Between the Folds

The piece in question

Just yesterday I was playing around with a piece of paper, folding it, watching it, seeing how it could move.  I showed it to a friend and said, "I guess what I'm doing is improvisational origami."

Just today I mentioned the same thing to a different friend.  She said, "Have you seen Between the Folds?  You have to..."

Here's the trailer for the movie -- it's on hold at the library.  Hopefully I'll get to watch it soon.  Can't wait, can't wait, can't wait...

Every day I am finding more and more of the pieces that connect math, art, dance, rhythm, science, expression and creative practice.  I think the reason I'm so excited is that I recently found my way back from a wrong turn I took with this inquiry. But, that's part of the process, too, no matter what the medium. After determining what I don't want or need to do right now, the missing pieces seem to be falling into place.

[Edit: This post seems to be getting a lot of traffic, for some reason. If you're interested, here is the follow up post I wrote after viewing the documentary.]

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!

Bal-a-vis-x: Rhythm is the Key Element!

"Bal-a-vis-x: Rhythmic Balance/Auditory/Visual/Exercises...requires focused attention, demands cooperation, promotes self-challenge, and fosters peer teaching."  

Wow, a program that uses rhythm, cooperation, crossing the midline, and visual tracking all in the pursuit of an integrated nervous system that is ready learn.  There are many similarities here to aspects of Math in Your Feet including rhythmic movement in concert with others, self-challenge, peer cooperation, and cross-lateral movements.  Sounds like a great idea whose time has come! 

A Moving Classroom: "Find Your Center"

"Nothing exists without a center around which it revolves, whether the nucleus of an atom, the heart of our body, hearth of the home, capital of a nation, sun in the solar system, or black hole at the core of a galaxy.  When our center does not hold, the entire affair collapses.  An idea or conversation is considered 'pointless' not because it leads nowhere but because it has no center holding it together." 

-- Michael S. Schneider, A Beginner's Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science

I just picked up A Beginner's Guide to Constructing the Universe.  I still have unanswered questions about what kinds of math are interwoven with a creative inquiry into percussive patterns.  That's not to say that I am confused.  I know exactly what kind of math the kids are learning, but I am sure there is more and I am searching for evidence.  Some days, like this one, I find I understand math more than I think.  So do you, by the way. 

Here's is a reposting of one of my very first pieces on this blog, The Power of Not Moving (October 13, 2010), which attends to the value of a single point of stillness and focus, the center, in the moving classroom.

The Power of Not Moving
So much of what we know about how the brain learns points to using all the senses -- moving, touching, smelling, looking, leaping, running, talking, writing, tracing, solving, thinking, responding, producing, revising, doing.

But, what about stillness? 

What about a moment of doing nothing except making your body balanced and quiet, ready for learning?

In Math in Your Feet we call it 'finding your center.'   This means, quite literally, to stop what you're doing and put your two feet in the middle of your square dance space.  Arms by your sides. Eyes on the teacher.  Mouth quiet.  In control and in charge of your body.  Ready and waiting for the next thing to happen.

One of the biggest concerns teachers have about bringing kinesthetic/movement/dance learning into their classrooms is that it's going to be chaotic and uncontrollable.  I hear it every time I lead a professional development session for teachers.  By the end of our workshop, however, they realize that just because it's movement doesn't mean that self-control is absent.  This is something I bring up with kids, as well.  Continuing to return back to your center is one of the ways that you can remind your body what it feels like to be in control.  And then, you are ready to move again.

Movement is crucial to helping children learn, even if it just means there is time in the day to get up and move around the classroom.  However, when you are using a lot of movement, children need stillness to counterbalance all the activity.  This idea is worked into the flow of each class I teach in an academic setting.  My formula has always been 3-5 minutes of moving, followed by 3-5 minutes of sitting and focusing on other things -- watching and responding to others' creative work, receiving information, clarifying content.  This kind of non-moving time, coupled with 'finding your center,' becomes a powerful counterpart to all the jumping, sliding, stepping, turning, talking, collaborating, resolving, and creating the students do the rest of the time.

Too Cool (Video)

Got this from Nic. Too cool.

Number Discovery

In retrospect, this amazing discovery started with parental neglect and erroneous math.  I freely admit I was at fault, but it all worked out amazingly well in the end.

It was all precipitated by an innocent question, yelled from another room: "Mama, what is half of 38?"

I did a little mental math: "Sixteen!" I yelled back.  In my defense I was in the middle of something very important.  That's why I didn't insist she figure it out herself, you see. 

When we were finally in the same room it took me a minute.  I was actually fairly impressed with her thinking.  Even though it was all wrong, she still had logic and structure to her reasoning.  Can you figure out what her 'rule' was?

[And, yes, at six she still writes many things backward; sometimes, as is the case here, she even writes from left to right.  Ah, the growing brain!]

Luckily, I had my wits about me and simply said, "Cool, look at that!  Hey, let's check your work with the Cuisenaire rods!"  This was a brave move since, prior to this moment, we have done absolutely nothing with 'taking away' or 'difference' in any formal way let alone using the rods.  Fortunately, the taking away part was so wonderfully obvious in this visual/tactile realm that I had no problem explaining it and the girl got it right away.  During this process I also noticed that in the intervening couple months between our first major experience with Cuisenaire rods and today, her ability to visualize and attribute amounts to the rods has become second nature.

Here the total number was 31, take away nine.  It was so very satisfying to physically take away a number and literally see the difference. 

Sorry this is so blurry, but hopefully you can see that we started with 37 and took away 15.  I'm not completely sure why we didn't start with the original number in question, 38, but like I said, it all worked out in the end.  Just look at it!  It's beautiful, don't you think?  By the fourth equation I asked her if she could figure out what would come next.  She guessed right for the last five equations, but wanted to check her work with the rods each time anyhow.

Today I am basking in the joy of unexpected discoveries and a growing mind. 

Geometry Discoveries

One of my most enjoyable creative endeavors these days is accompanying my daughter (age six) on her math learning journey.  Sometimes we play games, or work with pattern blocks or Cuisenaire rods.  Sometimes she wants to earn and save money so she can eventually run away with her best friend.  Whatever we do, she learns best through conversation and narrative. 

She also learns math without me.  There are plenty of times when she happens upon something 'by accident' by which I mean: things (books, puzzles, making supplies, marbles, tape...) left around the house, acted upon, forgotten for a while, ultimately to be rediscovered a month or three later and acted upon again, but this time in new ways.

This happens a lot 'round here. 

For instance, we have tangrams in almost every room of our house.  There's a magnetic puzzle book set in the car and approximately six plastic sets on the game shelf, of various colors and all mixed together. I have four sets of magnet tangrams that I got at the NCTM Annual Meeting in Indianapolis this past April.  They have been on the fridge in the kitchen since then with minimal interaction...until recently, and, last I looked, they were all in use in some kind of design or another.

Here's what I found my dear child doing with the set of car tangrams (inside for some reason) over the last few days.

By way of setting the scene, I should tell you that the kid generally initiates and works independently on any project she pleases during her afternoon quiet time, usually with the company of an audio book.  A few days ago during this time she informed me she was "making a math book."  I had observed her tracing tangrams, but was surprised to find that she was doing it in a deliberate way, and that she was also able to clearly describe her discoveries and observations to me. Here is the book, so far.  I am just a scribe here -- these are her words as she dictated them to me, except where noted:

"Four of these triangles that you see here can make a square.  If you pull these triangles apart you can see that they're little triangles.  But you can see on this page that they make a square."  [I see that she has drawn them as individual shapes, which, over the course of her illustration, merge together into her intended shape.]

"This rectangle you see is made up of a parallelogram and two triangles.  Really they're just shapes, but when you put them together they make a rectangle." [It looks like she's numbered the inside angles of the individual shapes.  It also looks like she is again showing the process individual shapes merging into the intended new shape.]

"You see the wheels of this bike as rhombuses but really they're squares turned so their points are facing up and down, and to the side." [She was gesturing this first, and at first she used the word 'flipped' to describe the orientation of the square wheels.  I focused her on the orientation of the corners to describe how the square was turned.]

"The square and the rhombus that you see here, their edges are both the same length.  The difference is a rhombus is a squished square, squished to its side.  The rhombus has two larger angles and two smaller angles than the square.  But the square has the same angles on each corner."  [These are actually tracings of shapes from the pattern blocks set we have.  The ruler markings was her idea for comparing the two shapes.  I supplied some new vocabulary in the form of 'angles' and made some observations about the difference by overlapping the two shapes, in an effort to help her observe and further articulate the difference between the two.]

Writing about all of this made me think to look up more about the van Hiele levels for geometric reasoning.  I need to spend more time with that to analyze how I'm interacting with future discoveries of hers.  I also think I'd like to focus more specifically on flips, turns and slides.  Maybe I'll be super ambitious and have us play around with flips and turns using both 2D and 3D shapes!  

What Do You Make of This? (Build What I Have, Redux)

In my recent post Conversational Math: Part Two I described a game that my daughter and I like to play with Cuisenaire rods called "Build What I Have".  It's really fun.  So fun, in fact, that my daughter asks for it every other day or so.
Here's a funny thing that happened today:

It was my turn to give directions.  To start, take a look at how my design looked when I was finished giving directions.  I should point out that there is only one layer of blocks laid out on the board we were using.  I should also point out that we hid our work from each other.  She is trying to replicate my design using only my spoken words as a guide.  Ironically, no gestures are allowed in this game!

Mine.

Here's how I started: "Take three light green rods and put them together so they are all touching and all perpendicular to the bottom of the board.  It should look like a square that is three units across.  Now add a fourth light green rod on top of the square.  It should be placed perpendicular to the other green rods and parallel to the bottom of the board.  It should look like a rectangle now."

So far so good!

Me: "Take an orange rod and place it on top of the green rectangle."
Kid: "Like a teter totter."
Me: "Yeah, I guess so! Now, take two white blocks and put them on top of each end of the orange rod" (etc.)

Then it was time for the big reveal...tah dah!

Hers.

Geez Louise, what happened?!?  Her orange rod IS on top of the light green rectangle, but it appears the kid built up while I built...up?  I suspect we were both right, but I am at a loss to explain why or how.  I think it is accurate to call the kid's work three dimensional, but so is mine because it is built with polyhedra and not 2D lines on a flat surface.

Hmmm....any ideas?