by Conrad Cook
Winsights No. 80 (January/February 2005)
Photography by Elan Sun Star
I taught my six-year-old nephew how to multiply and divide. I figure some of you may want to know how to do this:
(How to teach a six-year old, that is; but if you want to know how to multiply and divide, you’ll get that, too.)
1 — Motivate the kid to learn
In this case, I said, “Do you want to know a secret?”
“You’re not supposed to know how to do this.”
“Kids your age are not supposed to know this. Nobody your age knows this yet. Are you sure you want to know this?”
2 — Set up the representation
Begin with the known
I asked him if he knew two minus two. He said four. (I’d quizzed him more extensively earlier, so I knew he could add and subtract by counting.)
I asked him if he could write it. He wrote 2-2=0. He wrote the 2’s backwards, but got the form right. I praised him, but wrote the 2’s correctly above and had him write it correctly below, “because that’s important.”
Create the representation with knowns
He told me that “minus means take away,” so I asked him if he knew how to draw it.
He said, “No.” So I asked, “How many are you starting out with?”
He said, “Two!” So I told him, “Then draw two circles.” He did. I asked, “How many are you taking away?”
He said, “Two!” So I told him, “Then cross out two circles.” He did.
“How many are left?” I asked. “None!” he said.
“So what’s the answer?” I asked. “Zero!”
“And that’s what you got,” I said, pointing to his written equation. “Very good!”
Now he understands the representation.
3 — Explain the unknown procedure with the representation
I showed him how to say and write multiplication problems.
I told him that, to find the answer, you make as many big circles as the first number. Then, inside each circle, you make as many X’s as the second number.
In this way, I had him multiply 2 by 3:
2×3=_ [I wrote]
(xxx) [circle one]
(xxx) [circle two]
I had him count all the X’s and write that number as the answer.
4 — Practice
Praise correct behavior and correct incorrect behavior.
I gave him three or four more problems, which he did himself.
By the time I gave him 3×4, he was doing them in his head.
5 — Prompt autonomy
“Now you come up with one.”
He said, “Mmmm, five times six!” I was doubtful, but I said, “All right, you can try that one,” — and figured I’d keep an eye on him.
6 — Get social proof (praise success)
My nephew, with absolutely no intervention on my part, calculated 5×6=30. He won’t learn that in school for another year or two.
“Very good,” I told him. “Go show that to your stepsister.” [A fourteen-year-old.] “I think she’ll be impressed.”
The stepsister was dazzled, and let him know it.
Teaching him to calculate multiplication and division problems by counting (in his head or on paper) took about half an hour.
Even if I stopped here, he’ll be far more prepared for the next big leap in math, more comfortable when the school decides he needs to memorize his times tables.
If my sister (his mom) keeps having him play multiplication, he might have them memorized before the school gets around to it.
Sneaky, aren’t I?
When his Mom got home, she was just floored. Didn’t know what to say. It was beautiful.
Division, for you guys who want to teach kids using this model:
Example: 6 / 2 = 3
The kid draws as many big circles as the second number:
( )
( )
Then counts up to the first number, putting an X in each circle in turn:
“one, two, three…”
(xx)
(x )
“..four, five…”
(xxx)
(xx )
“…six!”
(xxx)
(xxx)
Then, have them count the number of X’s in one circle, and that’s the answer.
Note: This is more complicated. I had to, in the first two problems, move his hand for him and count aloud so he understood what to do.
Then I backed off and corrected from over his shoulder; then I was a useless spotter.
When I told him to come up with his own division problem, I told him to tell it to me first. My job is to make sure it’ll come out even.
We’ll get into remainders another day.