Fiona joined me for a cuddle in bed this morning. I'm not sure how (it happens a lot with this kid) but talk turned to math. Recently she's been using "the stacking trick" to add multidigit numbers, stacking them vertically and adding the different place value columns. After doing this sort of math mentally or with manipulatives for a couple of months, she thinks that doing it on paper like this is a sneaky fun shortcut and has been loving it. No regrouping, though -- just straightforward addition in columns. She was first introduced to it a couple of days ago and it's at the front of her mind. She started talking about it in bed.
"What do you suppose would happen if you added up your ones sometime and got ten?" I asked.
"Whoa!" She giggled at the thought of something so interesting.
"Do you think you'd have to squeeze a ten into the answer place?" I asked. I figured I'd get a clue from her answer as to whether she was ready to go further with this.
"I'd probably just put a zero," she said. Aha, I thought. I think she's ready.
"Cool!" I replied. "I think that makes a lot of sense. But somehow you'd have to fit that ten into the answer, right? You're on the right track. What we do is to take that extra ten and combine it with the other tens in the tens column. I'll show you when we go downstairs."
And so I did, right after breakfast. With coins and a whiteboard. And she got it. And extended the concept on her own into other scenarios, eventually ditching the manipulatives and merrily inventing her own problems to solve on the whiteboard.
I've paid careful attention to Fiona's evolving concept of place value over the past year or so. Her growth in understanding has been steady and uncomplicated ... and impressive for one so young. The cuisenaire rods (with the addition of a few ten-rods and hundred-flats from a base ten set) have been extremely helpful for her. Over the past few months she's gone from using them to imagining using them to merely thinking in place-value categories while performing mental math.
This week's progress feels more like a leap, though. She's now working easily with abstract symbols rather than tangible manipulatives or pictorial symbols. And she's applying those abstract symbols to more and more complex tasks -- yet without losing the mathematical understanding of what those symbols and their manipulation mean.
I wonder if it is just a coincidence that this week she also began reading dotted rhythms and compound time easily and with accuracy in music? Sometimes you can almost hear the synapses forming in this kid's head.