Units, attributes and four-year olds

From mrdardy in the comments recently:

Slightly off topic, but I wanted to share a conversation with my soon to be four year old daughter from this past weekend, We were on a long car drive and she was asking how far we were from our hotel. I replied that we were twenty minutes away. Later in the pool she was jumping to me from the pool steps and commanding me to back up some. I asked her how far I should go and she told me to be five minutes away. I said “Do you mean five feet away?” and she replied, firmly, that she meant for me to be five minutes away. I am wrestling with whether I think this is just charming and (semi) clever on her part or whether I need to start answering her pleas in the car with distances. Curious to hear some ideas on this.

I am happy to weigh in here.

Anna Sfard describes knowledge as participation in a discourse and learning as changes in that participation. That is, we can measure whether someone knows something only to the extent that they can talk in ways that adhere to the norms of other knowledgeable people. And when these behaviors change to conform more closely to these norms, we can say that they are learning.

Nowhere is this more clearly demonstrated than in the learning of young children.

The four-year old in question here (let’s call her “Little Dardy”) is trying very hard to participate in conversations about measurement. Measurement, though, is a challenging and rich domain. 

mrdardy outlines two scenarios in which the concept of how far comes up for Little Dardy. It shouldn’t be at all surprising—considering Sfard’s model—that she answers a distance question in the same way her father had earlier on. She has taken his example in using units of time to discuss how far something is.

My approach would not be to avoid using units of time to answer the question how far? After all, people do this frequently; it is part of the discourse of measurement.

No, I would use this tension to encourage Little Dardy to think about the two attributes in question here: time and distance. It might go something like this…

Little Dardy: (four years old) Back up, Daddy!

Daddy: This far?

LD: More!

D: Here?

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

D: Do you mean five feet away?

LD: No! Five minutes!

D: 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…]

LD: OK! There!

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

LD: Yes.

D: Do you know how long five minutes is?

LD: That far.

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

LD: No.

D: 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 Little Dardy 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 Little Dardy’s attempts to participate in the discourse of measurement.

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

A kindergartener on units [Talking math with your parents]

The following conversation took place in my house the other day. Tabitha (6) had been informed by her mother that she (Tabitha) needed to eat something healthy before eating a chocolate-covered donut. I was—and remain—ignorant of the origins of this donut.

I came in partway through the conversation.

Rachel: I’m going to cut you a small slice of this apple.

Tabitha (6 years old): Do I have to eat the whole thing?

R: The whole apple? No.

T: No, the whole slice!

R: Yes!

If you are unaware of the fun we have had with units around our house, you may wish to check out our discussion of brownies, and (of course) the following.

A fun game to play

Here’s a fun game to play with the nearest small child.

Offer her a slice of cheese. Produce said slice. Ask if she would like two. When she says yes, cut the slice in two pieces.

Observe the child’s (dis)satisfaction and discuss.

 

Cheese-pepperoni crackers

It was snack time in the Triangle household the other day. Tabitha decided she wanted crackers with pepperoni and cheese, baked briefly in the oven.

She asked for eight. I was skeptical that she was that hungry. I was worried about wasted food. I asked her about this.

Tabitha (five years old): I promise to eat a little bit out of each cracker.

Me: Wait. Say that again.

T: I’ll eat some of the crackers, and the ones I don’t want to eat all of, I’ll eat some out of each one.

Me: So you want eight, and you’re not promising to eat all of them, but you’re promising to eat part of each one?

T: Yeah. I promise.

I relented. And I got out the baking sheet and 8 crackers.

Me: Can you put those crackers out in nice rows?

T: What kind of rows? Three of three?

As we’ll see shortly, she was meaning to ask how long each row should be—should each row have three crackers? She did not seem to be asking also about the number of rows. Although I didn’t know that yet and was prepared to be dumbfounded.

Me: Whatever you can do. You choose. Make them the same size, though.

T: I made three rows and there’s two left. I can’t do it.

Me: Oh. So what are you gonna do instead?

T: Two of two?

That is, she is going to make rows of two, since rows of three didn’t work. She worked busily at this for about a minute, chatting to herself about her work.

Me: Now. How many pieces of pepperoni do you need?

T: Eight.

Me: How do you know that?

T: Cause there’s eight crackers.

Me: Oh.

T: But first you need the cheese.

Me: No! It’s pepperoni and then cheese goes on top.

I cannot explain my need to adhere to this principle. But I do stand by it.

Me: Here you go. Here’s ten pieces of pepperoni.

T: What?!?

Me: Are there gonna be any leftovers after you put them on there?

T: No. I’ll put two on two.

Here, she means to say that she will put two pieces of pepperoni on each cracker. Again, I am prepared to be amazed. I am thinking that she might mean that she’ll put two pieces on two crackers, and one piece on the others. She does not mean this.

Me: What? What do you mean?

T: Two on two.

Me: Show me. I don’t get it.

She proceeds to put two pieces of pepperoni on each of the first few crackers. The result is one of the funniest things she has seen in days.

T: Like that! [Laughs loudly and long]

Me: Keep going. I want to see how this works out.

T: But if there are, I’ll eat them…This is funny, isn’t it? Putting two pieces of pepperoni on each one?

Me: You gonna have enough?

T: Uh oh. No, no, no. Different idea. I need eight, sorry.

I did two and I only had three left. I mean I had two left.

This is pretty difficult to parse if you were not there. She has covered one of two columns of four crackers with two pieces of pepperoni each. This leaves her with two pieces of pepperoni in her hand, which she notices is inadequate for doing the same to the remaining column.

Me: So you need eight exactly?

T: And there’s a whole row left!

As a developing authority on the topic of one, I would like to point out the various units Tabitha is keeping track of in this work. She is counting (1) crackers, (2) pieces of pepperoni, (3) the number of pieces of pepperoni per cracker (a unit rate), (4) the number of crackers in each row (another unit rate), and (5) the number of rows of crackers. This is cognitively complex stuff.

Me: I understand. I totally understand. So you have ten pieces…

T: Oh! I’ll put eight…I’ll put one on each one and then eat the rest…and then eat the leftovers.

Me: How many leftovers are there gonna be?

T: Oh. I don’t know. Oh. I have an idea.

She removes the pepperoni pieces from the crackers, stacks them and counts in a whisper.

T: Ten…I mean no! One! I mean two! Two.

Me: Two.

T: I can eat two.

Me: See if this works out.

Tabitha proceeds to match the crackers with pepperoni slices.

T: Why don’t they call this cheese-pepperoni crackers?

Me: We could call them cheese-pepperoni crackers.

T: It’s cheese and crackers if there’s no pepperoni, so it should be called cheese-pepperoni crackers.

She finishes up.

T: I was right, two! Look. There’s two left.

Me: What are you gonna do with them?

She shoves both in her mouth dramatically and laughs.

And here is the result; ready for cheese and then the oven.

Decimal place value v. whole number place value

As the rest of my household heads back to school, I am getting down to the business of planning for my new semester. One instructional problem I am trying to solve is that of pushing my future elementary teachers to understand decimals more deeply. Whole number place value I have nailed down. But decimal place value is another story. The rules are so deeply embedded in them that it is very difficult to get them to question these rules, or to seek a better understanding of them.

I have collected substantial data over recent semesters (not research quality data, but consistent formative and summative assessment results) to demonstrate that my students operate on decimals using a combination of known rules and whole-number place value principles. They can switch back and forth between these fluidly and generate right answers to nearly any decimal task.

Which leaves me seeking questions that they cannot answer without digging more deeply.

I have a candidate question in mind: Why can we put zeroes after a decimal number without changing the value, but not after a whole number without changing its value?

 

It occurred to me last evening that I knew what my answer to this was, but that I lacked a wide repertoire of correct explanations. So I asked Twitter. Lots of good stuff followed. Read the full conversation on Storify.