Does the Earth have an end?

Talking Math with Other People’s Kids Month continues…

A while back, Rafranz Davis reported a conversation on her blog. She writes frequently about the adventures she has with her nephew Braeden. I asked, and she gave me permission to remix a conversation she and Braeden had about the ends of shapes—especially the ends of the Earth.

Rafranz and Braeden (8 years old) are spending some quality weekend time together when he asks a question.

Braeden: Does the Earth have an end?

 

Rafranz: Braeden what do you mean by “does the earth have an end”?

B: I’ve been meaning to ask you this question for a long time, at least 2 months. I’ve always wanted to know if the earth stops when you get around it.

Rafranz is a master at the art of mathematical conversation. She asks Braeden a question that gets him talking and thinking.

R: What shape do you think that the earth is?

B: I think that it’s a circle.

R: Really, why a circle?

B: A circle is round.

R: Hmm, interesting. So what shape is that basketball? (The nearby ball may have sparked Braeden’s thoughts)

B: It’s a circle.

R: What about a pizza?

B: It’s a triangle.

This is great! Miscommunication. Rafranz is asking about the whole pizza. Braeden is thinking about a slice of a pizza.

6.slice.pizza

R: I mean a whole pizza. What shape is a whole pizza?

B: It’s a circle

R: Why do you think that a pizza is a circle?

B: It’s round and has a center.

R: Earlier you told me that a basketball is a circle and a pizza is a circle. Are they the same?

Again—great move here. Braeden has identified the basketball and the pizza as being round, and therefore circular. Rafranz asks him to compare these two things and to look for differences. She is using Braeden’s curiosity to pursue some deep and important mathematical questions.

B: No, the pizza is flat. The basketball is round…like Earth. The pizza does start and stop when you get all the way around but the basketball can keep going around and around and around.

R: What do you mean around and around and around?

B: If you had a really long string, you can go around the pizza one time but a basketball, you can keep wrapping the string forever. I know why. The basketball is a sphere. (I had no idea that he knew this word)

R: What about Earth?

B: I think that earth is a sphere too and I don’t think that you can go to every single place on earth. I bet that you can keep going around and around and around.

So what do we learn?

Rafranz asks three simple questions at exactly the right moments in this conversation.

  1. What do you think?
  2. Why?
  3. Are they the same?

It turns out that Rafranz really didn’t know enough about Braeden’s original question to answer it the first time around. Those were sincere questions she asked, and they produced a genuine conversation.

Ultimately, Braeden knew that if you walk around the outside of a circle, your path comes to an end—you end up back where you started, having visited all locations on the circle. But if you do this on a sphere, it seemed to him that your path does not necessarily end up back where you started. It’s a lovely insight about the relationship between two-dimensional objects and three-dimensional ones!

Starting the conversation

If you are new to talking math with your kids, don’t worry about getting the timing right. Just start to make a habit of asking those questions. The first few times, you may not get much. That’s OK. It can be like introducing new foods—children need multiple exposures to new things before they accept them. The other question to add to this collection is How do you know?

Baking cookies

Talking Math with Other People’s Kids Month rolls along…

Jennifer Lawler wrote up the following conversation on her blog.

Jennifer is in the kitchen baking chocolate chip cookies when her son Ian (8 years old) wanders in and observes her methods. She has put three balls of cookie dough in a row, two balls of dough in the next row, and is beginning a new row.

Ian (8 years old): Are you going to put three in the next row?

Mom: Yep.

Ian: And then two in the last row?

Mom: Yep…How many cookies are on the tray?

cookie.doughSimulated cookie dough. Shout out to anyone who can ID the actual substance in this photo.

Ian: Ten.

Mom: How do you know that?

Ian: Three plus three is six, and two plus two is four, and six plus four is ten.

Mom: Hmm….my brain immediately puts the three and two together to make 5 and then adds the 5s together.

Mom: The recipe days it makes 5 dozen cookies. How many is that?

Ian: So that’s 5 12′s?

Mom: Yes.

Ian: 36? No…24 plus… No, wait. 60.

Mom: Ok, I made a double batch, so how many is that?

Ian: 120

Mom: And if there’s 10 on a tray, how many trays of cookies will that be?

Ian: 12

Mom: I have three cookie sheets, so how many times will I have to put each tray in the oven?

Ian: 12 divided by 3 is 4 – four times.

So what do we learn?

What I love about this conversation is that every question is an authentic one that someone baking cookies might consider along the way. I love that Jennifer keeps asking questions until she hits one that forces Ian to think, and I love that she offers Ian a different way to view the cookies on the tray (2 fives instead of 6+4). This last bit sends an important message—that math ideas are something we talk about, not just memorized facts.

Most of the time when people think about the math involved in baking, it’s the fractions. Fractions of a cup and of a teaspoon are fine. But we don’t actually do much math with them. If I need 3\frac{1}{2}, I usually measure 3 cups and then use the \frac{1}{2} cup measure. It’s counting the whole way. This is good, and it’s useful for helping children become accustomed to the relative sizes of fractions, and to the language surrounding them. But there isn’t as much mathematical thinking going on as when Jennifer asks Ian how many cookies are in 5 dozen, or to say how he knows how many cookies are on a tray with a 3-2-3-2 pattern.

Starting the conversation

Baking together is a great opportunity for asking how many? questions of various forms. Ask your child to put things in rows, or to count things that already are. Guess how many chocolate chips are in each cookie, and then in the whole batch. Compare to the expected number of raisins in an oatmeal cookie.

All along the way, listen to your child’s thinking and offer your own ideas. Make it a conversation.

Tabitha and I had a blast a while back arranging crackers and pepperoni in rows. Just like Jennifer and Ian, we predicted how things would come out and enjoyed talking and cooking together.

Legos

The dirty little secret about Legos is how very many pieces there are to be cleaned up after building. And how very ugly the clean-up battle can become.

I try to keep calm. I try to turn the clean-up battle into math talk from time to time.

Here is how it played out a while back, when Griffin was 7 years old (nearly 8).

I send Griffin upstairs to clean up a big Lego mess. A while later, he returns.

Me: Did you pick those Legos up?

Griffin (7, nearly 8 years old): Yes. Well…not all of them.

Me: What fraction did you pick up? More or less than half?

G: Three fourths. No. One fourth. No….I’d say one third.

One third is more than one fourth.

Me: How do you know that?

Griffin draws a picture like the one below.

legos.1

G: You see that it’s bigger.

Me: OK. Do want to hear how I know it’s bigger?

G: Yeah. How?

Me: If we had one big cookie and I had to share it with three people or share it with four people, I’d get more if I only shared with three.

G: Yeah.

Me: Which is bigger: three fourths, or two thirds?

He draws a picture like the one below (the original is lost to history).

legos.2

G: Three fourths is.

Me: How do you know?

G: [referring to his picture]: It would be here [indicating the heavy line pointing to the lower left] if it were two thirds, and there’s more shaded in, so three fourths is bigger.

So what do we learn?

It seems to be easier to engage children in comparisons than in precise computations. I have a lot more success with Which is more? questions than I do with Exactly how much/how many? questions. This is especially true with fractions, where precise computations are often quite tedious.

In any case, the simple follow up question, How do you know that? is a powerful one. That is where the conversation happens nearly every time.

At this point Griffin is accustomed to being asked this question, which means he is pretty good at doing so. It seems natural to talk about how he knows something. If you are just starting these conversations with your own child, expect to have to ask on 10 or 15 separate occasions before it starts to become natural for them. This is just like introducing a new food. Repeated exposure is key.

Starting the conversation

This is classic lemonade-from-lemons parenting. I knew Griffin would not have completed his assigned task. We could fight about it, or we could talk about it. I chose to talk about it, and save the power struggle for a bit later.

You can do the same. What fraction dressed are you? What fraction of your teeth did you brush? How much of your body was touched by soap? Et cetera. Have a little fun with the math first. Then make them finish the job.

Guess the temperature

This post is from last year on my math teaching blog. Presently we (along with much of the American Midwest) are in the middle of a serious cold snap. So I have edited and remixed it for the Talking Math with Your Kids audience.

guess.the.temperature

This morning’s situation. Colder air is on the way.

Enjoy.

And stay warm.

Griffin (8 years old in this story) and I play a little game called Guess the Temperature. It goes about how you would expect. We step outside on the way to his bus. I ask him to guess the temperature. If I don’t already know, I get to guess after he does. If I do already know, I don’t cheat; we just remark on how close his guess was.

In Minnesota, in winter, this means we get to study both positive and negative numbers.

Me: Griff, guess the temperature.

Griffin (eight years old): Two below zero.

Me: It’s three degrees above.

G: So I was off.

Me: Not by much, though. How much were you off by?

G: [muttering to himself, then loudly] Five degrees!

Me: How did you know that?

G: It’s two degrees up to zero, then three more.

Me: So what if it had been 10 degrees out, and you guessed 3?

G: [quickly] I’d be seven off.

Me: Right. How do you know that?

G: Ten minus three is seven.

Me: Nice. Subtraction. Do you know that you can always express the difference between your guess and the actual temperature with subtraction?

So in that last example, you subtracted your guess from the actual temperature. You could do that with your real guess today.

So three minus negative 2 is five.

G: [silent]

By this time we were nearing the bus stop. Griffin’s silence seemed a clear sign that he was ready to move on.

So what do we learn?

Two things are important in this conversation: (1) Griffin’s solution method, and (2) the connection to subtraction.

Griffin’s solution method. Griffin’s strategy is a common one for children to invent. He uses zero as the boundary between positive and negative numbers. To compare how much bigger a positive number is than a negative number, you have to cross that boundary. You have to go past zero, so it just makes sense to him to divide the distance into two pieces—the part to get to zero, and the rest.

We live near a major road—Arcade Street. We often use it as a boundary for our neighborhood. So the local recreation center is three blocks on the other side of Arcade; plus the one block to get to Arcade. That’s four blocks total. Talking in this way about everyday navigation supports thinking about temperatures, which in turn support thinking about integers.

Connection to subtraction. From years of teaching middle school (and—to be honest—college), I know that subtracting integers is tough going. The rules for solving 3-^{-}2 don’t seem to connect to students’ experiences with numbers.

Notice how quickly Griffin connects the 10° and 3° situation to subtraction, while not seeing that subtraction applies to the 3° and -2° situation we started with. Perhaps my mentioning that these are the same will lay the groundwork for him noticing it in the future.

In the meantime, we learn that learning subtraction is a lot like learning division. In a recent post, I showed how Griffin thought differently about division depending on the numbers involved and on the context for thinking about it. Now we see that this is true about subtraction, too. 10-3? No problem. 3-(-2)? Problem.

Starting the conversation

Children—all children—develop mathematical models of their worlds before they study them in school. Parents have opportunities to support this through conversation.

Talk about landmarks, way stations on your journeys. Mileposts, subway stations, bus stops, blocks…these are all opportunities to help children build the mental models necessary to think about zero as an important landmark.

Talk about distances. How many blocks is an example from our family life. How many subway stops came up for girl I observed in New York City last fall. How many pages did we read in our book is another example where subtracting endpoints is helpful.

Big Cheez-Its

There are now BIG Cheez-Its (U.S. only, it appears). The package claims that they are “Twice the size!” of regular Cheez-Its.

image

Naturally, I bought some a few months back.

I asked Tabitha (6 years old) and Griffin (8 years old at the time) what they thought. I started with Tabitha when Griffin wasn’t around so I could get her pure thoughts.

She put one cracker on top of the other and proclaimed, “No”.

image

I wanted to know why she thought that. I thought she might be mistaking side length for area. That is, maybe she was paying attention to the lengths of the sides of the two crackers rather than to the amount of cracker. So I asked about it.

She pointed to the uncovered part of the BIG Cheez-It and argued that this wasn’t enough to make another full regular Cheez-It. So she was paying attention to the amount of cracker.

A few minutes later, it was Griffin’s turn. He ran like a chipmunk with his two crackers into the dining room.

I imagined that this chipmunk would be nibbling the crackers next door and that our conversation would be at an end.

I was wrong.

He was in search of paper and a pen. He carefully traced each cracker, cut out the uncovered part of the BIG one and attempted to partition and reassemble this remainder on top of a tracing of the regular cracker, which it did not completely cover.

Sadly the cut outs are lost forever.

His conclusion: BIG Cheez-Its are almost but not quitetwice the size of the regular Cheez-Its.

So what do we learn? 

Notice the differences between the children’s strategies. Tabitha, the six-year old, worked with the crackers. She put one cracker on top of the other and tried to picture whether the leftover space made up a whole cracker. She was very concrete in her thinking.

Griffin, the eight-year old, worked with representations of the crackers. He traced and cut out squares of paper which he could manipulate with more precision than the actual crackers.

The two children reached similar conclusions.

Neither child used tools to calculate areas.

Knowing whether one cracker is twice as big as the other does not require measuring how big either cracker is.

All of this is very typical for young children. Younger children tend to work with the actual things they are comparing. They are what we call concrete thinkers. Older children begin to work with representations of the things (e.g. Griffin’s cut outs). They are more likely to be abstract thinkers.

Starting the conversation

Investigate advertising claims. Have a healthy, skeptical attitude towards these claims, and encourage your children to wonder about them, too.

Be forewarned, though! You may create critical thinkers who question your authority, too.

And you may end up spending a LOT of time trying to figure out whether Double Stuf Oreos are really doubly stuffed.

The meanings of division

I was talking with Griffin one day when he was in third grade.

Me: Do you know what 12\div 2 is?

Griffin (8 years old): 6

Me: How do you know that’s right?

G: 2 times 6 is 12.

Me: What about 26 \div 2?

G: 13

Me: How do you know that?

G: There were 26 kids in Ms. Starr’s class [in first grade], so it was her magic number. We had 13 pairs of kids.

Me: What about 34 \div 2?

G: Well, 15 plus 15 is 30…so…19

My notes on the conversation at this point only have (back and forth), which indicates that there was probably some follow-up discussion in which we located and fixed his error. The details are lost to history.

Our conversation continued.

Me: So 12 \div 2 is 6 because 2 \times 6 is 12. What is 12 \div 1?

G: [long pause; much longer than for any of the first three tasks] 12.

Me: How do you know this?

G: Because if you gave 1 person 12 things, they would have all 12.

Me: What is 12 \div \frac{1}{2}?

G: [pause, but not as long as for 12÷1] Two.

Me: How do you know that?

G: Half of 12 is 6, and 12 \div 6 is 2, so it’s 2.

Me: OK. You know what a half dollar is, right?

G: Yeah. 50 cents.

Me: How many half dollars are in a dollar?

G: Two.

Me: How many half dollars are in 12 dollars?

G: [long thoughtful pause] Twenty-four.

Me: How do you know that?

G: I can’t say.

Me: One more. How many quarters are in 12 dollars?

G: Oh no! [pause] Forty-eight. Because a quarter is half of a half and so there are twice as many of them as half dollars. 2 times 24=48.

So what do we learn?

Mathematical ideas have multiple interpretations which people encounter as they live their lives. As we learn more mathematics, we become better at connecting these different ways of thinking about ideas.

In this conversation, Griffin relies on three ways of thinking about division:

  1. A division fact is a different way of saying a multiplication fact. (12 \div\ 2 is 6 because 6 \times 2 is 12).
  2. Division tells how many groups of a particular size we can make (Ms. Starr’s class has 13 pairs of students).
  3. Division tells us how many will be in each group if we make groups that are the same size. (When he was working on 34 \div 2, Griffin put 15 in each group to start off with.)

We were just talking for fun, not homework or the state test. So I wasn’t worried about his connecting those ways of thinking. I was just curious how he would apply them to some more challenging tasks, such as dividing by 1 or by a fraction.

I was surprised by how difficult 12 \div \frac{1}{2} was for Griffin. Not because it is an easy problem, but because he could have applied his how many of this are in that? idea, or his multiplication facts idea. But he did neither and reinterpreted the task as twelve divided by half-of-twelve.

I was also surprised at the length of the pause he took for 12 \div 1. It makes sense in retrospect. After all, are you really making groups if it’s just one group? I imagine he had to think that through, rather than the number relationships involved.

Starting the conversation

When the opportunity presents itself—when you and your child are not under homework stress, not rushing to get out the door or find the dog’s leash; when you happen to be talking about number anyway—ask follow up questions. Even a simple set of division problems got a lot of good thinking out of Griffin. Problems involving 1, 0 and \frac{1}{2} are especially challenging.

Vary the size of the numbers.

Don’t worry about whether the answers are right or wrong.

Keep asking How do you know? and listening to your child’s answer.

Offer a few ideas of your own.

Quit before anybody gets frustrated or bored.

Short notes

I passed a lovely morning with some math teacher friends at the Museum of Math in New York City today. Sadly, I did not have Griffin or Tabitha along. We would have had a ball.

I have three observations…

One. There is lots for kids to do there. It is truly a bonanza of mathematical conversation starters. I left wanting to be hired on as “Docent for Talking Math with Your Kids”. This is because the math does not smack you in the face. Instead, the math in the exhibits tends to surface in the process of playing, experimenting and paying very careful attention to what is going on.

In short, you want to talk about these exhibits and you want to linger.

Two. Many of the exhibits are designed in ways that allow kids to play and to do math beyond the intended activity. The exhibit below, for example, kept two little boys (each 5 or 6 years old) busy for 20 minutes although the original activity was way beyond them.

The first five minutes, they were sharing each of the shapes equally, making sure each of the two boys had the same number of squares, and the same number of triangles, et cetera. The next 10 minutes, they spent arranging the shapes on the screen, marveling at the things this provoked on the screen beneath.

It was lovely.

It could have been a bummer, though. If someone had insisted on doing the intended activity, these kids could not have done it. If the shapes had been electronic instead of concrete, the kids would not have been able to play with them in such creative ways.

Well done, Museum of Math!

Three. On the subway train back to my hotel, I noticed a little girl—probably 4 years old—holding up four fingers as she sat between her mother and brother.

I moved to be in listening range.

It turns out, she had been told they would be on the train for five stops. She had been putting up one additional finger as the train left each stop.

Sister (4 years old): [She is holding up four fingers as the train enters the station] This is our stop!

Brother (8 years old): No. The sign says there’s one more stop.

Sister: But you said five stops.

Mom: One more, sweetie.

The train stops. People get off and on. The doors close. The train starts up again.

The girl keeps exactly four fingers up the whole time.