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.
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.
I had a surprisingly similar conversation with my 9 year old yesterday.
My wife and two kids will be running in a 5k fun run in September, and they’ve been doing a little bit of running to train for it. Yesterday they went to a local track to see if they could run one mile together. I timed their last lap around the track at 2:45.
On the way home I asked how long 4 laps would take if each lap was 2 minutes and 45 second. The first response was “8 minutes plus 180 seconds, which is 9 minutes and 80 seconds.” Sort of a curious mistake, actually.
I told him that he done something wrong and to double check his reasoning. His response went something like this:
“Well, 4 times 2 minutes is 8 minutes. 4 times 40 seconds is 160 seconds which is 2 minutes and 40 seconds. That’s 10 minutes and 40 seconds. Now we have to add 4 times 5 seconds, which is 20 seconds, so, oh, the total is 11 minutes.”
I’m struck by the similarity to an experience I had with my son, Zane, when he was five. We were driving to school (a long haul for reasons I need not go into here), and he piped up from the back, “I know how much 100 and 100 is!” I asked him to tell me and he said, “200!” without hesitation.
Dad: Okay, Zane. How much is 250 and 250?
Zane: (after a tiny pause): 300!
D: 300?
Z: (immediately) No, 500.
D: Okay. . . and how did you get that?
Z: Well, I thought that 200 + 200 is 400, and 50 + 50 is 100, and 400 + 100 is 500. (Pause). Do you want to know why I said 300 the first time?
D: Yes, I would.
Z: Well, I thought you were asking what 150 and 150 was, so I did that problem. Then I realized you’d said 250 and 250.
This led me to some thoughts similar to yours, Chris. It also hit me that even at five, he was both using thinking that I wouldn’t have arrived at for a long time afterwards, AND that he was also making mistakes I wouldn’t have made. The latter and former seem somewhat interconnected to me. It takes sophistication and a willingness to think outside the box to do what he was doing. But when you have the confidence to do that, you also don’t always feel the need to take time to be absolutely sure you’ve heard/read the question correctly, because your mind is racing ahead to the fun stuff. I didn’t make a lot of careless calculation mistakes in early elementary because for me, the whole point was to get the problems right (accuracy) and then, maybe by third grade, to do them all right faster than anyone else in the class. Increased confidence that I was really solid in basic arithmetic allowed me to increase my speed accordingly.
My son, however, got derailed at some point. Without wanting to point fingers, because I don’t know for certain whom to single out, he became more concerned with never making a mistake in math (and other areas) as he got older. And he became painfully slow. Always got very high scores in math, but if not for what was diagnosed as ADD/ADHD giving him leeway to take longer (and longer as he progressed through K-12), he might have well failed math for taking so long (to check and recheck, re-read, and generally turn himself into a pretzel to avoid any kind of error).
I think we are still caught in a culture that thinks that math = calculation = 100% accuracy at number crunching, and considers the latter superior by far to the sorts of thinking that your son and mine and many other children are perfectly capable of doing, given some breathing room.
That is so sad to read. Makes me think of how to encourage more exploration and experimentation (failure).
I think unfortunately our system breeds this sort of thing. Everything is either “right” or “wrong”. “Right” is good, smiley face or check mark. “Wrong” is bad, big red x. Add to that the pressure of fitting in with 30 other unique individuals desperately trying to guess what normal looks like and mirror it and it’s a bit of a disaster. I wish I had a solution that would work on as large of a scale as a school requires. This, a long with a zillion other small but significant reasons, is why I learn with my kids through life at home. We read and talk, a lot. Laundry baskets full of books go in and out of our home. The internet is in constant use but we also like to slow it down and dig through encyclopedias the old fashion way. We talk, we question, we play, we giggle at our strange ideas and we learn from our errors which really are just great ways to learn new questions. How do I know they’re learning? They tell me they are all the time. It’s in their games, their questions, their assertions. My kids had to teach me it’s okay, essential, to fail. They had to rekindle my thirst for knowledge, my love of ideas. There is hope for your son. One day he’ll start looking into something he enjoys, something that isn’t covered by the curriculum, something he won’t be tested on. He will spontaneously learn and he will love it and all those old creative juices and out of the box thoughts will come flooding back in.
Great response. Thank you.
This and the Short Waffles Conversation are really resonating with me today. I’ve got a 6th grader who is unusually weak in his times tables. I was working with his small group today when the need to recall ‘six times nine’ threatened to send him over the cliff. Watching his patience disintegrate, I asked him, “What’s six times ten?” He, of course, had that immediately at hand. Then I said, “Subtract six.” He had a mini-lightbulb-moment and eagerly finished his problem. This happened a couple more times, though he needed prompting each time. I can see that he’s pretty discouraged (on the edge of checking out), but the fact that he can be reeled back into the conversation so easily gives me a glimmer of hope that he can be encouraged to stay with it. Makes me wonder how many more kids there are out there who have never been encouraged to access their derived facts.
– Elizabeth (@cheesemonkeysf)