Those Eggo mini-waffles are paying off.

We had this conversation the other day…

Me:There you are, Tabitha. Two sets of waffles.

Tabitha(six years old): That’s 7. No…8!

Me:[washing dishes with my back turned to her] Right. Two sets of four is eight.

T:That’s not how I know.

Me:You counted?

T:No.

Me:Oh. How did you know, then?

T:Three plus three is six. And there’s 2 more.

Me:[Big smile and thumbs up for encouragement]

# So What Do We Learn?

I recently pushed Tabitha past the limits of her patience by asking about lobsters and half-lobsters. But in doing so, I was continuing to lay the groundwork—how she thinks about things is interesting to me. I want to know, I value and reward her thinking. So she talks about it.

When you consistently talk math with your kids, you will make progress. It may seem slow at times, but you’ll make progress.

Mathematically, there is something really wonderful going on here. She is trying to figure out 4+4, but it’s not a fact she has handy. So she thinks of 4 as 3+1.

Now it’s 3+1+3+1, which she rearranges as 3+3+1+1, which is the same as 6+2.

She uses a fact she knows (3+3) to find one she does not (4+4). This is an example of using *derived facts*, which Griffin did also in a recent conversation about the number of fives in an hour.

# Starting the conversation

Listen for the times that children announce how many things there are. Ask them how they know.

Another example: Griffin had his ninth birthday party recently at a local swimming pool. The cake was provided; the high schooler who brought over the cake asked me his age and proceeded to count candles from the pack. It was hot; the candles must have slightly melted into the container because she was struggling and took a good minute or two to dislodge the candles, leaving them on the table before disappearing.

She had left eight candles behind.

For a nine-year old’s cake.

Needless to say, this was a topic of great conversation among the children present. Somehow Griffin didn’t notice. But his friend from up the block, W, did. She asked me, “Hey wait! Why are there only 8 candles?” *I don’t know, *I replied, *but how did you know there were 8? *She gave me a funny look. *Did you count them one by one?* “No,” she said, “by twos…2, 4…”

It is that easy.

You just have to put up with a few strange looks from children sometimes.

For birthday cakes, we’ve always done binary numbers: the candles are put on the cake in a line, and we only light the ones that would be ones in the binary representation. We only have 6 candle holders, so I don’t know what we’ll do when I turn 64. My son was counting in binary on his fingers by kindergarten (age 5 or 6).

I wish that many of my high school students were as resourceful as Tabitha was here. I’m reminded (most recently by Michael Pershan in his Rational Expressions blog) of Polya’s directive to look for a simpler problem. I think that I model this behavior but I need to change my practice as a dad and as a teacher to make this habit more explicit.