The Map is Not the Territory

Math related conversations with my 8 year old tend to pop up unexpectedly. These are often around something she's making, and are often a gorgeous little gem of a surprise. Today was no different.

I found her creating a grid of tape on top of a piece of origami paper. She was coloring it in when she said, "You know, mama, patterns don't necessarily have to be colors in order."

Oh my gosh!

Me: What do you mean by that?

As you can see from the picture, below, she was coloring the whole page somewhat randomly, sometimes following the columns down, sometimes not. 


Her: Well, the colors don't have to be regular, they just need to be in the windows."

Me: You mean the windows are the places where the tape is not covering the paper, that's what you're looking at? It doesn't matter what color windows are, just that they're colored in?

Her: Yeah.

So here I pause to do a happy dance. My biggest discontent around patterns is that many kids grow into adults who think that "patterns" are only linear repetitions of colors. It is clear she has not internalized that particular reality.  The other reason I'm happy is that Prof. Triangleman once said to me:

"Math is when you say exactly what it is you want to pay attention to, focus only on that attribute and ignore everything else."

She was doing this!  Her pattern has 'windows' that are colored in, but the pattern is not defined by the colors themselves. She was consciously creating a pattern of windows (spaces between!) and consciously excluding the colors. I am thrilled to have caught a glimpse of this multi-layered attention in action.

This really goes to show you that it's worthwhile to keep your ears open while kid is focused on making or building something.  Even if it's after the fact, ask your kids to tell you about what they did, even if it doesn't look like much.  I mean, just look at the taped/colored piece again. It's pretty much a bunch of scribbles and it'd be super easy to pass it over, to think it was nothing special. In reality, though, there was so much thinking going on while she worked.

Here's what the piece looked like when she took the tape off:
Her: Oh.

Me: You seem surprised. Did that not turn out the way you expected?

Her: Why do you sound like a journalist?

I started working with six, seven and eight year olds this week.  Two more weeks to go.  To start things out, the summer program I'm working with requires me to create and ask my new students a few questions which I'll also revisit at our last class.

One is "How can you make rhythm with your feet?" The other, "How can you make a pattern?"  The predictable and unsurprising answer to that one? 

Colors.
Shapes.

And that's it.  That's all they got.

My dream is to move kids beyond one-attribute linear patterns.  You know, "red blue red blue" or "circle square circle square."  I think those are fair places to start, but based on my experience last summer, even when kids get into upper elementary, they still give the same two answers as the 6 year olds. 

It's a wasteland out there. We're literally wasting kids' time on AB patterns when we could be engaging them in some truly exciting, interesting and beautiful mathematical pattern-based play, analysis and reasoning.

On my board after the first three days I have written:

"How many different kinds of patterns can we make?"

So far:

Rhythm patterns, in our feet, in our hands

"Recipe" (algorithm) patterns (and there I've noted the beginning 'recipe' for our Pizza Clogging choreography which we'll extend next week with our own favorite pizza toppings in our feet.  I also read them the fabulous book How to Make an Apple Pie and See the World).

Nature's numbers: The first nine numbers in the Fibonacci sequence including the one that showed up in the apple star I 'magically' discovered.


Also, in the slices of paper pizza we've been designing. More magic and transformation for the primary set. (The more math magic the better, as far as I'm concerned.)

Of course we'll also look into linear patterns too, but before we design pattern units and make our beaded icicles we'll  read The Lost Button (a Frog & Toad story) and investigate the attributes in our bead choices (color, texture, shape, size). 

Because, when you have more than one attribute you get to think deeply about similarities, sameness and differences, another thing I don't think little kids are asked to do often enough.  With more than one attribute you get a chance to evaluate, analyze, think, talk, make, dance, sing, tap and clap mathematics.

I don't have a lot of time with these kids, but I hope that the world gets a little bigger and their eyes open just a little more to the beauty and structure around them.  Because how will  they come to know and love math otherwise?  These are the basics, folks.  Just like 'literacy' is way more than decoding written words, so too is math.  A visual, kinesthetic, aural and expressive mathematical literacy for all elementary students.  That's my dream.

I'm really enjoying math in the morning.



















There's something fresh and new and hopeful about mornings lately and, even though I'm not doing anything ground breaking, I'm really enjoying how connected everything seems to be these days, mathematically speaking. 

In mid-October I mapped out a basic math plan when it was clear the girl (now 7.5) needed and wanted more and different kinds of math challenges.  I decided to call the plan 'algebra' because, from what I've read, algebra combines a number of skills and concepts that we have to start learning anyhow at the primary level.  And, because the girl has often balked when I introduce new math stuff, calling it algebra motivates her to give it a try; 'algebra' is a big kid skill, and she really, really wants to be big.  

So, in the morning it's been fun to open my math folders and give my lovely child her choice of math activities.

Solve for x (conceptualizing equality and sameness, sums and differences) or math card games (3 digit mental sums and differences)?

Christmas themed beginner Sudoku puzzles or Factor Dominoes?

Growing patterns or Bean Soup (fractions, multiplication, division)?



















 "What's division, Mama?"

"You know, like when you wanted to see if you were halfway through your reader.  There were 96 pages and you figured out in your head that half of 90 was 45 and then you..."

"Oh, yeah."

Later that morning at the science museum she built this:



















"Hey Mama, look!  I got the water to cover the whole area!"

"Cool!  That's an example of division, too.  The water is being distributed evenly across the table."

So, here I am, trying to connect our morning math time to the bigger math picture that I'm constructing in my brain.  Like algebra, as I've mentioned.  What is algebra???  I did some algebra when I was in high school, but failed literally and horribly (although I aced geometry).  But why let that stop me?  After some research I decided that the concepts of balance and sameness, solving for unknown quantities, and growing patterns were all pretty darn interesting and relevant whatever we called it and away we went.

We began by building and analyzing growing patterns from some pattern starters I found.  Here's one she came up with on her own:

























And, when we do the occasional worksheet (the one below is skip counting/multiplication) I look for ways to extend the activity forward, even if I'm not completely sure I'm right.  The sheet below was for figuring out numbers of feathers on different numbers of hats, legs on rabbits, and petals on flowers.  She got the skip counting patterns easily, so I described the data another way by saying:

"The number of hats you have is multiplied by three feathers per hat which gives you the total number of feathers," while writing h x 3 = f.  The next two examples she talked herself through the whole thing and wrote it down.   But maybe it would have been better to say, "The total number of feathers you have is equal to the total number of hats multiplied by 3"??  Writing all this out has me thinking of another way I could have done that, but I think sometimes its okay for me to muddle through stuff like this.

























And, if skip counting is coming easily, why not challenge her to fill in the chart backwards instead? Ooooh, that was cause for consternation, but she had been saying "This is easy, this is easy, " all morning.  I know for sure she needs more experience with subtraction, I know being able to invert a procedure or concept is an important skill, and I also know that if something is too challenging she'll just give up.  So, here was an opportunity for deepening her experience with a skip counting chart and providing just the right amount of challenge at the same time. 











Filling in the charts backwards really made her think.  That's what she wanted, right?

Anyhow, it's nice to be entering a stage with her where we can sit down and 'do math' outright instead of trying to leave it around the house for her to discover, although I'm keeping that strategy in my kip for now!  More than anything, it's been lovely doing math, in the morning, with the winter sunlight streaming in, with a child who is finally (finally!) allowing herself to be interested and excited about exploring this mysterious and wonderful subject.

When I was in third grade I learned multiplication.  Well, really, it's more like I memorized the facts.  I learned about fractions too, but I never really understood them...at all.  To this day, I have only a cursory understanding of ratios and percentages.  I have higher hopes for my seven year old daughter, though, especially now that I know all of these subjects are related. 

My daughter and I have explored a lot of different aspects of math over the last year (mental math, sums and differences, lots and lots of geometry, fractals in various forms, mathematical stars, flexagons, functions...).  It's been great and we're still going to keep exploring the beauty and structure of patterns wherever and whenever they come up.  But, I recently came to the decision that though it might be a challenge for both of us, multiplication was next.  And I knew that my focus, for now, was not going to be about facts or memorization, it was going to be about comprehension. 

My daughter has basically understood the groupings concept of multiplication for a while now, well before she mastered addition and subtraction.  It's of the 'cookies on the plate' variety, except for one of my favorite lessons using multiples of threes to make stars.   Fortunately, I recently found a great card game from Let's Play Math (a free download, which I wrote about here) that helped me introduce the idea of multiplication as a number sentence, arrays, groupings, and measurement.  We've also started playing with this Primitives Application (which is really more about factoring, but for the elementary mathematician the visual groupings can't be beat).  I first found out about the primitives app from Maria Droujkova of Natural Math.  And, I recently purchased and hung Natural Math's Multiplication Models poster in a prominent place just so I can ponder all the wonderful information in it. If my kid gravitates toward it, all the better.

As I expected, soon after I hung the poster, I got inspired.  This little activity was completely influenced by the combinations portion of the poster pictured below:



































I was pointing it out to my daughter -- "Look, a cat-dog!  Let's call it a cog!"  I was having fun coming up with silly names for each combination and that's when it hit me.  I was actually understanding the process of combining something other than dance steps.

Then I thought: Grids. Arrays. Multiplication. Perfect. Let's make our own!

I made a quick grid that night and upped the combinations to 4x4.  I asked my daughter what animals she'd like and found some clip art that worked.

The next morning I ran into a roadblock.  The kid was not happy about cutting up all the animals in to heads and bodies.  Not.  At.  All.  I quickly changed that to 'tops' and 'bottoms' hoping for a less gruesome association to only partial success (she covered her eyes every time I cut an animal in two).

To insure that this activity didn't drag on, I did most of the cutting and pasting.  I suppose it would make a good fine motor activity, but I wanted her to focus on the combining rather than the preparation.



































We started with the pig 'top' and bear 'bottom' (hee hee) and called that one a 'Pear' and continued haphazardly from there.  In retrospect, I wish we had moved left to right and top to bottom on the grid, pasting parts and making up the names from there.  Giving them combination names was a good challenge but if I had another chance to do this again I would formalize the naming process by using the animal top to blend in with the animal bottom or vice versa. You can't name a pig top/horse bottom the same as a horse top/pig bottom, right?  They're different creatures altogether.  So, for example, Pig/Bear would be "Pear" and Bear/Pig might be "Big".

Although I wish we had filled in the grid in a more organized manner, it also did work out the way we did it, too.  I'd ask, what combination do you want to do now?  She'd choose and then I'd ask her to find the square where that combo animal had to go on the grid.  She didn't quite get it at first, but after a few combinations she had the hang of it. 

At the end I guided her through counting the rows and columns and finding the total number of combinations we had made.  I also focused on writing the answer in two different number sentences, as multiplication and again as repeated addition.  By the end, I realized just how dense this kind of activity is.  If I do it again, I'd plan for a large grid (maybe 6x6?) but I'd have us combine three animal tops and bottoms to start with.  Depending on the kid, you can add one or two more animals to the mix or even just (aha!) predict how many more combinations you would have with the addition of each new animal.  And, then you can make a connection to square numbers.  Cool!

Okay, I'm revising the list of math activity/topics in this post: growing patterns (algebra), square numbers, grids, arrays, multiplication, and combinations.  When your students are done with this deceptively simple activity give them a big pat on the back and then take them out for ice cream.  I also owe my daughter a big thank you for being a fairly willing guinea pig on what turned out to be a very tired day for her.

If you want another similar and super fun grid/combinations activity check out this great post from Yelena at Moebius Noodles called Mr. Potato Head is Good at Math.  It's fantastic.

And, don't forget to check out all the fun we're having over at the Math in Your Feet Facebook page!  Today we found the math in an incredible flower from Argentina!

Day One
Me:  I'm going to make a triangle out of three candies.  Can you copy it? Good! Now I'm going to make this triangle bigger by adding two more triangles.  Let's see if you can make the same as me.

Kid:  Look, Mama!  This side is red, yellow, red, yellow.  And the other side is yellow, orange, yellow, orange, and the third side is...

























Me:  Great!  So let's see if we can make another triangle just like our first two, except with different patterns. [Kid starts right in...]

[I've been thinking lately about the importance of modeling inquiry, especially in the math we do.  I want her to not just identify patterns, which she's good at, but also consider those patterns malleable to the whims of her own curiosity.  Also, if I ask a question that leads to my desired result it's generally a lot more successful than giving her a direction.]

Me: What would happen if we added on two more triangles the same size but with different patterns?

[With these sized candies it was easier to build the fractal structure if we made each 3-candy triangle out of three of the same color.  I also had to point out that the right and left sides of the top triangle had to be extended diagonally to make this work, which was a bit challenging for her to see and do.]

























Kid:  Oh look!  There's a triangle in the middle!

Me:  Let's take this big one apart and see how many different colored candies we used. These three columns each have six candies and the orange row has how many...?  How many candies is that all together?

























Me: I wonder if we could make another big triangle using the same candies, but different color patterns?














































Day Two
I have been waiting for months and months to use this sheet I found here.  It's meant for older kids, I think, but we adapted it just fine for the candy approach.

Patrick Honner's Moebius Noodles guest post about mathematical weaving has been in the back of my head for over a month and a half.  Mathematical weaving employs one of my favorite making materials - colored paper! - and I thought it would be fun to try with my seven year old.  I know the rudiments of weaving, but I wasn't sure how to get started, so yesterday I played around to try and figure out a few things.  It was actually sort of challenging, but I landed on some solutions and new grid games, so I thought I'd share.

I'm not done with my exploration, but what I have discovered so far is a perfect little unit for young children.  I am imagining that the weaving and the games would be completed in an enjoyable collaboration between adult and child over the course of a day or two.

I started by experimenting with loose 1" strips of paper as the warp (vertical strips) but soon found that much too unwieldy for even my adult hands.  The pieces were not connected to each other so they slipped all over the place and I had to use a lot of tape to keep the weft (horizontal strips) connected to the warp, which wasn't ideal.  So, I searched for some advice on how others have made paper weavings.  A quick Google search and I found this video (which is cool, but still too persnickety for the young ones) and this video which, although the cutting is somewhat haphazard, led me to a solution for how to weave paper without tape...

I first decided that a 3/4" width for vertical and horizontal strips made a more pleasing final product to my eyes than 1".  To make the vertical strips I folded a piece of paper in half and used my paper cutter to cut 3/4" strips from folded edge to about 3/4" away from the open edges closest to me.  Essentially, I was creating a paper warp that was still basically one piece of paper.



















As you can see, below, the horizontal strips weave in very nicely and don't need any glue or tape to keep them in place if you focus on pushing them gently, but snugly, downward.  For the young ones, at least, a basic over/under/over/under weave is challenging enough.  Using two (or more?) horizontal colors creates visual interest and perhaps even a conversation about the patterns you see: alternating colors both vertically, horizontally and diagonally.  You can also make a connection to odd and even numbers.  Yellow squares in the design show up 2nd, 4th, 6th... places.  Green squares are 1st, 3rd, 5th...

























The minute I finished the piece I thought - A GRID!  It's a grid!  The Moebius Noodles blog is very inspirational and a great source of grid games (my favorite so far is Mr. Potato Head is Good at Math) and I always have grids at the back of my mind these days because of them!  Here are some of the ideas I came up with using a newly woven paper math and one of my favorite math manipulatives -- pennies!

Adult: Oh look!  There are three different colors of squares in our woven grid.  I've got some pennies -- I wonder if we could make a square by putting pennies down on only one of the colors?

























Adult: That does look like a square. Let's count and see if there are the same number of little squares (yellow, blue, yellow, blue...) that make up each side?  There are!  How many little squares are there on each side?

Adult: But, wait! Look what happens when I push a corner penny in toward the center!  Yep, it lands on a green square!  Let's do it with the rest of the corners and see what we get.  Oh, lovely.  A rhombus.

























Adult:  The corners on the rhombus are on the yellow squares.  I wonder what would happen if we pushed them one square toward the middle?  Ooooh, look!  We have another square.  Is it bigger or smaller than our first square?  Each side on our first square was six little squares long.  This square has sides that are...three little squares long.  Cool.

























Another exploration:

Adult: Here's a little story about a tiny X who wanted to get bigger.  Can you help him figure out how to help the X get bigger?


























Or, how about the tale of some square numbers who also wanted to get bigger?  What little kid doesn't want to grow up?

























And, here's my favorite.  It's a 'let's make a rule' kind of game.  The first penny goes in the bottom left hand corner, and you start counting from there.  The first rule here (pennies) was two over, one up.  Each time you repeat the rule, you start counting from the last token on the grid.




















You're probably wondering about the buttons?  Well, that's a different rule: one over, one up.  Isn't it cool how they overlap, but not always?  Kids can make up their own rules after a little modeling or you can challenge them to guess a rule you made up and keep it going. 

And then, of course, the final thing would be to leave the pennies and the paper grid mat out to explore at leisure. 

I made a new game!  Well, maybe it's really more like an activity, a really fun activity where stars, factoring, combinations and geometry are all rolled up into one very beautiful package.  My seven year old and I both learned a lot in this first round of what I hope will be a very fruitful inquiry into stars and their use in elementary math learning.



















This whole activity is thanks to some recent interactions I've had with Paul Salomon of Math Munch and Lost in Recursion.  A couple days ago, Paul posted a photo of some amazing stars he designed and manufactured himself, using a laser cutter to and 1/4" plexiglass.  Aren't they cool?













I showed them to my daughter and although we both thought they were super cool, we also wondered what in the heck was going on there?  I noticed that the bottom row would fit into the centers of the top row, and I counted twelve sides and twelve rays/points in each star, but other than that I couldn't figure it out.  Paul was nice enough to explain it to me:

"The top row is every possible 12-pointed star. The bottom row is the cutout dodecagon that fits in the middle. Starting on the right we have a dodecagon (every line goes over one corner); then on the next one 2 hexagons (lines go over two corners); then 3 squares (lines go over 3); then 4 triangles (lines go over 4); then a single-pieced 12-pointed star (go over 5); then a a 6-line asterisk (lines go over 6).  Make sense?"

It did make sense, but only in a fuzzy sort of way.  I was still incredibly curious about what kind of math this was.  I thought I saw some geometry (shapes, right?) but I suspected it was more than that.  Here's what Paul told me:


Paul's patient explanation included a link to Vi Hart's video on doodling stars which was really helpful and a very cool star applet for playing around with different permutations (or is that combinations?) of points and lines. As always, I started thinking about how my daughter and I could explore these ideas together.  I have my own inquiry separate from hers, but it always seems to come back to one question: How can I use this new information with a seven year old in a way that is mathematically meaningful?

Tah Dah! 



















I used the star applet Paul recommended to make a 3 star into 6, then 9, then 12 etc. and then put them side by side on the same sheet of paper.  My thought was that it might be an interesting visual way to explore multiplication and groups (for example, 1 group of 3, 2 groups of 3, etc.) as well as part to whole.  Basically I had no specific ideas about how we were going to explore the sheet until we started, but I think, for the first try, it worked out rather well.  Here's how things went down this morning:

Me: We're going to do something with the number three.  There are three of one thing that we're going to add to another three and see what it looks like then.  Sort of like multiplication.

Kid: I already know how to do that.

Me:  I know, but this is a new and different way to think about it all plus we can use your new colored pencils!  So, let's look at this first triangle.  What can you find three of?   

She quickly found each point/corner, which she spontaneously marked with blue colored pencil, a move that completely influenced the way I guided the lesson from that point on.  I've used geometry words with her in the past and I asked if she remembered the word for what we were calling a corner.  She didn't so I had her write down the word vertex.  Then I wondered if she could  find something else there was three of in the triangle. She took her orange pencil and marked the sides, and then I had her write down the word 'edge'.  I thought we were done, but she suddenly found the interior angles and marked them with a pencil.  Awesome.










On the second star I said, "Now we're going to see what happens when two of the same triangles are put together.  What do you notice?"  She started counting individual points and found there were six.  To clarify the 'groups of three' I asked her to find one triangle to trace -- it took a minute but she finally figured it out and, after that she quickly found the second triangle which she traced out in blue.

Me: So, since there are now two triangles, we have two groups of three.  How much is that all together? 

She wrote down the number six inside the star using both the blue and orange which I was pretty happy to see.  I would have never even thought to prompt her in that direction, but it was a clear indication that she understood the number was a combination of two different numbers.  As you'll notice, she continued this practice all the way through the 18 pointed star.

Me: So, the next star is three groups of three.  Let's outline each triangle.  [This was challenging for her visually, but a good kind of challenge.]

Then it was time to move on to the next row.  Instead of tracing each individual triangle I suggested putting the same colored dot on each of the three points of any given triangle.  The first time she did this it took some concentration.  The second time she did a star this way it was no problem and she also started to notice that the colors went around the star in a pattern but, all of a sudden, she got suspicious...

Kid: Hey, wait a minute, this one is the same as the other one!

She started making marks while she counted the points and, sure enough, it was the same star twice.  It felt like the perfect imperfection for this lesson.  I always love it when kids discover anomalies or mistakes.  It means they're really paying attention. 









By the time we got to the 18 star she exclaimed: "All these stars are making my head hurt!" but I encouraged her to persevere.  After she found the first triangle (with the pink dots) I asked her if she knew enough now to predict the placement of each consecutive color.  She put down the orange and right away saw that, if you go in the same direction (in this case counter clockwise) three dots of a new color always go counter clockwise to the previous color. 

Even though I had said we could be done after the 18 star, we did get to the 21 star and it's good we did because I got to make another interesting mistake.  For this last star I wanted to show her a different color pattern I had noticed.  Since the previous star (18) needed six colors to highlight each individual triangle, I had her pick one more color for a total of seven.  Then I asked her to see what would happen if she just put down one color at a time in a sequence until she had used all seven colors. All was going smoothly -- she put down seven colored dots on the points starting with light blue, and I drew two little lines to show where the seven started and ended.  Then she repeated the same pattern (this time she went clockwise, I think) two more times.

It was at that point that I realized something was amiss; that we had visually shown three groups of seven instead of seven groups of three which was one of the main goals of this lesson.  A minor point, but one made much more obvious using the colors, and a result I am still puzzling over.




Here's the whole sheet where we left off:

























I think my pictures tell a pretty good story, but in the moment there was some major flow happening.  The colors are not just beautiful visual additions to the designs but also really effective in illustrating the structure, combinations and multiples within the stars. Overall, I'm pretty proud of myself for setting up and guiding this little exploration, but I would love (love!) to hear your feedback on this activity and any ideas you have on what we could do next. 

Malke Rosenfeld delights in creating rich environments in which children and their adults can explore, make, play, and talk math based on their own questions and inclinations. Her upcoming book, Math on the Move: Engaging Students in Whole Body Learning, will be published by Heinemann in Fall 2016.

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. 

Summer programming has given me some room to experiment with ideas that have been brewing over the last year.  Ultimately, I designed three new non-dance making activities that reinforced the Math in Your Feet ideas of creating a pattern unit using multiple categories of attributes.  My favorite by far, especially for its flexibility with a range of ages, is this paper patterns activity.

If you've scrolled down already you'll probably be thinking that it looks a lot like a paper quilt activity.  It is, sort of.  What distinguishes it from other paper quilt activities is that there is no pattern to follow, only some guidelines to direct the design process.  Because there's been a day or two at each site this summer where dancing was just not an option -- too hot, too crazy, too distracted by thoughts of swimming after class, whatever -- this quieter activity has been perfect at reflecting and reinforcing the dance work we've been doing on other days.

Just as with the dance work, the individually designed paper pattern unit is made up of four pieces (beats in the dance, squares in the paper design) with choices of various attributes.  The pattern unit is then repeated, or joined with a different design, to make a larger pattern, revealing complexity in combination.  Here's how I introduced the activity at my first summer site:

























The idea was to experiment with different four-square designs, using a combination of squares and triangles.  At that point the kids would pick their favorite pattern unit and then repeat that favorite four times in the sixteen square grid, as pictured above.

This turned out to be confusing for the kids at the first site.  They were on the younger side (ages seven, eight and nine), most of them emergent or beginning readers.  This, in itself, is a clue as to the importance of an activity like this. A pattern unit is comprised of various smaller parts that make a larger whole.  In math this is called 'chunking' and is a crucial skill for algebra down the line.  If a child has not yet learned to 'chunk' words, they will also most likely benefit mathematically and in their reading from the challenge this activity provides. 

Unfortunately, it was a crazy site, and I didn't have the time or the support to help individual children the way I would have liked.  Since this was the first time I had tried the activity, I didn't anticipate some of the issues that came up.  For example, the kids had too many color choices and ended up using all of them.  It was okay to have four, five or six colors in one, four-square design, but when it was time to repeat the pattern unit, it was much too hard for them to slide the design over or down to repeat it in the grid. 






















There was some interesting experimentation with the triangles and squares, though, which is always good in my book.




















Do you see what I mean about too many colors (design on the left, specifically)?




















Here are two kids who figured it out:




















Well...sort of. 




















By the time I got to my second site I had adjusted the activity.  I split the sixteen square grid into four smaller, separated parts, and when they had created a design they liked on that sheet I gave them the larger grid. This helped focus kids on the individual unit itself and then make the transition to the larger grid.
















The kids at this site were also older (nine, ten, eleven, even twelve).  This time around I decided to limit it to a choice of two colors which I think helped focus the activity tremendously.



















This design is a little out of the box, but it doesn't surprise me as this girl was the only one to use a combination of turns in the dance patterns she created.  Her brain was already 'there' if you know what I mean.,









































This final picture sums up why I love this activity so much.  Small, simple pattern units get combined to make something surprising, beautiful and mathematically interesting.  The elements of personal choice and action on the design process creates unique results for each child.  Just like in Math in Your Feet.

And the math?  Flips, slides, turns.  An inventory of attributes.  A problem solving process.  Grids.  Patterns. Attending to precision. Just like in Math in Your Feet!

Most importantly, the idea of pattern unit and the concept of chunking is reinforced throughout the entire activity.  And, if they can do it, great!  If they can't, you first and foremost find the beauty in their efforts.  It is this personal work, full of thought and energy, that becomes a self-generated incentive for moving forward to the ultimate goal.  Not only does this activity allow you to celebrate the individual efforts of each child but also makes it easy to immediately assess where the learning points are -- because the evidence is right there in front of you! 

I'm trying to figure out how to do all that on a budget.

One of my ideas is a beading project that will work well for both boys and girls in the younger and middle age groups.  I'm thinking about starting with both these books.