A quick plug for Estimation 180

Estimation is more than rounding.

Most of the time we don’t teach this, but it is.

Tabitha (8 years old) had a homework assignment the other night that asked her to imagine she had $100 to spend in a catalog, and to make a list of things she would like to buy from that catalog. She found the latest American Girl catalog and got to work.

There was a table to fill out with three columns.

1. Description of item
2. Actual cost of item
3. Estimate

A couple minutes later she asks, What’s the estimate if it costs five dollars? Should I write $5.01?

She has discerned that estimate means write down a number that is not the exact value.

But that’s not what estimation is about at all. Estimation is about finding a number that makes sense, and not worrying about whether it’s the exact value or not.

The image below seems to be going nuts on the Internet today (despite my exhortations to the contrary! Oh, Internet! When will you learn to listen to me?)

“Is this reasonable?” is a great estimation question. Rounding is one way to answer the question. But if a kid can quickly find a number that makes sense and it happens to be a precise number, then we probably haven’t asked a good estimation question. Rather than mark it wrong because the kid didn’t round, we should ask this kid a more challenging question next time.

What does a good estimation question look like? What would be more challenging?

I’m glad you asked.

Estimation 180. Thinking of a number that makes sense is much more interesting when you have to bring your knowledge of the world to bear.

Is 75 inches a reasonable answer for the difference between the father’s height and the son’s? Is 75 centimeters reasonable?

Oh! And here’s the first in a nice sequence about numbers of pages.

Mindsets, research and talking math with kids [#NYTEdTech]

This conversation happened in New York yesterday.

A view of New York City from the Times Center on Tuesday.

During a coffee break, I sat down on a white pleather sofa next to an older man.

Me: How has your day been?

Him: Good. You?

Me: Pretty good. Interesting.

What do you do?

Him: Retired.

Me: From what?

Him: I was president of [small New England college]. How about yourself?

Me: I teach math at a community college in Minnesota.

But I’m also working on a project. I work with future elementary teachers, so I have studied the mathematical development of children.

Him: Uh huh.

Me: And I want to use that knowledge for something else, which is this: I am trying to understand what knowledge parents need in order to support the mathematical development of their children.

Him: That’s important.

Me: Right.

[Short pause]

Me: Do you have grandchildren?

Him: Yes. They are 8 and 10.

Me: Oh nice! So their parents—your kids—are my target market.

Him: Yes. Their father is really into that. They use Khan Academy and all that.

—FIN—

If the end of that conversation makes no sense to you, I ask that you please, please, please spend the next 15 minutes over at my website, Talking Math with Your Kids. You might be especially interested in the research summaries, which demonstrate that young children need to talk about number and shape with their parents rather than (or at least in addition to) being sent to website, iPad apps and decks of flash cards.

Kids need mathematical conversation. And they enjoy it.

How tall is the hill? [summer project]

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.

In a few days, we will return to the topic of the State Fair Giant Slide and see whether these techniques generalize in my children’s minds.

Incommensurate 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.

On seeing this claim, I thought for sure that we were gonna have a We mean four times, but say twice sort of a situation on our hands. So I bought some.

And then I asked Tabitha (6 years old) and Griffin (8 years old) 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”.

I wanted to know the source of that. I thought she might be making the classic linear v. area error (i.e. interpreting twice to mean twice the side length). So I asked.

She pointed to the uncovered part of the BIG Cheez-It and argued that this didn’t constitute another full regular Cheez-It. Score one point for argumentation, but minus one for spatial visualization.

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

Nope.

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 quite twice the size of the regular Cheez-Its.

Volume perhaps?

Addendum 1

If the crackers are twice as big, but the mass of one serving is constant, and if one serving of regular Cheez-Its consists of 27 crackers, how many crackers should be in one serving of BIG Cheez-Its?

There are 14.

Addendum 2

If the area of a BIG Cheez-It is about twice the area of a regular Cheez-It (as Griffin confirmed), then the side lengths should be in a ratio of approx. 7:5 (a reasonable estimate of the square root of 2).

 

addendum 3

Notice the progression in the children’s strategies. The six-year old worked with the crackers. The eight-year old worked with representations of the crackers. Similar conclusions were reached; the child who worked with representations could manipulate those representations in order to achieve a greater degree of accuracy, and to investigate hypotheses that the child working concretely could not.

Neither child used tools to calculate areas.

Summer project

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 will be a summer project for us: Measuring stuff without putting a ruler next to it. I’ll report on our progress in this space.