# Units of measurement

This post is edited and remixed from a post on my other blog last summer.

Loyal reader Jim Doherty wrote in to report the following conversation with his 4-year old daughter Mo.

They are on a long drive to a hotel.

Mo (4 years old): How far are we?

Jim: We are 20 minutes away.

Later, having arrived safely, the family heads to the pool. Mo is practicing the fine art of jumping from the edge of the pool into her father’s arms. An important part of this art is to increase the risk by jumping greater and greater distances.

Tabitha reconstructs a jump of considerable size for illustrative purposes.

Mo: (four years old) Back up, Daddy!

Jim: This far?

Mo: More!

Jim: Here?

Mo: More! You need to be five minutes away!

Jim: Do you mean five feet away?

Mo: No! Five minutes!

At this point, Papa Doherty is flustered. Is Mo messing with him? Is she confused? Is he at fault for answering Mo’s earlier How far? question with a time rather than a distance? What should he do?

My hunch is that Mo is not messing with her father. Instead, she has taken his cue for talking about how far, and she is playing with it. This is how children learn—they hear something and they try it out.

Here is how we might turn this conversation into a bit more math learning. Imagine Jim’s next response this way:

Jim: OK. Tell me when I’m there. But then don’t jump right away; I want to ask you a question before you do. [Daddy backs up slowly…]

Mo: OK! There!

Jim: Right. Here’s my question: Do you think it will take you five minutes to get to me from where you are?

Mo: Yes.

Jim: Do you know how long five minutes is?

Mo: That far.

Jim: No, no. Can you think of something we do together that takes five minutes?

Mo: No.

Jim: It takes us about five minutes to read [INSERT TITLE OF FAVORITE PICTURE BOOK HERE] together. Do you think it will take that much time for you to get to me?

At this point, I have no idea how Mo will respond (which is what fascinates me so much about talking math with kids). I do know that pretty soon, she is going to want to jump, and that whether that’s right away or after a few more exchanges doesn’t really matter.

What matters is that she’s been asked to think.

This line of discussion lays the foundation for thinking about distances, times and their relationships to each other. It supports Mo’s attempts to participate in the conversation about measurement.

My conversation with Tabitha about the height of our hill last summer was in a similar spirit; we worked on the meaning of height when she asked me to lie down on the hill.

Griffin wanted in on the action. Here is his jump shot.

# Peeps

This is one of my favorite tasks in recent years. The idea is that we will compare two sets of Peeps. Are there more of one color or the other?

There is so much fun to be had counting Peeps. Now that Valentine’s Day is past, Peeps (a common Easter candy) are back in stores in much of the U.S. So here we go…

In the spirit of Talking Math with Other People’s Kids Month, I report to you conversations other people had about one of these photos, as well as one Tabitha and I had. This is truly, though, a task for all ages.

## Comparisons

Each of these conversations stems from this photograph.

### Liam

Kelly Darke reports this conversation with Liam, who was 3 at the time.

Kelly: Which box has more, the pink or the purple?

Liam (3 years old): Pink.

Kelly: Why?

Liam: Because I like pink.

Kelly presses on with the other photos. Liam offers a color preference each time; sometimes preferring pink and sometimes preferring purple.

This is fine. Liam is clearly not interested—or not ready—to make numerical comparisons. He is enjoying having a talk with Mom about comparisons. Another time, he’ll be ready. In the meantime, he has the idea that comparing collections of things is something people talk about. This increases the chances that he will think about comparing collections of things.

### “Brandon”

Luke Walsh reports the following conversation with his five-year-old son, whom we will call Brandon.

Luke: Are there more pink Peeps, or purple ones?

Brandon (5 years old): The purple is more because it is taller and they ate less.

Notice the difference between a 3 year old and a 5 year old. The 5 year old is using evidence.

Brandon has two arguments here. “Taller” is not a valid one. You could have one column of three Peeps and the taller argument would give you the wrong answer. It is more sophisticated than “I like pink Peeps” but it’s not really right. This is how ideas develop, though. Height is easy to observe, and it corresponds pretty well to size and age when comparing people. So it is commonly applied to quantities, too. As usual, this partially correct answer can lead to more discussion. Luke could ask, Will the taller arrangement always have more Peeps?

“They ate less” is insightful. Brandon seems to notice that the two boxes started with the same number of Peeps, and that if more have been eaten from one box, there are fewer left. The natural follow-up question here is, How do you know fewer purple Peeps have been eaten? and then Why does fewer purple Peeps being eaten mean there are more purple Peeps?

### Tabitha

Tabitha, who was barely six years old at the time, used Brandon’s first line of thinking.

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

Tabitha (6 years old): Purple.

Me: How do you know?

T: It goes all the way to the top.

A follow up task helped to push her thinking a little bit.

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.

I have used these Peeps photos to encourage discussions of number with fifth graders, with undergraduate education majors, and with middle school math teachers. Good times for all. With the older ones—and in a large group setting—we strive not to mention the actual number of either color of Peeps, and we strive to have multiple ways to describe how we know which is more.

You can download a complete set of four comparison photos by clicking on this link [.zip]. You can also just click on the photos below to enlarge them. Your choice. Either way, they are free for you to use to encourage math talk. Please report back what you learn.

# Book shopping

Math teacher mom (and long ago former student of mine), Megan Schmidt sent in the following report for Talking Math with Other People’s Kids month…

Her husband (who is not a math teacher) and three-year-old daughter—we’ll call her Marian—are playing “store”. Marian is trading coins and marbles for books and blankets.

Marian (3 and a half years old): I want to buy a book for mommy to read.

Dad: Pick one and I’ll tell you how much it costs.

M (grabbing a small book from her book shelf): This one is new. Mommy wants to read it to me.

Dad: That one will be 3 silver coins.

M: 1, 2, 3. Now I want this one (picks a bigger book)

Dad: How much do you think this one should cost?

M: 5 coins!

Dad: How come this one is three (pointing at the small book) and this one is 5? (pointing at the larger book)

M: This book is large, the other is medium.

Megan writes that Marian is quoting Dad here and that Marian’s fondness for the number five may have more to do with her response here than a certainty that five is more than three.

## So what do we learn?

Trading stuff is a fun game to play.

You don’t need all the fancy store equipment. A few coins and a few valued objects (here books) and you’re good to go.

There is so much opportunity to mention, discuss and ask about numbers. Fun, fun, fun.

While the idea that 5 is more than 3 is not at all beyond the grasp of a three-year old, I do love Megan’s tentative attitude here. It certainly is possible that Marian considers five more valuable than any number—that the large book should cost five coins because five is the best number, even if the medium book costs 23 coins.

## Starting the conversation

A beautiful part of this conversation is when Dad asks Marian, How much do you think this one should cost?

This question invites Marian to think about and to discuss numbers. It’s lovely, easy to do and is very low risk for both child and parent. It is low risk because there is no wrong answer. Marian is free to set her own price, but thinking about what that price ought to be engages her mind in a deeper way than does simply counting out the coins.

Don’t get me wrong: counting out the coins is a lovely activity too. But How much do you think this one should cost? is a brilliant conversational move that got even more thinking from a three-year old.

# [Product Review] Tupperware Shape-O Toy

You know this thing.

This thing has been around for many, many years. You may not know that it is officially the “Tupperware Shape-O Toy” but if grew up in or near the United States anytime since about 1960, you have encountered this toy. It is the rare math-y toy that is actually awesome in the ways it was intended to be.

(See discussion of the Multiplication Machine on this blog for an example of a math-y toy that is awesome in unintended ways. See your local Target for a wide selection of math-y toys that are not awesome in any way at all.)

We had some fun on Twitter last fall when a math teacher and father, Dan Anderson, invited speculation about which shapes would be easiest and most difficult for his $1 \frac{1}{2}$ year-old to put in the holes.

[Fun fact about Dan Anderson—if you heard last year about how Double Stuf Oreos are not actually doubly stuffed, it was his classroom that got the media ball rolling.] Anyway, here is his ranking—following his son Calvin’s lead:

And here is an amusing video of a cute kid playing with one. The parental participation in the play may be a bit heavy-handed but the spirit is right—encouraging and playful.

Notice that the triangle is harder for him to fit in than the square, and that it’s tough for him to distinguish the hexagon from the pentagon.

Tons of fun to be had with this classic!

# How young children learn about numbers

“As in other areas of language development, it appears children infer the meanings of [multi-digit] numbers using whatever experiences they can access.”

This is one of several conclusions a group of researchers at Michigan State University and Indiana University drew from their study of $3 \frac{1}{2}$ through $7$ year olds (pdf). (Read the Washington Post’s report on the research here.) In particular, these researchers were studying the place value knowledge of young children, trying to understand whether they learn multi-digit numbers logically through direct study or culturally through everyday experience.

Examples of Tabitha’s recent experiences with multi-digit numbers.

Their study made clear that children absorb a lot of information about multi-digit numbers through their everyday experiences.

These researchers provide compelling evidence that young children (as young as $3 \frac{1}{2}$ years old) connect number words (fifty-seven) to numerals (57). Children can use their ideas about these numbers to identify and to compare numbers.

Talking Math with Your Kids is a project based on this premise. Children don’t need iPad apps to teach about numbers, they need conversations about the numbers in their worlds.

If we are aware of the importance of these experiences, parents can provide more opportunities for children to think about these numbers. Some examples from this blog include Days to Christmas, The Biggest Number, Uncle Wiggily, and Counting by Fives.

# 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.

# Talking math with your daughters

The conversations we have with our children affect their thinking. Of course they have their own interests, but the conversations we initiate have an impact.

The New York Times’ Motherlode blog (subtitle, Adventures in Parenting—we’ll talk about the equating of parenting with mothers another time!) quoted a University of Delaware study a while back:

Even [when their children are] as young as 22 months, American parents draw boys’ attention to numerical concepts far more often than girls’. Indeed, parents speak to boys about number concepts twice as often as they do girls. For cardinal-numbers speech, in which a number is attached to an obvious noun reference — “Here are five raisins” or “Look at those two beds” — the difference was even larger. Mothers were three times more likely to use such formulations while talking to boys.

The researchers note that these differences are not intentional. They were observed in the course of free interactive play between mothers and their children.

The potential consequences are important. The researchers speculate in the abstract to their published research article:

Greater amounts of early number-related talk may promote familiarity and liking for mathematical concepts, which may influence later preferences and career choices. Additionally, the stereotype of male dominance in math may be so pervasive that culturally prescribed gender roles may be unintentionally reinforced to very young children.

So do Tabitha proud, OK? Go ahead and use the two of us as a model for talking math with your daughters.

# If it’s true for other language…

…it’s probably true for number language too.

University of Pennsylvania researchers have studied the quality of parents’ speech to their toddlers, and its relationship to the children’s vocabulary later on. “Quality” of speech was measured by how well an adult observer could guess a common word uttered by the parent, when the observer could see the parent and child, but with sound muted.

In the researchers’ words:

Strikingly, this parent-input quality difference at child age 14–18 months [about 1.5 years] signiﬁcantly correlated with the children’s vocabularies at 54 mo [about 4.5 years].

The ways in which parents were talking to their children at age 1 had an effect on the number of words the children knew at age 4.

## The way you talk to children like this…

Tabitha at 21 months (a bit OLDER than the subjects in the beginning of the study)

…has a profound impact on what they know when they are like this…

Tabitha at 54 months, the age of the subjects at the end of the study. She is on her way to her first day of Pre-K.

It probably goes for number words and shapes, too.

So let’s get out there and talk math with those little ones!

Count stuff, use number words at every opportunity, point out and talk about shapes. Start them young. And if you haven’t started them young, start now.

It’ll be fun, I promise.

# 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…

# A circular conversation

The following conversation took place about two years ago. It is probably the first one that made me realize how important it is to talk math with my kids. Near the beginning of the conversation I noticed myself making a choice between engaging her mind and moving on to other things.

That choice—and the knowledge needed to notice it, and to follow up on it—has become interesting for me. Through this website, I hope to share what I have learned about that, and to learn more through interaction with readers. So please send reports of your conversations to me. And get those questions to me, too. You can do both through the About/Contact page.

It’s Sunday morning. Summer has arrived. We are enjoying a beautiful morning on the front porch. I am finishing my coffee. Tabitha (four years old at the time) has finished her donut.

Tabitha: [four years old] Why don’t circles have tips?

Me: What do you mean?

T: Why don’t circles have tips?

Me: What do you mean by tips? What shapes do have tips?

T: Triangles and stars. Why don’t circles have tips?

Me: Well…that’s a good question. I guess that’s part of what makes them circles. If they had tips, they wouldn’t be circles.

T: But what if a circle did have a tip?

Me: Well, then it wouldn’t be a circle. I guess what makes a circle is that it’s round. If it had a tip it wouldn’t be round.

There is a pause, during which I realize that I have not really given Tabitha my all with that explanation.

Me: Do you want the real answer?

T: Yes.

Me: OK. Here’s the real answer. See this plate?

It’s circular. Its edge is a circle, right?

T: Some plates are shaped like a fishy.

Me: Right. Good.

But this one’s circular. There’s a point in the middle of the plate; that’s called the center. All the parts of the plate on the edge are the same distance from the center. If there were a tip, then the part at the end of the tip would be farther from the center than the other parts, so it couldn’t be a circle. What really makes a circle a circle is having all parts be the same distance from the center.

T: What if there were spines?

Me: What do you mean?

T: What if there were spines all around the circle?

Me: Well then the tips of the spines would be further from the center than the base of the spines, so it wouldn’t be a circle.

T: What if they were all around the circle?

Me: Still, there would be parts at the end and parts at the base.

Did you like getting the real answer? That answer about circles being round, that wasn’t really the real answer. Did you like the real one?

T: Yes.

There is a thoughtful pause.

T: What about carousels? They are circles and they have points.

Me: I don’t understand what you mean.

T: What about carousels? They are circles. They have horses on them; those are like tips.

Me: Oh. Right. The circle is just the edge of the carousel. The horses aren’t part of the circle.

T: Oh.

Me: What got you thinking about circles, anyway?

T: [points out the window]

Me: What are you pointing at?

T: [smiles]

Me: I don’t get it.

T: The tree!

T: The bark!

T: It looks like a circle.

Me: Do you mean if you cut the trunk, the bark around the edge would look like a circle?

T: Yes.

Me: And that circle would have tips?

T: Yes.

# So what do we learn?

There is a lot in this conversation. As is often the case, when the conversation began I had absolutely no idea what she was talking about. What in the world could she mean by “why don’t circles have tips?” I work each semester with college students planning to be elementary teachers. I preach to them the importance of patient listening and asking questions to better understand what their students are telling them.

This is a message I frequently need to take to heart.

Tabitha’s questions are about making a transition from what shapes look like to what makes them what they are. She seems to want to know what makes a circle a circle.

This takes place as she thinks about the cross section of the tree in our front yard.

She knows that this would look circular, but that it isn’t a circle. She identifies a property that the tree cross-section has that a circle does not-tips, or sharp points.

I started with a crummy answer. I basically told her that Circles don’t have tips because if they did they wouldn’t be circles. And I felt guilty right away.

So I offered her a real explanation. That explanation was based on the definition of a circle, which is The set of all points a fixed distance from a common point, called the center.

This explanation was one that the average parent may not have ready at hand, though. So what do you do if you don’t know why a circle has no tips (or whether a square counts as a rectangle, or whether it’s still a right triangle if it points to the left, or…)? You model good information-seeking skills. Try to agree on what the question is (What do you mean by tips? What shapes do have tips?) Then consult books and friends and neighbors. You must know someone who has taken high school geometry more recently than you have. Maybe you have an engineer in the family, or a math teacher up the block. Your library has a librarian. Any of these people would be delighted to help out a young child with a geometry question.

And now that you’re reading this blog? You’ve got a friend ready to help. Shoot a note through the About/Contact page; we’ll get you an answer ASAP.

# Starting the conversation

This conversation was Tabitha’s idea. The only thing I did here was listen and try to understand her questions.

We can all do that.

# Postscript

Take the time to read the comments. Other parents weigh in with some lovely ideas for additional directions one could take this conversation. The key is that there is not one right conversation to have with your kids. The key is to have that conversation by asking and listening.