The Map is Not the Territory

There are times when I doubt my approach to teaching and learning math with my seven year old daughter.  The approach relies heavily on having faith in a learning process that cycles through input-conversation-incubation-inspiration-output and back around again.

Every time I wonder if there's anything sinking in, or think that we're not doing 'enough' of the 'right' kind of math, I invariably run across jewels like these which put my mind and heart at ease...at least until the next fallow period.

A week ago we were both playing around with a nice Christmas gift of a felt board, colorful felt triangles and some awesome design cards.

















Yesterday I challenged her with some of these designs and watched her approximating and adjusting angles, creating strategies for reproducing the designs, turning and flipping the triangles, and predicting the growth of the triangle pattern in the upper left corner.


Today I found this on the making table, buried under some other projects.  It looks like she was drawing (mostly) isosceles right triangles, y'all.  Awesome! 


































Also of note, the child sat herself down today and, after weeks of conversations about what she might want to do with a pool of money collected from various sources (holidays, allowance, odd jobs, etc.), devised this budget. (Annotated translation below.)


































It says this:

I have $50
Spend $25
Save $25

Cocoa [hand-made stuffed kitty] spending $5 [Meaning she's putting aside that money for things it 'needs'.]
Biscuit [lovey kitty] spending $5  ["And then I'll have $15 to spend on myself Mama!"]
Things I need or want too expensive for me: my own ice skates

List of things Biscuit needs: 4 pairs of socks

Me: "Why four pairs?"
Kid:  "Because he has four legs, silly!"
Me:  "But pairs come in two...does Biscuit have eight legs?"
Kid: "Oh, right!  He'll need two pairs of socks!!"
[Me, thinking to myself: iconic multiplication!!]

Things Cocoa needs
__________________

We do all kinds of math these days -- reading living math books, games, design activities, drawing, noticing things when we go out, engaging in loooong conversations about the fate of her Christmas money and, yes, some workbook stuff.  Taking the long view, I suppose that what I want most for her is that math becomes both useful and personally meaningful for her.  I think today proves we're headed in the right direction.

It's been over five months since I pretended I was Tana Hoban, the photographer who found math everywhere with her camera and turned the pictures into wonderful books for children.  Since then our eyes have been WIDE open, finding math just about everywhere we look. The more math we see, the better we become at finding it; the more math we find, the better we are at understanding it.

As my six (now seven) year old and I traveled around town this spring we found lots of shapes and patterns, parallel and intersecting lines, even spirals.  It's been your basic geometry kind of math, but we've had some incredible conversations about what we see and find. 

Lately, though, our math eyes have become remarkably more advanced.  For example, my daughter saw a tetrahedron in ropes staked into the ground, steadying a young tree.   I started the spiral inquiry, but she's the one that started seeing them everywhere we went, even places we go to regularly.  She still notices spirals all the time.  Recently, she found math in the most prosaic of circumstances...a moment of recursion in the restroom mirrors (one on each wall) at a local grocery store.   I guess math is everywhere!

























As for me, my eyes have very recently been opened to stars.  I would have never recognized this particular star for what it is without the last week of exploration and inquiry under my belt.  There are actually at least four different kinds of stars in this picture.



















And, here's an 8 star I also found today.



















The stars and the rest of the photos are from our trip to the zoo and botanical gardens.  My daughter found some flowers that had dropped to the ground and shouted over to me, "Mama, look!  These have five petals!  A Fibonacci number!"



















These circles were near the carousel at the zoo.  I love it!  And, that reminds me that, although it was impossible to get a picture of it, the carousel platform was round on the outside, but actually created out of twelve trapezoidal sections leaving an interesting hole in the middle -- a dodecagon!  Geez, I was really impressed with myself for seeing that one.