The Map is Not the Territory

I have come to terms with the fact that different people see different things in Math in Your Feet depending on where they stand. Based on feedback from a wide variety of teachers I've had the honor of working with this summer, as well as my own perspective, here are some possible answers to the question:

What is Math in Your Feet, really? 

• Low floor, high ceiling (useful and interesting to diverse groups of learners and backgrounds)
• Geometry topics
• An in depth inquiry into mathematical patterns including explorations of transformations, symmetries, group theory and equivalence classes.
• An opportunity to use mathematical language in context. 
• A chance to build and strengthen spatial reasoning, what I call "the step-child of mathematics education".
• A chance to harness existing body knowledge (developed through being in the world) to strengthen understanding of mathematical practices and topics.
• Potential for developing new insights about previously familiar mathematics.
• Inspiration for mathematical question asking in fourth graders and (open-minded) research mathematicians alike.
• A major cognitive schema. The Source-Path-Goal Schema, to be specific. ALL of it.

WAIT! What?!?

Yep. A schema is a cognitive framework, essentially a mental frame that helps us organize, sort and classify sensory input into something that makes sense to us. The book Where Mathematics Comes From by Lakoff and Nunez makes a comprehensive argument for how the core schemas identified in cognitive science also come to influence the development of mathematical ideas. Lakoff/Nunez call the source-path-goal schema "ubiquitous in all mathematics" meaning: this is how we need to think when we do mathematics at ANY LEVEL.

We build MiYF dance patterns by asking:"Where are we starting [source] , where are we going [goal], and how are we going to get there [path]?" 

During this process we use the following categorical variables to create our patterns which help us think about ___location (foot position) and the body's trajectory (direction), and how exactly we're going to get from point A to point B (movement)

The Source-Path-Goal Schema ("ubiquitous in all mathematics") includes the following:
[Direct quotes are presented in italics here and can be found on p37 of the book.]

A trajector that moves, like these guys:



A source ___location (the starting point):



A goal--that is, an intended destination of the trajector. In this case, both boys are turning left toward their intended destination facing the back of the dance space.



A route from the source to the goal. The route of girl on left is a left turn. The route of the girl on the right is a right turn.



The actual trajectory of motion. 
The position of the trajector at that time.
The actual final ___location of the trajector, which may or may not be the intended destination.

In the picture below, the girls have reached their intended position, the front right diagonal of the square:


We had a fantastic time with embodied mathematics at Twitter Math Camp 2014, but on top of being highly engaging it also brought up a really important question among the math educators involved:

What is the role of embodied mathematics in our classrooms?

As we move this question forward together we need to remember that all learners (even adults) need experiences with the processes of math in multiple modes and settings. With these kinds of experiences, including the body-based ones, math learners are well supported to engage in mathematical content in meaningful ways.

Ultimately, in the early years of creating Math in Your Feet I didn't explicitly set out to build the program around the source-path-goal schema but I asked honest questions about what math is and how it's learned. These are the kinds of questions that can put us in a good place to start uncovering the hidden metaphors carried in our bodies. From there it's not too hard to envision the path toward using these ideas in creating meaningful, useful body-based lessons for classroom use.

There's so much more to talk about in relation to this topic, but I'll stop here for now. We have our "source" question. We can see the "path" ahead as well as the "goal". Let's get started! Together.

A riddle, by my newly 9 year old daughter, presented in 20 Questions style, at 7:00am in the morning.

Her: Mama! I have a great riddle!

Me: I'm trying to read the paper, I'll be there in a minute...

Her: But this is SO great.

Me: Okay...

Her: I'm thinking of something that is whole, but not whole.

Here is what my guessing revealed:

1. Not alive
2. Not an object, but can be (her emphasis)
3. Made of many parts
4. Parts look similar to each other
5. Not a reflection
6. It's in our house and other houses too.
7. We use it every day.

Her: Here's a hint, it relates to math.

8. It's an idea.
9. We use this idea.
10. The idea helps us do stuff around the house
11. Idea helps us do housework.
12. A specific kind of housework? Sometimes yes and sometimes no. Maybe.
13. Helps us clean.
14. Helps us clean inside and outside.
15. Helps mama think about our day

Me: Is it the calendar? [And, because I thought we had gotten to 20 questions I continued] Okay, just tell me!!

Her: IT'S A WHOLE FRACTION!!!  Like when you cut carrots into sixteenths [we have never ever counted carrot pieces]. Look...[she takes my pen and writes something like this]:

1 = 1/2

Me: But one isn't equal to one half, is it?

Her: No, but it's a whole half. It's a whole fraction but still a fraction of something.

Me: So you mean the 1 is whole, but when you divide in half, the 1/2 is a new whole?

Her: YES!!!  Can I watch Frozen?

I think this calls for a celebratory viewing of this wonderful video:

First off, what's the difference between this:


and this?




Which one provides the more meaningful learning experience with multiplication? 

I mean, just look how many ways we can experience and come to understand multiplication! Stunning. Given this reality, why would we want our students to only understand multiplication as a series of facts?

So, now, take a look and tell me the difference between this (done as part of a computer science education project):



And this (start around 0:25 and watch until at least 2:00):



Which one provides you the more meaningful experience with Hungarian folk dance traditions?  

Can you see what happens when the dancing's sole purpose is to be shoehorned into a formal mathematical framework (the sorting algorithms)? 

To me, Video #1 has some interesting footwork but the choreography seems stilted and out of context -- sort of like only ever memorizing multiplication facts.

This may just be my own particular sensitivity but I'm curious what you think. Bonus points for going to the dance/computer science project website, trying out one of the sorts and reporting back how or if the dancing helped you any more than the computer animation they provide.

I don't think I've ever done a post like this before but, then again, I don't think I've run across so many wonderful, thoughtful, helpful blog posts in such a short time.  In other words, there's been a plethora of fabulous thinking and writing showing up in my feeds, all of it related to math education in some way.

Here's what has inspired me in the last week or so:

From Michael Jacobs in Canada comes some really interesting thinking about connecting spatial skill development to math class in intentional ways.  He alerted me to the fact that my Mathagogy video was being shown at a meeting focusing on this topic. This chance reminder about spatial reasoning reminded me that as a dancer teaching math and dance and writing a book about math and dance I really want and need to understand this topic more deeply. All this got me thinking. I love it when that happens.  Here's Michael's post on spatial reasoning.
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Another chance encounter on Twitter was with Edmund Harriss and his post Rational Parameterisation of the Circle.  The images above are Lissajous curves and they made me wonder how they might be used as interesting choreographic prompts; I immediately started wondering how to turn all that math into a meaningful math/dance inquiry in the classroom.  Our conversation helped me realize that it might actually be a really cool project. Hopefully we'll be able to work out some more ideas about this while at Twitter Math Camp in July.
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"Today's little dance/math tidbit, an odd over even experiment ...Ultimately, I can only do the 13 beat phrase twice w/out naturally trying to even things out, partly because the music & dance work together so closely.  Putting the dance 'at odds' with the music can have some really cool results, but in the case of this experiment I may need to dial it back a little."  

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Then there was a lovely, detailed article in The Atlantic blog featuring an interview with Maria Droujkova about her vision for helping young children and their parents make math meaningful in their own lives.

“You can take any branch of mathematics and find things that are both complex and easy in it,” Droujkova says. “My quest, with several colleagues around the world, is to take the treasure of mathematics and find the accessible ways into all of it.”

I have gained so much from interacting with Maria through her Natural Math and Moebius Noodles projects, and am honored to be included as a contributing blogger on the Moebius Noodles site and also asked to co-author a book with her Delta Stream Media company.  Keep your eyes out for Socks are Like Pants, Cats are Like Dogs a book of games, puzzles and making activities around the ideas of variables, attributes, sorting and more which I am co-authoring with Gord Hamilton of Math Pickle.  Our manuscript is almost complete!
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This week there was also an excellent piece from Edward Frenkel, How our 1,000 year old math curriculum cheats America's kids.  Read it together with Maria's interview piece and the two together really pack a wallop in terms of creating a beautiful big picture of how important it is to include the exciting big ideas in math learning.

I had a very interesting conversation with my eight year old over math homework this morning revolving around the commutative property.  Interesting because of her thoughts and also interesting that, for the very first time, I backed away from mathematical correctness, and truly listened to what she had to say. It was fascinating.

The homework asked for factors of various two digit numbers. For 24, my kid put 2x12, 1x24 & 24x1.

I said, "Those last two are the same thing, what other factors can you figure out?"

The response was immediate and somewhat intense. She was convinced that 1x24 and 24x1 were different because that is what the teachers said.

I mentioned we had read about the commutative property in Beast Academy 3B but, sweetly, nothing could sway her loyalty to her teachers and her opinion about what she thought they had taught her.

It was at this point I thought back to all the things Christopher Danielson has written about Cognitively Guided Instruction and the wonderful modeling of his Talking Math with Your Kids project. These approaches show the worth of conversation around math with an emphasis on the adult really listening to what the child is thinking.

She continued. "See! One times twenty four is [pointedly counting] one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve...twenty four.  Twenty four times one is...[pausing, then saying emphatically] twenty four."

I nodded. "Oh, I see what you're thinking. The first way means you have to count by ones 24 times. The second way you just have to say 24 once."

In her mind it's the process of getting to the final answer that makes the two facts different. Never mind that she gets the same answer both ways. Never mind that she knows all about the "twin facts" on multiplication chart. Never mind that we're having fun finding different ways to memorize multiplication facts including sneaky guerrilla tactics. Nope. This is her reality and it's not going to budge by quoting official definitions.

All I said was,"You can put those two facts on the paper, but your teachers may want you to put some others as well."  In the end she found all the factors of 24, but wrote each combination twice (e.g. 6x4 and 4x6).

In the process of writing my new book, tentatively titled Meaning in the Making: The Body Learning Math, I've been doing a lot of reading and thinking about how the processes of doing and learning math are just as important as the product.  In this case, she can easily figure out factors of two digit numbers, but it's by watching her process closely and engaging in conversation about her thinking where we really get a glimpse into what she knows and how she knows it. Specifically, we can see how she is literally making and reasoning out her own meaning of how multiplication facts are combined. 

We only get half of the picture if we look at the final product/answer (double facts). I know how to watch for and identify understanding through the processes of making math and dance at the same time, but now I'm really learning about how it works with numbers, too! Fun stuff.

First, there was Amelia the hand-made doll and her essay "What Infinity Means to Me." That was early in 2013.

"Mama, is a curve a line?"

"What do you think?"

"I don't knowwwww...."

"Oh, come on, you asked the question, I bet you have some thoughts about it."

Showing me her Etch-a-Sketch and turning the knobs: "Okay, this is a line [drawing a straight line horizontally to the left] and this is a line [drawing a line vertically upward] but this is not a line [squiggling the line back and forth]."

"Why isn't that back and forth drawing a line?"

"Because it has to go in one direction. This curve is a line, but not when it goes back and forth..."

I left it there because the kid was sick, it was time to start resting and I know answers often rely on more questions about and interactions with the idea at hand. I'm sure this'll come up again at some point in the future. 

Approximation of original by the mother.  As users of Etch-A-Sketches will understand, the original got destroyed with an inadvertent shake.

I've just started a book project.  What's it about?  Well, for now, I think it's about digging into the whys and hows of learning math and dance at the same time. This may change, but that's where I'm at right now.

To organize and clarify my thoughts I've been doing quite a bit of background reading, having e-mail/online conversations with various wonderful people, and drafting some preliminary chapters in the blissfully serene Silent Reading Room at our local public library.  Tons of natural light, total silence except for the hum of the HVAC and the occasional cough.

One day last week I sat down to read and felt that old feeling coming on that I used to get in college. It signifies I am completely saturated and overwhelmed with thinking, reading and writing. It makes me a bit crazy, honestly. To combat it, I've learned to skim text and only read closely when I need specific information. Even better, go for a long, long walk with far away vistas.

I call that part of my brain my 'word-mind' and it is very clearly located in my head. 

Using my word-mind is a completely different experience from thinking with my body-mind.  I know, because after college I started dancing and was introduced to another part of the thinking equation -- the body thinks too, a phenomenon that is defined and described by studies in embodied cognition (a branch of cognitive science).

The body thinks, too.  This was truly a revelation to me.

Ultimately, there were years (and years) where I was required to use only my word-mind to learn. And then there were bunches of years (as an adult) where I learned and expressed myself solely through my body mind (dance and music performance, never wanted to get on the mic to talk). My body-mind thinks differently than my word-mind, but what I have come to realize is that, in addition to needing to find a balance between the two in my daily life, as a learner I learn best when I am using both at the same time. 

I was in Minnesota this summer, sharing my work with a group of teachers and teaching artists. That's when I met Christopher Danielson in person. Yes, an educator of math educators came to my workshop. I was a wee bit nervous but I needn't have worried.  Even though in my workshops we make dance and math for a large proportion of the time, Christopher was game to dance and dance he did.  He also took copious notes by hand at various intervals.  I was again a wee bit worried, but it turned out our 90 minutes of dance and math making had gotten him thinking.  In a good way. By engaging with the work of Math in Your Feet from the inside of the experience instead of simply watching, he generated many new questions that moved his thinking forward in new ways.

The next day we met for a breakfast conversation during which, in relation to one of the questions he had while dancing, Christopher mentioned a now defunct unit in the Connected Mathematics series that investigates transforming squares in a series of combinations and is related to the thinking in modern algebra.  Later, he sent me the book.  I meant to get started on it but life took over.  And then my book took over.

This week my word-mind gave out. I was literally a little dizzy from thinking in my head and was completely DONE with words. I do a lot with language these days.  I keep a blog, I'm active on Facebook and Twitter, and I edit a year-round, international online writing project for the Teaching Artist Journal.  So to have reached my limit with words is saying something.

I needed something completely different to do.

I pulled the Connected Mathematics book off my desk, sharpened my pencil and headed for the library.  I made my little square out of nice stiff paper. I labeled the vertices.  I combined series of turns and flips, filled in the table and investigated the patterns I saw.  For a good portion of the time I was completely frustrated and confused -- turns out I was turning clockwise when I should have been turning counter.  Apparently there is some convention among mathematicians that all rotations go counterclockwise.  They must have all been left handed and never learned analog (clockwise) time because this right dominated person kept seeing the "R" for rotate as meaning "turn RIGHT."

At some point, though, I was looking at my hands, turning, flipping, holding corners to orient them as I turned and flipped, and I realized I was in some kind of zone. I was talking to myself in words: specific conscious reminders to turn left when rotating, murmuring less conscious words while filling in the chart, a gentle narration as I talked myself through each new sequence of flips and turns and, eventually, the body and words merged and I was just doing.  I had fallen into a zone of concentration while making math that mirrored what often happens when I am learning something new in dance.

After making numerous combinations of transformations with my hands, I now have a new feel for what a square is and is capable of.  Both the word-mind and the body-mind had a part to play in this new understanding.

I hope it is obvious that this is not an either/or kind of story -- it's about both.  Understanding cognition and thinking and knowing is a vast endeavor and I don't know if we will ever fully understand the complexity of it all.  My focus is on what it means and looks like to learn math with the body as an equal partner and to do this I need to keep the body's way of knowing and thinking firmly in my sights.

Ultimately, what I am wrestling with is this:

We can see the body moving, but most of us have not yet learned how to look at and understand the knowing and learning happening while the body moves.  It's an inside process and not one that a child will necessarily be able to tell you about.  As more experienced learners the onus is on the adults to look for how children show us what they know and think with all their one hundred languages, including the body.

I'll leave you with a powerful paragraph found in the conclusion of the recent study Children's Gestures and the Embodied Knowledge of Geometry (bolding emphasis mine):


"Despite two decades of research on the embodied nature of cognition, constructivist perspectives continue to emphasize abstraction of knowledge from the physical engagement with the world ... In this way, constructivism orients our attention away from the body, concerned as it is with the construction of mental entities and representations...  

Once we acknowledge the body as the seat of knowledge itself rather than as a stepping stone to abstractions, it is possible to organize teaching differently.  In doing so, [this] brings forth the possibility of inquiring into such questions as, 'What might we do to recognize and understand children's knowledge expressed in modes other than speech or writing?'"

One thing you can do is to try learning math and dance at the same time yourself. I'm available to talk about how I can help, if you like.  Feel free to get in touch.

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

An excerpt from my newly published article for the Teaching Artist Journal:

"In the eight years since I first had the germ of an idea about the possibility of connecting percussive dance and math at the elementary level I have had more questions than answers about the nature of arts integration, specifically in relation to mathematics...

"After some years of sitting on this question, and a few more of actively searching for answers, I think I have finally come to an understanding about what is going on in Math in Your Feet. I am much clearer on how the math and the dance  interact beyond specific math topics and vocabulary.  These are answers about the really interesting, important connections between percussive choreography and mathematical thinking, moving well beyond memorization and procedure and into the real processes of doing math.  They are also answers about the choices I made as I brought my art form and the strange, beautiful world of mathematics together for young learners."

Ostensibly, the audience for this piece includes arts educators and teaching artists but I humbly submit that anyone interested in interdisciplinary teaching or learning, especially in concert with mathematics, may be able to take something of value from this piece. And, the same goes for math educators interested in how to harness what Seymour Papert called 'body knowledge' (in his seminal book Mindstorms) in a way that maintains an authentic learning experience in both math and dance.

You can read the entire article here.  This piece is the result of over two years of investigation, thinking, learning, exploring and question asking. I am curious to hear your thoughts, observations and, hopefully what new questions you might have.  Any feedback you wish to share will be extremely helpful as I begin to conceptualize and outline a much bigger writing project that will considerably expand on the ideas in this new article.

For some reason, I thought that when we stopped homeschooling and my eight year old started going to 'regular school' (albeit an independent community school) I would have less access to her mathematical thinking and learning.

Turns out I was wrong.  Two examples:

Walking to school.  The girl is holding her library card like it's made of gold.  She thinks maybe they're going to the downtown library that afternoon and she doesn't want to part with it.  We pass a brick wall built with some of the bricks on the diagonal, creating an interesting pattern.

She stops to notice.  "Look at this cool design!"

She notices something else. "Hey, the library card is the same width as the brick."

And, another thing. "This brick is two and a half....no, two and a fourth times longer than the card."

[All thanks, I think, to the great afternoon we spent last winter with the book Ten Times Better - read about our antics here.]

Second example, math homework, and she actually wants my company.  If you know our history, the fact that she invited me into her math world is truly a momentous occasion.

The homework is titled "Things That Come in Groups".  There are eight juice boxes in a package. She easily figures out how many juice boxes in three packages.

Next question asks how many juice boxes are in six packages.  And....wait for it...she says, "Well, there are 24 in three packages so there are 48 in six packages," and proceeds to write that equation in the spot it's requested.  The next question is about nine packages, and she continues the pattern of reasoning -- 72 total juice boxes.

I suppose I get delighted by little things but, no matter, this is wonderful to watch.  I know what kind of equation they were asking for, though.  It actually took her longer to figure out how to to write it the way they wanted (6x8=48) but no matter.  I love that she saw patterns and easily computed the answers in her head. And, because I know her math learning history, I remember her at age six developing this strategy as she made her very own game about doubles and halves.



"I love homework, mama."

"Why is that?"

"It feels so old fashioned and classic."

I look forward to more math homework because I think it will afford many more opportunities for watching her mathematical thinking expand and deepen.  I love watching kids learn, especially so up-close. Sigh.

What to do with a whole bunch of unused Cuisenaire rods?  Put them on the activity shelf at your kid's school, clearly labeled for play, that's what! 

 

 

 

 

 

Shhhh...don't tell my kid's teachers but today marks the inaugural installment of Marvelously Math-y Mondays.  Yes, I know I'm posting this on Saturday, but I just couldn't wait!

On MMM's I plan to bring something math-y to my daughter's 3rd/4th grade classroom to inspire wondering, noticing, questions and conversations.  I'm kicking things off in style with two variations on the classic multiplication table and one make-your-own version.

At first glance, this article about the value of reading aloud to older kids would not seem to connect to math learning. But, to me it does.  Here's the piece that really stood out:

“The first reason to read aloud to older kids is to consider the fact that a child’s reading level doesn’t catch up to his listening level until about the eighth grade,” said Trelease [a Boston-based journalist, who turned his passion for reading aloud to his children into The Read-Aloud Handbook in 1979], referring to a 1984 study performed by Dr. Thomas G. Sticht showing that kids can understand books that are too hard to decode themselves if they are read aloud. “You have to hear it before you can speak it, and you have to speak it before you can read it. Reading at this level happens through the ear.”

Did you catch that? "You have to hear it before you can speak it, and you have to speak it before you can read it."

I made a similar point while working with teachers and teaching artists in Minnesota last week when participants noticed how the math language was woven seamlessly into our dance work. 

This vocabulary development, I said, was initially an attempt to help kids pay closer attention to what they were doing while they created their dance patterns. I noticed that they became much better creators when they had the right words to help them identify their movement choices.  When I started developing the Math in Your Feet program (after about six years of teaching clogging and developing the Jump Patterns tool) I put all those words into a poster that became an ongoing resource in our classroom:


Here's how it all connects to the hearing/speaking/reading continuum.

In my classes at first we just dance, getting a sense of the new movement vocabulary and style of percussive dance in our bodies. I talk us through our dance learning by demonstrating with my own body and also saying the descriptive words out loud during our group warm ups and dance time.  Sometimes I point to the words I've posted while I'm talking. When students look confident with the dancing I ask them to verbally label the specific attributes of their dance patterns, using the Movement Variables chart as a resource. (Not incidentally, attributes are also a 'big idea' in mathematics.)

I add vocabulary words as we go along: yellow is dance, blue is math, and green are where dance and math ideas overlap.   This picture was taken about half-way through our work together.

In addition to being able to parse our patterns, we use tons of other math terminology while we choreograph with our teammates and also in whole group discussions.  This approach allows students to fully grasp the real meaning and application of these ideas which, ultimately, allows them to write and talk confidently about their experiences making math and dance at the same time. Teachers consistently notice an increase of 'math talk' in their classrooms during the time I am in residence. As in, "I couldn't believe how much math vocabulary they were using!"
Math is a language but it's not just about terminology; there is also a need to become fluent in playing with mathematical ideas, noticing patterns and structure, sorting and comparing, and reasoning out and understanding relationships. Similar to the point made in the article, linked above, these math ideas are ideally facilitated by an adult through conversation, play and exploration before bringing it to the page.

How do children learn their native languages? By first hearing and playing with the sounds of words, experimenting with the rules of the language, testing what effect those words have on the world and the people in it, and fairly quickly coming to understand that words are useful in all sorts of situations. It's the same for math learning.  Similar to literacy (not just decoding) a child needs to 'hear' the math first (talk it, do it, play with it, manipulate it, make it, see it) before they can abstract or 'read' the math (notation, symbols).  

Ultimately, what this means is that not only is it important to get a sense of the rhythm and the flow of the math/music/language first, but it is also really helpful to learn things in context.  Words mean something, language is a two-way street, and nothing means anything unless it has a use. You may be able to memorize a definition (or a math fact) but if you never get a chance to see, hear, feel or use a math idea in any meaningful, useful or fun way, it's really just a spectacularly sad example of 'in one ear and out the other'.

Luckily, there are many wonderful resources for creating contexts in which you can help children (or yourself!) learn math 'by ear' through games, puzzles, interesting videos, conversations, living math books and math art.  Here are a few of my favorites:

Moebius Noodles
Let's Play Math
Living Math (especially the list of math readers)
Talking Math with Your Kids
Math Munch
Math Pickle

Edit/Addendum: After I posted this piece I realized that I could have also very easily supported the idea of learning math by ear using examples from the last couple years of homeschooling a wonderful, bright but resistant learner in the ways of math.  I could have also made arguments and brought in examples from my experiences as a traditional musician and dancer (Irish flute and a variety percussive dance styles from Canada, the U.S. and the British Isles) where music and dance is learned and taught as an aural tradition, very literally 'by ear'.  

I started working with six, seven and eight year olds this week.  Two more weeks to go.  To start things out, the summer program I'm working with requires me to create and ask my new students a few questions which I'll also revisit at our last class.

One is "How can you make rhythm with your feet?" The other, "How can you make a pattern?"  The predictable and unsurprising answer to that one? 

Colors.
Shapes.

And that's it.  That's all they got.

My dream is to move kids beyond one-attribute linear patterns.  You know, "red blue red blue" or "circle square circle square."  I think those are fair places to start, but based on my experience last summer, even when kids get into upper elementary, they still give the same two answers as the 6 year olds. 

It's a wasteland out there. We're literally wasting kids' time on AB patterns when we could be engaging them in some truly exciting, interesting and beautiful mathematical pattern-based play, analysis and reasoning.

On my board after the first three days I have written:

"How many different kinds of patterns can we make?"

So far:

Rhythm patterns, in our feet, in our hands

"Recipe" (algorithm) patterns (and there I've noted the beginning 'recipe' for our Pizza Clogging choreography which we'll extend next week with our own favorite pizza toppings in our feet.  I also read them the fabulous book How to Make an Apple Pie and See the World).

Nature's numbers: The first nine numbers in the Fibonacci sequence including the one that showed up in the apple star I 'magically' discovered.


Also, in the slices of paper pizza we've been designing. More magic and transformation for the primary set. (The more math magic the better, as far as I'm concerned.)

Of course we'll also look into linear patterns too, but before we design pattern units and make our beaded icicles we'll  read The Lost Button (a Frog & Toad story) and investigate the attributes in our bead choices (color, texture, shape, size). 

Because, when you have more than one attribute you get to think deeply about similarities, sameness and differences, another thing I don't think little kids are asked to do often enough.  With more than one attribute you get a chance to evaluate, analyze, think, talk, make, dance, sing, tap and clap mathematics.

I don't have a lot of time with these kids, but I hope that the world gets a little bigger and their eyes open just a little more to the beauty and structure around them.  Because how will  they come to know and love math otherwise?  These are the basics, folks.  Just like 'literacy' is way more than decoding written words, so too is math.  A visual, kinesthetic, aural and expressive mathematical literacy for all elementary students.  That's my dream.

Every day this week we've been playing with math dice. Enthusiastically.

I'm not going to name the company because not only do I not review or endorse any product on this blog for money or power (not that they asked) but it is also quite easy to go out to your local games shop and get your own set of two 12-sided and three 6-sided dice. (The rules are also pretty easy to figure out: multiply or add the two numbers on the 12-sided dice and then roll the six-sided dice and try to find a way to make the target number using as many operations as you know.)

Did I mention the enthusiasm?


My newly eight-year-old is enthusiastic about many things but has always been a little standoffish with her affinity for math, probably because, I think, she perceives it as my 'thing'. So, it's been nice to be able to truly enjoy a math game together.  (It's been a while -- we were heavy into UNO a couple years back which was super fun.)  It's clear my kid is on her way to a happy relationship with operations, but there's something even more interesting developing...

I had always thought my girl was not what I would call 'systematic' or 'precise'.  I know for sure she is prone to intuitive leaps of connection or understanding and lots of messy tinkering, none of it looking either precise or systematic to my eyes.

As I've been drafting and revising this post I've realized that maybe she has been those things, I just haven't been able to see it.  And, as we've been playing the dice game I've watched her systematically running through different combinations of the six-sided dice (by moving the dice physically to different positions) and  reasoning to herself out loud as she thinks through the different ways to use the hand she's rolled.

I guess I always thought that precision in mathematical problem solving looked, well, neat and orderly and on paper.

Anyhow, I am not (too) ashamed to admit that I was wrong. I think she's been precise and systematic in her own way for a while now. In retrospect, I realize I've heard this kind of  'talking herself through' a series of moves or ideas before. Systematically. In math and in many other contexts. For years. In a messy, verbal, highly enthusiastic way.

Okay, so I'm a slow learner I guess, but pretty open minded all the same. I think it's worth considering that there must be a difference in the way children and adults go about their reasoning. Or, at the very least, that I have a deeply ingrained image of 'what it looks like to do math'. I'm going to keep thinking about all this. If you have any observations or resources to share on this subject, I'd be tickled pink.

In the end, I'm super impressed that not only is she beating the pants off me but she has also created her own strategy for combining operations to reach a target number.  And it's all her.  The only thing I did was bring out the dice.

So...who likes bedtime?  I, for one, am not generally enamored with detours from the normal bedtime routine.

But when my kid said, "Hey Mama! Want to hear this song I made up that helps you with your three times table?!" I decided to give her a little leeway so I could capture the moment.   What can I say, I'm a sucker for math!

The song starts: "There were three ice cream trucks at the corner of Circle Drive..."



My favorite part is 'on the corner of Circle Drive' since, of course, circles have no corners!  But, I'm pretty sure this was not intentional on her part.

Did see her looking off to her right to silently skip count the answers in her head?  We've been doing a lot with conceptualization of multiplication/division (arrays, multiplication towers, factor dominoes, scale, re-imagined factor trees, exploring the concept of units) but almost nothing with memorization.  

I'm happy to have this unexpected piece of evidence that she is thinking about and internalizing these concepts.  I'm also pleased to report a happy conclusion to bedtime!

So, the seven-year-old has been having a spate of self-initiated math lately.  First there was her 'map of angles' and then having her dolly write an essay on 'What Infinity Means to Me'.  So, I guess I wasn't too surprised to hear her from the other room giving her dolly another math lesson:

In her most patient, teacherly voice:

"I'm drawing a simple house, Amelia.  Everything in the house is mostly 90 degrees...you don't have to be exactly accurate but it just has to be good...a triangle window and here's the front porch....compare these two houses.  Can you fix this one?  Good job!"

[Her explanation to me when I asked her later what she was doing: "I drew a house without angles and with angles and Amelia had to fix the house without angles up!"]

After Amelia's success, she continued the lesson with this explanation:

"There are even angles in nature -- straight up and down trees, but some are even 80 degrees, slanting.  The old ones are 50 or 40 degrees."

Later, I got a look at her drawings:






































In the larger house I see her thinking through the angles all starting from the bottom left vertex/corner of the house, which is forward movement from her original representations in the Map of Angles post.  Below the big house is the 'house with angles' at the bottom and what I think is the 'house without angles' (all wonky looking) above that (I thought I saw a different drawing with the same ideas but that, apparently, has gotten lost in the shuffle.)

Another quiet moment found her exploring the structure of an isosceles triangle. 

"See, there are eight of these triangles on each edge [above] and fifteen squares on the bottom edge," she told me.  She also called the line she drew from the top vertex to the center of the bottom edge a "diameter" which she knows is how you divide a circle in half.

In addition to all our sidewalk math adventures over the last year, we've learned more about identifying and classifying geometric shapes in the Beast Academy 3A series but it's been a while since we did the polygons chapter.  We got through skip counting which was perfect and, after entering the perimeter chapter decided to take a break.  This drawing really shows me she's thinking very specifically about the length of each edge.





















And, finally, although this may seem more in the 'art' category, I know for sure that drawing three-dimensionally has all kinds of math involved in it, I just don't know what kind, lol!  Six or nine months ago she tried to sketch Platonic solids and really didn't do it very successfully.  I think her eye has come a long way:

Her milk box:

























An Asian ceramic bowl with some paper flowers in it:
























Her electric pencil sharpener:

















I love seeing (and hearing) the world through her eyes.