Thursday, August 21, 2014

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!

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.

Tuesday, August 12, 2014

"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. would you fill in the blank here?  

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

Would love to hear your answers!

Sunday, August 3, 2014

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.

Friday, August 1, 2014

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.


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!?


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!?

Wednesday, July 30, 2014

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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