A short waffles conversation

Those Eggo mini-waffles are paying off.

We had this conversation the other day…

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

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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.

How many fives in an hour?

Our local public library has a summer reading incentive program. Children keep track of the amount of time they spend reading, and when they reach 20 hours they get a prize. Some of the prizes are good, including a ticket to the State Fair.

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To keep track of their time, children get a chart. The chart has 20 individual hours, each represented by an icon. Half of these are circular, suggesting clocks, and half are rectangular, suggesting books. Each icon is broken down into five minute intervals. We were driving home one June Sunday afternoon after picking up Griffin and Tabitha’s summer reading charts.

Me: Griff, each hour on your chart is broken up into 5-minute chunks, right?

Griffin (seven, nearly eight at the time): Yup.

Me: So how many of those chunks are there in an hour?

G: (long pause) Sixteen.

Me: Why sixteen?

G: Well, I thought of 5 minutes like a nickel, and there’s 20 nickels in a dollar.

Me: Wow.

G: So I minused four, because it’s four less.

Me: Right. 60 cents is 4 tens less than 100 cents, though. So I think we need to…

G: (interrupting) Oh! RIght! So…it’s twelve. Twelve fives in an hour.

Me: That’s some really good thinking there, buddy. I wouldn’t have thought to do it that way.

So what do we learn?

If you are new to thinking about people learning math, it may be surprising that asking children to explain their thinking aloud often leads them to correct their mistakes.

Math is very often portrayed as a subject where things are either right or wrong with no in-between. This is not a helpful image of the subject. Indeed, there are many shades between these two extremes. Sixteen was a wrong answer; there are not 16 fives in 60. But underneath that wrong answer is some pretty sophisticated thinking.

When we figure out some new answer based on one we already know, this is called using derived facts. It’s a very useful mental math strategy and it should be encouraged at every opportunity.

You can only encourage it if you know it is being used. And that’s another reason we need to ask about process. We want to know how kids are thinking so that we can help them make that thinking better.

Starting the conversation

While mental math strategies are becoming more explicit in schools, many parents today did not learn many such strategies when they were in school. The emphasis for many parents may have been on (1) memorization of facts, and (2) paper-and-pencil computation. Therefore you may not know very much about derived facts, or more likely, you don’t notice that you use them.

If you have ever thought “58+9 is 67 because 58+10 is 68, and 9 is one less,” you have used derived facts.

Whenever a computation of some kids comes up in daily life, ask your kids to talk through their thought process. Model your own thinking for your kids.

In short, make everyone’s thinking part of the number conversation.

You and they will get better at it as you keep at it.