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.

An evening ride on the A train

I was in New York City for a conference recently. I arrived at JFK airport on a Monday evening and took public transportation into the city. I observed the following scene.

At about 8:30 p.m., a mother and her (roughly) 6-year old twin girls board the train and sit quite near me.

Mom opens the first girl’s backpack to check whether her homework has been completed.

Child 1: I did it. Oh…on that one, I forgot to write the 6.

[Child 1 proceeds to talk aloud and hold up her fingers]

These are my daughter's fingers. I did not photograph the children on the subway.

These are my daughter’s fingers. I did not photograph the children on the subway.

Child 1: Two and four is…1, 2, 3, 4, 5, 6!

Mom checks a couple of other things and puts the materials back in Child 1’s backpack. She opens Child 2’s backpack.

Mom: Why don’t you have a math book?

Child 2: I do! It’s Go Math!

Mom: Where is it?

Child 2: At school. Teacher says it never leaves school.

Mom: So you get worksheets?

Child 2: Uh huh.

Mom: What’s the point of having a math book, then? Oh well.

She flips through the worksheets, but does not discuss their content with the children. The worksheets appear to her (and to me) to be a disconnected jumble, which the child has completed.

About ten minutes later, the girls are acting up a little bit. Nothing major, but they are getting loud and silly. They are clearly getting on Mom’s last nerve. Mom has expressed to them how tired she is.

Another ten minutes pass, with the girls just barely keeping it together. They get off the train at their stop.

So what do we learn?

A major goal of my writing here and in my book is to help parents notice opportunities to support their children’s mathematical development, and to take advantage of these opportunities.

Traditionally as parents, we look over our children’s homework and keep an eye our for errors. We help our children when they are stuck. If everything is correct and complete, we move on. All of that is good. It is important to help our children when they are stuck. It is important to check their homework. All good.

I want to point out the opportunity this mother had when her daughter was talking about the neglected six.

Child 1 said, “I forgot to put the 6,” and then demonstrated the relevant addition problem on her fingers.

One way to seize this opportunity is to say, “Oh, good! I like that you showed how you know 2 and 4 is 6. What are some other ways you can make 6?”

That conversational technique will bear fruit with kids 95% of the time. When you notice an idea, praise the child for expressing it and ask a follow-up question, the child will nearly always answer that question. If the question suggests that there are multiple answers, the child will usually keep thinking beyond their first idea.

Continuing the conversation would provide the mother in my story with two things:

(1) Increased mathematics thinking on the part of her children, and

(2) less misbehavior later on.

Yes it takes energy to carry on a conversation of any kind with your children when everyone is tired. But it takes less energy to do that than to keep them quiet when they don’t want to be.

I understand, of course, that not everyone knows how to do this. That is what we’re working on here.

Starting the conversation

This is an example of starting a conversation by listening and asking follow up questions.

Some conversations parents initiate. Others children initiate. When children initiate them, the conversation only moves forward if we listen and ask questions.

I will restate an important conversational move in this scenario:

Oh, good! I like that you showed how you know 2 and 4 is 6. What are some other ways you can make 6?