"Kids your age are not supposed to know this. Nobody your age knows this yet.
Are you *sure* you want to know this?"

*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:

2x3=_ [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 3x4, 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
5x6=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.

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