The Map is Not the Territory

Moving Forward: The New 'Making Math' Page!

noreply@blogger.com (Malke) — Mon, 27 Feb 2012 01:54:00 +0000

So, you may or may not have noticed that this is a multi-topic blog.

I am a teaching artist who finds connections between math and percussive dance. I give kids the tools to explore space and time. I give their teachers the tools to bring body-based inquiry into their own classrooms. I started this blog to share my thoughts and ideas on creative (as in creating something), hands-on, hopefully kinesthetic learning, wherever it may happen.

This blog also documents my inquiry into what it means to 'do' math. I've been lucky enough to connect with a bunch of really smart people in the math ed world who have helped me toward my goal. When I first started the blog I had a hunch that there was more math embedded into the dance work we do in my program Math in Your Feet. I wanted to figure out what it was and how to explain it.

Sixteen months in, I finally feel like I've made some real progress! You can read about my big aha! moment about attributes here. You can also read about the moment when I uncovered an important connection between my art form and making meaning in mathematics, both of which incoporate moving patterns. It's probably my all-time favorite post. So far.

There is no way of anticipating how far I will go on this journey but recently I've begun to feel like I'm reaching another bend in the river. I am more fascinated than ever with what it means to think mathematically and how that skill is developed, especially in preschool and elementary age children. And, I'm a mom of a snappy six year old who accuses me of being a 'math mommy!' because, well, I am. I find inspiration in her learning, and within that process I find some of the answers I've been searching for.

In the last two months I've noticed a trend in my post topics. Nearly all of them outline some kind of hands-on project that has helped me, or my kid, or both of us, deepen our understanding of math. Not all of these activities are original, but every 'making math' post is a snapshot of our inquiry. Some of these activities were born out of some inspiration or question, but they are all easy to reproduce on your own and deliciously open ended (no wrong answers!). I've decided to put them in their own section in the hopes you might find them useful in expanding or deepening your own understanding of math. I plan to add to this collection over time.

I invite you to take a look at the new Making Math page and see if there's something you'd like to do with your students or kids. Most, if not all, of these projects can be easily done in some way by the youngers (usually with adult support) but also fully enjoyed by the olders, both in school and out. As Maria Droujkova of Natural Math and Moebius Noodles says: "'Make your own' is such a simple and powerful way to open mathematical depth. Like a game? Make your own!"

Make Your Own Attributes Matching Game!

noreply@blogger.com (Malke) — Sat, 25 Feb 2012 17:42:00 +0000

I have had attributes on the brain for a couple weeks now. Most of my thinking has been on how the practice of identifying similarities and differences (a closed process) really opens up when put into a design context.

So, given how much I've been thinking about it, it's no wonder that what I thought would be a ho-hum, pass the afternoon kind of game turned into an incredible brainstorm.

The kid and I were playing a Blue's Clues matching game. I was, truth be told, not 100% engaged, but it was a pleasant enough way to pass some time. At some point I noticed that this particular matching game was really quite tricky. You'd turn over two cards with green dogs on them but they were different in some very tiny ways.

Despite some very minor irritation that I might actually have to pay attention, I started to think about what those game designers were doing. Nothing like a little internal meta-conversation to help put the pieces together....

Click!

Attributes! On the matching game (brilliant, brilliant, game designers)! What if we made our own? Could we make our own?

"Do you wanna make our own matching game?" I asked the kid.

Needless to say, the answer was yes. We rushed around the house trying to locate the 3"x5" cards but, of course, they were nowhere to be found. I rounded up some other paper, the paper cutter, some shapes to trace, and the crayon caddy.

It was fast and furious, but in the midst of it all we managed to agree on my some parameters:

Geometric shapes only. This is (sneaky) math after all. We used tangram pieces, pattern blocks, some Cuisenaire rods, and some magnets from the fridge.

Each pair of cards has to be exactly the same. It's never too early to experience congruence, and what better way to do that then to be personally responsible for it? This is a key point in my still-developing argument (one that is based on my experience with Math in Your Feet as well as what I've been doing lately) -- it's one thing to observe congruence, it's another to have to be congruent yourself. I realized as we went along that the 'make a game' energy really motivated my kid to do this part up right. She paid a lot of attention to sameness in her designs.

Each pair of cards have to be different from the other sets in some way. We have yet to explore this fully, but this means that if you make make more than one set of triangle cards, the sets have to be different from each other in color or design or size.

It's this last point that created the most conversation as we created our cards together. According to the kid, I was altogether too boring in my designs. I would say, "But every design you make is colored in! They have to be different from each other, so I'm going to leave mine uncolored." This, apparently, motivated her to copy my 'boring' designs and make them 'more interesting' than mine!

So, we had a lot of levels of similar and different going on, including not only what we created but how we were doing it and what our personal design aesthetic was and how these creative choices were different or similar from each other. My brain hurts just thinking about it! See what you think:

All told, and in very short order, our prototype set of matching cards totalled fourteen pairs, which made for a very satisfactory game. I'd love to find a calmer time to for us to work more slowly, carefully, and thoughtfully on another set. She wants to make sets of matching games to sell at her ever-evolving lemonade-origami-bookmark stand in the spring, which I think might be motivation enough to take a closer look at how to generate similarity and differences.

If you end up making your own attribute matching game I would SO love to see what you create (please, please, please?!?).

Take Your Kid to Work Day

noreply@blogger.com (Malke) — Fri, 24 Feb 2012 19:00:00 +0000

In this case it was "take your kid to 'help' set up the space" day. For me, every new space presents its own unique set of challenges and it's nice to be able to set it up ahead of time. That way I don't have to drive 60 to 90 minutes, set up a new room, teach five classes of kids I've never met before, and drive home, all on the first day of a residency.

Here's what part of this one will look like on the first day (I was way too busy problem solving to take lots of pictures and, anyway, it's really a boring space without the kids):

Here's my dance board plus the student spaces for demonstrating to the class. Notice the carpet. In this case we have enough space, but the kids won't be able to hear the sounds their feet make. But that's okay, we'll still be doing awesome work.

Here's my kid on 'take your kid to work' day. She helped me put up the posters, move some chairs and then decided to play with tape. Lucky kid. I don't have pictures, but she made two really long lines of tape so she could do cartwheels down one and roll tape rolls down the other.

When I was helping her put the tape down I casually mentioned that we need to make them parallel so that they'd be the same distance apart all the way up and down the line. After that she took a closer look at the Turns poster (see first picture) and went over to the table to make a poster for her own dance and math class.

Translated into standard English, it reads: "Cat Dance. Cat Math. Phone Number. GO. Choreography. Performing Cats." Sounds fun!

You also might notice the effect the environment and conversation had on her illustrations. I especially like the square with arrows radiating out of it in all directions (probably influenced by the Turns poster). And, in the illustration of shapes, along with the circle, triangle and square are a set of parallel lines!

[A side note: Her spelling is making noticeable gains in the past week. I have to say that's a pretty good go at 'choreography' and 'performing' for a kid who, as recently as last week, skipped all the vowels when sounding out her words, lol! Isn't brain development spooky?]

She figures that six participants will be enough for the class to run. If you sign up she'll send you more information in the brochure she made on the car ride home.

Let's Play "How Much Change is in Mama's Purse!?"

noreply@blogger.com (Malke) — Wed, 22 Feb 2012 19:59:00 +0000

I was going through the Common Core State Standards for Mathematics the other day, to finalize a document that illustrates the connections between the CCSS and my program Math in Your Feet. I was really focusing on fourth and fifth grade standards, but while I was there I peeked in on the first grade standards as well. I was curious.

I've got a first grader of my own, and I shepherd her math learning. I watch her figure out how numbers work and relate to each other and I observe her make observations and formulate questions while experimenting with shapes and other manipulatives. So, I really liked looking at the CCSS to put all that work into perspective. The content standards were all pretty straightforward but I immediately started trying to figure out how I could really know if those words meant things my kid knows. What does her understanding look like in action?

Luckily for me, it turns out that a great way to assess a lot of what a first grader knows and can do, mathematically speaking, is to look inside my purse!

We were having a late lunch/early dinner at our local co-op. The plan was to eat a nice balanced meal and then hit the bulk bins for chocolate coconut candies after that. I was digging around in my purse to make sure I had enough money to get the promised after-dinner treats.

Digging...digging...and finally finding a handful of coins. That's when inspiration struck.

"Hey! Let's figure out if I have enough to buy dessert!"

Unfortunately my camera was at home with battery charging, but the following photos are a faithful re-creation of the independent thinking that followed.

Showing no hesitation she went for the quarters first. Lined them up, counting by 25, dollar by dollar. Got the dimes in line counting by ten, and finished out that dollar with nickles (by 5's) and pennies (by 1's). Added it all up in her head, no fingers (you go, girl!). The grand total? $5.07.

"Which coins make the highest stack?" I wondered.

We made stacks from tallest to shortest, quarters, pennies, dimes and nickles. And then I remembered attributes, and we compared each coin and ordered them thick to thin. Turns out the phrase 'one thin dime' could also almost be 'one thin penny' by a hair. You'll also notice that putting them thickest to thinnest lines them up biggest coin to smallest coin. Hmmm...I wonder how that happened?

What did all this dinnertime inquiry show me about my daughter's math skills in relation to the CCSS?

First, it's important to remember that using money is just one of many ways to represent numbers and understanding of numerical relationships. As an instructional tool, I think that using money is one of the best ways to develop numeracy. You're working with a whole unit of 100, for one thing, and anything less is a fraction of the whole. Also, the random sizes -- a dime is worth more but smaller than a nickel -- is a fabulous conceptual challenge, and think of all the different kinds of skip counting! Plus, money amounts are always changing depending on what you save, spend, or find on the ground, so there's always something new to figure out.

So, she can represent her numerical understanding through money and mental calculations, but although she likes to write out simple equations, we haven't done much with adding larger numbers on paper. After reading Peggy Kaye and Constance Kamii (which I've written about here and here), I'm fine with it that way for now.

As for the standards, most of 'how much money in the purse' challenge was related to numeracy skills but, overall, the activity showed me that my daughter is gaining mastery in multiple content ares in three of the four content domains:

Operations & Algebraic Thinking: "Developing understanding of addition, subtraction, and strategies for addition and subtraction within 20."
Numbers & Operations in Base Ten: "Developing understanding of whole number relationships and place value, including grouping in tens and ones."
Geometry: "Reasoning about attributes of, and composing and decomposing geometric shapes."

The other half of the CCSS for math is the Mathematical Practices section. The change in my purse also illustrated these goals in action at the first grade level:

"...adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy)."

In this case the mathematics was most certainly 'sensible, useful and worthwhile'! Although the change in my purse did not asses everything in the practices section, of course, it was helpful to have a spontaneous moment to see up close not just what she knows but how she goes about using what she knows. On the whole, I feel confident things are going in the right direction.

And, in case you're wondering, the chocolate coconut clusters were delicious!

You Can't Plan Perfect

noreply@blogger.com (Malke) — Tue, 21 Feb 2012 17:00:00 +0000

I l.o.v.e. when things fall seamlessly into place.

We've been talking about pentominoes for a week. This morning I had the thought that today would be the perfect day to find some.

Looking for audio books at the library, we run across Chasing Vermeer by Blue Balliett.

I say: "Hey, here's that book I was talking about. The boy carries a set of pentominoes in his pocket and uses them to solve a mystery about a missing piece of art. Wanna get it?"

We drive to the other side of town and find a beautiful set of lemon yellow overhead pentominoes for $4.

The kid can't wait and opens the pentaminoes bag in the car. Plays with them. Starts singing..."Is-o-be-ell...lo-o-o-ves...pen-ta-mi-noes!"

We're home and the kids pulls out the pentominoes and starts the book. I'm typing 'how to use pentominoes' into Google and the narrator starts to read..."About pentominoes..."

A few minutes later, the character Calder has just finished making a rectangle using seven pentominoe pieces and I look over and the kid has a rectangle in front of her. I go over quickly and count her pieces...eight!

You can never plan for something this wonderful.

Mapping the Familiar

noreply@blogger.com (Malke) — Fri, 17 Feb 2012 20:00:00 +0000

There was a bright, blue sky up above, and at ground level there was frozen fog. The result was a morning delicately dressed in sparkles. Perfect for a walk.

Down the driveway. Pick a piece of lavender, covered with millions of tiny ice crystals. Which way do you want to turn, left or right? Next intersection, right or left?

We walked through the fog, already burning off. We could feel the sparkles entering our lungs with each breath.

Busy road and cars zooming. Three choices, right, left, or straight ahead. "Mama, I used to be afraid of those trees, but I'm not any more." We pick tiny pine cones off the tips of the branches.

Next crossroads, "Mama, that's University Street down there, let's go there." Thinly frozen puddles alongside thinly disguised reading practice: "What does that P stand for, do you think? The one with the red circle and the cross over it? What does the rest of the sign say?" I ask.

Right turn down University and an old dog barking at us across the lawn. "Mama, I used to be afraid of dogs, but now I just don't like their licking."

Find two sticks, clap them together and then we're marching. Left turn. Right turn. Rose street! Find the stone pig, march on home. "I wanna make a map!" the girl says.

She wanted red and blue like the roads in the atlas. We recalled which way we turned, what we saw, remembered the crossroads, one block at a time. All of a sudden, she notices corners and the geometry of the street layout. "I always thought the blocks went in a circle. I didn't know they were squares." The things she loves go on the map (friends' house, stone pig) as well as the things that 'used to' frighten her. Her long battle with anxiety has its landmarks as well.

It may not be to scale, but when we go out again this afternoon to follow our map, the issue of scale might come up. I'm also pretty sure we didn't get all the streets on there especially at the end of our route, but next time we'll bring paper and make some notes as we walk.

Mapping the familiar twists and turns and landmarks of our neighborhood -- bodies first, memories second, paper and pencils third. I marvel at the human brain inside my six year old daughter's head that is so driven toward representation of her experiences and activities; driven toward it even though she is still just learning to decode print and write using 'the rules' .

She makes maps of other kinds, too. Sewing "patterns", also not to scale, but clearly a sequence of steps mapped out:

I recently read a fascinating article in the New York Times about teachers taking their young students on walking field trips as a way to develop literacy. This kind of activity is literally a step in the right direction. Without concrete, kinesthetic, physical experiences like these, no child can fathom the meaning behind the marks on the page or develop full mastery of the human brain's greatest gifts. The order needs to be sensory experience / memory / symbols, not the other way around.

Math Books for Cats

noreply@blogger.com (Malke) — Thu, 16 Feb 2012 18:44:00 +0000

I don't know for sure, but I suspect that every kid holds in his or her heart a metaphor or identity that can open the door to learning.

With my kid, you don't have to dig too deeply. If it's a narrative that includes cats, she's interested (except with the Clan series -- too many kitties getting hurt make her crazy). She is a cat, you know. She started out a kitten at the age of two and now she's a cheetah (to account for the size of her six year old self). Maybe you know a kid or two like her?

It's amazing how much you can learn through cats. Interested in pioneer life? No problem, there's the short chapter book called Pioneer Cat. For a cat-led adventure through gripping transitional moments in world history, try Time Cat by Lloyd Alexander. Flying cats? The Catwings series by Ursula K. Le Guin is just the ticket. Cookie's Week is perfect when you're learning to read. Cross-species communication is covered in Jean Craighead George's How to Talk to Your Cat. And, the cat book lover's ultimate cat book, Dewey: There's a Cat in the Library by Vicki Myron.

So, it probably comes as no surprise that I love any chance to work math into the cat ~~obsession~~ lifestyle. Here are three books that have a special place in my heart.

My very favorite cat and math book is Five Creatures, by Emily Jenkins. It's about the similarities and differences (attributes!) between the members of a lovely little family.

"Five creatures live in our house," it begins, "Three humans and two cats. Three short, and two tall....Three with orange hair, and two with gray." We read this book when my daughter was in preschool and it was fun for both of us to look at the pictures to see who matched each description. The categories of family attributes are not always straightforward, which makes this a wonderfully interactive read. I am planning to bring it out again and see how it looks through my daughter's now six-year-old eyes.

The tag line for One Hungry Cat by Joann Rocklin is "A greedy cat learns division." When we first found it we read it over and over and over, giggling every time. I'm going to go back to the library and get it again since we're heavy into arithmetic these days; it's got an introduction by Marilyn Burns, and a bunch of activities created by her to go with the story. We play (and love) another Marylin Burns-created math activity using Cuisenaire rods called "Build What I Have" so I have a feeling there are some rich resources there.

So Many Cats! is also a favorite. It's a straightforward story of a growing family of cats, but the rhyming text is musical and fun to read. If you're a cat lover or, indeed, a real cat, you will read this with great affection for every new cat that joins the story.

If you have suggestions for more cat and math books to add to our list, both the kid and I would be thrilled to hear them!

Hundreds Happiness!

noreply@blogger.com (Malke) — Tue, 14 Feb 2012 20:18:00 +0000

Okay, I'll admit that when looking at a hundred's chart I've had the thought, "What's the big deal?" I mean, the kid learns to count to one hundred and that's pretty much all there is too it, right?

I still had that thought when I read Julie's post over at Craft Knife about her love of hundred's charts and her kids' love of the game 'Roll to 100'. But, since Julie was so enthusiastic about it all and since we've been using dice for math games lately, I thought I might look into it. Maybe the kid would be interested in coloring in the hundreds chart that way?

She loved it! In fact we played the game twice! The second time we played she got all fancy and wanted to color in the chart using 'rainbow order.'

I think this must have been the first time she'd seen a chart like that. I know they did a lot with counting to one hundred in kindergarten; I may be wrong but I don't remember seeing a chart like this in the classroom. Anyhow, after a couple games with the dice I wanted to bump it up a little, and I remembered that Let's Play Math has a great post about twenty things you can do with a hundreds chart.

From there I found a link there to some printables, including partially filled in charts where the kids can finish filling in the numbers. Frankly, I was still dubious. I wondered, "How hard is it really, to fill in the chart?" I printed them out anyhow on the off chance I'd find a way to use them with the kid.

That was last night. Today the kid had a leftover Valentine sticker that looked like a gold medal and she was searching for something to 'win'. I said, "Wanna play some games with me?" and she said, "Why don't you give me a math test and then I'll win this gold medal!"

Perfect!

I said, "Well, why don't we do something with the hundreds chart again, but this time I have a couple different options for you. Do you want the one that's already filled in or one that's partially empty?"

She picked the partially empty one and, as we sat down, I did have the fleeting ~~panic~~ thought, "What am I going to do with this?!?" I started with the already filled in three and said, "What comes before three? Can you fill in those numerals?" Then I noticed the tens all went down the right hand column. I knew that she loved skip counting by tens, so we started there.

"What do you notice about this column?" I asked, and then the fun began!

Over the course of about 40 minutes, which is amazing for her, we alternated filling in columns and rows. Since she knows skip counting by fives, we did that column next, but quickly realized that even in the five column you don't count by fives! Obviously, I did not spend enough time with this chart when I was a kid.

Anyhow, after figuring out what really needed to happen we tried the one's column. I asked her, "This column starts with one, and the next number below is eleven. We know now that when we go down the column, each number is ten more then the one before, so what number comes next in the box below eleven?" It took a few minutes to establish an understanding of the pattern, and I used words like "Three tens and a one, four tens and a one, etc." to try and emphasize place value a little. She only put up with that for about a minute at which point she asked me to stop talking so she could figure it out herself!

Initially she found filling in rows easier, because all you do is increase by one, but I encouraged her to do columns too. At some point she exclaimed, "There's a pattern to this!" and happily filled out the rest of the chart column by column. About two thirds of the way to completion she gushed,

"I love this. I feel like I'm in the chart!"

In the end, when I look at the completed chart I'm still sort of thinking "What's the big deal?" but, watching her figure things out today, I know the meaning is in the creating. There were a bunch of different, mostly empty chart printables that I printed out, and I'll leave those lying around and see what happens next.

Symmetry Artist: Exploring Attributes Through Design

noreply@blogger.com (Malke) — Mon, 13 Feb 2012 18:41:00 +0000

I recently discovered how important the practice of identifying and describing attributes really is to math learning, especially at the elementary level.  Learning to discern similarities and differences develops mathematical thinking skills that can be used at all levels and topics within mathematics.

Despite the thrill of discovering how this kind of thinking is used in my program Math in Your Feet, I was left with lingering questions about the differences between identifying attributes and using attributes in a design process. I suppose they are two sides of the same coin, but I can't help thinking that being able to choose from an inventory of possibilities is the preferable skill-building activity in the long run.

I tested my theory on myself and my six year old daughter by using the Symmetry Artist at Math is Fun. Maybe you've seen this before? Check out all the attributes you can choose from to make your symmetry designs:

You can choose between reflection or rotation symmetry, with six to eight choices in each category.

Your pen has five choices.

Seven thicknesses.

And many, many colors.

There's a lot to choose from here, but the tool is easy to use, so you can easily change your mind and start over, or take out your last move or series of moves.

While you experiment and play around you are also noticing relationships between the lines and shapes you make and observing the structure of the final design. The addition of color serves to increase the complexity and interest in both the process and the product.

I suppose the difference between identifying attributes and using attributes is that one is a more closed process than the other.  I'm thinking that with a set of attribute blocks, for example, although you are learning to discern differences and similarities, there are really only right and wrong answers. The benefit of this kind of activity is that you are actively using math vocabulary: thick/thin, large/small, circle, square, triangle, edges, etc. This is all very useful but, as I said, a somewhat closed process in terms of inquiry.

In contrast to 'compare/contrast/identify' there's the process like the one you use in Symmetry Artist. Instead of simply identifying attributes, you are using this skill in context while thinking mathematically in an active way -- you are actually 'doing' mathematics. By this I mean you are asking questions ("What would happen if I started my circle here? What would the same design look like with nine iterations instead of four? How'd that pentagon get there?"), experimenting with and analyzing your 'answers' (designs), erasing your answers, starting over, printing out the answers you like, asking more questions.... This is the creative process in action; it is also mathematics in action.

I love Symmetry Artist because it is a beautiful and fun way to play around with, learn about, and compare how lines, shapes, and iterations interact within these two symmetries.  Below I've put just a few of the designs my daughter and I recently made.

As a first grader, my kid uses this tool primarily for exploration.  I started by explaining the different categories, but not much more, and she jumped in from there.  I, on the other hand, being aware of just how many choices I had, jumped in but soon got overwhelmed with too many questions which led to too much erasing and/or starting over, resulting in pretty much nothing to show for my efforts! That didn't stop me from noticing a few things, however. Take a look at what we did, see what you think and then go try it yourself!

Here's my daughter's first design. Notice that she set it for rotation, 'eight', pen, one color, and a medium thickness. She 'drew' with abandon, and I was thrilled with what resulted, mostly because I would have never thought to do it that way! As you'll soon see, my initial approach was a bit more measured.

Her second design was inspired by the first. "I want to draw a sun!" she said. I don't know if you can see it, but she started with 'four' red, then 'nine' red, then 'nine' yellow.

Here's the one I made as an example to show the kid what you can do with the tool. I used 'five', circle, and some variations in thickness and color. I love the almost-pentagon in the center where the five big circles cross.

The one thing during my experimentation that really thrilled me was that I noticed a difference between the process of drawing a design with a line of symmetry compared with what happened while using 'two' rotation. Can you see it?

I know what it looks like, and how to describe it, when a two-person team transforms their percussive patterns using both kinds of symmetries, but there was something about using Symmetry Artist that made it stand out to me in a different way. You are actually drawing one design and the multiples show up automatically. There was something about seeing it happen in real time (moving patterns!) that I would not have noticed if I had been drawing it by hand, one reflection or rotation iteration at a time. If I had been doing it that way (pencil and paper) I probably would have noticed something else altogether.

This is all just more evidence that you really do need multiple opportunities to observe and work with a math concept in a number of different situations to really understand it. In the end, I think that both the act of identifying attributes and and the act of using attributes in a design process (choreography, visual art, tangrams, etc.) have something to offer each other.

Because I am a teaching artist, dancer and musician, I am coming at math education from a different direction than many. I am also a big picture learner, so putting math in context makes a great deal of sense to me. What I am still thinking about are things like: When is it important to just teach math as math? When do you move on to reflection on and representation of the math you've discovered during an open-ended exploration? These are the questions that quite literally keep me up at night, and one reason that it's taken me so long to get this post the way I wanted it. I suppose I may be overthinking things, but I really am curious about all this.

What's your experience with attributes? I'd love to hear what you think about all this!

Open Ended

noreply@blogger.com (Malke) — Fri, 10 Feb 2012 19:27:00 +0000

I just realized that a lot of my posts lately have been about making mathy things out of open ended materials: straws, pipe cleaners, craft sticks, rubberbands, paper, glue. It's made me think about when I teach Math in Your Feet, how fourth and fifth graders almost can't believe it when they've made up their own eight-count pattern out of nothing more than three categories of movement-based attributes. One child even said to me, during our end-of-week reflection time: "I didn't know I could make anything." Heavy sigh.

This ad from 1981 reads:

"Have you ever seen anything like it? Not just what she's made but how proud it's made her? It's a look you'll see whenever children build something all by themselves. No matter what they've created."

There's a lot this ad says to me, but the very most important thing is that:

It is imperative that we give our children, and our students, a chance to explore and discover at least part of what they learn ON THEIR OWN, ideally with open-ended materials and perhaps a very casually introduced question. This doesn't mean that the learning environment becomes chaotic, or that teachers aren't needed. Structure and forethought are still useful but, in this case, they become the background against which children employ their own desires to the materials at hand.

We need to trust that kids can figure things out for themselves sometimes, and that the best learning often happens as a result of a question and the self-motivation to find an answer that suits them in the moment.

Shut the Box, aka Beat the Sailor: Fun with Number Combinations

noreply@blogger.com (Malke) — Wed, 08 Feb 2012 23:33:00 +0000

"Wanna play UNO?"

"No!"

"How about Shut the Box?"

"No! I want to go to finishing school!"

"I'm not sure we have one around here."

"But we can do it at home."

"True. I know! Every fine young lady needs to know how to outsmart a sailor." Noooo...there was nothing at all off color implied here, it's just that... "Shut the Box is the sailor's game, you know. Part of your finishing could be to learn how to beat them at their own game!"

"Why?"

"Because fine young ladies also need to learn to be smart and strong and know how to think for themselves."

"Okay! Let's play!"

And, in case you didn't know, Shut the Box is fabulous for ~~practicing combinations of ten~~ [Edit: That's not what I meant! I meant finding different ways to make the number you roll with the two dice. Duh!]. And then you add up the leftover points, which can go into the twenties if you haven't yet figured out all the strategies. (She's been doing all this in her head, sometimes with fingers for the larger numbers.)

And...once she is really strong with tens we can move on to a game that didn't quite work a few weeks ago -- 'Tic Tac 15'. The stronger she gets at adding, the more adept she is at skip counting and figuring out her own answers for questions that require multiplicative thinking.

For instance, we were at the grocery store the other day, walking through the wine section (I know, I know, this all has a sordid cast to it, but really, it's all circumstantial and quite innocent). There was a case of bottles and a sign that said "$10.99 a bottle". I didn't even notice it until, unbidden, she started counting, "Ten, twenty, thirty...."

She was trying to figure out how much the whole case cost! Next time I'll try to keep my wits about me and mention that $10.99 is actually very close to $11.00 and then I show her the fun in skip counting by elevens!

p.s. In case it isn't clear, I'm taking the role of the sailor. I'm also the coach -- I've been walking her through the strategy of shutting the higher numerals first (sort of like in UNO) so you don't get stuck with those points at the end, but on my turn I play more erratically so that she wins. For now, anyhow. At some point we'll move on to timed rounds or something even more cutthroat. It is a sailor's game, after all.

A Big Discovery

noreply@blogger.com (Malke) — Mon, 06 Feb 2012 20:11:00 +0000

The kid and I were playing around with tangrams the other day and -- all of a sudden -- I had an Aha! moment. It was a chain of connected thoughts and realizations which provided clarity around an answer I've been hunting down for the last fifteen months. Yeah, I know, this big! Here's how played out:

I recently read a post by Shelisa at Think Magnet which I found at the Love2Learn2Day Math Monday Blog Hop. The post was about a brilliant game Shelisa plays with her kids every time the family sits down to a meal.

I was completely impressed with the game because not only did it grow out of a spontaneous observation by one of her kids (my favorite kind of learning) but she was then inspired and able to turn the moment into a wonderfully complex and endlessly interesting activity.

I can't really do justice describing the game here; you'll have to go see for yourself and you'll be glad you did! But, the upshot is that the game is all about attributes and learning how to discern differences and similarities. It is also about just how inspiring a playful approach to learning can really be.

For the most part, I've been following my own kid's interest and questioning related to math over the last six months, an inquiry which has focused on measurement, maps, number patterns, and geometry discoveries. Reading Shelisa's post I started wondering how I might incorporate it more directly into our own math explorations.

So there I was, later that day, mulling all this over at a semi-conscious level in the back of my brain while sitting on the kitchen floor with my daughter. We had just re-discovered a whole box of tangrams, with four or five puzzles all mixed together.

"Look Mama!" the kid exclaimed, "I made a parallelogram out of triangles! And I can use two more triangles to make it into a rectangle!"

At the time I was playing around with building squares and I decided to start building onto her rectangle.

"Oh, look!" I said, "I can make a big square out of two triangles. And I can make a second square out of four smaller squares. If we make a third square and add it on, we'll have made your rectangle even larger! How can we make the third square differently from the first two?"

And, in my mind, I said: We can play around with position, size and color. Adding attributes like these increases complexity and interest...

And that's when it hit me. How many ways can you make...? That's the question I had been asking the kid about the number twenty or the number ten. And here I was asking a similar question (How can you make the third square different from the first two?) about combinations of shapes, and colors, and sizes.

Attributes and combinations.

I ask this type of question in Math in Your Feet too: Which of these movement variables will you choose as you create your four beat pattern? How can you make your second pattern different from your first? To help them find their 'answer', students use a tool I created called Jump Patterns to choreograph their own foot-based percussive dance patterns. Jump Patterns are created by working with the elements of a percussive dance step (the related math term being 'pattern unit'). Take a look at the chart I use in the classroom:

Students use this chart to experiment with what I have always called Movement Variables, except now I realize that, in the math sense, they're a collection of....attributes!

For the classic set of attribute blocks it's five shapes, three colors, two thicknesses and two sizes.

In Math in Your Feet kids choose from five foot positions, six types of movement, and six directions to create each of the four beats in their Pattern A, and then they make choices again when they choreograph Pattern B. There are two other categories to work with as well, including tempo and starting position.

Here's why it's a big deal:

The math topics in Math in Your Feet have always been clear to me: concrete, kinesthetic experience with spatial reasoning, combinations, congruence, transformation, reflection, and rotation. These are the things kids and their teachers know are being taught. They've always been the 'selling points' of the program because they are easily recognized and understood as 'math'.

But, after years of working out connections between these topics and percussive dance I knew there was more going on than simply learning 'about' math. My hunch was that there was some real mathematical thinking going on, I just needed to figure out where it was occurring and how to explain it.

This hunch was one reason I embarked on my current math quest (chronicled in this blog over the last fifteen months). I've been working hard to figure out what it means to 'do' math. And now, my new understanding about the nature and purpose of identifying attributes in elementary math education has provided at least part of the answer:

When children are able to identify the elements (attributes) that they have used in their creative dance work they are thinking mathematically. When they use this understanding to analyze and critique others' creative work they are thinking mathematically. Attributes are 'charactaristics of an object or a shape' and I'm confident that this definition is broad enough to include moving pattern units as well.

I am struggling at the moment to articulate my thoughts about the differences between identifying attributes in a traditional math education setting vs. choosing attributes in a creative or design process. But, no worries! After more than a year of asking, surfing, conversing, reading and learning, I feel like I have won some kind of prize! And, thanks so much to everyone who is sharing this journey with me.

p.s. If you want to learn more about Jump Patterns, here's a link to an article I wrote that details the whole story of what they are, how I developed the tool, and how they are used in the classroom.

Math in Action: Catapults!

noreply@blogger.com (Malke) — Mon, 30 Jan 2012 20:02:00 +0000

When my daughter was in preschool, one of the moms (who happened to be a science ed specialist) did a project with the whole class. It was brilliant. She taught the kids how to work a catapult (made with a plastic spoon, I think) and, over two years later, my kid still talks about that day.

So, I was not too surprised when the girl said recently: "I want to make a catapult!" A quick Google search turned up instructions for one made out of craft sticks and tape. Now that is my kind of project. Historically I've not been too handy with 3-dimensional construction, but I'm up for anything that requires tape!

We started by making three congruent triangles and followed the directions through the next step. Suddenly, recognition!

"Hey look at what we've made!" I exclaimed. "It's a...?"

"Tetrahedron!" the kid laughed.

Not surprising -- triangles are the strongest shape, and it was fascinating to see how strong they really were, even given the modest construction materials.

We used a simple origami box we had lying around as the catapult cup, and wondered if a wider arm would fling objects further.

We love our new catapult and have had many hours of enjoyment flinging unifix cubes around the living room. Now that is math in action, lol!

Friday (Video) Fun: Gregory Hines on the Arsinio Hall Show

noreply@blogger.com (Malke) — Fri, 27 Jan 2012 22:44:00 +0000

Nic Gareiss at Dance is Music continues to be my source for all manner of percussive dance inspiration!

Despite all my more recognizable math postings lately, and my never-ending enjoyment of accompanying my 6yo on her math journey, this kind of dance (not tap dance, actually, but related forms) is really where my interest in math all started.

Over time, performing and, especially, teaching percussive dance made me wonder if there was math in what I did. I simply asked the question and now kids get to learn math topics and exhibit their mathematical thinking and choreographic prowess through their bodies and foot-based dance patterns. Here's the whole story, if you're interested.

Addendum to 'Knowing Numbers'

noreply@blogger.com (Malke) — Fri, 27 Jan 2012 01:53:00 +0000

After posting Knowing Numbers & UNO Update yesterday there has been a development around subtraction, the addition of division (word play intended), an observation about the thinking-backward mind, and the arrival of the perfect book at the perfect time.

Today, the kid asked for another math test on the drive home from the Columbus Children's Museum. I gave her equations: 18+2, 16+4, 13+7, 10+10.

"Hey, those are a lot of different ways to make twenty!" she exclaimed.

"How many more ways can you think to make twenty?" I asked, thinking only of additional sets of addition. Here's what she came up with: 22-2.

We stopped to get gas. We went into the service station with me quizzing her: "Twenty take away one is... twenty take away two is..." and came back to the car having easily figured out the pattern all the way to zero.

I think that all of this has been made possible because her brain spontaneously clicked into counting backward a couple weeks ago. I am triply convinced that this is a brain development phenomena because earlier this week she wrote five sentences 100% in reverse (right to left sentences and words plus all letters written backward). I had never seen backward writing like this from her of this magnitude. And, just tonight, she was working on singing the alphabet song backward. This is not a coincidence. A supportive learning environment is key, but sometimes you just have to be patient for the brain to be ready.

Back in the van she asked, "Do you think I'm ready for the next level?"

"Sure," I said, with mind racing to figure out what the next level was, "What's 33 take away four?"

Done. "Do you want a puzzle to solve?" I asked, thinking of something Raising a Happy Child had written in comments to yesterday's post.

"Yes!"

We'd been doing multiplication using visual groupings, why not invert the process? Voila! The kid's first introduction to division. "You and three friends have 20 candies. If you share them equally, how many candies will each of them get? You can use pictures like we do when we play Star Count, you know, make circles and put in the dots."

She started easily, but it took a little work. No matter, we've got mental and visual math firmly in place now, and she is so proud of herself. I think she likes the feeling of a good challenge -- solving a puzzle that's hard enough, but not overwhelming.

And, to close, here's the book that came through the library system to my door tonight:

That's right, Young Children Continue to Reinvent Arithmetic, Implications of Piaget's Theory by Constance Kamii.

I am on some strange yet wonderful journey of discovery. What's next?!

Knowing Numbers & UNO Update

noreply@blogger.com (Malke) — Wed, 25 Jan 2012 23:51:00 +0000

I've been watching my six-year-old daughter explore math all fall and now into the winter. At the same time, I have also been reading books about how K-2 kids learn math, and growing my own 'math eyes' (as Maria Drukova of Moebius Noodles and Natural Math says).

It's been fascinating to watch my kid's understanding of numbers develop over the last six months, even with a somewhat hands-off approach. She started this summer with a lot of comparing and measuring. Then she started asking lots of questions about numbers and shapes, mostly on her own time. Sometimes I'd be there during the process, as in this line of questioning about dividing a number in half. Other times I'd run across the results of her thinking later in the day, as with a geometric study she initiated with some tangrams.

There's a book I've been reading, a few pages in bed each night before I turn off the light. It's called Games for Math: Playful Ways to Help Your Child Learn Math from Kindergarten to Third Grade, written by Peggy Kaye.

In this book, Peggy shares all sorts of fun math games that require no more than some paper, a pencil, some dice to roll, tape, egg cartons, beans...you get the picture. The best thing about this book, besides the very easy-to-implement, high-interest games is how she puts the games into context with stories about working with specific children. I really, really like 'hearing' her have math conversations with children; the activities are pretty simple but the important thing is how she models mathematical reasoning and problem solving strategies while she plays the games with each child.

My favorite games so far are Star Count and Lots of Boxes. Both involve a pen, some paper, and a die to roll and both focus on understanding groupings of numbers as a way of practicing counting and introducing multiplication at the same time. My kid has been asking questions that show she is thinking multiplicitively, but she is not quite ready for or interested in sitting down to discover the patterns in the multiplication chart, so these games are perfect for her right now. Also, How Many Boxes is also a great introduction to perimeter and area since it is played on graph paper.

A couple weeks ago the kid surprised me with some spontaneous backward counting from 100. It was wonderful listening to her find enjoyment in figuring out the pattern, but I didn't really understand the importance of this new skill until last night. That's when I read a section where Peggy Kaye talks about the kinds of number skills kids need to have to learn how to add and subtract. These include:

- Counting forward from one (I keep trying to remember to start counting from zero, but starting at one is just too ingrained)
- Counting forward from a number other than one, like five and going from there
- Counting backward from ten or twenty or one hundred
- Counting backward from a random number
- Skip counting twos, threes, fives, tens...

My kid has been spontaneously skip counting since the early summer, and counting forward for longer than that. In terms of the girl's new penchant for counting backward, it seems this new skill of hers is really important for mastering subtraction, something we haven't done a lot of.

What she and I do do is a lot of adding numbers together with daily UNO games but, just today, the whole 'adding up the UNO score' activity seemed like it had lost it's challenge. When I lose, she whisks the cards over to her space, spots the tens easily and without prompting, adds up 10's and 20's without pause, and takes the extra cards and finds a final total. Sensing a turning point, I suddenly had an idea for another way to challenge her. Here's my new thing:

Instead of adding up the numbers, take all the cards you've won from your opponent and use them as digits. The object? Make the absolutely largest number you can. A five and a one can make 15 or 51, for example. Or, there's this hand I lost to her this afternoon:

Ha! It's probably the largest UNO score EVER!! And, it's a great way to introduce hundreds, thousands and millions as well as 'which number is biggest' on the grandest possible scale. The cards become great manipulatives, so once she gets comfortable reading the numbers I've made (while modeling my thinking process), she'll get a chance to figure out how to move them around to make 'the biggest number' she can.

But back to counting backwards. I don't really understand my child. She is quite independent in her learning, and often resistant to formal instruction. But she really likes it when I give her spelling tests!? And, today, when we were at the library, she asked me to give her a math test. Um, okay.

I eventually obliged and put together a list of equations. As I've said, we haven't done much in the way of subtraction, so this little test was actually a good way to assess where she is right now. Plus, with my new understanding of counting backwards? Wouldn't you know it -- there she was counting backward to figure out the answers!

She also wanted some multiplication problems so I used a strategy from Peggy Kaye's Star Count game to illustrate each problem. "Two groups of two", "three groups of four", "five groups of one" are the way we say it. She got her answers by counting the dots, but I can imagine if she keeps this up we might be ready for coloring in the patterns in the multiplication chart by spring. I really want to do square and triangular numbers with her using visuals but, as always, I just need to be prepared to jump in with it when there's an opening.

In closing, as I was in the process of wrapping up this post, the kid wanted another math test. I learned a lot from seeing how she answered the questions. Next time I'll put the addition, subtraction and multiplication problems in their own sections, not all mixed together. But the biggest thing is that I'm going to get the Cuisenaire rods out again so we can solve the subtraction problems together for a while. 30-2 was just a little too big of a 'problem' for her, but if we can look at it with the rods I think the idea will click into place without much fuss. My final lesson? Math is definitely a morning thing. No more math at 5pm!

[Edit, 1/26/12: Here's what happened the very next day with the subtraction and the counting backward. When it rains, it pours.]

A Work in Progress: Paper Quilts & Hexagonal Rotation Designs

noreply@blogger.com (Malke) — Mon, 23 Jan 2012 02:04:00 +0000

I am in the process of developing a paper quilt project for kids using a hexagonal design built from triangles. Here's a picture of what my kid and I have done with paper quilts in the past:

The design incorporates lots of little triangles, but the end result is pretty square. Not that there's anything wrong with that, but wouldn't it be cool if a beginner paper quilter could create something more circular using triangles?

Yes!

Here's the book that was the inspiration for my hexagon-based project, as detailed in a recent post:

It turns out snowflakes are hexagonal! But they look so different from what we usually think of as a hexagon...

source

There are endless tessellating design possibilities here which would be fun to explore with paper in the future, but I've got something else on my mind at the moment....

In her snowflake book, Paula Nadelstern shows how to design snowflakes using a "60˚ triangle" unit. She doesn't say equilateral, but I think that's what she means. I read through her book and tried to get the gist of how she makes these beautiful fabric images; it appears the whole thing revolves around one single, accurate triangle template.

Here's what happened when I thought I could get away with a random black-line triangle pulled of a google search, one that had every appearance of being an accurate equilateral triangle:

Not good! Seems to be missing a few degrees. So, I tried again. In her book, Paula gives a formula that was helpful for constructing an accurate triangle. The smallest one had a vertical length along the center axis of 2 3/8" and a horizontal length, to both the right and the left of the center axis of 1 3/8". It worked! Here's a picture of how I marked it out. I cut out the triangle template using an exacto knife.

I made the template out of cardstock because it was all I had on hand. I made it what I'll call the 'reverse' of a traditional-type template so I could easily see where I was putting the triangle on the patterned paper.

As you can see, below, I was pretty successful at getting the same part of the repeating design in the same place on each triangle using this 'see through' template.

Here it is, starting to come together:

And here is what it looked like when all six triangles had found their places:

Yay! I thought it looked really good at this point, but I wanted to see if I could give it just a little more shape.

What I noticed in Paula's designs is that the snowflakes get their form by focusing the design at the edges where the triangles meet; one edge has half the design, the adjacent edge has the reflection. Focusing on the 'spokes' (for lack of a better term) that are created where the edges meet (and create lines that radiate out of the center of the hexagon and through each vertex) ultimately creates the snowflake's design. I think!

I had just enough time to try a small experiment to see if I could highlight the 'spoke' portion of the hexagon. After making a couple more measurements and lines I cut a new template out of the leftover 'insides' of the triangle template.

I think the final design looks more like a flower or a star, but I'm still pretty pleased.

I think I'll try stripes next time and see what happens!

While I was in the process of cutting out and placing triangles, my daughter got really excited about the intermediate shapes I was creating. For the K- 2 set I think a great start to the project would be to give them some patterned paper triangles and some glue sticks and let them explore what kinds of designs they can make, observe others' work, and help them find ways to describe what they see. Later, you could bring in some inspiring hexagonal designs (both your own and others') to serve as models for the rotational design element. Once you spend some time deconstructing the examples it would be time to give the kids more materials and start the design/observation/reflection process all over again.

I believe that, even with the younger elementary kids, the experience of putting six duplicate, congruently patterned triangles together to make a hexagon has the potential to reveal the complexity of this shape in a new and inspiring way. I, for one, am seeing hexagons in a whole new light.

For middle and high school students the challenge of building the perfect template for such a project could easily catapult this whole investigation into some really cool mathematical inquiry. (Plus, I like the idea of giving them a choice, at first, about whether to use a 'pretty good' black-line template or the measuring method and let them compare the results!)

Finally, to make this a viable project for multiple children I really need to find a easier way to prep a large amount of paper triangles. Does anyone out there know what kind of cutting tool might be able to do this? I'm planning to play around with this some more, so stay tuned!

Inspiration Strikes: Hexagonal Rotation Designs

noreply@blogger.com (Malke) — Sat, 21 Jan 2012 19:18:00 +0000

Inspiration can strike at any moment. Even mundane moments like flying in and out of the library to pick up a book on hold and then running home to make dinner.

Here's what I found on my way to the 'hold' shelf:

These are quilted snowflakes. Snowflakes are hexagonal don't you know and just LOOK at these beauties. I've always thought of a hexagon as a bit cumbersome, but these take my breath away.

I snatched up the book and devoured it. I have no intention of quilting anything, but I am handy with paper and I think that somewhere within all this complexity is a (much simpler) project for younger kids, and possibly a fun construction challenge for older ones.

In my next post I'll share my first attempts (one failure and one more successful) at finding bilateral symmetry within a paper/fabric print and using it to create the rotational symmetry of something resembling a snowflake, or a star, or...err, well it looked nice, whatever you want to call it.

Until the next post when I share my modest efforts, here's a quick peek at the glorious visions of Paula Nadelstern's kaleidoscope quilts.

Blog Carnival over at Let's Play Math!

noreply@blogger.com (Malke) — Fri, 20 Jan 2012 18:28:00 +0000

Hey y'all, I've been included in the "Math Teachers at Play" blog carnival over at Let's Play Math!

This particular carnival features "a smorgasbord of ideas for learning, teaching, and playing around with math from preschool to pre-college" and is illustrated with some great examples of living math books.

I'm pretty excited to be included in my first math blog carnival but least you think I have my math act all together, let me assure you this is one big adventure I'm on. My blog has been, and continues to be, a vehicle for me to figure out for myself what math is. I've got a pretty firm grasp on the math we use in my program Math in Your Feet, but I'm a big picture person and I've always wanted to find a way to put our dance work firmly in context on the mathematics map. I am also perpetually curious about how people teach math at an elementary level.

Luckily I've got a six year old daughter along for the ride. I use her interests to frame my inquiry and, by working at the pace of a first grader, I am much less intimidated about re-learning math than I might be otherwise.

Working with my daughter has actually been, to use a new math term I've recently learned, a kind of recursive, looping process. She shows me, through her drawings, and conversations, and number play, how she is thinking and then I go find great resources like Let's Play Math, or Living Math, or Natural Math, or Love2Learn2Day, and see if I can figure out how to support and deepen her questioning.

In the process of finding a new math game or book for the kid I also find some new bit of information for myself about a math topic or an instructional approach or theory. I spend part of my days with kid math and my nights reading everything I can get my hands on, from research studies about cognition in mathematics to books about archetypal relationships between mathematics, arts, science and nature. Then I'll go to bed only to wake up to hear what my daughter's got going around her head, and the process cycles around again.

I hope you will take some time to check out the Math Teachers at Play blog carnival! If you are interested in doing and making more math with your own children or students it's really just as simple as finding an interesting math-y story book or game and going on from there. My biggest lesson in the last year or so is that I don't have to know everything, I just need to keep asking questions! That's really all there is to it. Have fun!

And the Award for Favorite Platonic Solid Goes to....

noreply@blogger.com (Malke) — Mon, 16 Jan 2012 16:00:00 +0000

After constructing all the Platonic solids out of straws and pipe cleaners (some with my daughter's help) our favorite platonic solid is hands-down the icosahedron. I made this one with one green pentagon end, and one yellow pentagon end. It 's strong and sturdy, spins well, like a top, and is fascinating to look at from every angle!

Here's another view:

In a past post, I observed that illustrations of these solids are rarely to scale. I wondered how a icosahedron would compare to the unwieldy dodecahedron I made. Here's a family portrait of all the platonic solids, all constructed with six inch edges. See what you think:

Yup! Looks like a family, to me (Papa solid, Mama solid, sister solids, baby solid, lol)! But, clearly, the icosahedron, despite more faces and the same number of edges, is smaller than the dodecahedron. For what it's worth, I guess I have my answer about the scale issue.

Hmmm...I wonder what I should try next? And, here's a question: if the kid's not involved in the making, would my obsession be considered parental neglect, do you think, or could I just call it 'enriching her learning environment'? I'm thinking about one of these (source), something truncated or stellated...

Solids and an Observation on Scale

noreply@blogger.com (Malke) — Sat, 14 Jan 2012 19:39:00 +0000

We've been exploring platonic solids around our house lately. We've made them using toothpicks and marshmallows. We've made them with straws and pipe cleaners. We're even in the process of building a house using straw and pipe cleaner cubes as the building blocks.

Both myself and the kid are learning a lot through the making process, but also from the the fact that we've filled our living environment with the structures. Because they're so open, there are all sorts of relationships within the solids themselves that we observe by simply walking through a room on the way to somewhere else. This is one more great example of environment being an effective teaching and learning tool.

The kid is definitely interested in all this, but me? I have to say that I'm a bit obsessed. Questions keep coming up, and they are loud ones that are demanding answers. Luckily, an office supply store near us had bags of 100 brightly colored straws on sale for $1.00 a bag, so we've got enough supplies to go on for a while.

Here is the dodecahedron I made from 6" straws and some pipe cleaners. Twelve pentagonal faces, 30 edges and 20 vertices. (I know because I counted, multiple times. Sometimes the things that seem the easiest on paper are actually the hardest to figure out in real life.)

And here it is compared to three other platonic solids: a cube, a tetrahedron, and an octahedron, also made with 6" straws and some pipe cleaners.

Wowsa. Not only is the dodecahedron much bigger, it's also really obvious that its not strong enough (using these materials) to hold its shape.

Here is what happened when I cut the straws in half and made another:

It supports itself much better and feels and looks, well, more solid than the first. Here it is in relation to the other solids:

Interesting! Remember, three of the solids shown are made with six inch edges and the dodecahedron is made with three inch edges.

Have you ever noticed that when you see illustrations of platonic solids that they all appear to be about the same size? Here's an example (source):

Or, this (source):

I never would have observed this if I hadn't built a 3D model. Score another point for hands-on learning!

I wonder what will happen when I try the last platonic solid, the icosahedron? Even though it has eight more faces than the dodecahedron the faces are triangles. I'm thinking that even with using 6" straws the end result won't be as cumbersome. Wish me luck!

Making Math: Big Dreams of Building

noreply@blogger.com (Malke) — Fri, 13 Jan 2012 15:58:00 +0000

We are going to build a cottage. That we can live in.

We are building it out of straws and pipe cleaners. We started with one cube and attached another to it. You should have seen the kid's jaw drop -- it was like magic. And, did you know that after the first cube you don't need to make a complete cube ever again?

I quickly figured out my methods, but it was hard for my six year old to keep up with the precision. To attach the straws use half a pipe cleaner folded at each end, then bend it into a 90 degree angle and slide the straws on.

Unless of course you are attaching your cubes together. In that case you make one folded end for the open straw, and slide an unfolded end into the attachment point.

But that's just what I discovered and, honestly, half the fun was in trying to figure it out! I suspect there are a lot of right ways to go about doing something like this. If you decide to try building like this I'd love to see pictures of what you make!

In the end, the kid and I worked as a team for a while and then she left to rehearse her show which opens with the song "The Moral of the Soggy Hat" (an original number). I imagine this building activity continuing mostly on mama power, but that's because I have ulterior motives.

I get to do a ten-day version of Math in Your Feet this summer for some inner-city programs for kids ages 6-12. In addition to rhythm and pattern in the feet and hands, we'll be exploring what else you can do with a pattern unit, which includes building! I can imagine, over two weeks, the older kids will have built something really cool. And if they get really good at it, I might suggest trying to construct something out of tetrahedral units just for fun! But I think I should try it out first, just to get a sense of what the pitfalls might be.

As for my girl? She's all for building something she can live in, even if it's made from drinking straws. "But Mama," she said, "Will I have any privacy?"

Straws, Solids and a Possible Parthenon

noreply@blogger.com (Malke) — Thu, 12 Jan 2012 01:46:00 +0000

I'm trying to brush her hair. We've got places to go, people to see. She darts over to the straw/pipe cleaner tetrahedron and brings it back to me.

"Mama!"

"What?"

"This tetrahedron does NOT have equal sides."

"What do you mean?"

"Well...[positioning the tetrahedron so that one triangular face is toward her, and pointing through to show the two other triangular faces coming together in the back]...you see these sides are two together, but this one...[losing steam]"

"Those sides are called faces. I think what you're showing me is that there are two faces that look like they're opposite each other, and that there's a third in front that has no match?"

"Yes."

"Well, 'equal faces' or 'equal sides' does not mean there is always a match. What it means is that each face is the same shape and the same size. [Note to self: How should I really be describing the sides of regular polyhedra?] For a tetrahedron that means that each face is a triangle shape. Let's look at the cube...all its faces are squares of the same size, that's what equal means in this case. It doesn't matter whether there is a match for each side...

"Here, look. Let's count the number of faces of each solid you made with marshmallows and toothpicks... [counting together, four...six...eight] It doesn't matter how many faces there are, just that they're all the same. Hey! I know, wanna try making an octahedron out of straws to go with our tetrahedron and this cube?

"Mama!"

"What?"

"Let's make a model of the Parthenon out of straws and pipe cleaners!"

"Uh, okay...maybe we should finish building these solids first? So then we'd have some building blocks to work with?"

"It'll be math and history at the same time!! That'll be really fun!!"

"You're absolutely right! [rummaging] Hey, here's that book I got at the library book sale, on buildings of the ancient world. Wanna see? Here's the Parthenon."

[Kid, looking at the book, flipping through the pages.] "Or we could build a castle. But what would we do for the round towers?"

Never did get her hair brushed. Didn't really get to building the Parthenon either, but we did build the octahedron and get out the door, eventually.

This is the first time we've tried making structures with straws and pipe cleaners. The beauty of platonic solids made with a unit length of 6" or so is that because they're not really, well, 'solid', you easily look through them to the other side which is what prompted the whole exchange, above. You can also play around with them more easily, experimenting with how they fit together and observing how the relate to each other. Just as exciting is having them around the house with us, keeping us company. We've been admiring the toothpick/marshmallow sculptures and solids from last week as well.

After making all these wonderful structures, it's also been wonderful to share our space with, well, space. And, I am reminded over and over, that just because I look at something doesn't mean I understand it. Even for me, as an adult, physically constructing these solids is bringing to light a whole new understanding of structure, relation and order. As always, this is an amazing (and fun!) learning journey for both of myself and my daughter.

Marshmallow Math: Solids & Sculpture

noreply@blogger.com (Malke) — Fri, 06 Jan 2012 19:39:00 +0000

Given my daughter's resistance to formal lessons, it really was the perfect storm. A supervised math activity of new concepts thinly disguised as 'something to do while mama makes dinner'? Yes, please!

It came about as a sort of semi-premeditated accident: It was time to get dinner ready and there was a lot of prep work to it. I had the mini-marshmallows and toothpicks at the ready, as well as buy-in from the kid who had seen this post of the amazing, huge, Serpinski-esqe giant tetrahedron built by the kids at Almost Unschoolers the day before.

I laid out some newspapers on the kitchen floor and poured the marshmallows into one bowl, the toothpicks into another. Completely off the cuff I said, "Here's how you can make a square base, and then build up from there." I built about one half of a cube and then said, "I really want to make tetrahedrons, but I don't have time now. See what you can do."

The kid jumped in enthusiastically. She started by making one cube, no problem. She added on another cube, then two more to make a four-cube base. She built it up a level, and up a level again.

"I'm an artist! I'm making a sculpture! Let's put these piece under glass, like in a museum. It's marshmallow art." Despite this enthusiastic chatter she was actually a little worried that the kids at Almost Unschoolers would be mad at her for copying their work. This led to some conversation about copying vs. learning the basics so you have the skills to express your own personal vision, and then, thankfully, our attention was diverted.

The tower/sculpture of cube units started to lean.

These marshmallows are tricky! The kids at Almost Unschoolers made it look easy, but it's not! There are all sorts of structural problem solving challenges to overcome. I suggested turning her 'sculpture' on its side (pictured below) so it would be a little more sturdy. We counted how many cube units made the structure.

When she was done, thinking toward my goal of tetrahedrons, I casually mentioned she might try a triangle base. After some experimentation she started making this:

It has an interesting twisting quality to it and is my personal favorite of all her 'marshmallow art' pieces.

By this time, dinner was in the oven and I had time to experiment myself. I made a tetrahedron, then attempted a larger one using four tetrahedral units. It was a pretty strong structure, but only after I figured out that if you're going to connect the tetrahedral unit patterns, you need to share the same marshmallow where the vertices make contact. (An aside: Spell check said that is was 'vertexes' instead of 'vertices'. What do you think?)

She worked on another sculpture and then...lo! The girl created a tetrahedron of her own, then played around and discovered, on her own, how to create an octahedron!

The picture below shows all of today's creations. The 'sculptures' which she created first, are in the back row. The geometric/platonic solids, created at the end, are in the front. What's interesting to me, in terms of my daughter's learning process, is that it is more apparent than ever that she really wants to explore and discover new things on her own first and then, when she's had her fill, she comes more easily and willingly toward 'the point' of the lesson.

Whatever you call this afternoon's explorations it was a fun, full hour of inquiry. It was also the first time, except for origami, that I've willingly pushed us into 3D math. I'm learning right along with her, and it's an amazing journey.

Teaching Math in Your Feet...Without Me!

noreply@blogger.com (Malke) — Fri, 30 Dec 2011 18:28:00 +0000

The program Math in Your Feet was developed as a five-day artist-based residency, led by me, the percussive dancer. I created the program in collaboration with Jane Cooney, an elementary math specialist in Indianapolis, IN. I hammered out the dance/math integration and dance class activities through sheer repetition, trial, and error in classroom after classroom. In 2006 I was lucky enough to also have the opportunity to develop a teacher workshop through association with the Kennedy Center and Clowes Memorial Hall at Butler University. I've been doing both student residencies and teacher workshops ever since.

Meg Mahoney, an elementary dance specialist in Seattle, read the article I wrote about the development of Jump Patterns as a teaching tool in my program Math in Your Feet (published in the Teaching Artist Journal, April 2011). As a big fan of her work I was thrilled when she told me she was going to try it out with her own fourth and fifth grade students! Even more amazing, she has never seen the program in person or on video. True, she is a fabulous dance teacher, having worked for fifteen years in an academic setting, but still, it take a lot of guts to commit six weeks of your school year to something brand new like this.

In a recent post on her blog, Meg says:

The article unwraps the dance/math residencies Malke Rosenfeld teaches in public schools. The fact that she shares her methodologies with classroom teachers for use in the classroom lit a spark for me. Even without being a step dancer myself, maybe I could lead my dance students through the jump pattern curriculum!

Apparently she was quite successful! Read on...

I’m about to begin Week 5 of 6, finishing the jump patterns with my second set of 4th & 5th graders (six lessons per group), and we’re all enjoying it. Malke’s outline provided lots of material to work with, and I’ve worked the pacing & focus of instruction for each lesson to fit my ELL learners & my circumstance. The movement variables are broken into malleable chunks, and we’ve explored the math-related concepts of precision, congruency, reflection, and turn symmetry, with students choreographing patterns in teams of 2 and 3. In addition to integrating dance & math, there’s a problem-solving (choreography) component that parallels the "workshop/conferencing” structure that my students are familiar with through Writers Workshop, allowing me time to confer with & jump-start individual students. In addition, there’s a spatial arrangement that supports classroom management (personal dance spaces for each team — wow, what a concept!). Add in some dance videos to “mentor” the kids in their choreographic process & journaling questions to provide feedback on what students are learning, and it’s no wonder we’re all engaged!

A perfect Math in Your Feet moment, courtesy of Meg's blog. These boys are in the middle of a 270 degree turn. Nice!

My first groups of 4th & 5th graders finished the unit before the holiday break, with some of them performing their patterns, both congruently & in mirror symmetry. They nailed the precision steps they’d created, even without the support of their personal dance spaces, and their peer audience was able to talk about what they were seeing with insight and new vocabulary. What a pleasure to watch…

In this description of how how she used the Math in Your Feet curriculum, Meg shows us all that the program is flexible enough to be adapted by teachers in a number of key ways:

- Easily reproducible outside the artist-residency setting
- Engages and inspires both learners and teachers to learn and grow in exciting new ways
- Creates opportunity for new insights into topics and practices in both math and dance
- Encourages fluency with new math and dance vocabulary, in context
- Allows exploration of, and play with, math concepts in a dynamic, physical, choreographic process
- Adaptable to ELL learners
- Adaptable to the particular circumstances/expectations of a school and/or district

I'm continuing to work away at producing the Math in Your Feet curriculum guide and instructional DVD so that any elementary classroom teacher, or PE, dance or music specialist can reproduce the same energy and engagement that an artist-based residency provides. In a way, the Math in Your Feet program is poised to be more effective in this new form because, as a teaching artist, I am just a short-term visitor who really knows very little about the individual learning needs and goals of each student, let alone the specific circumstances and culture of individual schools and districts.

Based on Meg's success, and the successes of teachers who have taken my professional development workshops to learn more in-depth about the methods and content of this program, it is not hard to imagine hundreds, maybe thousands, more children jumping (and sliding, stepping, turning, hopping...) their way through math class led by their very own teachers!