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.

Making eight

I am writing a book. In the process of doing this, I come across homework assignments that parents find frustrating, and that they share on social media. These almost always get me thinking, and they frequently lead to math talks with my children.

This past weekend was one such instance.

Worksheet: "The whole is 8. One part is 8. What is the other part?"

Talking Math with Your Kids is not a place to hash out the details of whether this is a well written question, or whether this was an appropriate homework assignment for this child. We can discuss that on Twitter if you like, or through my About/Contact page.

Talking Math with Your Kids is about taking opportunities to have math conversations with our children. In that spirit, I share the conversation we had in our house.

Out of the blue, I asked Tabitha (7 years old) if I could ask her a math question. It was maybe Saturday afternoon. We had nothing special going on.

Me: Tabitha, can I ask you a math question?

Tabitha (7 years old): Yes.

Me: If I have eight things, and seven of them are in one hand, how many are in the other?

T: That’s not even a math question! That’s too easy!

Me: OK. But will you answer it anyway?

T: One.

Me: OK. What if I had five in one hand?

T: And you still had 8?

Me: Yeah.

She spent a few moments thinking.

T: Three.

I had a couple other questions, which I asked and she answered. The next day, I realized that I didn’t know how she knew that second one.

She was getting ready to brush her teeth on Sunday evening when I asked whether she remembered the previous day’s conversation. She did.

Me: How did you know it was three?

T: I counted.

Me: Like this? Five, then six, seven, eight?

T: Yeah. And that’s three. But actually, I kind of already had it memorized.

Me: Oh yeah? How did you memorize it?

T: Huh?

Me: Did you try to memorize it? When I want to memorize a phone number because someone told it to me and I don’t have a pen handy, I say it over and over to myself. Did you do that with 5 + 3?

T: No! I just have counted it out a lot of times.

Now, I should also mention that I asked Tabitha, If I had 8 things, and 8 of them were in one hand, how many would be in the other? She replied Zero without much hesitation. This If I have this many in one hand, how many are in the other formulation is probably less clumsy than the If this is one part, what is the other part? formulation on the original worksheet. But the intention is the same.

So What Do We Learn?

The kind of problem Tabitha and I were working with is called Part-Part-Whole. For young children, this is different from the standard “takeaway” problem because there is no “taking away”. I didn’t eat, lose, destroy or give away any of my eight things in these problems—I just have some in one hand and some in the other.

Because Part-Part-Whole involves a different way of thinking, it’s a good idea to practice some of these problems. It helps children to build a better understanding of addition and subtraction relationships if they see all the various ways these relationships appear in their worlds.

Tabitha herself pointed out an important principle of Talking Math with Your Kids: Many things that you hope to remember, you can remember by encountering them frequently. Tabitha has never sat down with flash cards to memorize her single-digit addition facts. Yet she is in second grade and is starting to feel confident with them.

She and I talked about familiarity—how maybe learning 5 + 3 is a little like learning the name of someone you see in your neighborhood. You don’t recognize the person as being the same person the first few times you see them. But eventually, if you see them frequently enough, you do recognize them, and you might introduce yourself. Pretty soon, you know their name. And if you just can’t seem to remember it? That’s when it’s time to drill yourself. That’s when you repeat the name over and over and over.

Starting the Conversation

Ask the questions I did. This is an easy conversation to have. If your child isn’t confident with addition and subtraction facts, ask about six in one hand instead of jumping to five in one hand. 

More broadly, look for Part-Part-Whole opportunities to talk about. This is an important interpretation of subtraction, and one that is often neglected. Examples include apples (Our fruit bowl has 8 apples—5 are red, how many are green?), pets (There are 8 pets on our block—5 are cats, the rest are dogs. How many dogs?), et cetera.

An evening ride on the A train

I was in New York City for a conference recently. I arrived at JFK airport on a Monday evening and took public transportation into the city. I observed the following scene.

At about 8:30 p.m., a mother and her (roughly) 6-year old twin girls board the train and sit quite near me.

Mom opens the first girl’s backpack to check whether her homework has been completed.

Child 1: I did it. Oh…on that one, I forgot to write the 6.

[Child 1 proceeds to talk aloud and hold up her fingers]

These are my daughter's fingers. I did not photograph the children on the subway.

These are my daughter’s fingers. I did not photograph the children on the subway.

Child 1: Two and four is…1, 2, 3, 4, 5, 6!

Mom checks a couple of other things and puts the materials back in Child 1’s backpack. She opens Child 2’s backpack.

Mom: Why don’t you have a math book?

Child 2: I do! It’s Go Math!

Mom: Where is it?

Child 2: At school. Teacher says it never leaves school.

Mom: So you get worksheets?

Child 2: Uh huh.

Mom: What’s the point of having a math book, then? Oh well.

She flips through the worksheets, but does not discuss their content with the children. The worksheets appear to her (and to me) to be a disconnected jumble, which the child has completed.

About ten minutes later, the girls are acting up a little bit. Nothing major, but they are getting loud and silly. They are clearly getting on Mom’s last nerve. Mom has expressed to them how tired she is.

Another ten minutes pass, with the girls just barely keeping it together. They get off the train at their stop.

So what do we learn?

A major goal of my writing here and in my book is to help parents notice opportunities to support their children’s mathematical development, and to take advantage of these opportunities.

Traditionally as parents, we look over our children’s homework and keep an eye our for errors. We help our children when they are stuck. If everything is correct and complete, we move on. All of that is good. It is important to help our children when they are stuck. It is important to check their homework. All good.

I want to point out the opportunity this mother had when her daughter was talking about the neglected six.

Child 1 said, “I forgot to put the 6,” and then demonstrated the relevant addition problem on her fingers.

One way to seize this opportunity is to say, “Oh, good! I like that you showed how you know 2 and 4 is 6. What are some other ways you can make 6?”

That conversational technique will bear fruit with kids 95% of the time. When you notice an idea, praise the child for expressing it and ask a follow-up question, the child will nearly always answer that question. If the question suggests that there are multiple answers, the child will usually keep thinking beyond their first idea.

Continuing the conversation would provide the mother in my story with two things:

(1) Increased mathematics thinking on the part of her children, and

(2) less misbehavior later on.

Yes it takes energy to carry on a conversation of any kind with your children when everyone is tired. But it takes less energy to do that than to keep them quiet when they don’t want to be.

I understand, of course, that not everyone knows how to do this. That is what we’re working on here.

Starting the conversation

This is an example of starting a conversation by listening and asking follow up questions.

Some conversations parents initiate. Others children initiate. When children initiate them, the conversation only moves forward if we listen and ask questions.

I will restate an important conversational move in this scenario:

Oh, good! I like that you showed how you know 2 and 4 is 6. What are some other ways you can make 6?

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.

waffle.array

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.