Griffin (eight years old) and Tabitha (five years old) were discussing the day’s activities. The feature activity had been making brownies with Mommy. This occurred while Griffin was out the house.

Griffin: How many brownies did you make?

Tabitha: One big one! Mommy cut it up.

I have emphasized elsewhere the importance of the unit; that one is a more flexible concept than we might think.

A few years back, one of my elementary licensure students was trying to understand a conversation she been part of. In this conversation, an early elementary-aged child could not give a substantive answer to the question, What does five mean?

I suggested to my student that this was a pretty abstract question and I wondered what exactly would constitute a good answer to it.

And then I decided to show her what I meant by this. I made a video (embedded below) in which Griffin (who was five at the time) and I had the following conversation.

Me: Can you count out five blocks?

Griffin (five years old): [He does so flawlessly while mugging for the camera] What’s next?

Me: Is five a number?

G: Yes.

Me: What does five mean?

G: I don’t know. Do you?

There you have it. He knows what five is, but cannot articulate what five means.

There is more fun to be had here, though.

Me: Is one a number?

G: Yeah.

Me: Is zero a number?

G: Mmmmm…No!

Me: Why do you say that?

G: Because it’s…not necessarily a number. It’s not big…I don’t know. It’s just not a number.

Me: How about uh…

G: It’s not like any other number.

This is very common. Another math (and physics) teaching dad, Casey Rutherford, discovered this recently when discussing what 10 takeaway 10 is with his five-year old daughter. She said zero. I asked him to pursue the question of whether zero is a number with her.

@Trianglemancsd I stand corrected. Apparently zero is not a number.

Having settled the status of zero, I ask Griffin about one half.

Me: Do you know what one-half means?

G: No.

Me: Do you know what it means to have half of a cookie?

G: Yeah…sort of.

Me: Is one-half a number?

G: No.

Me: Why do you say that?

G: No…because….No. It is not a number

Me: What is a number?

G: A number is like 2, 4, 6, 8, 9, 1…stuff like that.

One hundred. A billion. Three hundred and ninety nine. Those are numbers.

Me: But not zero.

G: No.

Me: And not one half.

G: No.

Examples. Griffin tells me what a number is by offering me examples of numbers. Maybe this is because he has no other way of talking about the meaning of words? Maybe he always offers examples as the sole means of explaining what a category of objects is.

I investigate this hypothesis too.

Me: So I’m gonna ask you one more silly question.

G: OK.

Me: You’re playing with blocks there, right?

G: Yeah.

Me: What is a block?

G: A block is something that’s made of wood and it can be colorful or just plain. And you can build stuff with them. And it’s a toy that has…

There it is—the abstract nature of numbers.

What is a number? All he can give is examples.

What is a block? He is full of characteristics of blocks, uses for blocks, categories into which blocks fit, etc. He has a robust and explicit scheme for sorting blocks from non-blocks. He has no such thing for numbers.

My wife began a tradition that she now regrets. After turning out the lights in the kids’ shared room at bedtime, she would tell the children to picture something in their minds. This was typically a soothing or amusing scene related in some way to the days’ events.

As time has passed, the picture has become more elaborate, the children’s expectations higher. Tabitha in particular can be demanding. I was told the other night that the picture I provided was not long enough, for instance.

Most of the time, when it’s my turn to provide the picture, I begin to speak without a clue what the resulting image will be. Sunday was one of those nights.

Me: Picture that you are riding in the car and you come to a sudden stop because there is an elephant parade. A very long one consisting of 27 elephants marching one directly behind the other.

Now…the strange thing about this elephant parade is that each elephant is a bit smaller than the one in front of it, and each one has an additional leg.

So the first elephant is the usual elephant size and has four legs. But the second one is a bit smaller than the first and has five legs. The next is a bit smaller still and has 6 legs.

Like I said, the pictures have gotten elaborate and expectations are high. Also, I mentioned that they are not preplanned, right?

The elephants with an even number of legs have the same number of legs on each side; even the ones with many legs. Like a centipede.

But the elephants with an odd number of legs have the same number of legs on each side and one in the middle.

So you sit in the car watching this very strange parade go by. Picture that.

Griffin: (8 years old) How many legs does the last one have?

Let’s pause for a moment. This was his question, not mine. Real world be damned, this is a habit of mind thing.

The number of legs on the last of the fictional elephants is not a number that matters in any way. Griffin’s intellectual need arose from narrative. This is a stupid story, but a story nonetheless. Number is the driving force in the plot and the boy wanted resolution.

Me: Oh. Good question. Lemme see…The first elephant has 4 legs. Then 26 elephants later is 26 more legs, so 30 legs on the last one I guess.

Griffin and I play a little game called Guess the Temperature. It goes about how you would expect. We step outside on the way to his bus. I ask him to guess the temperature. If I don’t already know, I get to guess after he does. If I do already know, I don’t cheat; we just remark on how close his guess was.

In Minnesota, this means we get to study integers.

Me: Griff, guess the temperature.

Griffin (eight years old): Two below zero.

Me: It’s three degrees above.

G: So I was off.

Me: Not by much, though. How much were you off by?

G: [muttering to himself, then loudly] Five degrees!

Me: How did you know that?

G: It’s two degrees up to zero, then three more.

Let’s pause for a moment here. You know how I just won’t shut up about CGI (Cognitively Guided Instruction)? It’s because they’re right. Children know mathematics before it is formally taught.

Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

Griff pretty much has this nailed down and is making progress on grade 7. But no one has formally taught him how to subtract integers. He reasons his way through a problem by making sense of the relationships in the context. He can find 3-(-2) without knowing keep-change-change.

But it’s not just Griffin. CGI demonstrated that children—all children—develop mathematical models of their worlds that precede instruction, and that instruction sensitive to these mathematical models is better than instruction that ignores them.

Back to our conversation.

Me: So what if it 10 degrees out, and you guessed 3?

G: [quickly] I’d be seven off.

Me: Right. How do you know that?

G: Ten minus three is seven.

Me: Nice. Subtraction. Do you know that you can always express the difference between your guess and the actual temperature with subtraction?

So in that last example, you subtracted your guess from the actual temperature. You could do that with your real guess today.

So three minus negative 2 is five.

G: [silent]

By this time we were nearing the bus stop. I had offered this tidbit as an intellectual nugget to chew on, rather than a lesson I expected him to absorb. But that is what it means to have instruction be sensitive to children’s mathematical models.

I don’t want to give the impression that Talking Math with Your Kids always goes well. It does not. There are bumps in the road for sure.

I was having breakfast with Griffin (8) and Tabitha (5) over Thanksgiving weekend.

Griffin: It’s exactly one month until Christmas.

Tabitha: I knew that!

Me: That’s right 30 days.

T: So it’s zero weeks.

Me: How many weeks?

T: Zero

Me: In 30 days?

T: Oh! I thought you said 3 days.

G: So how many weeks is it?

Me: What do you think?

G: Well…five sevens is…er…six fives is 30 so seven fives is 35…Five weeks and five days! Is that right?

Me: Tell me how you got that.

G: [long pause]

Me: [long pause]

G: [frustrated] Just tell me if it’s right!

Tears followed shortly afterwards. I suggested we move on and talk about something else. Griffin cursed Tabitha’s name for making him wonder how many weeks this was. Emotions ran high and I told him we needed to discuss it later.

An hour later…

G: [on the couch, addressing me as I came down the stairs] Is it four weeks and two days?

Me: Did you do two sevens are 14, then double? Or did you do five sevens are 35, so five weeks are five days too much?

G: I did the thing with 14.

The most important message I can send my kids is that they can make sense using what they know. Their minds and the world are the arbiters of right and wrong, not me. In not telling Griffin whether he was right the first time, I was reinforcing my insistence on this principle.

I knew what his mistake was and I knew that he would find it if he stopped to think. Five sevens are 35 because seven fives are 35. And 35 is 5 bigger than 30. But that means five weeks is five days too big, not five days too small. I knew he could figure that out himself. Further, I knew that if he could not, we could talk about it and that he would learn from this conversation.

I also know that it is important to compromise. In the second part of the conversation, I knew that he knew he was right. But I knew that prodding too hard on the emotional wound recently inflicted could quickly lead to trouble.

So I compromised. Rather than have him tell me what he did, I offered him choices. I essentially asked him whether he had started from scratch (finding four sevens), or adjusted his previous answer (five sevens are 35).

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