Short notes

I passed a lovely morning with some math teacher friends at the Museum of Math in New York City today. Sadly, I did not have Griffin or Tabitha along. We would have had a ball.

I have three observations…

One. There is lots for kids to do there. It is truly a bonanza of mathematical conversation starters. I left wanting to be hired on as “Docent for Talking Math with Your Kids”. This is because the math does not smack you in the face. Instead, the math in the exhibits tends to surface in the process of playing, experimenting and paying very careful attention to what is going on.

In short, you want to talk about these exhibits and you want to linger.

Two. Many of the exhibits are designed in ways that allow kids to play and to do math beyond the intended activity. The exhibit below, for example, kept two little boys (each 5 or 6 years old) busy for 20 minutes although the original activity was way beyond them.

The first five minutes, they were sharing each of the shapes equally, making sure each of the two boys had the same number of squares, and the same number of triangles, et cetera. The next 10 minutes, they spent arranging the shapes on the screen, marveling at the things this provoked on the screen beneath.

It was lovely.

It could have been a bummer, though. If someone had insisted on doing the intended activity, these kids could not have done it. If the shapes had been electronic instead of concrete, the kids would not have been able to play with them in such creative ways.

Well done, Museum of Math!

Three. On the subway train back to my hotel, I noticed a little girl—probably 4 years old—holding up four fingers as she sat between her mother and brother.

I moved to be in listening range.

It turns out, she had been told they would be on the train for five stops. She had been putting up one additional finger as the train left each stop.

Sister (4 years old): [She is holding up four fingers as the train enters the station] This is our stop!

Brother (8 years old): No. The sign says there’s one more stop.

Sister: But you said five stops.

Mom: One more, sweetie.

The train stops. People get off and on. The doors close. The train starts up again.

The girl keeps exactly four fingers up the whole time.

Counting fingers

Counting fingers

A while back I met a mathematician. He is the husband of a colleague. He found my Talking Math with Your Kids project fascinating and asked repeatedly for additional examples of the conversations I have had with Griffin and Tabitha.

He referred to my work as brainwashing, using the term with great delight.

He shared a story of a young child who, when asked Do you have more fingers on your right hand, or on your left hand? responded without counting, but by matching the fingers thumb-to-thumb, index-to-index, et cetera.

The child invented one-to-one correspondence! my mathematician friend exclaimed with pleasure.

In a sense this is true.

There are things that we tell children. And there are ideas they have on their own, without knowing that anyone has had these ideas before. These really are inventions.

Children can invent more than we sometimes suppose they can.

In any case, this mathematician friend of mine was very curious to know what Tabitha would make of this story. I promised him I would ask. Here is what happened.

We were lying on the bed one evening, having just finished a book and with a few minutes left before beginning the remaining bedtime rituals.

Me: Tabitha, I want to ask you a question.

I told her that I had met a mathematician who was curious to know what she thought about something, and that this something had to do with an interesting answer that another child had once supplied to a question.

Me: The question asked of this child was, “Do you have more fingers on your left hand or on your right hand?”

Tabitha (six years old): That question doesn’t make any sense!

Me: But it’s the question that was asked.

T: But it doesn’t make any sense. Look.

[She counted the fingers first on her left hand, then on her right]

T: 1, 2, 3, 4, 5…1, 2, 3, 4, 5.

Me: So it’s the same on both hands.

T: Right, so the question doesn’t make any sense.

Me: OK. But that’s not how the child answered it. The child did this.

hands

Above, you see what the child did originally.
Tabitha re-enacted it later for the purposes of this post. We regret any confusion.

Me: The question I want to ask you is, what do you think the child was thinking?

T: Oh, I know what she was thinking!

Me: Really?

T: Yeah. It’s the same. If they all touch it’s the same number.

Me: I wonder if that would work with toes.

Tabitha proceeded to demonstrate that it does in fact work with toes.

feet

Me: Ha! I was thinking about comparing the fingers on one hand to the toes on one foot.

T: Well, it would be hard because the toes are all squished together.

We spend a few moments playing with our fingers and toes, trying to match them up, noting their relative cleanliness, and then we get on with the rest of our evening.

So what do we learn?

The technique of asking what a child thinks of an idea is a powerful one. I use it in class all the time: What do you think the person was thinking who got a different answer from you? How do you think Brianna knew to do that?

Asking children to evaluate and comment on the ideas of others helps them also to think about their own thinking.

The specific idea we discussed here is that of one-to-one correspondence. We discussed this in the recent conversation about holding hands at the farmers’ market.

Starting the conversation

This is an easy one. It doesn’t depend on your child providing an idea or knowing any particular fact of mathematics. Sometime soon, you will have a quiet moment together. Maybe it will be at the end of an all-out living room danceathon, or after reading a big pile of books. Tell your child about the mathematician’s question. Show your child the answer that so impressed the mathematician and ask, What do you think the child was thinking?

I had this same conversation with a highly precocious three-year old recently. She insisted that you needed to count the fingers in order to be sure. We had a fine time doing that. Tabitha was within earshot of the conversation with a wry smile.

Holding hands at the market

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 Leila 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.

Number of children = number of hands

Number of children = number of hands

So What Do We Learn?

An important thing to notice here is that there is only one number word in this whole conversation. I say the word two. That’s it.

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. The fancy math term for what we are talking about here is one-to-one correspondence. It refers to the fact that when two collections (A and B) have the same number of things, we can match them up; one thing from set A and one from B, with no leftovers in either collection.

The mind-blowing part of one-to-one correspondence is that it’s true the other way around. If we can match up with no leftovers, then the sets are the same size. Even if we don’t know how big either set is. That is what this conversation works with—comparing sizes of sets without stating the size of either one.

I am quite certain that Tabitha pictured Yusef holding the hands of two children, leaving one who held Natalie’s hand. That left one hand unheld, available for a bag. I do not think (although it’s possible) that she thought 4 hands minus 3 children leaves one.

She matched kids to hands in her mind. One-to-one correspondence.

Starting the Conversation

Listen, and notice when your child is comparing two quantities. maybe they are equal (as in this case), maybe they are not.

As in this conversation, you don’t need to discuss actual numbers to compare two quantities. More, fewer, same…there are many times your children use these words. Follow up with some what if questions and see where they lead.

Beyond the Conversation

One-to-one correspondence, and its implication that we can compare two sets without counting either one, was the idea that Georg Cantor exploited in the late 1800’s to prove that some infinities are “bigger” than others. Cantor demonstrated that there are just as many whole numbers as fractions (because we can cleverly match them up in one-to-one correspondence), but more real numbers than fractions (because it is impossible to match them up in one-to-one correspondence; any attempt you might make will leave out some real numbers).

Five year olds do not need to know this. Nor do you, probably. Middle school kids will find it fascinating. Cantor’s argument is accessible to most high schoolers (but would never occur to them, nor to me—it’s a brilliant insight).

Marilyn vos Savant is wrong when she writes, “Math doesn’t enlighten us the way literature, social studies, or art appreciation do.”