Time Zones

Griffin is 13 years old and seems to be coming to the end of that early adolescent phase of rejecting everything those around him hold dear. Engaging him in math talk has taken more finesse in this phase of life.

Mostly it has involved giving him responsibility for things that involve making calculations. When he was little, we could talk collaboratively about how many tangerines are in a 3 pound bag and discuss whether this would be enough to last the family a week. Now I tend to put him in charge of getting enough tangerines to last us a week. He still has to do the same thinking, but he’s in charge.

This is not enough tangerines for a week at our house. (By the way, which is more?)

From time to time, though, we still put a mathematical idea up for discussion, and as he ages through adolescence, these conversations happen a bit more often. Yet he is still wary. Nevertheless, I persist.

We have been watching the Olympics, and we have wondered about which events are happening as we watch them, and which ones happened earlier (yet somehow happened “tomorrow”!)

Griffin was thinking about time zones, and about their implications for traveling as we wrapped up an evening this week, and made preparations for the next day.

Griffin (13 years old): So they’re 14 hours ahead of us?

Me: Yes.

G: You’d get a lot of jet lag, huh?

Me: Yeah. Maybe not as much as it looks like, though. Maybe it’s just 10 hours’ worth, going the other way.

There is a bit of a puzzled silence.

G: Wait. Really?

Me: Yeah. Well, plus a day.

G: Wait. Is this one of your mathy talks?

Me: Absolutely not.

If you’re reading this, Griff, I’m sorry (sort of). I am totally busted.

Me: Yeah. 14 hours ahead is the same as 10 hours behind, right? Just going the other way.

G: But the day would be wrong.

Me: Yeah. You have to add a day, but you don’t get jet lag because the day changes, you get jet lag because the time of day does.

G: Maybe.

He returns to packing his lunch. I go back to whatever I was doing. Putting turtles in boxes, probably.

A couple minutes later…

G: So the east coast is 23 hours behind us?

So What Do We Learn?

Keep trying. Opportunities to talk about numbers, shapes, and patterns present themselves. Seize them and do not stop. Ask questions, think out loud. Don’t worry about whether any particular conversation goes anywhere. Just keep at it.

Nights of camping

The following conversation took place in the run-up to our annual summer camping trip recently.

Rachel has no interest in camping, so this ritual is all mine. I started the little ones young with a one-night trip within an hour from home so that we could come home if it’s a total disaster. As they have aged and we have developed our routines, we have gone further afield, exploring wide-ranging Minnesota state parks for two-night stays. We added a weekend fall trip, too.

Last summer, the kids began to ask why “we only go for two nights”.

Ladies and gentlemen, when the kids ask that question, you know you’re doing it right.

So this summer we are expanding to three nights. Tabitha was thinking about that change the other day.

I am straightening some things on the front porch, sweeping and tidying. Not thinking about anything in particular.

Tabitha (7 years old): If we’re going for three nights, is that 2 days and 2 half-days?

Me: Yes.

A few seconds pass.

I realize that I have an opportunity here.

Me: How did you think about that?

T: Every night is a day, except the last one, when we go home.

Me: What if we went for a whole week’s worth of nights? What if we went camping for 7 nights?

T: Easy. Six days.

Me: And?

T: Two half-days.

Me: OK. Ready for a hard one?

T: Yeah!

Me: There are 365 days in a year. So what if we went camping for 365 nights?

T: [slowly] Three…hundred…sixty…four!

Me: Nice!

T: I can even do 400.

Me: You mean 400 nights of camping? You know how many days that would be?

T: Yeah.

Me: All right. Tell me.

She does.

Later, she is in the shower. I am not-so-closely supervising nearby. I get an idea.

Me: Tabitha, what if we wanted seven days of camping?

T: How many nights?

Me: Right.

T: Eight. Am I right?

Me: I can’t trick you at all, can I?

T: Ask me another!

Again, a sign that things are going well. Contrast with her claim a couple years back, “Sometimes I don’t want to tell you about numbers because it’s just going to turn into a big Daddy math talk!”

I have to think hard to dig up something that will be more challenging for her.

Me: You want a hard one? A really hard one?

T: Yes!

Me: Last year, we went camping twice. Altogether, we camped 4 nights. How many days did we have?

T: Three…five…

It turns out that Griffin is lingering in hallway outside the bathroom. He chimes in.

Griffin (9 years old): Four.

Me: Two days, and four half-days.

G: Right. That’s four.

Me: But she’s thinking about it as four half-days, since they aren’t attached to each other. I can see an argument either way.

This summer’s trip was to Lake of the Woods in the far northern reaches of Minnesota.

Griffin posing with an oversized walleye statue in Baudette, MN

So what do we learn?

It may surprise some readers that I have filed this conversation under Algebra.

Like many of the other algebra posts, we are not using x or y, or making graphs or solving for variables. Instead we are thinking about a relationship, and about what that relationship looks like for a wide variety of numbers.

The relationship we are working with here is a simple one: one less. Tabitha had noticed that the number of full days we camp is one less than the number of nights we camp. She had even generalized the idea—notice that she didn’t count the days individually. She said, “Every night is a day, except the last one.” This answer doesn’t depend on any particular number of days; it works for all numbers of days.

What I did in this conversation was help her to apply this idea. By asking her about a wide range of numbers of days, she got to feel the power of her generalization. That is algebra.

The other important part here was continuing the conversation while she showered. Thinking in reverse is an important mathematical skill. We had started with how many days do we get with a certain number of nights? I moved us to how many nights do we need for a certain number of days? The fancy math word for the relationship between these two questions is inverse.

Starting the conversation

Camping trips, vacations, trips to grandma’s house…these are all opportunities to have the conversation we had. If your child doesn’t ask about it, you can ask your child. We are going to grandma’s house for three nights—how many days will you have to play with your cousins while we’re there?

More generally, there are two Talking Math with Your Kids moves I want to emphasize.

  1. It took me a moment to notice that Tabitha had offered me an opening for conversation. I was thinking about something else at the time. When I noticed it, I put those other thoughts aside to talk, ask and listen. That part of the conversation took probably 2 minutes. We can all spare 2 minutes to get our kids’ minds working. We just need to notice the opportunities.
  2. I followed up later on. Following up is good for two reasons: It lets you and your child examine an idea more deeply, and it helps cement memory of the conversation. We remember something we revisit multiple times better than something we only think about once.

Doll years

Out of the blue on our recent camping trip, Tabitha had an announcement for me.

Tabitha (6 years old): I am 12 in doll years and Griffy is 16 in doll years.

Her brother Griffin is 9.

T: So how old are you in doll years, Daddy?

Me: Well, how do doll years work?

Photo Oct 12, 1 34 29 PM

T: Well, I’m 12 and Griffy’s 16.

Me: Is it twice as old? Then I would be two times as old, so nearly 86.

My birthday is coming up next week. This has been a point of discussion around the house recently.

T: No! It’s 6 times!

Me: You’d be 36 then.

T: No. I am 12 in doll years.

Me: Oh! Six years older not 6 times as old!

T: Yeah.

Me: Then Griffy is 15, not 16. And I would be almost 49.

So What Do We Learn?

Children build lovely and complicated imaginary worlds. For a long time, Griffin and Tabitha would play “creatures” together. Whole societies of stuffed animals, dolls and plastic figurines rose and fell. These societies had celebrations and tragedies. There was Creature Christmas that could take place at any time of year. Also a Creature State Fair. Et cetera.

Combine this parallel creature/doll universe with learning about the passage of time and pretty soon doll years are going to pop up.

Griffin and I talked about tortoise years and dog years a while back. At the time, Griffin was 8. He was comparing life spans of tortoises to those of humans, as we do with dogs to generate the 7 dog years per year comparison that is commonly known.

Tabitha is firmly grounded in comparing by counting and addition, as is appropriate for a 6 year old. Somewhere between third and sixth grade, children transition from always comparing by addition and subtraction to being able to compare by multiplying and dividing. This difference is what Tabitha and I are discussing in this conversation. She says Six times but means Six more.

Starting the Conversation

Listen for the comparisons your children make. Here, Tabitha compared ages. But heights, dollar amounts, number of Tootsie Rolls in a candy dish, et cetera; all of these are possible comparisons that children will naturally make. Ask a follow-up question. How do you know? is a good place to start. What if? is a lovely follow up. For example, What if there were a newborn baby in our family; how old would it be in doll years?

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.

bookawocky_215

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.