Units of measurement

This post is edited and remixed from a post on my other blog last summer.

Loyal reader Jim Doherty wrote in to report the following conversation with his 4-year old daughter Mo.

They are on a long drive to a hotel.

Mo (4 years old): How far are we?

Jim: We are 20 minutes away.

Later, having arrived safely, the family heads to the pool. Mo is practicing the fine art of jumping from the edge of the pool into her father’s arms. An important part of this art is to increase the risk by jumping greater and greater distances.

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Tabitha reconstructs a jump of considerable size for illustrative purposes.

Mo: (four years old) Back up, Daddy!

Jim: This far?

Mo: More!

Jim: Here?

Mo: More! You need to be five minutes away!

Jim: Do you mean five feet away?

Mo: No! Five minutes!

At this point, Papa Doherty is flustered. Is Mo messing with him? Is she confused? Is he at fault for answering Mo’s earlier How far? question with a time rather than a distance? What should he do?

My hunch is that Mo is not messing with her father. Instead, she has taken his cue for talking about how far, and she is playing with it. This is how children learn—they hear something and they try it out.

Here is how we might turn this conversation into a bit more math learning. Imagine Jim’s next response this way:

Jim: OK. Tell me when I’m there. But then don’t jump right away; I want to ask you a question before you do. [Daddy backs up slowly…]

Mo: OK! There!

Jim: Right. Here’s my question: Do you think it will take you five minutes to get to me from where you are?

Mo: Yes.

Jim: Do you know how long five minutes is?

Mo: That far.

Jim: No, no. Can you think of something we do together that takes five minutes?

Mo: No.

Jim: It takes us about five minutes to read [INSERT TITLE OF FAVORITE PICTURE BOOK HERE] together. Do you think it will take that much time for you to get to me?

At this point, I have no idea how Mo will respond (which is what fascinates me so much about talking math with kids). I do know that pretty soon, she is going to want to jump, and that whether that’s right away or after a few more exchanges doesn’t really matter.

What matters is that she’s been asked to think.

This line of discussion lays the foundation for thinking about distances, times and their relationships to each other. It supports Mo’s attempts to participate in the conversation about measurement.

My conversation with Tabitha about the height of our hill last summer was in a similar spirit; we worked on the meaning of height when she asked me to lie down on the hill.

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Griffin wanted in on the action. Here is his jump shot.

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.

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?

Summer project (3 of 3)

We went to the State Fair last week.

Photo Aug 25, 10 54 32 AMHaving wondered about the height of the Giant Slide, and having developed a technique for measuring things, we needed to collect the information required to answer our original question.

The only problem? No one but me wanted to ride the slide. This is a big change from previous years.

I went up alone in the name of mathematics.

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There are 104 steps to the top.

I asked a young woman employee how many steps she thought there were. She said 108. I told her my count and she was ready to believe it. I asked a young man employee how many steps he thought there were. He had no idea.

How can you work at the top of that thing and not be curious how many steps there are?

In any case, should someone wish to check my work next year, I got 1 set of 20 and 3 sets of 28.

I also took a kebab stick from Griffin’s dinner along with me. I broke it off at the height of one step partway up. I checked it against another step further up. Then I taped the stick into my notebook when we got home.

May not be actual size on your screen.

May not be actual size on your screen.

So then Griffin and I sat down one morning to finish this off. Recall our guesses of 40 and 45 feet.

It was a fairly conventional conversation, so I’ll just list the bullet points instead of trying to reconstruct our exact words.

  • I asked him to estimate the length of the stick, which he did—4 inches.
  • He measured the stick with a ruler—4\frac{1}{2} inches.
  • He suggested a calculator was in order.
  • I suggested that this would not be happening.
  • We sought to find 104\cdot4\frac{1}{2}.
  • He naturally subdivided this into 104\cdot4+104\cdot\frac{1}{2}.
  • His first answer to 104\cdot4 was 408; on further reflection he got 416.
  • I was a useful resource for remembering intermediate results (such as the 116).
  • Half of 104 was easy for him.
  • We ended up with 468 inches.
  • He knew we needed to divide this by 12.
  • I modeled an intelligent guess-and-check strategy for doing this by asking him to guess. I did the multiplication. You can see the results below.

For the record, I spoke aloud while doing these. E.g.

For the record, I spoke aloud while doing these. E.g. “Two 45s is 90; ten 45s is 450, so 540” Et cetera

Upon completion of our analysis, Griffin wanted to know how high the Sky Ride is.

Success.

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.)

square

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.