New Year? New Books!

It’s official! I will spend 2015 writing about my approach to combining math and dance in the Math in Your Feet™ program! Meaning in the Making: Embodied Mathematics in the Classroom will be published by Heinemann in 2016.

Meaning in the Making is for any K-6 classroom teacher, music or P.E. specialist, or home educator interested in understanding how and why to bring dance or movement into their math teaching. The book describes, analyzes and illustrates the learning that happens at the intersection of math and dance in the Math in Your Feet program and usefully extends these methods to help bring the moving body into any math classroom in a meaningful way.

The book (in print and e-versions) will include links and QR codes that take readers directly from descriptions of students engaged in various aspects of their math/dance making to video clips of the work referenced in the text. These video clips are part of a larger online book companion website that will house even more resources for facilitating Math in Your Feet with your students.

In the long term, I hopes to create an online learning and peer support community for teachers and parents learning to facilitate math/dance and moving scale math work with their own students.

Also, my book, Socks are Like Pants, Cats are Like Dogs, co-authored with Gordon Hamilton of Math Pickle will be published in 2015! This book, for young children (up to about grade 3), their siblings, and their adults, explores the concepts of numerical and categorical variables through games, puzzles and making. A project of Natural Math and Delta Stream Media.

Sign up here to know when both books become available for purchase!

The Hundred-Face Challenge [C-rods and Constraints]

It was "Math with Malke" day in my daughter's 3/4 class! This activity was inspired by Simon Gregg's ongoing Cuisenaire Rod work with students; it was one recent conversation in particular about some "faces" that had shown up during one of his classes that got me thinking.

Because the kids' exposure to the C-rods was limited, I wanted to give the third and fourth graders a short intro to the rods before presenting the project.

I built a sequence of simple investigations that led up to the big challenge. Kids were split into (semi) random pairs; each group got a bag filled with a random amount of C-rods. I asked them to open the bags and find one of each color. 

While they were sorting, I noticed that the groups were naturally ordering the rods as they went through their bags. I paused the class to have everyone look around the room at how others had set out their 10 rods. We did a quick list of our noticings:

- 1 cube different between each

- Different sizes

- Each block is 1 number away from each other

- Looks like stairs
- Looks like a graph chart
- Looks like a sail
- Different colors

To give them some tactile experience with the lengths/amounts of the rods I had them pick up white, red, light green and yellow rods and put them behind their backs. They faced their partners and gave a series of commands: "Show me...green!" "Show me...white!"

As a final introduction before the face making I modeled adding the rods together, white to orange. Then came the challenge! 

Make a face that "adds up to 100" or as close as you can get. 

Constraint #1: Use only the rods in you bag. 

That's it. Off they went! Some groups added as they built and found a need for paper and pencil to keep track.

Some kids went immediately for the tens which made for easier counting.

Some kids built first, counted second and added or subtracted rods as needed.

Some kids just made awesome faces. Me: "Hmmm...that looks like it's more than 100. What are you going to do?" Kid: "I guess we'll take off the hair."

Some faces were closed (all rods touching), some were open.

I like this humble little guy:

Then it was time to pause for some more noticing:

- We used 10 rods to make 90 

- Some groups built on top of other blocks

- We changed plans a lot

- We had to add carefully

- Added by 10s

- We counted as we went along

- Everyone used at least one 10 rod

- Some people built and then counted

And the final challenge! Make a second 100 face different from the first. For example, if your first one was open, find a way to have all rods touch each other. If you had a closed face, make an open one.

Constraint #2: No matter what you do, the face needs to be balanced like a human face.  I illustrated this step by step on the board. [And, yes, I know that a human face is naturally balanced, but I wanted to make the idea explicit. It's one thing to see the balance/symmetry, it's another thing altogether to actually make it.]

When everyone was done with face #2 I said: 

"We’ll go around the circle and you can tell the group just a little bit about how you made this particular face.  What changed about your strategy from the first time you did it? Was there something you had to do differently, something you guys talked about that was a challenge while you made it? Or something that came to mind while you were building it."

Here are few of the conversations we had:

Group 1:
Girl: We started by just making a symmetry face and then counted them up and figured out we needed 24 more. So then we basically just added the hair with 24. Then it wasn’t perfectly symmetrical so we kind of made it the same on each side.

Me: And did it add all the way up to the hundred or did you get close?

Girl: It equals to the hundred.

Me (to boy): Is there anything else you want to add?

Boy: Well, no.

Me: I saw you put the pencil down [the middle of the face). Want to show them what you did? [He puts pen down the center.] Do you see that there’s a red on each side, an orange on each side?

Group 2:
Girl 1: At first we had a round face and we had eyes, a nose and mouth in it. But this time we made the background of the face ten each so we have 90 here and then we used twos to make the face.

Group 3:

Girl 1: Our first face was open and then we made another face but then Malke told us to use smaller pieces…

Me: [Laughing] Because why? Why did I give you that challenge?

Girl 1: Because we finished only using 10s, 9s…and the last face wasn’t as symmetrical. So we decided to do four browns in the middle and two blacks on either side. But then if you counted them up there was only 80 on the bottom.  So what we had to do is 4 + 4 is 8 plus 2 is 10 so we could only get up to 90 and if you let us use the 10 we could have made feet.

Me:  [Laughing] But that’s the thing, you have to work within the limits.

Group 4:

Boy 1: Well we started off again with a round face but then…

Boy 2: But then he was already making this other face on the ground and we just added some more to it.

Boy 1: One of the newer things was we took away the triangular nose and  put in these two [white blocks] and we added these red things.

Me: Does it add up to 100?

Boy 2: Mmm hmm. We got hair and you can [to other boy – can you hand me your pen? Thank you…laying the pen vertically down the center] it actually does go down and there are two teeth and two red things on either side.

The conversation reveal thinking about sameness and differences and emergent thinking about symmetry and balance. The teacher and I were both extremely happy about the activity, engagement and conversations in the individual teams and the class as a whole.

Next time I want to be make the constraints well, more constraining, like: "use only rods one through six."  I think this would also be a good to do again with this same group so we can deepen and extend the idea of balance.

New Dice Game: Variation on "Race to 100"

There's a K-6 class (20 kids total) at my kid's school and I know their teacher Jen is always looking for activities that are adaptable for this wide age range. Plus, she's game for just about anything! That's why, when I came up with a variation on the "Race to 100" hundreds chart game the other day I just knew her class would be the perfect testing ground.

Race to 100 (or whatever other number you choose) is essentially adding or multiplying three dice then flipping a coin to determine whether you have to halve that number (tails) or double it (heads).  When I tried out the basic rules w/ my 9yo she got bored quickly which made me wonder what *could* keep a kid interested in playing?

It was fun to hear this afternoon how Jen's class took the game and ran with it -- so many different rules and variations!

Below is Jen's write-up on her class blog (password protected) of all the wonderful adaptations her class came up with on their first "roll" through the game.  She's going to do this again next week some time, so stay tuned for the update!

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1. Historically, dice is the plural of die, but in modern standard English,dice is both the singular and the plural: throw the dice could mean a reference to two or more dice, or to just one. In fact, the singular die (rather than dice) is increasingly uncommon.

Today the kids played variations of dice/coin flipping games.  I gave them a basic idea of how they COULD play and it was up to them to come up with the rules that fit their ability.

The Basics:  Choose three dice.  Add or multiply them in any order.  Flip a coin.  Heads means double your answer.  Tails means half your answer.  Add your score.  First one to 100 wins.

They had choices in dice as well:  your typical 6-sided dice, 10 sided dice, 12 sided dice, and we had one 20-sided dice.  As I walked around, here are some of the rules they made up:

K and B were multiplying a 6-sided dice, a 10-sided dice, and a 20-sided dice.  They flipped the coin:  doubled their total if heads flipped, halved their total if tails flipped.  The originally planned to play to 1000, but changed is to 3000 after B won too quickly.

A and O were adding a 12-sided dice, a 12-sided dice and a 6-sided dice.  They decided that if heads flipped on the coin, they would add 1.  If tails flipped, they would subtract 10.  They added a rule that said you should subtract 20 if your dice total was more than 10.  They played to 100.

M and C were adding their two 12-sided dice.  For the coin, heads meant +10 and tails meant -10.

Z and E added 2 6-sided dice and then multiplied their answer by a “multiple of 10″ dice.  They played to 1000.  Coin flips:  heads +10, tails -10

S and L added 3 dice (2 six sided dice and one “multiple of 10″ dice).  Coin flips:  heads x2, tails divide by 2.  They decided to play to 900.  L was having bad luck though.  When I asked how, they told me that he kept having to divide his answer because he kept flipping tails!!!

L, E, and A were playing with three dice, but the third dice, a “multiple of 10 dice” only came into play if someone flipped heads.  Apparently on E’s first turn, “she got to add 90 to her score!!!!”

The Littles played as well, rolling and adding 2 6-sided dice.  Flipping coins with heads meaning +1 and tails meaning -1.  They played to 100 and used a hundreds chart to keep track of their score.

What were you trying to do? Listening in to kids think while they're creating something new

I talk to kids all the time in my math and making classrooms about what they are doing, the choices they're making and why. But for some strange reason it never occurred to me to record these conversations...until recently. Discovering a voice recorder on my iPhone has spurred me toward documentation in a big way.  Here are my first two recorded conversations, both around a Weaving Algorithms session I did with some 4th graders today:

4^th grader: I’m trying to get to not do the same pattern twice.

Me: So not even the same color pattern? There’s like random colors across the top [the warp]...? And what did you do the weft [weaving across] part?

4^th grader: I had one continuous pattern. I did yellow, orange, red, purple, blue because they were connected in the color wheel and then I did green, and the other shade of green and then the other blue…

Me: …and then you started again.

4^th grader: …and I tried not to make the same weave pattern…I tried not to make that touch two other things from the same color but this touched like three yellows.

All I've got to say for this one is: It takes a LOT of thinking to create an intentionally random design or pattern. I'm impressed!

Here's a conversation about a weaving design on the completely opposite side of random:

Me: Can you tell me why you’ve got your calculator out?

4^th grader: I was trying to figure out how many castle wall patterns…

Me: Can you tell me the numbers of the castle wall patterns?

4^th grader: 26 which would probably be 13 on the bottom…

Me: Okay, the orange ones?

4^th grader: 13 orange ones and 13 blue ones?

Me: And you call them castle wall patterns because…

4^th grader: Because they kinda look like castle walls…

Me: Like the turrets. Okay, cool! Anything else?

4^th grader: The pattern was over over, under under, over over, under under...

Me: So there was an orange and a blue that went over over and you took another orange and blue pair and went under under…nice!

I've got many new thoughts about how to use the voice recorder to help me talk and listen to kids. For now, though, it's very clear that in the noisy rush of large noisy classroom it is too easy for me to rush kids or inadvertently cut them off.

Here are a few other lovely designs from the day:

 

 

18 cubes + 6 colors + 1 elastic = Countless Hours of Math-y Fun!

We recently bought a What'zit*. A What'zit is officially a fidget toy. But, as my 9yo said, after playing around with it for a day: "It's a math toy, Mama. It's sort of like a Rubik's cube."

Here's what it looks like.  Actually, I'm calling this one Rectangle 0.

There are only 18 cubes but we have had a huge amount of fun working within these constraints! My first question was to see how many balanced/even designs I could make. I quickly noticed factors and multiples in every design.

Here's Rectangle 1:

Here's Rectangle 2:

My kid saw me making Rectangle 2. The next day she said, "Mama, I'm going to make a square like you did.  Oh, wait...this is harder than I thought [counting the cubes] 1, 2, 3, 4, 5...1, 2, 3, 4, 5, 6...[repeating the count just to make sure]. It's not a square. It's actually a rectangle."

The color patterns in this thing are fascinating too. For example, here's a 3x6 rectangle.

Here it is again. What's the difference?

The colors highlighted the number patterns. Above there are 3 sixes, grouped two different ways. These two pictures are of the same cube, but from different angles:

  

My kid spent a happy day experimenting with different configurations, like these:

We also made different kinds of zig zags:

And then did some double zig zags:

I love how she off set this!

FUN!

*Like any other product I talk about on this blog, I discovered this toy on my own am sharing our experience in the name of math fun and discovery. I never review products for compensation of any kind.

Pakora Math [#tmwyk]

My kid LOVES the vegetable pakora appetizer at a local Indian restaurant. Yesterday she had some serious dental work done and I sweetened the stressful day with an offer of pakoras for dinner.

Our plan was to get one serving of pakoras to eat there and one to take home to share with the papa. We talked on the way to the restaurant that we needed to find a way to split the two orders evenly between three people.

Our in-house plate of pakoras came first; there were eight of them.

"So," I said, "How many should we start with?"

9yo: Two for me and two for you.

[Eating commences. Yum!]

Me: So we have four left. What should we do now? If we take one more each, there'll be two left on the plate and...Oh wait, I just realized we don't know how many pakoras there will be in the take out box!

9yo: Probably eight.

Me: So how could we split 16 Pakoras evenly between three people?

9yo: [Turning slightly and looking to her left for a couple seconds] Five and one third.

Me: I wonder how you got that?

9yo: Well, 5 times 3 is 15 and then you split the last one into three pieces.

Me: But what if there are only 6 pakoras in the take out box? How will we split 14 pakoras so it will be fair for all three of us?

9yo: Well, 4 times 3 is 12...

Me: But that leaves two. How would you share those last two pakoras between three people?

9yo: Well you could cut them in half and each person gets a half.

Me: What would you do with the last half?

9yo: Give it to me? <sly grin>

Me: Well...but what could you do to share that last half fairly?

9yo: Hmmm. Cut it into thirds?

Me: Cut the last half into thirds? That would make what, exactly?

9yo: Um....really tiny pieces?

Me: It would make sixths!

[Take out order arrives!]

Me: Let's see how many are in this box! There are EIGHT!  Great, what do we do now? We've each had three, let's take two more each ... Okay, now Papa has his five in the box, and there is one left. Wanna cut it into thirds?

9yo: Sure! ... well they're not really equal ... I'll take the biggest piece!
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I love how we ended up talking about two ways to share between three people. First, many pieces into three shares. Second, only two pieces into much smaller shares. Both fractions, but of slightly different natures I think. In my mind "fractions" refer to really small pieces. But that's obviously not the case with my share of five whole pakoras!

During the entire conversation I felt really proud of myself! Helped by projects such as Talking Math with Your Kids and also a recent video showing two boys figuring out how to share sausages, the whole pakora conversation just flowed. Good modeling is definitely the key to learning how to 'talk' math.

And, as we were waiting for our check we also got a bonus #dswyk (Doing Science with Your Kids)!

Me: What do you think is happening there? 
9yo: There's a rainbow on my hand! Lol.

"Doing math means a lot of different things including..."

I LOVE this Ignite talk from Annie Fetter of The Math Forum.  Here's why:

"[My mom] didn't think of herself as someone who did math ... Why is it that intelligent people who are good at sense making and good at problem solving feel no affinity for school mathematics? If our students could do the things my mom could do we would be ecstatic. As math educators we need to make sure students and grown ups understand that doing math means a lot of different things including making incredible, beautiful art."

I especially love this because in her talk Annie points out all the math doing and making in which her mother engaged.

So...how would you fill in the blank here?  

Doing math means a lot of different things including...

Would love to hear your answers!

What is the role of embodied mathematics in our classrooms?

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.

New Questions at #TMC14

Try as I might to focus this post on other things, I think I first need to address the fact that up until TMC14 I considered myself a math outsider.  I've been working at the intersection of math, dance and learning for a decade now. Quite truthfully it's been a lonely road. My initial collaboration with an elementary math specialist was the foundation and the frame for Math in Your Feet, but I essentially was in charge of every other detail. For over six years I worked in individual classrooms for a week at a time, experimenting, revising, questioning my motives.  The past three years have included some really amazing, deep collaborations with some incredible math educators. But up to now none of this reality has changed my sense of being on the outside of math world looking in.

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At Twitter Math Camp I had the opportunity to co-lead a morning session on Embodied Mathematics. And I was lucky enough to have lots of people interact with both Math in Your Feet and other versions of body-scale math learning after hours.

In between all that I got to be a learner. I learned from the questions people had after engaging in MiYF lessons. And, maybe because I have been thinking about number lines for a while, I had new questions during Steve Leinwand's keynote at TMC14 when he asked: What is 5 + -9? and then showed us a number line.

I wondered why we would have to start at zero. I wondered what a number line is really for. I asked myself, why would we use it at body-scale just to get an answer if we could do that easily on paper? Why is the number line generally presented horizontal to the learner? [Then Steve showed the elevator version.] Why couldn't it be on the diagonal? What are the assumptions about the learning that can happen on number lines?

Most importantly, half way through Michael Pershan's afternoon session on the complex plane, when I had reached my limit with that math, I asked myself one final question:

What would I learn if I tried Steve's equation at different starting points other than zero?

I tried it and it was FUN to ask that question. I saw a pattern but didn't understand what I was looking at. Turns out I didn't have to figure this out on my own. Turns out I had an ally in the pursuit of understanding the meaningful use of body-scale number lines. Turns out Max Ray keeps a roll of tape in his bag for every classroom visit. My brother from another mother.

Blue tape can change your relationship to the space you're in. And when you put down blue tape and start asking questions, other people show up in a curious state of mind.

So we put down blue tape and along came Christopher with a gleam in his eye. And then we had a crowd and everyone was engaged in questions about working on a body-scale number line.

Which way should we face? What were our assumptions about its use? If we're going to use this tool at body-scale, why should we use it the same way we do on paper? What new insights might we have using the constraints of the body and the base metaphors created by living and moving in a body through space? Why is Malke asking so many questions!?

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The next day, during Dan Meyer's keynote about who makes up the #MTBoS and #TMC14 I turned to Kate Nowak who was sitting beside me and said, "I'm not really a part of this though." And then immediately gave myself a huge mental kick in the butt. WHY!? Why would I say that after a day and a half of some amazing interactions with some amazing, inspiring math educators who obviously all saw me as part of the group with something to offer math learning?

In the end there were two experiences I had at TMC14 that have inspired me to revise my outsider narrative:

1. I had lots of new questions related to conceptualizing a body-scale investigation of number lines. If I am engaging in mathematical conversations and debates, I am a math person.

2. At TMC people engaged with my math/dance work and had new questions and insights of their own. Creating an environment for asking new math questions makes me a math person, even if (especially if?) it's in the unfamiliar mode of embodied mathematics.

I am a math person. I am a dancer. My questions and inquiry reside at the overlap of those territories. I can be a math person and a dance person at the same time!

_________________________

On the LONG, LONG, LONG drive home from Jenks, OK I came up with some new questions for myself.

How can I keep the energy of inquiry and collaboration going all through the year?

How can I generalize the Math in Your Feet approach to body-scale math learning so it's useful in many different kinds of body/movement/math activities?

What's next!?

Lessons from the #BlueTapeLounge [TMC14 Recap #2]

TMC14 (Twitter Math Camp) was an amazing experience for me. Christopher Danielson and I collaborated on a morning session on embodied mathematics, juxtaposing familiar hand-based manipulatives with body-scale math/dance learning. You can read the Storify of our morning activities to get a sense of what we did.

In the evening, after dinner, some folks hung out in the back lobby of the hotel which included a little uncarpeted room with an ice machine and the path to the hotel pool. It's also where I decided to put down some blue tape and see what might happen.

What happened was that some of the folks from our embodied mathematics morning session taught other folks the math/dance steps they had choreographed earlier in the day. I was incredibly touched to see my math teachers/dance learners start by orienting their new dance partner to the square, showing them the many different ways they could move around that space and THEN teach the dance step.

The slide show and the videos below will give you a great idea of some of what went on, but here are the lessons I learned during those two amazing evenings.

1. You don't know what math you can learn while dancing or at body-scale until you actually do it.

2. Even experienced math teachers can have new mathematical challenges and insights at body scale.

3. It's useful and interesting to learn someone else's math/dance step, but even more interesting (and mathematical) to make up your own.

4. Dancing mathematically can lead to all sorts of new questions, but it doesn't always make for a dance that really works. This one was great, but the final version was better.

5. You can have a ton of new math questions while dancing, but sometimes you need to take the time to let your body catch up so whatever you're trying to express in the dance can look the way you want it to. 

The video below shows us still in practice mode. The final video (#6 below) shows obvious gains in fluency (practice makes permanent!)

6. When you make up your own math/dance step you start having new and very mathematical questions. Max wondered if you could make up a 4-beat pattern that would look beautiful when danced in all four quadrants of the coordinate plane.

The slideshow below is sequenced to illustrate the progression of people coming to the Blue Tape Lounge to experience and know mathematical patterns within a percussive dance system. THANK YOU to all who participated and observed these two amazing nights' worth of inquiry.

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