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.

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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.

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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”.

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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.

M&Ms

Dessert is a good time to get the children’s attention for a little math talk.

A few weeks back, a smallish serving of M&Ms was about to be given to each child, from a large one-pound bag.

In keeping with my assertion that a day should never pass without asking my kids at least one how many? question, I asked Griffin to choose the size of the serving (but unbeknownst to him that this was the purpose.)

Me: Give me a number between 10 and 20.

Griffin (eight years old): What’s the point?

Me: I won’t tell you until you choose.

G: I won’t until I know why.

This is my own doing. I have long told both children that people need to have reasons for asking you to do things, and that satisfying these reasons is more important than following directions blindly. This is an important element of problem-solving and critical thinking. It does have consequences; I understand this.

Me: Tabitha, pick a number between 10 and 20.

Tabitha (five years old): Twelve.

Me: OK. That’s how many M&Ms you each get for dessert.

G: Oh, then I pick 20.

Me: No. The first number I heard. That’s the one I’m using.

G: You should use the biggest.

Me: Nope. The first.

T: Next time, I should choose….thirteen.

This is beautiful, is it not?

I love the realization that things had not worked out for her maximal benefit. I love that she knows some thinking needs to be applied to the situation.

And I love dearly that the result of this thinking is an increase of a single M&M. Griffin comes to her rescue.

G: No, Tabitha! It’s between 10 and 20!

T: Oh. I should choose…nineteen.

So what do we learn?

This was totally devious on my part, and I do not recommend that you behave this way with your children. We do learn, though, that strategic thinking with numbers is something to be learned. The strategy of thinking through the biggest possible number within the given constraints is not obvious to young children. Looking for a bigger number is a prerequisite to thinking hard about the biggest possible number. 

We also learn, of course, that I am a horrible person.

Starting the conversation

Again, I do not encourage you to manipulate your children in this way. Although in my own defense, neither 12 nor 20 M&Ms is such a bad deal for 5- and 8-year olds near bedtime.

The pick-a-number game is fun for lots of things, though. Taking turns (whoever gets closest to the number I wrote down gets the first turn) is a classic example, but you can think up lots of your own. After the picking, talk about the selection. What would have been a better choice, knowing what you know now? What would have been a worse choice? Why did you pick the number you did? Et cetera. Listen to your children’s strategies and share your own.

Summer project (2 of 3): Measuring the Hill

This post follows up on a conversation I had with Griffin a while back about the height of the Giant Slide at the Minnesota State Fair.

Our house in St Paul sits on top of an odd hill; higher than others around it. Historical reasons for this are murky but it makes the place easy for guests to find. One of my least favorite tasks in all of my domestic life is mowing the hill.

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For a while now, the precise height of this hill has been the subject of family speculation. One recent lazy summer afternoon, Griffin (8 years old), Tabitha (6 years old) and I found ourselves hanging out on the hill with not much to do.

Me: How tall do you two think the hill is?

Tabitha (6 years old): Five feet.

Griffin (8 years old): I don’t know.

Me: How about this: Which do you think is taller; me or the hill?

T: The hill.

Me: Wait. I’m six feet tall. How can the hill be 5 feet tall AND taller than me?

G: You’re six feet, one inch.

Me: Right. Even so…

T: Oh. I don’t know how tall the hill is, but I think it’s taller than you.

Me: Why?

T: Lie down.

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T: See?

Me: Yeah, but just because it’s longer than me doesn’t mean it’s taller than me.

Tabitha seems puzzled by this distinction. Griffin is standing on the sidewalk at my feet.

Me: Look at Griffy’s eyes. Is he looking up or down at my eyes right now?

T: I can’t really tell.

I stand up, right next to Griffy, who cranes his neck back to look me in the eye.

Me: Now?

T: Ha!

I lie back down on the hill.

Me: So how come there’s a difference?

T: You’re lying down now, so that’s not really how tall you are.

Me: So how can we decide whether I am taller, or the hill is?

Nothing much occurs for the next minute or so. We are distracted by butterflies, the edible nature of clover flowers and other wonders of Minnesota’s too-short summers.

Me: Hey! Let’s try this. Tabitha, you go to the top of the hill.

She does, and she stands there, looking down on me with a self-satisfied smile on her face.

Me: OK. So you plus the hill are taller than I am. What about just the hill?

T: I don’t know.

Me: Lie down.

She does, although it takes a few tries to achieve the desired position by which she can look at me from roughly the level of the top of the hill.

Me: Are you looking up or down at me?

T: I can’t tell.

Griffin takes his turn at the top of the hill. He, too, is unsure.

Me: So how can we be sure?

T: You know, Daddy, I don’t really need to know this.

Me: You’re right. You don’t. Nor do I, really. But I have always been curious how tall the hill is. Aren’t you?

G: We could measure a step, then use the number of steps to figure out how tall it is.

I obtain a tape measure.

We determine that each step is 7 inches tall. We notice that the bottom step is shorter than the rest and measure it at 5 inches. Griffin laboriously counts the steps, finding that there are eight of them, plus the smaller one.

G: So what is that altogether?

Me: What? You can do this.

G: Do you know whether you are taller than the hill?

Me: Actually, yes I do, even though I don’t know exactly how tall the hill is.

G: If I figure it out, will tell me whether I’m right?

Me: Yes.

G: [Far too quickly for me to be convinced he has run any computations at all] OK. The hill is taller.

Me: How do you know?

G: Hey! You said you would tell me!

Me: That’s part of doing the math!

G: OK.

A long, thoughtful pause ensues.

G: Eight eights is 64, plus 5 is 69. So you are taller.

Me: But you need eight sevens, which is 56.

G: Oh. Right. Plus 5.

Me: Yes…?

G: Tell me.

Child, please.

Me: Seriously? You can do 56 plus 5.

G: 61.

Me: Yes, and I’m 73 inches tall.

Tabitha, despite her protestations about not needing to know, has been paying attention all along.

T: You’re taller than the hill?

Me: Yes. See? I told you it was interesting.

G: You knew you were taller?

Me: Yes. But I didn’t realize it was by a foot. I thought it would be only by a few inches.

G: How did you know?

Me: Because I look down—only slightly—but I look down at the top of the hill.

So What Do We Learn?

Tabitha was using a wrong idea that often shows up in children much, much older. She was thinking that wherever my head was in space corresponded to my height. Older children will display this when they say that the length of the diagonal of a 1-inch by 1-inch square is 1 inch (it is not; the diagonal is quite a bit longer than the sides of the square; about 1.4 inches.)

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Because my head was not at the top of the hill while I was lying on it, she took this to mean that the hill was taller than me. No amount of reasoning with her—and probably no amount of picture drawing—would help. She needed to see it. That Griffy could nearly look me in the eye when I was lying on the hill, but had to look WAY WAY up when I was standing was convincing to her.

This was a way of making an abstract comparison concrete and meaningful for her.

Griffin, by contrast, needed to be pushed to think about strategies for measuring. And this required time on task. I did my best to keep stringing each of them along in the conversation. I had the luxury of a lazy summer afternoon. I promise you that I don’t turn every empty space in our lives into a math conversation. But I make sure I do turn some of them that way.

Thinking about stuff in this context is much different from doing math homework. Homework has time pressures; homework requires neat handwriting and complete sentence explanations lots of times. Lazy summer afternoons have no such requirements. They offer possibilities not constraints.

Starting the Conversation

Talk about measuring stuff. Wonder about it. Don’t do it for your kids. Talk about how it could get done. Compare things (bigger, smaller, taller, shorter) before measuring. Think of ways to make your children feel those comparisons.

And make sure you measure something with your kids where you do not know the result. Help them see that measurement answers interesting questions that are otherwise unknowable.

Summer project (1 of 3)

The Minnesota State Fair is a fabulous event (Twelve days of fun ending Labor Day!). Rachel and I love the Fair, and we have passed this love along to our children.

Griffin must have been thinking about the wonders of the State Fair as summer slowly (oh, so slowly!) unfolded on our fair state. He asked a question at breakfast one recent morning.

Griffin (eight years old): How tall is the Giant Slide?

Me: Good question. I would guess…40 feet. What’s your guess?

G: 45 feet.

OK. That’s a mistake. We should have written our guesses down privately to avoid influencing each other. Oh well.

Me: Let’s look it up.

Google returns nothing useful. It does return this awesome video, though, which we watch together.

Me: I found lots of information mentioning the Giant Slide, but nothing on its height.

G: Measure it yourself, then!

Me: Good idea. How should we do that?

G: We’re gonna need a lot of tape measures put together.

This has been a summer project for us. In the next post, I report on an intermediate activity we did. Our results will follow that.

The unit is the thing that you count

Griffin (eight years old) and Tabitha (five years old) were discussing the day’s activities. The feature activity had been making brownies with Mommy. This occurred while Griffin was out of the house.

Griffin: How many brownies did you make?

Tabitha: One big one! Mommy cut it up.

So What Do We Learn?

What makes this more than just a funny story is that Griffin and Tabitha are clearly counting different things. They are talking about different units.

When we make cookies, everyone agrees on the unit; we know what one cookie is.

But brownies are different. Tabitha seems to think that a brownie is the thing that comes out of the oven. Griffin seems to think that a brownie is what you eat in one serving.

One brownie according to Tabitha.

One brownie, according to Griffin

I have emphasized elsewhere the importance of the unit; that one is a more flexible concept than we might think.

Fun follow-up question: Does the thing in this video count as one brownie?

Starting the Conversation

Anytime there are things in groups—or things being cut—is a good time to talk about units.

Grocery stores usually have express lanes where you have to have Ten items or fewerAsk your child whether someone with a dozen eggs could use that lane. What about someone with 12 apples in a bag? What if the apples are loose?

When your child asks for two slices of pizza, take one slice, cut it down the middle, smile wryly and ask whether that’s OK.

In all of these cases, the central question is What counts as one? Play with that question.

Also, watch that video together. It’s a ton of fun.