Cocoa Puff or Cocoa Puffs: The language of nothing

In honor of Tabitha turning 11 this week, here’s a conversation from 6 years ago. 

We have a little family tradition. When we go grocery shopping the weekend before your birthday, you can choose one box of any cereal you want-no restrictions. In the weeks and months leading up to the grand event, much time is spent in the cereal aisle weighing the advantages of the various sugar-laden options.

The week before turning five, Tabitha nearly dropped the ball. She just grabbed the first box of anything at hand. I don’t remember what it was, but it seemed out of character for her. I reminded her of the cereals she had been coveting as recently as the previous week.

She went for the generic Cocoa Puffs.

I steered her towards the real deal. If you’re only gonna eat ’em once a year, you might as well have the sugar-addled bird bouncing off the box in front of you, right?

A sugar-addled bird

One morning shortly afterwards, we had this conversation:

Tabitha: Do I have Cocoa Puffs or Cocoa Puff in my hand?

Me: Well, you have four Cocoa Puffs.

T: [with only one in her hand now] Do I have Cocoa Puffs or Cocoa Puff?

Me: You have Cocoa Puff.

T: [huge smile] Right!

Me: [with empty hand displayed] Do I have Cocoa Puffs or Cocoa Puff in my hand?

T: [silent but smirking]

Me: Well…Is it Cocoa Puff or Cocoa Puffs?

T: [continued silence]

Me: I have zero…

T: [bigger smile]

Me: …Cocoa…

T: Puffs!

Me: Yeah. Isn’t that weird? If you have one, it’s Puff; if you have none it’s Puffs.

T: I knew that.

Me: Of course you did.

T: No! I knew that; I was showing you that [you had zero] by not saying anything-zero words!

So what do we learn?

Children listen carefully to language patterns. They do not learn a native language like a second language in school. The rules are not carefully explained to them one at a time.

Instead they listen, speak, get corrected, and try again. All of this can be tremendously fun for child and parent alike.

It is an odd quirk of English that zero is plural, grammatically speaking. We talk about having one child, but zero children. More commonly, we use no instead of zero, as in My neighbors have no children. The grammar is the same either way; saying My neighbors have no child sounds funny to our ears.

Starting the conversation

In discussing place value, zero is sometimes called a place holder. To understand that, children need to understand zero as a number. They need to understand that zero can legitimately answer the question, How many are there?

We talk a lot about zero in our house. You can too. Ask your children, “Would you rather have one cookie, two cookies or zero cookies?” Ask who has more of something, even when one of the people has none.

The Mad Hatter in Alice in Wonderland gives an example of this. The March Hare offers Alice “some more tea”. When Alice says she can’t possibly have more, since she hasn’t had any yet, the March Hare replies, “It’s very easy to take more than nothing.”

Another silly language game we play in our house is this. If you look in the pantry and see that there are three cookies left, you can report this in the following two ways: (1) “I checked the cookies; there are three left,” and (2) “There are three cookies.” If, however, there are no cookies in the pantry, these two ways of reporting the sad fact become: (1) “I checked the cookies; there are none left,” and (2) “There are none cookies.” We like to treat none as a number. There is no good reason for this; it is for personal amusement purposes only.

Postscript

Tabitha again chose Cocoa Puffs on this, the week of her eleventh birthday. She is enjoying them, but she has also stated the obvious—they look like rabbit poop.

February is “Talking Math with Other People’s Kids” month

You won’t be hearing much from Griffin and Tabitha this month. Instead, you’ll hear from other children and their parents who have talked math and have shared their conversations with me.

It will be a ton of fun to get a peek into these other households, and to see how frequently ideas and questions about number and shape come up in life with young children.

I would love to hear your own reports, and to gather a collection of stories representing diverse families, cultures, languages and experiences. Shoot me a note describing conversations you have participated in or witnessed. I’ll feature as many of them here as I can.

Let’s kick things off with an example of a father and his five-year old daughter, and how Twitter helped them talk a bit more math than they otherwise might have…

Andy is a dad in Minneapolis. Let’s call his daughter Martine. Andy tweeted me on Friday (January 31).

Here is how this sort of thing would go in our house.

Martine (5 years old): If tomorrow is February first, does that mean today is February 0th?

calendar

Dad: Yeah, I guess we could call it that. If we do, what would yesterday have been?

M: February negative one.

Dad: Oooo. Nice! And what about the day before that?

Et cetera. At some point, the conversation would go somewhere else. Or if she’s still interested, I might give it a twist with a question like this.

Dad: So if today is both January 31 and February 0, and if tomorrow is February 1, shouldn’t it also have a January name?

I would be probing Martine’s double-naming idea for each day. And then…

Dad: Hey! I know! If today is both a January and a February day, then tomorrow should be both a February and a March day, right? What is tomorrow’s date in March?

As I mentioned, the conversation may very well have broken down by this point. But these what if questions are the things that turn a cool observation into a conversation. That conversation is where we turn kids’ minds on.

Dad Chris Hunter suggested that first follow up question: What about yesterday?

Andy asked Martine about that on Saturday.

Martine said that the day before February 0 would be February negative 1. Andy reports—and this is important—never having explicitly discussed negative numbers with Martine. No number lines, no backwards counting past 0.

But surely they have talked about the weather. Below-zero temperatures have been as common as snowflakes in Minnesota this year. Talking about the weather may have planted the idea. Then the calendar was an opportunity to make a connection.

All of this leads to two important ideas about talking math with kids:

  1. It’s not a conversation until you, as a parent, participate. Martine noticed something (Jan. 31 could be Feb. 0). Andy turned it into a conversation when he asked about the previous day.
  2. These conversations are facilitated by availability of objects. Turning the calendar became a learning opportunity for Andy and Martine. No calendar, no conversation.

You can read our full Twitter conversation here.

And you can read about other conversations facilitated by objects in these previous posts:

Days until Christmas

Each day, Griffin (9 years old) has taken great joy in setting the Countdown to Christmas cube calendar the kids recently received from my father and stepmother.

On Thursday, I was doing my end-of-semester grading from home and noticed after he had left that he had neglected to set it on his way to catch the bus to school. So I did.

christmas.06

My wife got Tabitha out the door later in the morning while I was at work in my basement office.

Later, I noticed that Tabitha (6 years old) had modified it.

christmas.6

Here I will pause to give readers a day or so to consider why. What might a 6-year-old have in mind that would cause her to remove the zero from in front of the six?

I asked her about it later on and we had a lovely conversation. I’ll report on that soon.

The biggest number

I do not recall the beginning of this conversation, but I do recall that we were eating pizza at the dinner table when Tabitha anticipated my turn in the dicussion.

Tabitha (6 years old): I know what you’re going to say, Daddy. “Counting never ends.”

Me: I suppose that sounds like something I would say, yes.

T: What’s the biggest number, though? Googolplex?

Quick tutorial. A “googol”—spelled that way—refers to this number: 10^{100}, or “a one followed by a hundred zeroes”.

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

It is, of course, a very big number. Far too big to be practical in any meaningful sense. The very idea of such a large number having a name is fascinating to children. Most children (in my experience) encounter one googol in their social interactions with other children. The googol does not appear in the Common Core State Standards.

A “googolplex” is 10^{10^{100}}, or 10^{googol} or “a one followed by a googol zeroes”. You cannot write this number out in standard form.

You may Google googol for lots of interesting characterizations of how extremely silly this very large number is.

For example, you will not live for one googol seconds.

Indeed, the universe has not existed for one googol seconds (not even by the greatest estimates of its age—not even close).

You get the idea.

Me: Well, like you said I would say, counting never ends, so no googolplex is not the biggest number.

T: If you counted by 10,000 could you ever get to googolplex in your life?

Me: No.

T: If you counted by 11,000?

Me: No.

T: 12,000? 13,000?

Me: No. Even if you counted by googol, you couldn’t get to googolplex in your lifetime.

T: Well, what if you counted by googolplex?

Me: Well sure. It would the start of your count, wouldn’t it?

She decides to demonstrate this (Side note, we have been counting by various numbers of late).

T: Googolplex.

She smiles broadly, congratulating herself for successfully counting to what she has perceived to be the largest number.

We discuss further the existence of a largest number. Then Tabitha makes a claim that takes us in a different direction.

T: Eventually, numbers just go back to the beginning.

Me: So if you keep counting, you get to zero?

T: No.

Me: One?

T: No, Daddy! Don’t you remember there are numbers before zero?

So what do we learn?

Big numbers are fun. Boy howdy are big numbers fun. Children love to talk about the biggest number, and whether one exists. There is all kinds of lovely thinking going on when they ask these kinds of questions.

Talking about big numbers often leads to talking about infinity. If there is no biggest number, it is because numbers go on forever. The only thing Tabitha has experience with that goes on forever is a loop. She drew on that loop metaphor in imagining that numbers go back to the beginning eventually.

Starting the conversation

Listen for the biggest number talk. It often surfaces when children are comparing their athletic prowess (I can jump 2 sidewalk squares! I can jump 100 sidewalk squares! Pretty soon, someone is claiming to be able to jump googol or infinity sidewalk squares.)

When it surfaces, support it. Play and explore with your child. Answer questions. Ask questions. Talk about it and have fun. Look stuff up together when the questions go past your own knowledge. Shoot me a question here at Talking Math with Your Kids if I can answer any of those for you.