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.

Every day this week we've been playing with math dice. Enthusiastically.

I'm not going to name the company because not only do I not review or endorse any product on this blog for money or power (not that they asked) but it is also quite easy to go out to your local games shop and get your own set of two 12-sided and three 6-sided dice. (The rules are also pretty easy to figure out: multiply or add the two numbers on the 12-sided dice and then roll the six-sided dice and try to find a way to make the target number using as many operations as you know.)

Did I mention the enthusiasm?

My newly eight-year-old is enthusiastic about many things but has always been a little standoffish with her affinity for math, probably because, I think, she perceives it as my 'thing'. So, it's been nice to be able to truly enjoy a math game together. (It's been a while -- we were heavy into UNO a couple years back which was super fun.) It's clear my kid is on her way to a happy relationship with operations, but there's something even more interesting developing...

I had always thought my girl was not what I would call 'systematic' or 'precise'. I know for sure she is prone to intuitive leaps of connection or understanding and lots of messy tinkering, none of it looking either precise or systematic to my eyes.

As I've been drafting and revising this post I've realized that maybe she has been those things, I just haven't been able to see it. And, as we've been playing the dice game I've watched her systematically running through different combinations of the six-sided
dice (by moving the dice physically to different positions) and reasoning to herself out loud as she thinks through the different ways to use the hand she's rolled.

I guess I always thought that precision in mathematical problem solving looked, well, neat and orderly andon paper.

Anyhow, I am not (too) ashamed to admit that I was wrong. I think she's been precise and systematic in her own way for a while now. In retrospect, I realize I've heard this kind of 'talking herself through' a series of moves or ideas before. Systematically. In math and in many other contexts. For years. In a messy, verbal, highly enthusiastic way.

Okay, so I'm a slow learner I guess, but pretty open minded all the same. I think it's worth considering that there must be a difference in the way children and adults go about their reasoning. Or, at the very least, that I have a deeply ingrained image of 'what it looks like to do math'. I'm going to keep thinking about all this. If you have any observations or resources to share on this subject, I'd be tickled pink.

In the end, I'm super impressed that not only is she beating the pants off me but she has also created her own strategy for combining operations to reach a target number. And it's all her. The only thing I did was bring out the dice.

Her: Mama! I discovered another infinity! Half of two is one. Half of one is one-half. Half of one-half is one-half of one-half. Half of one-half of one-half is half of...

Me: Hey, I totally get it, but you wanna hear an easier way to say it?

Her: Yeah.

Me: What's half of two?

Her: One!

Me: What's half of one?

Her: One half.

Me: What's half of one half?

Her: I don't know.

Me: Ummm...you know how when we bake muffins?

Her: Yeah.

Me: Well, when we use the measuring spoons...[I talk a little more but it becomes obvious she's not getting it and I'm not equipped at 6:30 in the morning to think clearly. I abort the mission]...okay, here's the easier way to say it: Half of one is one-half. Half of one-half is one-fourth. Half of one-fourth is one-eighth, half of one-eight is one-sixteenth...

Her: So it gets smaller as it gets bigger!

What's remarkable to me about this particular conceptualization is that one, it mirrors thinking of done thousands of years ago by philosophers -- but she has never heard those paradoxes. The second thing is that, obviously, she's not done much with fractions and yet, the idea of half-ness and size is fully there.

And all this before 7:00am. Any time is math time, right?

Something shifted in my newly eight-year-old in the last week. It shifted in terms of how she understands math and science and in how she sees herself.

It started with a morning thought she had about the universe. It was such a big idea I knew I needed my smart friends on Facebook to help us out. At first, she was not interested. "No one cares what kids think," she said. My reply? "No way! That is not true. I have tons of friends who really value what kids have to say and who are interested in what kids think. Why don't you tell me your idea again and we'll see what they have to say?" She finally agreed. This is what she dictated and what I posted on my personal Facebook page:

Here’s how I came up with this theory. I saw the globe and was wanting to go on vacation and then I thought they should have a map of the universe but then I thought it would be impossible. But then this theory came to me. Since the universe doesn't have an end it must be a sphere and that is why no one has ever reached the end of it.

You know a tube of toothpaste with toothpaste in it? Well, it might be like the universe with all the planets in it and the universe might be shaped like a sphere and part of something bigger. Just like the tube of toothpaste is part of our bathroom which is part of our house which is part of our town which is part of our city which is part of our country which is part of our world which is part of outer space. So, those images get bigger and bigger just like the universe is in something bigger.

Seventy (70!) comments and twelve hours later, many of my wonderful math, science and artist friends had shared their thoughts, conceptualizations, and experience with us. Frankly, it was mostly over my head, but I did start to understand 4D geometry a little better. As for my daughter, she noticed that there can be different well-conceived theories out there that do not necessarily agree with one another. Also, that adults don't always know everything, but if you want to know more about something you keep asking questions. And that some of the reputable theories out there on the universe match her own thinking and visualizing. The fact she accepted the uncertainty of it all was a HUGE leap forward. But that's not all...

She had been working on and off for a few days to fully understand her own ideas about the shape and boundaries of our universe. Unfortunately, a lot of the physics concepts offered in the FB comments were hard to conceptualize at the elementary level. We were at the library a day or two later and she wanted to do some research - she found a book about the universe we hadn't read before and we started to read it. It was at that point I had a thought: thinking about the structure of the universe is also thinking about infinity.

We've done some reading and thinking about infinity in the past few years, on and off but, until now it's been sort of a fuzzy concept. Here is one of her musings from this winter titled "What infinity means to me" written by her little dolly. "It is sewing that has never been done. It is also cloth that can never be done. And a mountain that reaches on forever."

At that moment in the library I remembered the TED-Ed video called "How Big is Infinity?" and thought it might help her in her quest to clarify her questions and theories. The video utilizes set theory to help visualize the idea that "there are an infinite number of infinities of different sizes." In the end, I think that being able to visualize infinity via numbers set in a really fabulous animation helped her settle into her own thinking about infinite universes (the core of her personal theory). See what you think:

After she watched it she said, "I want to solve one of those unsolved math problems! But I'll work on my theories [about the universe(s)] first."

During all this Maria from Moebius Noodles asked Isobel if she would allow her thoughts to be published on the Moebius Noodles blog. My kid had to think about that for a little while. After all, these were her ideas in question. But, I reminded her about our conversations over the preceding few days that many good ideas stand on the shoulders of the work someone else has done. That's how good ideas are born -- from the seeds of past ideas and discoveries. "Not only that," I said, "but Maria is really interested in what kids think especially about math and science. She wants more kids to be able to share their interesting ideas." So, my kid consented, and even drew a picture of her idea. Her post is up now over at Moebius Noodles.

And, today, I noticed the shift. Subtle, but in a 'who is this child?' kind of way. All of a sudden, out of the blue, my daughter, the one who coined the term "math mommy" spoken in a derisive tone, says: "You know, I think math is actually pretty interesting." Not only that, today the math conversations seemed to flow much more freely:

Me: "Oh look, the Venus fly trap has caught some bugs!"
Her: "I know. I've seen it multiple times."

Her, looking at the pile of library books I brought home: "At dinner will you read me The Cat in Numberland? It's such a puzzling book." (And we read it all the way through together tonight. We've read it a couple times before -- it's amazing how much more we both understood now, after going through first and second grade math together.)

Her: "The picture of the [hexagonal] chip on the package is bigger than the chip in the bag."
Me: "How much bigger?"
Her: "Oh, about two times bigger." (She even helped me as I worked to figure it out for myself on graph paper. More on this in a future post.)
I know this is a long post, but it's been a pretty remarkable week where my kid had an idea and the adults took it and ran with it. That's got to be super empowering if you're newly eight and realize your thoughts and ideas have weight within the larger world. I thank all those who participated in this pretty astounding adventure with us.

She's not the only kid to have had these kinds of thoughts and ideas, but right now what I'm celebrating is just how great my friends were to have taken her ideas seriously which, in turn, showed her just how relevant she is to the larger picture of life. If you're a parent I think you'll understand just how grateful I am.

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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:

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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It's really no secret I love tape. I should have bought stock in it long ago. In my mind, tape is the ultimate open-ended, the world is your oyster, creative, hands-on learning and making supply. I mean, just look at how versatile it is: You can change an environment in an instant! Need kids to be able to visualize diagonals while creating foot-based percussive patterns? Voila!

Are you six and want to visualize a trajectory to the moon? Tape's for you!

Even clear packing tape can be harnessed in the pursuit of art making and invention:

Check out the endless ways tape can be employed in the interest of math, art, kinesthetic exploration, invention and education -- and then consider submitting some examples of your own! The Tape Chronicles are housed over at the Math in Your Feet website. Hope to hear from you!