeating whole). However I absolutely have to manipulate time, consciously and/or unconsciously as both things.

I think that duality is at the heart of choices about phrasing, for instance. You have to feel a note as a discreet duration, an almost word-like gesture. But you also have to feel it in relation to a pulse, even if there is no pulse, or you are trying to erase any implied pulse--it's still in reference. One does not have to consciously think in terms of fractions to do any of that. However, learning to read notation, and practicing manipulating that duality (playing behind, on, or in front of the beat for instance) can often increase one's control over it.

Finally, analyzing, generalizing and theorizing about that same duality can open up new options, or allow the translation of one dynamic application (actual playing) from one musical context to another.  In that sense theory is both a stimulus to new ideas and means of translating an idea for different contexts. I think theory plays those roles in most disciplines--a generalization that makes possible the prediction of as yet unknown dynamics. 

Me: The problem with using music notation to teach fractions is precisely what you just explained.  The written notation is the quantity, and the actual musicality/music playing is the quality.  What people don't get about mathematics is that 1) it’s not just about the notation (same for music) and 2) it's really about the quality of the quantities.  If you focus just on "how much" you get a dead language or, at the very least, information devoid of meaning. Which is why so many people hate math – for many reasons, and I’m speaking very generally, math education has essentially been bleached of its meaning.  

Nick: I’m actually not that interested in the idea of using musical notation to teach fractions. That seems boring and highly inefficient. What I am interested in, and I think kids might be as well, is working with music and fractional ideas at the same time to make interesting things. Perhaps those things would be musical, perhaps mathematical, perhaps visual, perhaps both.   Undoubtedly one would learn various things in doing such work, but I think it is the making that is what is appealing to me (and perhaps students) and that any real insights and learning depend on the impulse to explore in order to make. 
 To sum things up, Nick’s thinking about the nature of a mathematical idea (in this case fractions) in relation to his art form is exactly the kind of thinking process I engaged in to find a meaningful overlap between math and percussive dance in Math in Your Feet. It is also a perfect example of how it is possible to teach parts of math starting with your own experiences inside another discipline.  

Where to go next? One of the best music, math and movement resources I’ve run across is from Ellen Booth Church. The activities described in this article describe perfectly how the core concepts of mathematical activity can be connected to musical and movement activity.  Ellen is an early childhood specialist, so her ideas are based on that age group and applicable up to about grade 2, but I think this is the perfect starting place for anyone interested in making these kinds of connections in their own classrooms. 

Monday, October 7, 2013


I've been thinking intently for the past week. I've finally come to the conclusion that the challenge we face when bringing dance or movement into the picture during math time is not necessarily related to creating meaningful and effective learning experiences for our students, although these are certainly important concerns.

No, the issues we collectively need to address, before we can even start that process, are our deeply held beliefs about what math is and what dance is.

If math learning means number facts, right or wrong answers, learning algorithms, memorizing procedures, and experiencing math topics in isolation from one another then this video makes perfect sense to me. (I love the energy here, but question the assumptions.)

Or, this -- a very strong example of non-dance movement but, again, with the ultimate goal being memorization of math facts.

If, on the other hand, we can come to not only accept but truly understand the following vision of math making and math learning:
"Mathematics is a highly creative activity.  Mathematicians solve problems, but they also pose problems. They inquire. They explore relations. Investigate interesting patterns and craft proofs.  They present their ideas to the mathematics community and those ideas hold up only when the logic of those arguments are accepted. They don’t have a wise one who they line up for to check their answers with a red pen." - Cathy Twomey Fosnot (excerpted from this Context for Learning video)
...and if we can at least consider, as I argued recently, that the body is more than a drawing tool...

...maybe then we could come to accept (and eventually understand) how body knowledge is different from but not inferior to what we see as 'real learning': verbal and written discourse and reasoning abstractly through the medium of notated language.  If we could do this then perhaps eventually we could create some clarity on how the body can be more than simply the handmaiden to the goals of other disciplines, specifically math, in educational settings.

I'm still thinking on all of this, and it's for sure a good kind of think, but I do wonder sometimes if I'm setting the bar too high. I'll leave you with what I know:

- Kids love to move.

- Kids love to move, but there are different kinds of moving and different kinds of learning-while-moving.

- In her book, Smart Moves: Why Learning is Not All in Your Head, Carla Hannaford said, "Learning, thought, creativity, and intelligence are not processes of the brain alone, but of the whole body."

- There are ways to bring dance and movement into math learning and still maintain the integrity of both disciplines. My recent article in the Teaching Artist Journal goes into further detail about how this can come to be.

Saturday, October 5, 2013

"An Object to Think With"

The point of this post is to push back a little on the idea that simply being out of one's desk and on your feet will create what Seymour Papert termed 'body knowledge'. Specifically, my goal, now and in the future, is clarification of what it looks like to combine movement/dance with math learning in meaningful ways.

This post is the result of conversations inspired by this video from Simon Gregg showing an activity he did on factors and primes on the playground with his 8 and 9 year old students.

After he posted it, there was a Twitter conversation between myself and Christopher Danielson which I Storified here. This led to my post Starting the Conversation: Meaningful Movement and Math Learning (including a video using modern dance to illustrate statistics concepts). A day later there was a blog response from Simon and, finally, here is my response.
First of all, I really don't buy the idea that just by being on one's feet means there is opportunity for body knowledge. Simon said in his blog post:
"Another thing - one Malke and Christopher didn't mention - to focus attention, it might work well if some kind of game was involved at some point."
This sort of makes my point all on it's own -- Simon's Year 4 students were on their feet but there were a lot of distractions and, as I'm sure he knows, no opportunity for focused movement. Meaningful movement is created when the focus of the lesson is on a body task and the mathematical concept at the same time – the two together create the focus for each other.  In addition, both need to be developmentally appropriate. What Christopher described doing with the school yard hundreds chart (noticing pathways and the distances between numbers by standing and moving on the chart) highlight, among other things, important numeracy skills but I am looking at it as a movement activity as well.  The kind of body activity Simon and Christopher proposed in concert would, I think, be better suited for 5, 6 and 7 year olds and probably provide very little physical challenge for the upper elementary students in question.

So, if you're interested in creating more focus during 'stand up' math learning, what kind of movement skills can you embed into a hundreds chart activity like the one shown in Simon's video? What is the most important concept you would want your students to understand about primes? When you think of prime numbers, does your mind go to the intervals between them? If so, is the hundreds chart itself really the best visualization for that particular idea? What exactly is the big idea connected to prime numbers? And, if you were going to create a movement game what would be the point of the game physically and mathematically? 

If you are having trouble thinking of answers to these questions I think I know why. There is a perception I think many people hold about using movement in concert with math learning -- that they are primarily thinking of the body as a drawing tool. 

My standpoint on all this is deeply tied to what I see as a huge qualitative difference between mathematical representation and modeling and more literal illustrations of mathematical procedures or definitions (watch this video where the dance is little more than animated illustration for basic ideas of statistics). For one thing, definitions and procedural concerns are not math and dance is much more than literal interpretation. I know I need to develop my argument a little more but, for now, my point is that you are not creating body knowledge if you can just as easily illustrate (draw or identify) a math idea (like a right angle) using your eyes or a pencil and paper. If you can, there is probably no need to do a similar activity with your body.

Your body is an expressive vehicle and, as Papert intuited, an "object-to-think-with" capable of deep knowing that is rarely called upon or valued in academic learning. The body can be the perfect math manipulative, the perfect tool for children to use as they explore and experience the big ideas underlying mathematical activity – if used properly. Simon mentioned Dienes' principle of multiple embodiment. Here's what I took away from an article about Dienes' work I found on the Rational Number Project site regarding the use of math manipulatives (bolding emphasis and brackets mine):
“When individual activities [in this case with the body] cease to be treated as isolated actions and start to be treated as part of a systematic pattern of activities, the student begins to shift from playing with blocks to playing with mathematical structures [that's what we want] Yet, when concrete materials have been used in instruction, more concern is often given to the "concreteness" of the materials than to the "activeness" of the activity - as though the abstraction were from the materials rather than from the structure that must be imposed on the materials.”
This takes me back to my original point. If using math manipulatives does not automatically connect you to the underlying mathematical meaning then being on your feet is not a guarantee of building ‘body knowledge’ in mathematics or any other domain. It goes the other way, too. Just because you’re a skilled mover does not necessarily guarantee you will be good at math. The meaning is created through thoughtful construction of activities or even whole sequences of investigations where both modes, mathematical and movement, come to influence the understanding of the other. Easier said than done, bu it is totally worth the effort, so let's keep thinking and talking.

In the end, you may disagree with me and that's okay -- as Christopher said during our recent Twitter conversation, “I would love to hear you argue back."

Wednesday, October 2, 2013

Starting the Conversation: Meaningful Movement and Math Learning

I had an interesting conversation on Twitter today with Christopher Danielson about body knowledge (Papert) in relation to mathematical knowing and learning. I've Storified it now but here's what showed up in my in box later in the day -- perfect timing, as they say.

Some thoughts before you watch:

1. This is the first dance/math video I've seen that I've not been grumpy after watching it. I think I also finally understood statistics. That being said...

2. Could these concepts be illustrated effectively in some other way, meaning without the moving human bodies?

3. What, if anything, have we learned about dance in the process of watching this video?

4. What, if anything, could be learned by turning this into a dance lesson?  Would we understand anything more or differently after creating a dance based on the ideas in this choreography?

5. Would the dancing make sense without the text on the screen before and after the dancing?  Would the math in the dancing make sense without the text?

If we are serious about using movement, or a specific dance form, in our math classrooms, I think it's worth thinking and talking about these kinds of questions. I'm not sure I have good answers for all of them, and I will weigh in, but I'm curious to hear your perceptions and thoughts first!

My First Counting Circle!

It's crazy how much energy I get from teaching.  To be honest, as a freelancer, I'm not in the classroom on a full time basis, but even when I'm physically exhausted from daily driving on a residency week (two to three hours round trip for five days in a row) I am still mentally exhilarated from working with kids as they engage in making math and dance at the same time.

Now that we're not homeschooling, I've started volunteering in my daughter's 3rd/4th grade classroom during math time, 30 minutes with each group, and every time I volunteer I leave abuzz with energy. I am not a math teacher, per se, but in addition to my work with Math in Your Feet I've spent the past few years with my daughter focusing on mental and conversational math; essentially, learning math through games, making projects, and finding examples of math in the real world.  Now that she's in school I am still so curious about what comes next and about how kids other than my daughter make mathematical sense of the world.

Here's a confession: I used to be afraid of numbers. Every time numbers would come up in conversation I'd get a tight, anxious feeling in my chest and do my best to change the subject.  After a couple years of remediating myself by learning math along with my daughter, I have found that whole numbers are actually funny little friends who I might actually understand. I like having new friends, so this is a fabulous turn of events.

Earlier this month I made an investment in some math ed books including one called Number Sense Routines. I didn't think I'd have a chance to use it, I was just curious, mostly because I like reading about helping kids learn math.  Then I got wind of Sadie Estrella's work (she's @wahedahbug on Twitter) with counting circles. I read her stuff  and made the connection between what she is doing and my new book.  It's great to have both -- I love watching Sadie teach and it really helped me jump right into my very first counting circle today!

One of the points made in the book (it's called 'count around the circle' there) is that ground rules are important to make sure the quick thinkers don't take over the activity.  I reminded my kids that we needed to all wait until it was our turn, that we should count along in our heads, and to allow someone 'think time' if they needed it.  Also, that we would be doing a counting circle every time we get together.

Oh. My. Gosh. Not only was it an incredibly focused, almost meditative, experience but after five minutes of what seemed like a very simple task, it was like a light came on and I could actually see where the kids are on their own paths to numeracy.

First, since we we've been working on making our own Sierpinski Triangles we've been talking a lot about infinity and the powers of three. So, I thought for my first counting circle I'd do something with intervals of three. From previous interactions, I figured the kids could already do the normal kind of skip counting by threes (3, 6, 9...) so I decided instead to start at 7.  For both of the groups, things went very well until we reached the transition between 19 and 22, and at every similar transition up to 60.  This is where I was able to closely observe kids' strategies for figuring it out.  If they didn't know the next interval, most counted on their fingers or whisper counted by ones.  For one of the groups, as you can see in the picture, I had to visually draw individual 'jumps' and talk out the adding.  For example: "We're on 39, let's count...40, 41, 42 [drawing each individual jump as I spoke]."

In general, the counting was okay up to about 19 but required some real thought after that. Assessment wise, it showed me clearly that many of them have not yet internalized number bonds to 20, let alone transferred and generalized that knowledge to larger numbers.  This is thrilling news for me.  In part because I got such a interesting picture of where their numeracy is in a way I would have never have seen otherwise.  And, frankly, also because although I'm not really a math teacher, I LOVE TEACHING MATH! It is so fascinating to watch kids learn math.

I have a lot more to say about my hour or so at school today but I'll end with one small vignette. There was one boy who was quiet during the counting circle but when it came to his turn (small groups meant everyone could have at least three turns) it was clear he was not following along.  After counting, we went on to our Sierpinski triangle project.  Here's what he did:

I know it's not what you were expecting, but I see so much thinking in this picture. First of all, he was a natural at approximating midpoints. Second, when he first drew the little blue triangle at the top I had no idea what he was doing. I initially tried to redirect him toward dividing each iteration in the order we had discussed but while I was talking he drew a line across and just under the top little blue triangle and all of a sudden I saw what he saw. I encouraged him to continue on his line of inquiry and this is what resulted.

I was so thrilled and told him I never would have seen that kind of structure or pattern if not for him.  He told me he wanted to be an inventor when he grew up and I said, well, that's part of what inventors do -- they take things we think we know and turn them into something new.  It might be cliche to say it this way, but he was so proud of himself that he practically threw off sunbeams.

I think maybe next time he'll give the counting circle a go.


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