# 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.

# [Product review] The bathtub

Talking Math with Other People’s Kids month continues…

Today we pay tribute to the family bathtub, and its profound contribution to family math talk over the centuries.

Don’t laugh! Is yours more perfect?

Dad and loyal reader Jon Hasenbank reports some math talk at bathtime with his own 5 year old son, whom we will call Isaiah.

Isaiah is in the bathtub, having a lovely time. He has stacked his bath-toy Elmo on top of his bath-toy Cookie Monster.

isaiah (5 years old): Look! His eyes are peeking out!

He did not report further details to me.

But he did demonstrate an important principle of talking math with your kids—It’s not a conversation until you, as a parent, participate. When Jon turned Isaiah’s observation into a wondering, he set the stage for some good math talk.

The bathtub is great for this!

Tabitha has complained about the depth of her bath in the past—always that it is not deep enough. “It’s not even one foot deep” she has wailed as her toes stick out of the water. “Is it one hand deep?” I have asked. And—as with Jon and Isaiah—we have been off and running on a lovely exploration of measurement.

# February is “Talking Math with Other People’s Kids” month

You won’t be hearing much from Griffin and Tabitha this month. Instead, you’ll hear from other children and their parents who have talked math and have shared their conversations with me.

It will be a ton of fun to get a peek into these other households, and to see how frequently ideas and questions about number and shape come up in life with young children.

I would love to hear your own reports, and to gather a collection of stories representing diverse families, cultures, languages and experiences. Shoot me a note describing conversations you have participated in or witnessed. I’ll feature as many of them here as I can.

Let’s kick things off with an example of a father and his five-year old daughter, and how Twitter helped them talk a bit more math than they otherwise might have…

Andy is a dad in Minneapolis. Let’s call his daughter Martine. Andy tweeted me on Friday (January 31).

Here is how this sort of thing would go in our house.

Martine (5 years old): If tomorrow is February first, does that mean today is February 0th?

Dad: Yeah, I guess we could call it that. If we do, what would yesterday have been?

M: February negative one.

Et cetera. At some point, the conversation would go somewhere else. Or if she’s still interested, I might give it a twist with a question like this.

Dad: So if today is both January 31 and February 0, and if tomorrow is February 1, shouldn’t it also have a January name?

I would be probing Martine’s double-naming idea for each day. And then…

Dad: Hey! I know! If today is both a January and a February day, then tomorrow should be both a February and a March day, right? What is tomorrow’s date in March?

As I mentioned, the conversation may very well have broken down by this point. But these what if questions are the things that turn a cool observation into a conversation. That conversation is where we turn kids’ minds on.

Martine said that the day before February 0 would be February negative 1. Andy reports—and this is important—never having explicitly discussed negative numbers with Martine. No number lines, no backwards counting past 0.

But surely they have talked about the weather. Below-zero temperatures have been as common as snowflakes in Minnesota this year. Talking about the weather may have planted the idea. Then the calendar was an opportunity to make a connection.

All of this leads to two important ideas about talking math with kids:

1. It’s not a conversation until you, as a parent, participate. Martine noticed something (Jan. 31 could be Feb. 0). Andy turned it into a conversation when he asked about the previous day.
2. These conversations are facilitated by availability of objects. Turning the calendar became a learning opportunity for Andy and Martine. No calendar, no conversation.

And you can read about other conversations facilitated by objects in these previous posts:

# Playlists

Parenting is a tremendous amount of work. Within that work are beautiful moments of love and joy. For Tabitha and me, these moments often involve music. We had an impromptu dance party in the kitchen the other night that began with my putting on some music to do dishes by.

When Griffin was born, I began maintaining playlists. Each year, I collect songs that the kids liked, or that I was listening to, or that reminded me of them in some way. Some years I remember to burn these to CDs to share with family members. But I never delete them.

That first playlist is titled “Griffin year 1”.

Do you see the math here?

Tabitha (5 years old at the time): Are you done with my year 5 playlist yet?

Me: Yes. I finished that when you turned 5. Now I’m working on your year 6 playlist; I’m collecting a bunch of songs during the year and it will be done on your birthday.

T: Why isn’t this my year 5 playlist?

Me: Good question. Well…your first playlist I started before you turned one…

T: When I was zero years old.

Me: Right. Then when you turned one, I started your year 2 playlist. That’s what it means to be 1 year old; that your first year is over and you’re in your second year.

So when will I work on your year 10 playlist?

T: When I’m 9.

Me: How do you know that?

T: I don’t know. I just do…

So you’re working on Griffy’s year 9 playlist now? [Her brother Griffin was 8 years old at the time.]

T: Will you still be working on them when I’m an adult?

Me: I would gladly still work on them when you’re an adult. I don’t know if you’ll want me to at that point, but if you do, I will.

T: Oh, I will. Hey! Can you play my favorite song about the flower?

And so began the dance party.

## So what do we learn?

There is an important idea about counting and measuring here. During your first year, you are zero years old. Something that measures within the first inch on a ruler is zero inches long (plus a fraction).

This is not obvious by any means. If you have ever been frustrated by the fact that the 1900s were the 20th century, or that ours is the 21st, you understand the problem.

## Starting the conversation

These are fun things to talk about. Almost always, going back to the beginning is helpful for making sense of things. So ask your child about 2014 being in the 21st century, and why they think that is.

Or maybe start making an annual playlist. You won’t regret it.

# 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.

# Multiplication and rectangles

I want to suggest a lovely post by somebody else.

It is written by a math teacher who converses with his niece (who is 7 years old) about rectangles and multiplication. As an example, the rectangle below shows that 6×3 is 18. Or is it that 3×6 is 18? That becomes the focus of part of the conversation.

The girls’ parents look on as the discussion unfolds.

At one point, the math teacher stops the mother who is trying to intervene to help the child see that 4×3 is the same as 3×4. And this leads to the lovely sentence in the blog post:

I understand that it is not obvious to non-teachers that not every encounter with mathematics needs to reach “fruition.”

What he means by this is that children can learn from thinking about math, even if they don’t end up with the right answer, and even if they do not experience the full story (here, that multiplication is commutative, which means AxB=BxA for all possible numbers).

Another fabulous math teacher, Fawn Nguyen, told me, “I dare say that it’s not obvious to teachers also.”

Finally, non-math teacher parents may be interested to learn that—consistent with Fawn’s observation—a regular piece of feedback I get from math teachers on my writing here is how impressed they are by my ability to not worry about Tabitha and Griffin getting right answers.

# 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.

# M&Ms

Dessert is a good time to get the children’s attention for a little math talk.

A few weeks back, a smallish serving of M&Ms was about to be given to each child, from a large one-pound bag.

In keeping with my assertion that a day should never pass without asking my kids at least one how many? question, I asked Griffin to choose the size of the serving (but unbeknownst to him that this was the purpose.)

Me: Give me a number between 10 and 20.

Griffin (eight years old): What’s the point?

Me: I won’t tell you until you choose.

G: I won’t until I know why.

This is my own doing. I have long told both children that people need to have reasons for asking you to do things, and that satisfying these reasons is more important than following directions blindly. This is an important element of problem-solving and critical thinking. It does have consequences; I understand this.

Me: Tabitha, pick a number between 10 and 20.

Tabitha (five years old): Twelve.

Me: OK. That’s how many M&Ms you each get for dessert.

G: Oh, then I pick 20.

Me: No. The first number I heard. That’s the one I’m using.

G: You should use the biggest.

Me: Nope. The first.

T: Next time, I should choose….thirteen.

This is beautiful, is it not?

I love the realization that things had not worked out for her maximal benefit. I love that she knows some thinking needs to be applied to the situation.

And I love dearly that the result of this thinking is an increase of a single M&M. Griffin comes to her rescue.

G: No, Tabitha! It’s between 10 and 20!

T: Oh. I should choose…nineteen.

# So what do we learn?

This was totally devious on my part, and I do not recommend that you behave this way with your children. We do learn, though, that strategic thinking with numbers is something to be learned. The strategy of thinking through the biggest possible number within the given constraints is not obvious to young children. Looking for a bigger number is a prerequisite to thinking hard about the biggest possible number.

We also learn, of course, that I am a horrible person.

# Starting the conversation

Again, I do not encourage you to manipulate your children in this way. Although in my own defense, neither 12 nor 20 M&Ms is such a bad deal for 5- and 8-year olds near bedtime.

The pick-a-number game is fun for lots of things, though. Taking turns (whoever gets closest to the number I wrote down gets the first turn) is a classic example, but you can think up lots of your own. After the picking, talk about the selection. What would have been a better choice, knowing what you know now? What would have been a worse choice? Why did you pick the number you did? Et cetera. Listen to your children’s strategies and share your own.

# 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

## 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.”

# More fun with board games [Reports from the field]

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.

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

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: