The Map is Not the Territory

Okay, so we've walked past these two covers about a million times but today I saw them in a completely different light.

This winter/spring we've been doing a LOT with number multiples and conceptualizing multiplication and division.  Last week my mind moved toward the inevitable: fractions.  Although shivers go down my spine every time I think about fractions I'm still resolved to figure it all out for the sake of my seven year old, if not myself.  It's been sitting in the back of my mind so I guess that's why this cover caught my eye and brought me to a dead stop.

Can you see it?  Fractions and multiples!



































And, a little further on, this beauty: an 8-star and some fractions!




















What I really want to know is who designs these?  I want that job.  To see some other cool round things we found on a walk last spring, go read my post Channeling Tana Hoban: Juxtaposition Edition.   Now that was one amazing day for circles!

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!


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!)

After a mild winter we had a lovely and quick blooming spring which allowed us to get out and about earlier than we might have otherwise.  In March I posted about an outside adventure where we discovered a veritable treasure trove of circles in juxtaposition with other shapes.  I think this might have been the origins of what I've started calling 'sidewalk math'.



Sidewalk math is fun because, generally, all you have to do is keep your eyes open.  If you've got a camera to record your observations, all the better.  This is not necessarily an original idea; the photographer Tana Hoban has a whole series of books with photos of the math all around us.  Her camera is the eye through which we can notice math in the physical world.  There are also the engaging Math Treks developed by Maria Droujkova of Natural Math.

For us, sidewalk math is a combination of these two approaches and has turned into a large percentage of our first grade math classroom.  It capitalizes on my daughter's propensity to notice everything, fulfills her need for movement while she learns, and bypasses her resistance to formal lessons.  It's also an opportunity for us to make observations and pose questions in a collaborative way, which is an approach that works for both of us.  For example, on a recent walk my daughter notice a crack in the sidewalk that initiated an hour-long in-depth conversation and exploration into the nature of triangles as we traveled to the hardware store and back.

(And it's apparently it's sticking with her: As I'm writing this my daughter calls down to me to report that she and her dad saw "seventeen triangles on their way home from the park this morning....did you know that part of an arrow is a triangle?!" )



But, in this story, sidewalk math plays another role, that of salvaging my initial attempt to introduce functions to my young daughter.  You can read about my first attempt here where she was wholly and unequivocally unimpressed with my presentation of the subject and took matters into her own hands.  I ended the post wondering what to do next.

I was understandably thrilled when I came across the book A Game of Functions by Robert Froman.  It's part of the Young Math Series from the 1970's and is out of print.  A quick Google search found copies available for purchase between $17.00 and $115.00!!  Luckily, my husband works at a university with a very comprehensive library and I got my hands on a copy.  I read it to my daughter one morning.  She wasn't having a great day, but she didn't protest, and we got through most of it.  I let the idea sit, waiting patiently for an opportunity to put the ideas into action as the book suggests. 

The book starts out with an introduction to the idea of 'function', as in 'whether we go the park this afternoon is a function of the weather -- if it rains this afternoon we will go shopping, if it is nice we will go to the park' (I'm paraphrasing here).  Or, as in this example below, how long it takes you to run around the outside of your house depends on on whether you crawl, walk or run. How quickly you go is a function of your mode of movement.















You find a nice big area and draw a line across and a line up.  Lucky for us I had sidewalk chalk on me and we were at a park with a parking lot that looked almost like graph paper!










Who knows?  This journey is full of adventure and surprises.  It's not always smooth sailing, but we're learning a lot, her and I.  And, one thing's for certain, there is more sidewalk math in our future.