Not 'Just Shapes'

So....we've been doing Sidewalk Math on and off since, hmmm, February?  Three months or so.  It's been me, mostly, getting all excited about what we're finding when we walk around town, taking pictures, oooh-ing and ahhh-ing.  I have been having me some serious fun, let me tell you.

The kid (now a freshly minted seven) on the other hand...  She's been there with me, she's been interested, participatory, but now?  Now, when we go out, I am not allowed to pull out my camera.  Ever.  I am not allowed to initiate any math conversation whatsoever.  Sounds horrid, except for the fact that now it's the kid who is finding math everywhere.

She ooohs over every single triangle she sees -- a scrap of fabric, a crack in the sidewalk, something on a wall.  In negative space, she sees a triangle of ceiling made by a partially closed door. She even made me this triangle love note.  Even the message inside is written to take the form of a triangle.

"Look Mama!  When I folded the square [paper] I got two triangles.  Look!  Look inside!"

"Looks like four triangles to me," I replied.  I've never seen someone so excited over four triangles.

She ahhhhs over concentric circles.  She finds them everywhere -- even inside her cucumbers (where she also sees little stars, which, I manage to add over the adulation, are constructed with nature's numbers, Fibonacci, right?  Three, five, eight...the cucumber star is three....)  Seriously, this child has her eyes wide open.  We're in a rest room.  The baby changing fold out thing on the wall has circles, with circles inside them!  Concentric circles on the sidewalk.  At the bottom of the public pool.  Everywhere.

And let's not forget spirals.  I was making an extra effort a month ago to look for them and wasn't too successful, to tell the truth.  Also, the kid kept getting them mixed up with concentric circles.  It took a week or two of clarifying, on and off, to help her see the difference and... SNAP!  The light went on and now she sees them everywhere -- in fences, on light posts, on clothing, all around the house.  She draws them.  She finds small ones, big ones, mostly man made but also a vine here and there, or when we look really closely at how a plant is growing.  It's crazy.  Spirals are everywhere.

Oh, and composite shapes -- rectangles made out of squares, squares made out of triangles, squares made out of squares, trapezoids made out of squares and triangles.  And curves!  Let's not forget them.  And parallel lines!  Right angles!  The plaid fabric is a grid!  You get the picture, right?  I'm exhausted!

Today she said, "I'm going to make a book of all these shapes.  I'm going to take pictures with my camera and Papa will print them out for me."  She wanted me to have nothing to do with it.  Sniff.  But, still.  Isn't it cool that this whole thing has become completely, totally hers?! 

In the back of my head I am always observing my own mind-noise that says, "What's the big deal?  They're just shapes.  Kids learn about shapes in preschool."  But my answer always circles (spirals?) back to the fact that the longer we keep our eyes open for it, the more familiar she has become with the possible variations and anomalies (size, rotation, tilings, etc.) that can occur in 'basic' shapes.

(An aside: It's been hard to verbally explain the difference between a rhombus and a square rotated from its traditional position, but sidewalk chalk saved the day!)

Despite the doubts, my sense is that all the conversations we've been having over the last few months will eventually make geometry on the page that much more meaningful, and maybe even easy.  (I can also imagine that this kind of activity also serves the larger goal of pattern recognition in any area of math she attempts.)  It will all make sense because she will have seen those shapes in their natural habitats, and by identifying them and talking about them she will have made each shape hers and hers alone.  And, as the structure of the universe continues to emerge in front of her very own (and open) eyes, how much more fun will her world be to play in, explore, put together, and then take apart again?  A living math book, our world.  Not 'just shapes', y'all.  No indeed.

Check back with us in three or four years, maybe seven, and we'll see if I had it right.  Something deep inside me, though, says that the more we actively do and see and investigate the structures and forms math takes (like the curvy wall constructed with straight bricks?  Or, the sidewalk lines that look vaguely algebraic?) the more we can actually know math when we meet it in a more formal setting.

What I do know for sure right now is that the kid already considers herself 'good at math' and has added mathematician to her future career choices along with actress, singer, dancer, geologist, archaeologist, astronomer, writer, illustrator and cartographer. 

Have fun everyone!  If you haven't heard yet, math is everywhere!

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!