Two Sides of the Same Coin

So I’m working on a book.  While I work on another book.  This other book is a project through Moebius Noodles and Maria Droujkova’s publishing company Delta Stream Media. I’m collaborating with Maria and Gordon Hamilton of Math Pickle to create original puzzles, games, and making activities exploring numerical and categorical variables … for young kids!  It’s super awesome extremely cool.

And, today?  Today, our conversations in relation to the variables book helped me clarify something I’ve been thinking about on a LOT of different levels for literally YEARS.

This level
What’s the difference between using the body to illustrate mathematical ideas and using the body to create and express an understanding of mathematical ideas?  

This other level
What’s the difference between using body knowledge (ala Papert and his gears) and creating body knowledge?

And finally
What’s the difference between identifying properties of an object (say, a piece of art, or an insect) and actively choosing from an inventory of attributes to make your own?  

llustrate, use, identify  <======>  Create, express, choose

Each of these questions needs its own, more specific treatment.  My observation today is simply that each pairing seems to create a similar tension in my mind.  These are all active words, but the nature of the activity is qualitatively different depending on which side of the learning process you're on.

Today the words fixed and flexible came up in relation to how we are approaching the activities in the variables book.  

Fixed
Puzzles and games that focus on identifying properties happen within a fixed structure.  Using body knowledge to understand a set of gears assumes that you are using a certain set of body experiences, created at some point in the past.  Illustrating math ideas using the body means there is a predetermined goal for the activity and that the outcome needs to look a specific way.

Flexible
Learning vocabulary and language in context and, in math, using multiple strategies to solve a problem are both process oriented and context dependent.  In making, having a large inventory of ideas/things/skills from which to choose and create your own novel ideas (like a dance step) is an open-ended investigation.

“Fixed” and “Flexible” are not judgments; they are inverses of each other.  They go both ways.  Just like you need to compose and decompose numbers to see the full relationships embedded in those two activities, so do you need to identify and use properties, build and use body knowledge, and illustrate and express mathematical ideas. 

Fixed: The parts of learning (anything, really) that are perhaps learning objectives that are easier to identify in an assessment, but still crucial.  Some call this skill building.

Flexible: Relates to the processes of learning which (as anyone who has tried arts integration, project based learning, or focusing on mathematical practices may have experienced) are much harder to nail down when tasked with assessing such activity.  Some call this fluency.

You can't have one without the other.

“Without skill there is no art. The requisite variety that opens up our expressive possibilities comes from practice, play, exercise, exploration, experiment.”  --Stephen Nachmanovich, Free Play: Improvisation in Life and Art

 
Thoughts, feedback, pushback and conversation are always welcome. 

Bright, Brave, Open Minds: A Problem Solving Kaleidoscope | An Open Online Course for Parents and Teachers!

Bright, brave, open minds: A problem solving kaleidoscope led by Julia Brodsky and Maria Droujkova is a two-week long open online course in problem solving for parents and teachers of 8 and 9 year old children.

About the Course:

Why: Preserve children’s divergent thinking. Develop their critical thinking and problem solving skills. Contribute to making a book about young problem solving.

How: Provide a variety of insight problems to young children. Introduce complex, open-ended, and ill-defined problems as a way to teach problem solving skills. Recognize the feeling of being stuck as a necessary step in problem solving. Let children face deep, multi-dimensional problems. Meet other adventurous parents and educators. Contribute to research in math and problem-solving education.

Who: The course organizers are Julia Brodsky, former NASA astronaut instructor and math and science teacher, and Dr. Maria Droujkova, math education consultant. The course participants are families, adventurous teachers, math clubs, playgroups, and other leaders of groups doing problem solving with young kids.

What: In the first week of the course, we will discuss settings and practices for introducing problem solving to young children. You will look at topics from Julia’s math circle, and discuss how to teach them. In the second week, you will gather your kids and their friends in a casual math circle, try out the topics, and then answer a few questions about your experience.

When: Sign up via the Moebius Noodles website by December 2nd. The main course activities will happen December 2nd through 16th. Expect to spend several hours a week doing course tasks.

Where: Main course activities will happen at our online ask and tell hub. The page you are on now will be updated with major links and news. Organizers will send you summary emails. We will have two live meetings online, at the beginning and at the end of the course, and post their recordings to this page.

The Butter Knife and the Infinity Knife

Recently, my eight year old spontaneously discovered an activity I've seen in the Moebius Noodles book and yet another version of infinity (read about others here, here, and here).

Scene 1: The Butter Knife

Waiting for the bread to toast. She picks up the butter knife off the counter and places it vertically over a design on the lid to the butter container.

"It turns into an arrow!  What a cool design."

She continues to play around with "cutting" the lid's design in half, one side the real the other the reflection.

Scene 2: The Infinity Knife

The kid calls to me from the other room to tell me she's cutting triangles in half into smaller and smaller pieces, trying to see how many times she can cut them.  When I ask her to show me she picks up another piece of paper and starts again.  She cuts one big triangle in half equally, then one of the halves in half again...

"It has something to do with infinity," she says, as the triangles get smaller and smaller and smaller. "I need something different [than the scissors she's using] -- a tiny knife like scientists have -- to cut infinity stuff like this."

................................

I've been reading books by former colleagues and students of Jean Piaget who have carried his work forward and made it accessible to the rest of us. My take away so far, beyond the idea that all of us construct our knowledge by assimilating new information into what we already know or think we know, is that children think when they have something real, some phenomenon, to think about. Whether it's a butter knife or an infinity knife or whatever else, it's the interaction between the child and the object/idea that inspires the child to think -- what Eleanor Duckworth calls "the having of wonderful ideas." 

"There are two aspects to providing occasions for wonderful ideas," she writes in her book of essays The Having of Wonderful Ideas. "One is being willing to accept children's ideas. The other is providing a setting that suggests wonderful ideas to children --different ideas to different children --as they are caught up in intellectual problems that are real to them." [page 7]  

Whatever else we do as teachers and parents, I think we need to find multiple ways to allow kids the freedom to discover the world on their own terms, in their own ways. At the same time we need to create the time and space in our teacher/parent brains to catch our children and students in the middle of discovering something brand new (to them). Having wonderful ideas, as Duckworth says, is "the essence of intellectual development."  Honestly, being able to observe and even, sometimes, to interact with my kid when she's in the middle of a new thought is pretty much the prize of parenthood -- it is such a gift to see this happen up close.

Second Math-y Monday...

The first Math-y Monday was a big hit!  The kids in my daughter's 3rd/4th grade classroom are still working on their archetype times tables.  I'm hoping I'll be able to take some good photos of their work. Stay tuned.

For the second Marvelously Math-y Monday I am bringing in an extra Multiplication Models poster from Moebius Noodles that I had lying around (and the poster itself can be purchased here -- they're seriously gorgeous in real life).



And, because one of their teachers introduced them to magic squares last week (cool, huh?) I thought I'd bring in this book to add to their library:



It's going to be a great Monday!

Learning Math by Ear

At first glance, this article about the value of reading aloud to older kids would not seem to connect to math learning. But, to me it does.  Here's the piece that really stood out:

“The first reason to read aloud to older kids is to consider the fact that a child’s reading level doesn’t catch up to his listening level until about the eighth grade,” said Trelease [a Boston-based journalist, who turned his passion for reading aloud to his children into The Read-Aloud Handbook in 1979], referring to a 1984 study performed by Dr. Thomas G. Sticht showing that kids can understand books that are too hard to decode themselves if they are read aloud. “You have to hear it before you can speak it, and you have to speak it before you can read it. Reading at this level happens through the ear.”

Did you catch that? "You have to hear it before you can speak it, and you have to speak it before you can read it."

I made a similar point while working with teachers and teaching artists in Minnesota last week when participants noticed how the math language was woven seamlessly into our dance work. 

This vocabulary development, I said, was initially an attempt to help kids pay closer attention to what they were doing while they created their dance patterns. I noticed that they became much better creators when they had the right words to help them identify their movement choices.  When I started developing the Math in Your Feet program (after about six years of teaching clogging and developing the Jump Patterns tool) I put all those words into a poster that became an ongoing resource in our classroom:

Here's how it all connects to the hearing/speaking/reading continuum.

In my classes at first we just dance, getting a sense of the new movement vocabulary and style of percussive dance in our bodies. I talk us through our dance learning by demonstrating with my own body and also saying the descriptive words out loud during our group warm ups and dance time.  Sometimes I point to the words I've posted while I'm talking. When students look confident with the dancing I ask them to verbally label the specific attributes of their dance patterns, using the Movement Variables chart as a resource. (Not incidentally, attributes are also a 'big idea' in mathematics.)

I add vocabulary words as we go along: yellow is dance, blue is math, and green are where dance and math ideas overlap.   This picture was taken about half-way through our work together.

In addition to being able to parse our patterns, we use tons of other math terminology while we choreograph with our teammates and also in whole group discussions.  This approach allows students to fully grasp the real meaning and application of these ideas which, ultimately, allows them to write and talk confidently about their experiences making math and dance at the same time. Teachers consistently notice an increase of 'math talk' in their classrooms during the time I am in residence. As in, "I couldn't believe how much math vocabulary they were using!"

Math is a language but it's not just about terminology; there is also a need to become fluent in playing with mathematical ideas, noticing patterns and structure, sorting and comparing, and reasoning out and understanding relationships. Similar to the point made in the article, linked above, these math ideas are ideally facilitated by an adult through conversation, play and exploration before bringing it to the page.

How do children learn their native languages? By first hearing and playing with the sounds of words, experimenting with the rules of the language, testing what effect those words have on the world and the people in it, and fairly quickly coming to understand that words are useful in all sorts of situations. It's the same for math learning.  Similar to literacy (not just decoding) a child needs to 'hear' the math first (talk it, do it, play with it, manipulate it, make it, see it) before they can abstract or 'read' the math (notation, symbols).  

Ultimately, what this means is that not only is it important to get a sense of the rhythm and the flow of the math/music/language first, but it is also really helpful to learn things in context.  Words mean something, language is a two-way street, and nothing means anything unless it has a use. You may be able to memorize a definition (or a math fact) but if you never get a chance to see, hear, feel or use a math idea in any meaningful, useful or fun way, it's really just a spectacularly sad example of 'in one ear and out the other'.

Luckily, there are many wonderful resources for creating contexts in which you can help children (or yourself!) learn math 'by ear' through games, puzzles, interesting videos, conversations, living math books and math art.  Here are a few of my favorites:

Moebius Noodles
Let's Play Math
Living Math (especially the list of math readers)
Talking Math with Your Kids
Math Munch
Math Pickle

Edit/Addendum: After I posted this piece I realized that I could have also very easily supported the idea of learning math by ear using examples from the last couple years of homeschooling a wonderful, bright but resistant learner in the ways of math.  I could have also made arguments and brought in examples from my experiences as a traditional musician and dancer (Irish flute and a variety percussive dance styles from Canada, the U.S. and the British Isles) where music and dance is learned and taught as an aural tradition, very literally 'by ear'.  

A Shift in the Universe

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.

p.s. If you want to comment just be aware they don't seem to always show up in this template, so try either a) adding a '?' at the end of the address and hitting return to reload, or b) just clicking out and trying again.  Sorry for the hassle -- some weird Blogger bug I think.  

Full to the Brim

I've got so many amazing things happening these days.  Not only am I facilitating my daughter's path to math and bringing the Math in Your Feet program to students and teachers, but I've got a lot of other fun projects going on as well, all related to the theme of math and making. Take a look!

My TEDxBloomington Talk: Jump into Math!

It was an amazing process and, finally, an amazing day at TEDxBloomington.  (There I am in the green/yellow shirt at the end of the day when all the speakers, crew and organizers made their way onto the stage.)   

My talk will be online in the next couple months alongside the talks from Drew Ramsey, MD (one of psychiatry’s leading proponents of dietary change to balance mood, sharpen brain function and improve mental health), Ryan Germick (Google Doodle team lead), Eric Deggans (TV/Media Critic for the Tampa Bay Times and author of Race-Baiter: How the Media Wields Dangerous Words to Divide a Nation), Robert Einterz, MD (whose efforts have literally saved thousands of AIDS inflicted people in Kenya) and many other incredible people with incredible stories.  I was in fine company and it was an honor to share my my work in this way.

The Tape Chronicles Project
Oh Happy Day! The Tape Chronicles Project has finally launched!  I've had this idea for a couple years and am so excited to see what we can build. Here's the basic description:

Tape! The ultimate open-ended, the world is your oyster, creative, hands-on learning and making supply. Check out the endless ways tape can be employed in the interest of math, art, kinesthetic exploration, invention and education. 

Check out the Tape Chronicles page on my website and please consider submitting examples of your own!!  

Teaching Artist Tool Shop
As part of my participation in the new collective, Teaching Artist Tool Shop, I created a video about what success looks like in my teaching of dance and math and in a moving classroom.  You can watch the video What Success Looks Like in My Teaching on my website.  From the feedback I've received, the ideas in this video can be applied across disciplines.

Moebius Noodles
In January I was invited to contribute to the Moebius Noodles blog!  I'm so honored! Read my first two contributions Thinking in Threes and Hidden Math: Book Edition and then spend some time over there in math adventure land!

Math in Your Feet Facebook Page
Oh, we're having so much fun over there.  It's the place where I share all sorts of cool percussive dance videos, incredible visual math art, links to interesting ideas about teaching and learning math....all of which would never really make a great blog post on their own.  We all hope you'll join us!

Learner Dancer Teacher "Math Explorer"

It's been a year since I read Contance Kamii's Young Children Continue to Reinvent Arithmetic, Implications of Piaget's Theory.  I'd probably understand it better now, what with first and second grade math under my belt.  Well, my daughter's math anyhow.

I was just reminded of Kamii's work (and how much I want to re-read it) when I opened another gem today: Young Mathematicians at Work: Construction of Multiplication and Division by Catherine Twomey Fosnot and Maarten Dolk. I got wind of this book via a math education reading list constructed (ha!) by three educators-of-math-educators I know of via Twitter: David Wees, Christopher Danielson and John Golden.

Here's a really nice section from the Preface of Fosnot and Dolk's already very exciting book:

"Like all human beings, mathematicians find ways to make sense of their reality.  They set up relationships, they quantify them, and they prove them to others.  For teachers to engage children in this process, they must understand and appreciate the nature of mathematics themselves.  They must be willing to investigate and inquire -- and to derive enjoyment from doing so.  The book you hold is primarily about that -- how teachers and children come to see their own lived world mathematically, their journey as they pursue the hard work of constructing big ideas, strategies and mathematical models in the collaborative community of the classroom."

I love this and I can't wait to dig into the book.  I think the reason I'm so excited about it is that I've always been interested in how people come to learn and know things but I didn't have an outlet for this passion until fairly late. When I was very young I wanted to be a teacher.  As I grew older I thought, "I can't possibly be a teacher -- there is nothing I know and love well enough to teach."  And so I spent many years trying to do other things.

Then I found percussive dance.  I was 26 and a half years old.  A year or two in I realized, not only was I pretty good at dancing but I was also able to break down steps in multiple ways for various types of learners in a way that was useful to them.  And now, 'x' years later?  Who am I?  I'm someone who has built a bridge to the math for students who might not otherwise ever glimpse the other side, but I'm not sure I'm a math teacher yet.  I think the best description of who I am and what I'm doing is the one Maria Droujkova gave me when she welcomed me to the Moebius Noodles team last week as "the creator of the Math In Your Feet program and a math explorer...."

Now that is a title I can wholeheartedly adopt.  I don't know if I'll ever be able to call myself a math teacher or if that is really my purpose in the midst of all this wonderful, beautiful, gorgeous math that I'm finding. On the other hand, Fosnot and Dolk say that (math) teachers "must be willing to investigate and inquire -- and to derive enjoyment from doing so."  I think that is a pretty apt description of what I am doing as I bring math to young people, through percussive dance or other making-math ventures.

Over the last year or so I've found plenty of opportunities to listen in on the conversations happening between a wonderful group of math educators via blogs and Twitter.  They're often talking about my favorite topics: thinking, learning, teaching and...math (and all possible meanings of that word). This kind of conversation, about the facilitation of actual learning, feeds my intellectual heart and brain.  It also ignites my enthusiasm for the elementary math that I never really learned the first time around.  It's the best possible remediation: using my adult brain and my lifetime of experiences to explore math on my own terms and interests....and then figure out how to best facilitate the learning of others. 

That's me: learner, dancer, teacher and...math explorer. Up, up and away! ;-)

A New Venture for a New Year!

Hey friends!  I am thrilled to announce that I have been invited to be a regular contributor over at the Moebius Noodles blog.  Not heard of Moebius Noodles yet?  It's a fantastic resource for adventurous, meaningful math experiences for young children and their adults and has been an incredible resource in my own journey to math.

My first post Hidden Math: Book Edition is up now!  It includes two of my favorite children's books that have way more math in them than I first realized.  There's also a new post just today by Yelena on a similar theme.  Her post is called Math Goggles #1: Math-y Librarian and it highlights all the cool math she and her son found inside a beautiful library in her city.

Truly, I don't know where I'd be without Moebius Noodles (and other great resources like Let's Play Math and Living Math).  In the last year or so I would never have grown my math eyes or moving past a life-long math insecurity that started back in kindergarten with a bumblebee matching worksheet.  It was so utterly alarming and confusing to my five-year-old self that I can still remember that moment like it was yesterday. 

Today, however, relearning math slowly but surely along with my seven-year-old daughter has softened the edges of that memory. And, even better, I know that my strategy of focusing on first-order experiences, hands-on making, math games, living math books and sidewalk math adventures is bearing fruit.  Just today my seven year old (who is currently building competency with number multiples) stated confidently that not only does she "likes math" but that she is "good at it, too!

Yay for having fun with math and thanks to Maria and Yelena at Moebius Noodles for inviting me to join the adventure!

Weaving Inverse Operations, Multiples & Frieze Patterns

It's been a super exciting few days in mathematical weaving land here at our house.  This weekend I figured out some ways to facilitate basic paper weaving and grid exploration for the youngers, riffing off Patrick Honner's Moebius Noodles guest post.   Last night I did a little more searching for how others have managed the logistics of paper weaving and found this fabulous example of warp management which, in turn, inspired some incredibly productive inquiry on my part. Ultimately, all this will turn into something I can do with my seven year old but, for now, let me show you what I did!

Here's how I started the morning, with multiple colors of paper sliced down to 1/2" strips.  Previously, I had tried my hand at weaving with 1" and 3/4" strips, but both were a bit too chunky for my tastes (although they're perfect for the young ones).  My first attempt at the new approach started with gluing the top of each strip down onto a piece of paper, with a space between each consecutive strip of paper.

Below is some experimentation with keeping the weft (horizontal) snug, in yellow, and a little looser, in green.  I love the look, but for a child's mathematical inquiry, I think it's better to keep both the warp and the weft snug, so the final design is as mathematically accurate as possible.

Since I now had some flexibility with the number of vertical strips I started wondering if the number would affect the ultimate design.  Curious about working with threes I started with a warp of nine strips. At this point I was just playing around to find a design I liked.  There's a basic reflection from top to bottom and left to right in each design.

Here is another multiple of three, a six-strip warp -- a frieze pattern, I think!  The design is made possible because of the two-color warp.

It's also where I started of thinking about inverse operations.  Weaving technique requires the use of some combination of overs and unders.  Row 1, from right to left: [3-over, 1-under, 1-over, 1-under].  Row 2 & 3: both the inverse of Row 1, but with different colors.  Row 4: repeating Row 1, but in a different color.  Then repeat!  Even now it seems like magic.  I can't believe I figured this one out.

Why do inverse operations matter?  From what I've read, and the number work I've done with my daughter, I know you can't really fully understand addition until you also understand subtraction.  Same for multiplication and division.  Add/subtract and multiply/divide are each two sides of same process.  It seems simple to our adult brains but it can be a very hard concept for a child to grasp fully.  I'm thinking that a focus on creating a weaving algorithm and it's inverse might really be a supportive numeracy effort. 

Here's another multiple of 3 warp:

After this I tried a warp of four strips.  Each woven 'unit' is made up of three horizontal yellow strips.  I love how the red and the yellow are rotations of each other from left to right, and reflections of each other from top to bottom. 

Also, notice that the second unit is the inverse of the first.  Instead of having the first yellow strip go 1 under, 3 over, the second unit starts 3 over, 1 under.  Interestingly, I usually weave from right to left, but when faced with an inverse, I found myself weaving from left to right.  I did it every time.  It was quite fascinating to watch myself in this process, which is why I suspect this would be really great for kids.  There's so much to learn and understand as you work to create a visually pleasing design. 

Here is a multiple of four, an 8-strip warp.  In this case, it's a basic over/under weave, but the two colored warp gives it some real variety.  I love this one.

Another four multiple, this time putting together the previous two ideas: the weaving algorithm of the first and the two colored warp of the second.  I wasn't trimming the edges on my designs at this point, which I think affects how your eye sees the patterns.

Now I was curious to see what it would look like all one color.  I think using green in both warp and weft brings out the structure in the weaving in a whole new way.  I like the green one untrimmed.

And finally, onto multiples of five.  This one is nice...

But this one is by far my favorite!

The first row is double strips that go [1 under, 2 over, 1 under, 1 over] and the second row is its inverse.  So simple yet very effective.  A ten strip warp is interesting because it's comprised of two, five-strip units.  A nice juxtaposition of odd and even.

It was such a fun day!  I think I understand the symmetries and the inverse weaving at this point.  I'm not sure how or if building a warp out of multiples affects things.  If you have any insights or see anything in my post that is mathematically shaky please feel free to correct me.  And, if I can clarify anything, please do let me know if you have questions.

p.s. I've got a new Facebook page where I'll be sharing links to cool math activities I find and some other things I'm doing with math, making, dance and rhythm.  Hope to see you there!

Mathematical Weaving, Part 1: Young Children & Grid Games

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

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

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

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

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

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

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

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

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

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

Another exploration:

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

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

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

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

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

I have some more questions about how to facilitate Patrick Honner's activity with slightly older children (first and second grade-ish).  One of my thoughts is that there is a basic algorithm for weaving that is a combination of overs and ups.  The design in the picture at the top of this post starts on the first line (weaving right to left) as 'two over, one under'.  The next line is different: 'one over, two under' and then the next two lines are actually the inverse of the first two.  Since my seven year old is already a fairly competent weaver, I think giving her some examples of how different combinations of over/under interact with each other would be a good place to start.  I'm also curious whether my daughter would be interested in the mathematical modeling at this stage in the game.  She's still a do-first, map-it-second (maybe) kind of gal.

p.s. I've got a new Facebook page where I'll be sharing links to cool math activities I find and some other things I'm doing with math, making, dance and rhythm.  Hope to see you there!

My Guest Post Over At Moebius Noodles!

I'm beyond thrilled and honored to have been asked to contribute a guest post at the Moebius Noodles blog.  If you've not heard of Moebius Noodles, here's their wonderful description:

Moebius Noodles is an off-the-beaten-path travel guide to the Math Universe for adventurous families. A snowflake is an invitation to explore symmetry. Cookies offer combinatorics and calculus games. Floor tiles form tessellations. The games in “Moebius Noodles” draw on these rich properties of everyday objects in ways accessible to parents and kids, even babies. As the world turns into a mathematical playground, it transforms, one family at a time.

Moebius Noodles is the brainchild of Maria Droujkova of Natural Math, both of whom/which have been instrumental to my growing understanding of mathematics and what math really is all about. Although I am quite comfortable with the math topics that connect to the math and dance we do in my program Math in Your Feet, I knew that there was a way of thinking and seeing math that I did not fully understand.

Over the last year or so following Maria's accounts of math clubs, math treks and, especially, the Moebius Noodles project I have found my way to math.  And, I am not ashamed to admit that, as an adult who is remediating herself from years of math anxiety, I have learned more about real math from an approach created for small children than I ever did in high school!  My new confidence and understanding is largely due to how Moebius Noodles models math and math learning for parents and families -- one small observation, question, inquiry or game at a time.

Here's the link to my Moebius Noodles post!  And, while you're there, don't forget to follow or subscribe to their blog.

Playing Math Every Day from Moebius Noodles

If you haven't heard of Moebius Noodles, I highly recommend you check it out!

From the Moebius Noodles blog:
We are creating an advanced and accessible math book for young kids and their parents, called “Moebius Noodles” and an online knowledge exchange hub to support it. It’s an off-the-beaten-path travel guide to the Math Universe for adventurous families. A snowflake is an invitation to explore symmetry. Cookies offer combinatorics and calculus games. Floor tiles form tessellations. The games in “Moebius Noodles” draw on these rich properties of everyday objects in ways accessible to parents and kids, even babies. As the world turns into a mathematical playground, it transforms, one family at a time.

The most recent post has a fabulous menu for 'playing math every day' for the week of November 28 through December 4.  Activities include playing ball outside and then exploring a type of fractal called Apollonian gasket, fun subitizing activities, and starting a math journal. 

Try it out and let me know what your favorites are!