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!

New Math Game: Change Your Rule

This post is in two parts: The first part outlines the basic flow and structure of a new game idea which, I think, has the potential to support the conceptualization of math concepts such as combinations, permutations, patterns, variables/attributes and reflection symmetry.  

I know that's a tall order so in the second part of this post I briefly discuss my reasoning, pose a few lingering questions and then ask for feedback.

Part One: Basic Flow of the Game
The idea for this new game started with a picture:

Photo via: http://www.fotomat.es/arte-combinatorio/

I immediately thought how cool it would be to have this kind of mathematical art/visual inspriation in a living or learning environment, something to hang out with that evokes noticing and wondering.  In this case, a real-life shelf with spheres and cubes just begs for interaction, something to play around with and ponder. I want one! In the process of thinking about how to get this kind of installation in my own house, a new game was born.  Here's the basic idea:

Make/build a rule.  Find a way to change it.

These are the game pieces. Set 1 includes two shapes, four colors.  Set 2 includes six shapes and six colors.

Rule Change #1: Make a rule/pattern. Using the same shapes/colors how can you make the next iteration different from the first.  Does order matter?  How many different combinations can you make?

Rule Change #2: Make a rule/pattern. Change one element of each shape to make a new pattern.  For example, orange turns into red, or square turns into circle.  For each new design change only one variable for each piece.  How many times can you change the rule? How many new rules/patterns can you make from your starter rule?  Can you ever get back to the original rule/pattern?

Rule Change #3: Build your pattern then reflect the design. What do you have to do differently when you build the reflected pattern?

But what happens if you only use one color and shape?

How many combinations can you make using three of one shape and one other shape?  Would it change things if each of those circles were different colors?

On another line of questioning, what larger design can you make by building your design line by line using the same shapes/colors in each line?

Part Two: Thoughts and Questions

Even though Part One seems like a lesson flow, it's really meant to be more of a general framework for exploration.  My intent was to provide a little structure, some basic 'rules' and a lot of room for inquiry.

I'm wondering if it really feels like a game, or is more of an activity?  If it's going to feel more game-like, does it need more structure?  A timer? Some 'change your rule' cards?  Or, maybe, some cards that say how many different attributes to use in the pattern?  How many different rules would be enough to create a sense of chance? What other 'change rules' would you include?

Based on trying this out with my own kid this morning, if you were to do this more as an activity/lesson perhaps it would be helpful to have some really easy starter examples, like all one color, or two different colors to get the ball rolling in a positive direction.  I asked my daughter to build a pattern with three pieces but then gave her the 'change rule'.  She didn't really like that and we left it there.  The game pieces and board are still out, though, hanging around.  I'll see if she wanders back over.

Questions I still have include: When combining or creating permutations, how much does the second attribute matter?  For example, this first design (two colors, one shape) looks like it combines in exactly the same way as the second design (two colors, two shapes):

Another one of my goals for this game/activity was to also explore the ideas of attributes/variables in design -- how well do you think it does that?  I intentionally did not use pattern blocks -- only one attribute/thing to change -- and instead made my own (Set 1). What would pull the attribute/variable idea out a little further?

I hope it's clear that I am hoping to dig into the brain trust that is my modest but wonderful readership.  That means you by the way, so please if you have any thoughts about all this, I'd love to hear your ideas.  

Things I would love feedback on: key ideas about combinations/permutations at the elementary level, thoughts about the game structure, ideas for other 'rule changers' to add into the mix, and any other thoughts you may have.  Thank you!!

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