## Flashback Friday: Half and Double

When my son was about six years old we would “play math” when we were in the car, or getting dinner ready, or just “playing.” I remember asking him what 7 x 8 was, and hearing a reply that went something like this: “Well, 7 x 8 is the same as 14 x 4, and 14 x 4 is the same as 28 x 2, so. . . 56!” It was not quite what I expected, but then I recalled a brief conversation we had had some time earlier involving the number 12.

(Paraphrased)

Son: “3 x 4 is 12, and so is 6 x 2.”

Me: “Yes. . .?”

Son: “Does that always work?”

Me: “What?”

Son: “That if you double 3 you get 6, and half of 4 is 2.”

Me: “Well, what do you think?”

I could see the gears turning as he thought about it in his mind – picturing the four groups of three changing into the two groups of six. I don’t recall the resolution of the discussion, but apparently it was enough to satisfy him so that it became a strategy that he began to use on a regular basis. I am certain that I did not encourage it as a “method,” just an interesting occurrence.

A few weeks after the “7 x 8” question, I asked it again, expecting that he might remember “56” from his previous “derivation.” Here’s what I got: “Well, 7 x 8 is the same as 14 x 4, and 14 x 4 is the same as 28 x 2, so. . . 56!” He showed exactly the same enthusiasm for “solving the puzzle” that he had a few weeks earlier, not recollecting that he had, in fact, solved it before.

Contrast that with the child who has memorized 7 x 8 = 56, and would be able to repeat it back in the blink of an eye. Who has more Enduring Mathematical Understanding?