My older kids seemed to intuit the concept of place value very easily. I feel like I sort of missed watching it happen. I just knew they got it. Having talked with some parents whose children haven't easily developed that firm grasp of place value, I resolved to pay more attention to Fiona's number-sense learning to see if I could catch it happening. Fiona loves numbers, but it's clear that she hasn't really got an inkling about the place value thing yet. For instance, 9+7 and 10+7 are problems of equal difficulty for her, and she solves them both by counting up.
Last night she was doing some basic addition and subtraction problems in the Miquon Orange book. When the problems stray into things like 5 + __ + 1 = 8, she likes to use the cuisenaire rods. She had a few rods upstairs, where she was working away on the floor beside her dad. But suddenly she showed up in the living room, where the main cuisenaire collection was. I asked her what she needed.
"Oh, I want to figure out how many ones make ten. I don't have enough rods upstairs," she said.
Our base-ten set, which matches our cuisenaires in dimensions and colours, is all mixed in with the cuisenaires. For whatever reason, to figure out the answer to her question she used the ten-rods to represent ones and a hundred-flat as a ten. Was this because these were closer at hand, or because they were easier to manipulate, or because she's already getting this at a deeper level than I imagined? I really don't know. Anyway, she measured them out. She laid rods side by side until they matched the length of one side of the flat, then counted them.
"Ten ones makes ten!" she announced.
"Cool, you figured it out," I said. "So can you figure out how many ones it takes to make seven?"
"Seven," she replied, not missing a beat.
"How many ones in twenty-nine, then?" I asked.
"Twenty-nine!" she giggled.
I hadn't realized that she would need to count out how many ones in ten. She seems so clever with numbers -- I thought that would be obvious to her by now. But it wasn't. It was a sudden burning question she had to test out. And it wasn't "how many 2's in 8" or "how many 1's in 7" that she chose to examine. It was that very particular relationship that our place value rests on that she wanted to test out concretely. She quickly proved to herself that there was a relationship there between units and tens. And then she correctly and logically generalized it.
I can't help but think that this little exercise she was driven to carry out is going to form one of the keystones in the foundation of her place-value understanding. My, it is such fun watching children learn!