Zero=half revisited

A few weeks back, Tabitha asked Why are zero and half the same? I was curious to know whether that conversation had affected her thinking in any way. So I asked.

Me: Tabitha, do you still think zero and half are the same? Or have you not thought about that in a while?

Tabitha (six years old): I think…Half isn’t a number. I mean, it’s made of numbers put together, but it’s not a number.

Me: What is a number?

I love this question. How people answer it can be revealing. I asked a version of it of Griffin when he was in Kindergarten.

T: is a number.

Me: Oh?  is a number, but not one-half?

T: Yeah. But it doesn’t really get used.

Me: What do you mean by that?

T: Well, people say, 1, 2, 3, 4, 5, 6, but not .

Me: Oh. So when we count count, we skip over ?

T: Yeah.

We are both silent for a few moments, thinking.

T: Zero, too. People don’t count starting at zero. They say 1, 2, 3…

Me: Yeah. Isn’t that funny?

T: It should go half, zero, 1, 2, 3…

It seems clear that has indeed been thinking about that conversation. She is struggling with the betweenness of ; that it expresses a number between 0 and 1.

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Planting Seeds with Tabitha (or, The Pigeonhole Principle)

We were planting seeds the other day. Indoors. This is Minnesota, after all.

Over the course of many years of gardening I have worked out a system. Yogurt containers, potting soil and these lovely clear IKEA containers.

The IKEA boxes are a recent innovation. They keep soil moisture high (yet have enough volume to allow the plants to breathe), and they let me move plants inside and out according to the ever-varying spring weather (it was 80° on Sunday this week, and it snowed on Wednesday).

Sorry for the digression. Back on task.

We were planting tomato seeds by poking holes into the soil, placing one seed in each hole and covering the seed. We had discussed how deep to make the holes; that the depth corresponding to Tabitha’s first knuckle is not at all the same as the depth corresponding to my own, et cetera.

Tabitha (six years old): How many holes should I put in this one?

Me: Five. Put one in each hole.

I hold out my hand with several seeds for her to take.

T: But there’s more seeds than holes.

Me: So what?

T: So then they’ll be crowded.

This is her line of reasoning, not mine. I had not been at all concerned with trying to offer the precise number of seeds she would need. I had simply shaken some from the pack into the palm of my hand.

But since she started it, I develop a plan. I am going to do my best to get her to state the pigeonhole principle.

Me: But what did you say about the seeds and holes?

T: There are more seeds.

Me: And what are the consequences of that?

T: You said the plants wouldn’t grow as well if there are two in the same hole.

So close! She is using the pigeonhole principle, but I cannot quite get her to state it.

So I do.

I tell her about pigeons and pigeonholes.

We proceed to a lovely (and thoroughly uninformed) discussion of the mechanics of sending messages by carrier pigeon. She wonders, for instance, about how to send a message to your friend, since the carrier pigeon’s unique skill is to fly home from anywhere, but not vice versa. We deduce together that you must need to borrow your friend’s pigeon.

Oh, and those tomato seeds? Brandywine.

Dot-to-dot

Another one from the archives.

About a year ago, Tabitha was doing a dot-to-dot drawing.

This is not the one she was doing. It is an example. Not a very good one, either.

Things were going well. She got to 11.

Tabitha (five years old at the time): I don’t know what twelve looks like.

Me: It’s a 1, then a 2.

With this tip, she cruised through the teens and got to 20.

T: What does twenty-one look like?

Me: A 2 and then a 1.

T: And twenty-two?

Me: A 2 and then a 2.

T: So twenty-three is probably a 2 and then a 3.

This got her going on to 29.

T: I don’t know what 30 looks like.

There are a lot of interesting things going on in this exchange. Among them:

Sometimes in mathematics, we need to live with new notation before picking its meaning apart too carefully. See also fractions, functions and derivatives.

Numeration and number language do not develop hand-in-hand. Tabitha knows the number language; she can count past thirty. She has not learned how to read or write numbers that high.

Patterns are powerful tools in mathematics. Tabitha’s experience in the teens gave her powerful intuitions for the twenties.

Peeps math with Tabitha

After the Peeps photo session last week, I test drove my images with Tabitha (six years old).

Me: Which are there more of in this picture? Purple Peeps or pink?

Tabitha: Purple.

Me: How do you know?

T: It goes all the way to the top.

Piaget would be proud. Tabitha’s focus is on one dimension, rather than on overall quantity. So let’s test that hypothesis. Does she really believe that’s all that matters?

Me: What about in this picture?

T: Purple.

Me: But they both go to the top in this one.

T: This one (purple) has full rows, and this one (pink) has holes.

Me: Interesting. You know what I see? I see that if you moved that last bunny on the bottom row up to the next row, you’d have two rows of three and an extra bunny, while the purple has three rows of three.

T: Yeah.

Me: OK. One last one. What about this picture?

T: Purple.

Me: Because it goes to the top?

T: Yeah.

Me: Look carefully, though.

T: Pink.

Me: Why?

She proceeds to count 9 pink bunnies. I correct her and have her count over. She again counts 9 pink bunnies. I show her that if you move the two top purple bunnies into the second row, you would fill that row. She is uninterested and we move on to other things.

Pumpkin muffins

My wife Rachel made pumpkin muffins last night. Her contribution to the world of baking is the chocolate chip pumpkin muffin. I feel The Honest Toddler would approve. I know that Tabitha does.

Tabitha told me about a dream she had last night. In the dream, there was only one pumpkin muffin left.

Later in the morning, Tabitha counted the muffins in the tin.

She got eight.

Me: Wait. Count those again?

Tabitha (six years old): [Points to her mouth] One. [Points to first muffin in tin] Two, three, …

Et cetera, ending at eight.

Tabitha: So really, we had eight muffins left.

Me: I see, last night when you dreamt there was one, there were really eight.

T: Yes.

Me: That must be reassuring to you.

T: Did you eat one this morning? Then it would be nine.