Here are a few examples of how the math is going these days:
Scene 1: The grocery store
Seven-year-old is pushing cart around the store, narrating as she goes: "Go forward, now one quarter turn to the right, now go forward, parallel park. Okay, now turn half way around, go straight, one quarter turn..."
She sounds like what I imagine kids are doing when they programmed their Logo turtles in Seymour Paper's classrooms. We've never discussed quarter turns or half turns but there they were, helping her guide our cart.
Scene 2: Rest room at City Hall
Her: Mama! Look at the math on the wall!"
Me: What kind of math do you see?
Her: Look at the designs...let's figure out the perimeter!
Me: What kind of designs do you see?
Her: [silence, moving away from the scene]
Me: When I look at it I see a large square with green tiles in each corner and one in the center.
Her: Why do you always have to take pictures?
Scene 3: The Park
My kid is playing with a younger kid. Both are on a 1970's era climbing structure. The other kid's mom calls to the younger child to be careful and climb down. My daughter replies, "Don't worry, it's one hundred percent safe!"
We read a little about probability in G is for Googol which is where (I think) she first heard about this idea. She's also been reading the daily weather reports in the newspaper which are full of both percentages and probability. It's clear to me she's playing with these two ideas and trying to figure out how they work.
I am always a little tickled when I overhear or observe my daughter applying or identifying math in new ways. But over the last year I have come to wonder why she so clearly wants math to be all her own, separate from me. After thinking about this on and off for the last year (over which time she has really come to see herself as "good" at math) I think it is partly that she is such a fiercely independent learner. But I also think there is something more than personality at play here.
Based on my own math learning experiences these last few years, I can tell you that learning math is personal. I'm reading Seymour Papert's The Children's Machine and he is brilliant at specifically and concretely illustrating how real learning is a series of personally relevant connections. I think his is a theory that can be applied to many subjects, but it totally makes sense within the context of learning math. I've also read in different places that we all have what I would call differently constructed math schemas -- we all see math differently. The challenge of the math teacher is to teach from where the student is rather than require the student to assimilate the teacher's mind map of whatever math topic is being explored.
So, what I've ended up doing is creating situations for her to explore math within an actively hands-on, visual, and often narrative-based context. This approach started out as my way of dealing with a learner who was resistant to instruction, but quickly became a wonderful opportunity for me to re-conceive what math is, as well as where and how it can be learned. Basically, I had to rearrange my concept of math to fit my particular learner.
In our first grade year the learning happened during conversations about the math we saw walking around town (which I started calling "sidewalk math"), reading living math books, playing lots of math-y games, and strategically placing math manipulatives around the house. (I'm still sort of in awe at the independent work that went into this shape study using tangram pieces when she was six.)
In her second grade year we've done more math sitting down at a table using more recognizeable math manipulatives, but with the same approach as last year; our math has been hands-on, very visual, with a lot of room for personal aesthetic and narrative contexts.
I made a conscious choice to pause our math progression this year in about mid-March. I knew we could have kept going, but my instinct (and my work schedule) told me she'd be fine. And I was right.
What I've observed in my daughter over the last seven weeks or so is that within this 'void' of math lessons she has begun processing her learning by applying and using the math we've done together since mid-August. There's been a veritable flood of daily self-initiated math activity, thinking and conversation. The three examples above are just a fraction of how she has been playing with the math she knows (or is trying to figure out) by applying and using it in a variety of different settings.
I did have some thoughts in March about seeing how far we could get into third grade math but now I'm glad I didn't push it. It's been more than worth it to make the space and time for assimilation, that special kind of deep learning that happens unconsciously, below-the-surface. You have to be patient for this kind of 'proof'' to bubble to the surface, but if you keep your eyes open, kids will show you every day what they know. Even though we could have gone further in math this year, it's clear that the math we did encounter and explore is now truly all her own. And, I think this is the best possible outcome for my particular second grader.