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.

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.

**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.)

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.