Saturday, November 24, 2007

Odds have middles

Generally speaking, we've got a good thing going on with math at our house. We don't have a ton of manipulatives and educational games (cuisenaire rods, base-ten add-ons, and pattern blocks, mostly) but we have kids who, at least through the K-6 levels, enjoy discovering and implementing things about numbers, operations and relationships between them. I don't quite know how we ended up with this lovely math tradition, but we did.

Photo left: the face Fiona gets when she suddenly thinks her way to the solution to an equation using an "easy path," some mental math trick that helps her solve it easily. In this case, 2 x 15 became two 5's plus two 10's, and that was easy.

Fiona's curiosity about math continues to be boundless. I swear, this kid thinks about numbers non-stop. Probably in her sleep. On our trip home from Calgary she awoke from a nap and said "I just figured something out. Nine is half of eighteen."

A couple of nights ago Sophie was doing some multi-step word problems using ratios. She'd sorted out all the conceptual stuff and was down to the final arithmetical step. As often happens once the fun of formulating the equations is done, the arithmetic was less interesting. She needed to find a fifth of ninety, but her attention was wandering. "Use the division the other way around," I prompted her. "Try to figure out how many fives in ninety. You know how many fives are in a hundred."

"That'd be twenty," piped up a little voice beside me. Not Sophie. A littler voice. It was Fiona, delighted to be "helping" Sophie with her math.

Fortunately Sophie found it funny.

And then this morning, fresh out of bed and with her math mind churning, Fiona explained to me that odd numbers are the ones "with middles." I asked her to explain.

"Well, like five," she said. "If you count five things, the 'three' one is in the middle. Same for seven, nine, eleven, thirteen, fifteen, seventeen, nineteen, twenty-one and twenty-three. All odds have middles."

This observation wows me more than any of the other clever math problems she's solved. Why? Because she's observed a property that some numbers have but not others, described it to herself, developed a hypothesis about what is common to all the numbers that have it, tested the hypothesis in her head and correctly generalized and described the pattern.

All while still wearing her favourite long-outgrown size 2T snowflake pyjamas.


  1. Math happens similarly at my house. It is exciting to see the intuitive process at work. I can't imagine learning math by rote!

  2. Way to go Fiona! The part of your post I find interesting though is the fact that Sophie found it funny when Fiona was "helping" her. Emily could use a little bit of whatever personality trait it is that allows her to find the humour in that. I assume it's confidence - something which Emily lacks a bit in regards to math. Will often helps, with good intentions. Emily however feels that he is doing it to intentionally show her up . . . sigh.

  3. Debbie, it was close, but I think the gap between Sophie and Fiona is just big enough that she was able to see the humour. If "how many 5's in 20" had been the sort of question Sophie would have had to think to get, if it was the answer to the question she was struggling to finish, I think she would have been upset. But it was just a simple "clue" on the way to another answer for Sophie, and not the sort of question she'd struggle to access the answer to.

    We've had similar issues, with the tables turned, between Noah and Sophie, such that for a long time Noah refused to do math with Sophie within earshot, because he just knew that she was thinking the answers to things I was prompting him with. Even that was too much for Noah back when he was lacking in math confidence. So I know the dynamic you're speaking of!

  4. "Odds have middles."

    What a great way to think about it. How creative your kids are with math. I really enjoyed this post!

  5. While my daughter is not as apt as yours, she too seems to enjoy figuring out math problems.

    I had to smile when reading this post because I, too, have always thought of odd numbers has having a "middle". Way to go, Fiona! :-)


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