Counting by fives

The time I spend doing the dishes is frequently productive. Tabitha (6 years old) or Griffin (9 years old) will often linger nearby playing and talking. Sometimes they talk to me. Sometimes they talk about math, as on a recent evening.

Tabitha (6 years old): 5, 10, 15, 20, 25, 30…

She continues until 170, at which point she becomes bored.

T: Dad! I can count by hundreds. Want to hear?

Me: Yes. Yes I do.

T: One hundred, two hundred, …

She continues to nine hundred, when she pauses. I wait a few beats to see what will happen. When nothing does, I ask.

Me: What comes next?

T: Ten hundred.

Then…

T: Hey Dad, when do you get to one thousand?

Me: Awesome. Ten hundred is one thousand. That’s the word for ten hundreds.

T: What comes next?

Me: One thousand one hundred.

T: One thousand two hundred, one thousand three hundred,…

She continues. She reaches one thousand nine hundred.

Long pause.

T: Two thousand. [She smiles broadly.]

Later that evening, she is doing her weekly homework, which involves counting by fives. It is not clear from the directions whether she is to start over for each kind of five, or whether she is to continue counting all the way down the page.

count.by.fives

Tabitha decides to continue the counting, and this seems the more appropriate challenge for her.

She gets to 95.

T: How do you write one hundred?

Me: A one and two zeroes.

T: How do you write one hundred five?

Me: You tell me. You try, and I’ll tell you whether it’s right.

T: One-zero-five.

Me: How did you know that?

T: Replace the zero with a five. How do you write one hundred ten?

Me: What do you think?

T: One-one-zero.

So What Do We Learn?

The first activity was about number language. Tabitha was following the patterns in number language and puzzling over the places where the patterns break down. One hundred, two hundred, three hundred…how does one thousand pop out of this sequence?

The second activity was about writing numbers (or numeration). Tabitha had no trouble counting through one hundred five, but she wasn’t sure how to write it.

A common error is to write this: 1005. The thinking goes like this: One hundred is 100, so I’ll write that, then put a 5 at the end.

Starting the conversation

Any time a child asks if you want to hear her count, the correct answer is yes. Then listen. Listen for the challenging bits (the teens are difficult, as is anything following a ‘9’—such as 19 to 20, or 49 to 50). Be ready to help if they get stuck but also to let them think it through and try for themselves. Then talk about how they knew to do what they did. Talk about their thinking.

Post-Halloween Math Talk

File this under Talking about talking math with with your kids.

Waiting for the school bus this morning, the two adults and three children discussed last night’s Halloween events.

The neighbor girl, W (9 years old), announced that her brother, E (six years old), had gotten 90 pieces of candy for his trick-or-treating efforts. Griffin (9 years old) announced his haul of 51 pieces.

Me: Did E count each Nerd as one?

Image from Wikipedia

W: Oooo…maybe he did!

P (who is W and E’s father): We were at a house last night that had a bowl with a Take one sign. E went up, then came back and announced that he had taken three.

We told him he had to put two back.

He smiled and said, It’s a package of three!

I love this boy!

I thought for a moment about how various Halloween candies are packaged.

Me:Whoppers?

P: Yeah.

whoppers

Image courtesy of Free Photo of the Day

I am not proud that I know this sort of thing. But on the rare occasion that my extensive candy knowledge is useful, I am not going to hide it either.

So what do we learn?

We learn that there is always a follow-up question, and that the follow-up question can bring out fun stories and ideas.

The conversation could have died after E’s 90 and Griffin’s 51 pieces were announced. But I got fun stuff by asking exactly what was being counted.

We have had fun with the question of what counts as one before, when Tabitha and I talked about Eggo mini-wafflesfor example.

Starting the conversation

North American residents probably don’t need my help here. Your children probably know yesterday’s candy count cold. Ask whether the Nerds (or Whoppers or Smarties or…) count as one piece.

If Halloween isn’t a thing where you are, keep an eye and an ear open for when your children are counting things that are packaged in groups.

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?

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.”

Learning to count

I am fascinated by watching children learn to count. There are many surprising twists and turns kids take along the way.

Even more surprising, perhaps, is that what seem like crazy mistakes to us adults are completely sensible attempts at getting it right for kids.

For example…

  • In English, the pattern that occurs in the teens is complicated. “Thirteen” doesn’t sound very much like what it is: three plus ten, while “Fourteen” does.
  • Likewise, the names for the “decades”: “Twenty” means two tens and “Thirty” means three tens.
  • But once you get to twenty-one, the pattern is regular until twenty-nine.
  • We start counting at 1 (not 0), but we don’t start the decades at 21 or 31, so kids following the 1, 2, 3 pattern will often skip 20 and 30.

If you put all of this together, you might expect a typical young child who is counting “as high as I can” to:

  1. Have trouble in the teens
  2. Skip 20 in favor of 21
  3. Have more success in the twenties than in the teens, and
  4. End the count at or about twenty-nine (since the word thirty is not very predictable from the previous language patterns).

Here goes…

More fun with board games [Reports from the field]

A reader sends in a report of playing Chutes and Ladders with a 5-year old.

Of particular interest is his observation of the difference between how we structure hundreds grids in classrooms and how the Chutes and Ladders board is structured (each, notice with a ten-by-ten grid). I had not thought about that before and now have something new to play with.

hundreds.chart.1

Standard hundreds grid. Left-to-right, top-to-bottom.

Chutes and ladders board. Snaking back-and-forth from bottom to top.

Chutes and ladders board. Snaking back-and-forth from bottom to top.

Oh. Here is an important thing about Talking Math with Your Kids: When you notice differences like these, see them as opportunities to talk, ask questions, wonder and even to argue good-naturedly. It is not a problem that these grids are set up differently; it is an opportunity.

Here is rjbrow‘s report. Enjoy.

I try to play games whenever I can with my kids (ages 7, 5, and 2). Great practice with turn taking, understanding rules, making decisions etc. But like you mentioned here, games involving numbers provide a great opportunity to talk math with your kids.

I’m glad you mentioned Chutes and Ladders here too. I had written this game off because in playing it with my two older boys, frustration would ensue by the abundance of chutes that would prolong the game and sabotage their progress. After reading through your Uncle Wiggly post, I played a game of Chutes and Ladders with my 5 year old yesterday and found a couple of really nice opportunities to talk numbers.

First, each time he traveled a chute or a ladder, we had the conversation of which way to go on the new row. For example, he’d land on space 51 and travel the ladder up to space 67. So we’d talk about whether his guy should be traveling to the left or to the right. Jake figured out that if he was on space 67 he should be pointing his guy toward the 68 since it was bigger than 66. We’d also practice reading the numbers out loud. Like you mentioned, Jake can count to 100 but recognizing the written numbers is not necessarily the case. This was good practice.

You also mention that learning to count can be messy. We have a write-on number grid at home that counts 1 to 100. It’s really nice at looking at number patterns when you count by 2′s, 5′s, 10′s, etc. The numbers on this grid wrap to the next row so that all the like digits in the ones place line up vertically. Very nice. We’ve done some counting and pattern recognition using this grid. The difference with Chutes and Ladders is that the numbers wind back and forth on the way up the board. This was a shift to the counting we had done on the other grid. While this provided some nice conversation about how to work our way through this board, it did point out some messiness in how we present elementary number concepts to kids.

I hope my description makes sense. I have some pictures of our activity here:
http://brownmathwbl.blogspot.com/2013/08/chutes-and-ladders.html

Now who has a Chutes and Ladders board I can borrow to play with Tabitha and Griffin?

Uncle Wiggily

Tabitha was 3\frac{1}{2}  years old when we were playing a game of Uncle Wiggily.

In case you are not familiar with the game, I’ll briefly describe it. Uncle Wiggily is a board game with 100 spaces along a twisty path. Players draw cards; each card has a number and a brief poem. Perils and bonuses are judiciously spaced along the path. Uncle Wiggily is approximately 10% more complicated than Candy Land (which is to say, not very complicated at all!)

Tabitha: (Drawing a card for her first turn-it’s an 8) Got one Daddy!

Me: Mmm-hmmm.

T: What is it?

Me: Can you guess? Look closely.

T: (Quickly and with a big, eager smile on her face) Six!

Me: Good guess. It’s eight.

T: Oh!

Me: Can you count to eight?

T: (Bouncing her piece along the path, ending near the henhouse on the farm-themed board) One, two, three, four, five, six, seven, eight. By the cluck-cluck house!

Me: My turn. (Drawing a card-it’s a 10) What card did I choose?

T: Ten!

Me: Good. (Testing a hypothesis, I skip eight as I count) One, two, three, four, five, six, seven, nine, ten.

T: (Oblivious) My turn.

So what do we learn?

Learning to count is messy. Many things we might expect to be true about how children learn to count are not true at all.

We might expect children to learn the numerals (8) at the same time that they learn the words (eight). They do not. Notice that Tabitha counted flawlessly to eight, but did not recognize the symbol “8”.

We might expect children to learn the numerals in order, with all multi-digit numbers coming only after mastering the single-digit numbers. They do not. Tabitha recognized “10” but not “8”.

When I counted to ten, I intentionally left out eight to see whether she would notice. She did not. She could count to eight, but didn’t notice when it got left out on the way to ten. Mathematics is logical and orderly. The ways children learn mathematics are not.

This conversation came from a short video I made one day. I watched this video a year and a half later, when Tabitha was 5. After watching it, I immediately went into the kitchen where she was having a snack and counted: 1, 2, 3, 4, 5, 6, 7, 9, 10. She smiled and asked Why did you do that? (referring to counting in her ear). Then, a moment later, she said, Hey! You skipped eight!

Starting the conversation

Play games that involve numbers. Uncle Wiggily is great. So is Chutes and Ladders. Or Hopscotch. Any game that involves counting and reading numerals will give you the chance to practice these early number ideas.

While you’re playing, ask your child what number he drew, and what number you drew. If he doesn’t know, have him guess. Don’t worry about precision or correctness. Model good counting for your child. Help him count out some of his turns and let him count incorrectly on others. Have fun and don’t worry too much if he gets bored before the game is finished.

Have fun with it. Whatever you need to do to stay engaged in a couple of rounds of Uncle Wiggily is worth the effort. You can see the effort I invested in keeping myself entertained; I formulated a hypothesis about whether she would notice my own incorrect counting and tested that hypothesis.

Don’t get carried away with the hypothesis testing, though. Children do need models of correct counting. They won’t be damaged by a few experiments, of course. But you don’t want to become an unreliable source of knowledge.

The unit is the thing that you count

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 of the house.

Griffin: How many brownies did you make?

Tabitha: One big one! Mommy cut it up.

So What Do We Learn?

What makes this more than just a funny story is that Griffin and Tabitha are clearly counting different things. They are talking about different units.

When we make cookies, everyone agrees on the unit; we know what one cookie is.

But brownies are different. Tabitha seems to think that a brownie is the thing that comes out of the oven. Griffin seems to think that a brownie is what you eat in one serving.

One brownie according to Tabitha.

One brownie, according to Griffin

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

Fun follow-up question: Does the thing in this video count as one brownie?

Starting the Conversation

Anytime there are things in groups—or things being cut—is a good time to talk about units.

Grocery stores usually have express lanes where you have to have Ten items or fewerAsk your child whether someone with a dozen eggs could use that lane. What about someone with 12 apples in a bag? What if the apples are loose?

When your child asks for two slices of pizza, take one slice, cut it down the middle, smile wryly and ask whether that’s OK.

In all of these cases, the central question is What counts as one? Play with that question.

Also, watch that video together. It’s a ton of fun.