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.
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.
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.