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.

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One-to-one correspondence

This business of Talking Math with Your Kids isn’t about cute kid stories. Sure, my little guys are cute (the cutest, in fact). But that’s not what this is about.

This is about choices. More precisely it’s about noticing the moment when parents have a choice between (1) talking math with their kids, and (2) not talking math with their kids.

It’s also about identifying what a parent needs to know in order to (1) notice that there is a choice, and (2) make the choice to talk math.

Here on the blog, I’m documenting the moments when I make the choice to turn an everyday conversation into one that gets my children thinking about math.

Let’s use the analogy of sports. My children are woefully ignorant of the world of organized sports. We don’t watch much on TV, we don’t head out to ballgames very often. I play Ultimate once a week May—November, but they only come along a couple of times a year. They love to run and bounce and jump and exercise in a hundred different ways, but neither one could name a single Vikings player, nor tell you precisely which sports are presently in season.

This is because many times over we have made a choice not to turn the conversation to sports. It’s not because they couldn’t understand sports, or because they or I have a great distaste for sports; it’s just not how my wife and I spend our time and energy in interacting with our children. We make choices.

I argue that the same is true for math.

Every single day, parents have opportunities to talk math with their kids. Every day, we can help our children notice and wonder about number, shape and likelihood. It’s not difficult. It’s a ton of fun. But it does require a bit of knowledge and it requires a bit of awareness.

For example…

I take both kids grocery shopping pretty much every weekend, and I have since each was an infant. It’s a routine for us in which Mommy gets some quiet time around the house and I get some extended time around town with my little ones. This time of year, the excursion includes the farmer’s market. (Which, by the way, if you are ever in St Paul on a Saturday morning, you must attend; it’s one of the best in the country for sure.)

There is a tremendous amount of construction in the area right now, so the walk from where we park is circuitous and requires sharing a short stretch of street with an occasional slow-moving automobile.

Me: Can you guys grab my hands please? A car is coming.

Tabitha (five years old): We each get a hand!

Me: Yeah. Good thing I only have two kids, huh?

T: Yeah, if there were more kids, there wouldn’t be enough hands. Like Yusef [our next-door neighbor who has three children].

Me: Oh, right. Good point. What if Natalie came too, though?

T: Then there would be an extra hand to carry a bag. You don’t have that.

Me: Right. Sorry. That means you’re going to get hit by the bag a few times. At least it’s not full yet, though.

[pause]

Me: Do we know anyone with fewer children than hands?

T: Dawn!

Me: Good. I hadn’t gotten to her yet. I was thinking about Jenn, but she has Wynne and Emmett; and I was thinking about Addie, but she has August and Leo. Then I thought about Jimmy, but he has Laila and Otis. I hadn’t thought about Dawn yet. She just has Mateo, doesn’t she?

T: Yeah. So she would have an extra hand for a bag.

I argue that I made a choice here, and that it’s one all parents can learn to make. When Tabitha noticed that each child got their own hand, I chose to value and reinforce this observation. When my daughter noticed number, I chose to make a conversation out of it. I responded to her observations and asked follow up questions. The result is that we talked about one-to-one correspondence on the way to the farmers’ market. There’s only one number in the whole conversation; the rest is about whether Set A (children) has more or fewer members than Set B (parental hands available for holding). This is a remarkably sophisticated idea.

Yet, there is nothing particularly sophisticated about my end of the conversation. My ability to reinforce and extend Tabitha’s number noticing behavior has nothing to do with having passed graduate exams in Modern Algebra and Real Analysis (which I only barely did, anyway, and in which a certain number of pity points may or may not have been awarded).

But I am still exploring what a parent does need to know in order to notice these moments, and in order to use them to continue the conversation.

I do know that, as with sports, there is a habit that gets set. By now, my children notice number and shape in the world, and they bring the conversations to me. But that’s the result of years of enculturation.

My kids know that quantity and shape are interesting things to talk about. I’m not sure they know they know this is called math.