Thinking in Threes

My seven year old daughter and I are really quite a team.  One of us will share an idea or an observation and all of a sudden the other of us will be inspired into action.  For example, she found a triangle in this clover last week and I immediately knew what we could do with it.



And, although she sometimes eschews direct participation in projects I think up, my kid is generally always around as I'm making something.  In this particular case, I chatted with her about what I was doing while I glued and pasted, and she made a lot of observations, which is good enough for me.

Isn't it cool?!?

It's a (dried) clover Sierpinski triangle!  We picked clovers, flattened and dried them in a sketchbook, and finally found some time (and a glue stick) to paste them down.  I don't know about my kid, but I am really enjoying all the different types of Sierpinksi triangles we've made over the last few weeks: out of candies, with our straight edge and ruler, with our colored pencils, with money, and now this!

It's fun to find math, wherever we go and it's even more fun to make math out of the things we find. 

More Than Red, Blue, Red, Blue

First, a little context.  I've just finished four days of hands-on workshops for a summer camp of kids ages five to twelve.  Our hour-long, mixed age classes have been full of rhythm and patterns in the feet as well as exploration of other kinds of patterns including ones we can find in nature (Fibonacci numbers, hexagons in bee hives, etc.).   I've also told some stories about squares who are completely bored with their straight edge/sharp corner existence who want and need a change. (See a version of these stories in a recent post called Scissor Stories: Tales of Transformation.)

This kind of summer programming up in the city finds me working with kids whose backgrounds I know nothing about.  I don't know where they came from or where they're going, so I mostly just try and go with the flow, try to meet them every day where they're at instead of where I think we should be. 

Although grant sources require some 'evidence' of learning or growth, my job is to do the best I can in four days.  To the kids I describe our work as using and making and understanding patterns of all kinds, and we spend most of our time doing just that; instead of explicitly talking about patterns, we're just trying to make and use them.  As a result, kids may or may not have absorbed new vocabulary to use when it's time to assess their learning on the final day.  I think my goal is simply that they have a reserve of experience to call on when they're back in school learning math with their paper and pencils.

But, like I said, I still need to ask them questions at the beginning and at the end.  Here are the questions I asked this week.  I'm still not sure that the first question is a helpful one for assessing learning but, as you'll see, it did provide me with some very interesting information:

Question One: "What is a pattern?"

First of four days, summer camp children ages five to twelve: 
Something that is ABB and it keeps going...
Two or more things on paper...putting it on paper over and over.
Square then rectangle, keep going.
Repeats.
Something that gets put together.
Different colors.
Numbers -- 1, 2, 1, 2...

Last day:
Numbers, shapes
Rhythm
Nature
Sequence, repeating.
Heel, heel, toe, toe... (a clogging step)
Circle, circle, square, hexagon (We studied a hexagon as one of our nature's patterns inquiries.)
Take things and put them in order.
You know it's a pattern when it's the same, something that repeats, like red, yellow, red, yellow...

Question Two: How can you make patterns with your feet?

First day:
Different kinds of shoes and socks.
Switch shoes.
1,2,3 on one foot, 1,2,3 on the other foot.
Bang them.
Mix with other people's feet.
Big foot, small foot...
(Alternate) movement.
Switch shoes.
Number of taps.

Last day:
Steps, slides, jump, turn.
(Use directions) left right front back diagonal.
(Put your feet) together, split, crossed.
Big, small, big, small (movements)
Heel, heel, toe, toe (a clogging step)
Sounds
Use toes, heels, kicks.

It's heartening to see that the answers to the second question showed much more understanding after four days of dancing.   As for the first question, it's clear to me that, even after four days of playing around with patterns, the idea of pattern these children (and pretty much every other child I've seen this summer) have internalized is a very narrow conceptualization limited to colors and shapes that repeat in a linear way one after another, almost always "on paper." 

Driving home today (it's over an hour each way) I had a lot of time to think.  I was thinking of all the ways we can harness kids' love of doing and making in the elementary years to the goal of engaging in an deep and meaningful exploration of patterns of all kinds.  And that, in the the process of this kind of exploration, kids would get a chance to represent and experiment with this oft perceived 'simple' concept in a multitude of ways: 2D on the page (so many ways to do this), 3D with their bodies as well as all the great math building materials out there, and even 4D using time and rhythm. 

While I drove I also had a lot of questions in my head: How can kids learn to see patterns in numbers if all they know is 'red, blue, red, blue'?  How can they understand what patterns are if they don't have personal experience with constructing them, taking them apart again to explore the pieces, and transforming them into something completely different?

I am certain it is possible to provide deep, meaningful, artistic, open ended explorations of patterns at the elementary level.  We do it all the time in Math in Your Feet with percussive dance; I am starting to understand how I might move this approach toward other mediums.  (To see a really nice curated collection of math art, go to the Math Munch blog.)

I have a lot more thinking and learning to do on this topic, but for now I'm clear on one thing:

Almost every one of the 180 kids I encountered this summer, no matter their age or their dancing ability, were unable to identify or describe patterns outside the standard textbook context.  I think they can handle more.  Not only that, I think they want more. 

Like Bees to Honey

I'm at my final program site up in the city for this summer.  Every day we work with Jump Patterns for a while, learn some clogging and then it's time to sit down for a while and explore other kinds of patterns.  Nature's patterns, to be exact!

A couple weeks ago, at a different site, the kids got so excited about finding a 'nature's number' (Fibonacci!) inside an apple that I had to go out and buy a bunch of lunchbox apples and cut them in half so everyone could have their own apple star the next day.  I've never seen kids so excited about numbers, or apples for that matter. 

I did my apple 'magic trick' again yesterday, but for today I decided to focus on shape and design in nature.  First I taped out a pretty respectable hexagon on the floor in the middle of our dance space.

When it was time, I asked the kids if they knew what shape it was.  The older kids knew, but not the youngers.  (At this site I have five to twelve year olds in the same class.) 

I asked how many sides a hexagon has, but before I'd allow an answer I started asking for volunteers to stand at each of the sides.  "One child, one side," I said, "Who wants to stand on the next side?  Okay, two children, two sides..." and counted up from there. 

I picked the little kids on purpose since they had been the most challenged with the dancing. 

"Okay," I said, "Everyone put your arm in."  I guess I wanted there to be some sense of division of the space.  Tomorrow I'll got back and put down more tape to show some of the internal structure of this hexagon.

"How many corners does a hexagon have?  One child per corner...Oh look!  Six sides, six corners!"

Then each kid got their own pattern block hexagon, or two trapezoids, to add to the tiling I started.

"What can you find in nature that looks like this?" I wondered aloud.  Only the last class (with more older kids in it) knew right off the bat.

Then they got their very own taste of local honey.  It was a bit too strong a taste for most kids, but still a good new thing to try.

And then, much to my delight, something wonderful happened.  My first class was waiting to go on to their next workshop and were hanging around in the space.  They naturally found their way to the hexagon on the floor, made a circle around it and started playing song and chant games.  This is exactly the kind of thing that always happens when I put down tape where its never been before.  It changes the space and kids notice.

Another example from earlier in the morning: I was creating a set of parallel tape lines as I measured out the sides for the hexagon.  A little boy came over and jumped over the width and then the length.  I wish everyone would put tape down on the floor and then start the camera rolling to record all the awesome things that happen when kids discover this restructuring of their world.

This much I know: kids are to tape as bees are to honey.  Or maybe that's bears to honey?  Anyhow, here's the video of the girls playing 'Little Sally Walker' around the hexagon waiting for their next class: