This is a second installment on a series I kicked off in a previous post.
To summarize, we’ll be tracking the development of certain math ideas, through a typical student’s math learning. The later ideas usually completely supersede the earlier ideas. I suggest that there is nothing wrong with this process, and that each stage can be completely developmentally appropriate. The danger comes from regarding the latest of these ideas as Gospel Truth rather than simply one that’s developmentally appropriate for you (or, in some cases, the one that’s developmentally appropriate for our civilization at its current level of development.)
When you multiply a number, it gets bigger.
This one is a natural idea for children to acquire as they learn about multiplication. When they later encounter things you and I might call counter-examples, they might dismiss them rather easily at first.
Multiplying by one? Easily dismissed. “Not really multiplication. You leave the number alone. You are not scaling it up, you’re not making a multiple. You’re not multiplying, you’re doing nothing.” (I’m stating these in the language of adults – kids have their own versions.)
Multiplying by one third? Also easily dismissed. “That’s not really multiplication. That’s more like division. You are dividing into equal thirds.”
Yet sooner or later, this notion falls by the way side, and is replace with something like: when you multiply a number, it gets bigger, stays the same, or get smaller, depending on whether the other number is more than one, equal to one, or less than one.
Later in school, this notion falls by the way side, also, when students learn about multiplication with negative numbers.
These new conceptions of what you can expect from multiplication are not additive, they are inherently disruptive. And that’s natural. They are just as natural as a small child’s conception that the moon follows him around, to be replaced later with a notion that the moon stays put but that it somehow looks as if the moon follows people.