BatBoy has been exploring odd and even numbers for awhile, just categorizing them in his head and finding patterns. He came to a conclusion a month or two ago that odd numbers had 'middles' and even numbers didn't. I've tossed in the idea lately that the pieces of even numbers can be paired off; in essence, they are divisible by two. But, that didn't (yet) disagree with the 'middles' theory and BatBoy seemed to disregard it. In the past week, however, BatBoy has encountered two troubling numbers: zero and one.
One is even, he insisted, because it has no middle. And what about zero? What is zero? What would be the middle of zero?
For awhile, I tried to disagree that one was odd. I tried to explain that one is its own middle, to force the model. Then, I tried to explain that having a middle wasn't the 'real' definition of an odd number, that "we" define odd numbers by their inability to be divisible by two without remainder.
I could have gone on that way, and eventually, BatBoy would probably have come to accept that he was wrong and understand the "right" way. I am glad that instead, last night, I took the opportunity to recognize that this was not a misunderstanding of mathematics; it was a difference in definition. And I said this explicitly: Using BatBoy's definition of 'odd', 1 is indeed problematic. SpiderGirl, who understood my explaination, said, no, 1 fits, 1 is odd. And they were both right. It comes down to definition. We talked about things in math that are defined arbitrarily before we can begin to work with them. We talked about zero and one often being special numbers. We could have also talked about convention in math and how that helps us communicate.
The discussion ended up being much richer than it would have been had I stuck with the conventional definition as the "right" one. We discussed mathematics, rather than just communicate a concept. Did we come to a conclusion about the number one? I don't think so. I will know in the weeks to come as the topic arises again.
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