Permutations in Your Feet: #miyfeet Primary Project Day 1, Part 2

The collectively created permutations using (mostly) steps and jumps.

[I am working to adapt Math in Your Feet to the primary grades. Day 1, Part 1 is here.]

Introductions

I have a weird name. Mall-key. Like the "key to the mall." Because of this I open every workshop with kids by teaching my name. I do it through clapping, because I also need to quickly assess multiple things about each new group (focus, interest, attention, beat competency, group-ness, etc.)

I clap my name because it's an easy way to introduce our upcoming work. I can clap my name with two sharp, hard claps, or two soft swishes from rubbing my hands together. I can clap my hands and then my legs ("up, down") or I can make two deep sounds by hitting my chest. And then I ask my new friends if they have any ideas for how I can say my name with claps. Whatever I do, my intention is to illustrate, from the get go, that there are many ways to do one thing, and that it's fun to experiment with ideas.  

How it started

I also tell them that any time I make a pattern forward, I like to figure out how to do it in reverse (or backward, or opposite) and then we have fun thinking how to "say" my name forward AND backward. Because this was my first session with 2nd graders I had no idea how things might go. So, imagine my excitement that this idea of opposite/reverse/backward actually became a theme that influenced the course of the class.

We started with me introducing two movements -- two-footed jumps and single footed steps, both done in "center" which I also described as "middle."

Me: What do you think would happen if we put two jumps together and then two steps? What do you think would happen?

[Collective shrugging of shoulders. I turn off the music as ask again: "What do you think would happen?” Kid in blue, in front of me, jumps twice, steps twice, but I don't notice. I ask the question again, and kid in blue dances JJSS again, but my back is turned and I don't notice him. He keeps doing it through the conversation, but I still don't notice!]

Boy: Um, it will be four?

Me, to the group: It will be four what?

Boy: It will be four...movements.

Me: Ah, four movements! What else could it be four of? Is there any other ways to describe the four?

Girl: It could be a rhythm.

Me: It could be a four ... what do we call what makes the rhythm?

Girl: Beats?

Me: Yeah, a four beat rhythm. Let's try it together. [Burst of inspiration] Actually, do you guys just want to try it in your little partner pairs? And see what two jumps and two steps look like and feel like?  I'll walk around and see what you’re doing.

[Kids work.]

Me: Who wants to show me jump jump step step? Which team wants to show me?

[At this point I'd just like to point out that I ASSUMED that because I asked what would happen if they put "two jumps and two steps" together that I would get a single answer. Silly me.]

Kid in blue: Can we do any pattern?

Me: ANY pattern? Did you come up with something else?

Kid in blue: Yeah...J, S, J, S

Me, walking over to our little white board: Wow. You know what, it's not up here on this board. This is from the first graders and look! Now I have a fourth pattern to put on here. You guys dance it and we'll say it and dance it with our hands while we're sitting.

Everyone says it with me: Jump, step, jump, step [rehearsing the steps with their hands].

Me: Who else wants to show what they've done? Who has JJSS? Boys do you want to show us? ... and they even used the same feet! Now who would like to play around with some ideas about different ways to combine jumps and steps?  Who would like to take a few minutes to do that and see what you come up with? [All hands go up.]

[2-3 minutes of experimentation]

Me: So, the first thing I want to see if anyone came up with a new combination that isn't on the board yet. And the second thing I'm going to do is ask you, and I want you to think about this ahead of time, what kinds of things you noticed about what we did today ...

Now who has a combination they want to show us? Let's turn our faces and bodies toward our friends [two girls in the back of the space dance SJJS but not in tempo so I ask the group to help me keep a steady beat while the girls dance. But there is suddenly considerable muttering in the room and I ask...]

Me: Did anyone else come up with that combination?" [The whole left side of the group raises their hands and continues to exclaim.]

Me: Shhh, sh, sh!! You guys, look at this!! In the other group [first graders] there was JSSJ. You guys were doing the reverse of that! It's the opposite. It's so COOL that you all discovered the SAME thing independently!  Okay, one more but it has to be a different pattern from that.

Two new girls dance SSJJ.

Me: And you know what? Our class pattern was JJSS but now you did the opposite [writing on board SSJJ.] We have discovered so many patterns out of steps and jumps, it's amazing!

Wrapping Up

Me: I asked you at the beginning to notice something about this space...now I'd like to hear about what kinds of things you noticed about what we just did today.

Dancing
Rhythm
"Well, we did do a lot of two plus twos."
"All of the rhythms had four beats.”
Opposites
Movements
[Me: and what were the movements we did?]

Jump
Step

[Me: And does anyone remember the direction we were doing the movements in? What do we call it when we're right here [gesturing to middle of my board]. Were we moving forward and back?]

"No, we were staying in the middle."

What I learned
I learned that even though second graders still have challenges coordinating their bodies, they can stay in rhythm as a group, work productively in teams of two, and do original (to them) mathematical thinking. On the first day!

p.s. I would love to hear your thoughts about the words opposite, reverse and backward. And, after this workshop, I somehow came across (w/o searching for it) the idea that inverses are important to permutations. If you have thoughts on that, I'd love to hear those too!

The Chart of Possibilities: #miyfeet Primary Project Day 1, Part 1

I had my first workshops with K-2s today! In all I'm working with about 80 kids, most of them first and second graders, every Monday through mid-March. My investigation focuses on the following initial questions. I'm sure I'll have more:

 How can the Math in Your Feet program be adapted for meaningful use in primary grades (K-2)?

What are the particular challenges and abilities of these younger learners (socially, physically, cognitively, mathematically) in the math/dance making setting?

What part(s) of the original MiYF program can/should be used with younger children?

How much of the established activity sequence and mathematical inquiry can be used in lessons with K-2s? 

How can we focus on sameness and change (symmetry) with kids where Left and Right is still a murky concept? 

Here's a conversation I had with one lovely kindergarten student about what she was doing on the reflection sheet I provided:

Me: So can you tell me about your picture?

Kindy girl: [Pointing to each little picture in the big square] There’s me, and there’s Oliver and then there’s you on that board over there. Then, that’s that camera thing over there. And that’s the thing that you were dancing on. [Pointing to the word] and there’s clogging.

Me: So that’s what *I* was doing!

Kindy girl: Yeah.

Me: Are you going to write anything else?

Kindy girl: No I was just learning new words that I haven’t learned yet.

Me: Awesome, thank you!

Here are a few more representative K-2 kid drawings (both from kindys, for some reason). Lot's of kids drew the layout of the space and/or themselves inside their square dance spaces.

I wrote a lot of new words on the board. "Center" (used in context with the less formal "middle") was a big hit with many.

Another thing that was true for all three groups I worked with (and also really similar to how upper elementary students think): When kids start to learn a few patterns in their feet, their interest IMMEDIATELY goes to making up their own. Crazy exciting.

At the end of the K-2 workshop I asked: 

"When you came in you noticed a lot of things about the space and you noticed a lot of things about my equipment. But now I'm wondering if you could tell me what you noticed about what we did."

Kid 1: Jump Jump Step Step

Me: Oh, so we did some movements! Cool!

Kid 2: We did a lot of jumping and clapping.

Kid 3: Jump jump clap clap, step step clap clap

Me: So what word up there says what that was? Does anyone know what it's called when you do something in order and you can keep doing it?

Lots of kids at the same time: Patterns!

Me: A pattern! Let's all clap that! [clapping out syllables] but we can also clap it in a new way (whisper, sliding hands). What other ways could we clap that word? (responding to a child's movement) Oh! We can go up and down! Let's all do that. Good!

Kid 4: Oh I know one! (flips hair) "Pa.." (touches nose) "tern" [We all do that as a class.]

Me: What else did you notice about what we did today?

Kid 5: Well I was thinking we could do jumping jacks and then then we could step.

Me: Okay you're thinking of other patterns we could do! Do you want me to show you something?  THIS is my Chart of Possibility. Next time I visit I will put it up. It will show you some more possibilities for making your OWN dance patterns. [Cheers from the kids.] Right now those possibilities are covered [with tape] but you will start to see that we will have all sorts of options for making new patterns in our feet in a way that a tap dancer and a clogger does.

Kid 6: Can we UNcover one of them now?

Me: Tell you what. Next week I will ceremoniously uncover the new words we are going to use!

Stay tuned for weekly installments through mid-March, including Part 2 of this post where 2nd graders investigate permutations out of their own curiosity! I can already tell it's going to be an incredible adventure.

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!

....................................................

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.