Multiple Representations

In their spontaneous answers, two children in the course of less than 20 seconds show three strikingly different ways of thinking about the same arithmetic question. I asked:  “If Simon, the drummer, plays ten beats on the big drum and I am clapping two for each of his drum beats, how many beats will I clap altogether?”  Steven immediately answered “20.” I responded with, “How do you know?”

Steven: You could tell that Jeanne would be doing one more than Simon.   

And if Simon was going to do 10, that means that Jeanne had to do …

Leah: Twice that.

St: Twice that.

L: Because she did one in the middle.

St: Because she did one in the middle. So it had to be 20.

Steven’s immediate response, “You could tell that Jeanne would be doing one more than Simon,” is as if he is viewing just a single slice of the continuously going on, cumulating events.  Leah, interrupting Steven’s conclusion, “…that means that Jeanne had to do…” introduces the arithmetic with, “Twice that.”  Now putting time into space, Leah adds, “Because she did one in the middle.”

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