# Counting in downtown Saint Paul

I had my first book event today—for How Many? at Subtext Books in downtown Saint Paul. Lovely people, great little independent bookstore. You should buy some books from them.

We had a small but loyal crowd that included a three-year old and an eight-year old. The three-year old was charming, as all three-year olds are, and today she answered all yes/no questions in the affirmative. She and I talked about shapes and eggs and money. It was good times.

But I really got to get into the head of the eight-year old.

We discussed the grapefruit page below, and the unsolved mystery of whether there are exactly six grapefruit—the ones we can see directly—or more than that with at least one hiding underneath, possibly reflected in the surface of the bowl.

We moved on to the next page, which is where the real fun began.

My eight-year old conversation partner looked carefully, thought for a while, and announced that there must have been more than six grapefruit on the previous page because there are more than six on this page.

I asked, “How do you know?” and it turns out he was visually pairing the grapefruit halves on this page. He used his fingers to show me the pairs he made, but he was having trouble keeping track of their number. So when he came out with more than six pairs of grapefruit halves on this page, that meant there must have been more grapefruit in the bowl.

We flipped pages back and forth several times while sorting this out, and he finally concluded that there were six grapefruit on both pages. Children rarely have math tasks that connect this way, but they expect that the tasks should connect. It was delightful to watch this expectation play out.

Next up was the avocado page.

He thought for a bit and decided there were “seven point five avocados”. I thought I knew how he knew—same as the grapefruit—but I asked to be sure, and I was wrong.

“Three fives is fifteen, and then divide by two.”

It took a few more exchanges to extract that dividing by two makes sense here because there should be half as many whole avocados as there are half-avocados. Of course this is brilliant and important mathematics, and it arose in the context of making sense of a meaningful counting situation. Also notable is that three fives was a fact he retrieved quickly while three fours (of grapefruit halves) did not seem to occur to him.

The lesson here is that children are brilliant. They build math out of their everyday experiences, and when you offer them opportunities they apply the math they know to make further sense of their worlds.

Another lesson is that my new book—titled How Many?—is out. The best price and free shipping are at Stenhouse.com. If you read it with children, please report back and maybe leave a review at Amazon.

# Time Zones

Griffin is 13 years old and seems to be coming to the end of that early adolescent phase of rejecting everything those around him hold dear. Engaging him in math talk has taken more finesse in this phase of life.

Mostly it has involved giving him responsibility for things that involve making calculations. When he was little, we could talk collaboratively about how many tangerines are in a 3 pound bag and discuss whether this would be enough to last the family a week. Now I tend to put him in charge of getting enough tangerines to last us a week. He still has to do the same thinking, but he’s in charge.

This is not enough tangerines for a week at our house. (By the way, which is more?)

From time to time, though, we still put a mathematical idea up for discussion, and as he ages through adolescence, these conversations happen a bit more often. Yet he is still wary. Nevertheless, I persist.

We have been watching the Olympics, and we have wondered about which events are happening as we watch them, and which ones happened earlier (yet somehow happened “tomorrow”!)

Griffin was thinking about time zones, and about their implications for traveling as we wrapped up an evening this week, and made preparations for the next day.

Griffin (13 years old): So they’re 14 hours ahead of us?

Me: Yes.

G: You’d get a lot of jet lag, huh?

Me: Yeah. Maybe not as much as it looks like, though. Maybe it’s just 10 hours’ worth, going the other way.

There is a bit of a puzzled silence.

G: Wait. Really?

Me: Yeah. Well, plus a day.

G: Wait. Is this one of your mathy talks?

Me: Absolutely not.

If you’re reading this, Griff, I’m sorry (sort of). I am totally busted.

Me: Yeah. 14 hours ahead is the same as 10 hours behind, right? Just going the other way.

G: But the day would be wrong.

Me: Yeah. You have to add a day, but you don’t get jet lag because the day changes, you get jet lag because the time of day does.

G: Maybe.

He returns to packing his lunch. I go back to whatever I was doing. Putting turtles in boxes, probably.

A couple minutes later…

G: So the east coast is 23 hours behind us?

## So What Do We Learn?

Keep trying. Opportunities to talk about numbers, shapes, and patterns present themselves. Seize them and do not stop. Ask questions, think out loud. Don’t worry about whether any particular conversation goes anywhere. Just keep at it.

# How Many? An invitation to #unitchat

Make Math Playful is an unofficial slogan here at Talking Math with Your Kids. An important part of play is that there is not one right answer. Through Which One Doesn’t BelongI showed a way to make geometry playful. Now with How Many? I’m working on a way of making counting playful.

The idea has grown out of the TED-Ed video I did a while back, and the more I play with it, the more I see it in the world around me. My goal is to help parents, teachers, and especially children see it too.

Most counting tasks tell you what to count. Whether it’s Sandra Boynton’s adorable board book Doggies, or Greg Tang’s more sophisticated The Grapes of Math, the authors tell you what to count—or even count it for you.

How Many? is a counting book that leaves possibilities open and that seeks to create conversations. Creativity is encouraged. Surprises abound.

The premise is simple. Every page asks How Many? but doesn’t specify what to count. Each image has many possibilities.

An example. How many?

Maybe you say two. Two shoes. Or one because there is one pair of shoes, or one shoebox. Maybe you count shoelaces or aglets or eyelets (2, 4, and 20, respectively). The longer you linger, the more possibilities you’ll see.

It’s important to say what you’re counting, and noticing new things to count will lead to new quantities.

Another example. How many?

A few possibilities: 1, 2, 3, 4, 6, 12, 24, 36. What unit is each counting? Maybe you see fractions, too. 2/3, 4/6, 3/4, 1/12….others? What is the whole for each fraction? The number 3 shows up more than once—there are three unsliced pizzas, and there are also three types of pizza. Are there other numbers that count multiple units?

All of this leads to two specific invitations.

Let me come talk with your students.

(It turns out my schedule filled very quickly, and I’m no longer seeking new classrooms to visit right now—thanks to everyone for your support!)

If you are within an hour of the city of Saint Paul and work with children somewhere in the first through fourth grades, then invite me to come test drive some fun and challenging counting tasks with your students. I have set aside November 17 and 18 and hope to get into a variety of classrooms on those two days. Get in touch through the About/Contact page on this blog.

I’ve been using, and will continue to use and monitor, the hashtag #unitchat, for prompts and discussion of fun and ambiguous counting challenges. Post your thoughts, your own images, the observations of your own children or students, and I’ll do likewise.

How Many? A counting book will be published by Stenhouse late next year.

# Birthday Chocolate

Today is my birthday. Griffin (12 years old) gave me three chocolate bars as a gift. He gave me candy because he is deeply aware of its value in life. He gave me dark chocolate because he knows it’s my preference.

He is frequently disturbed by how slowly I eat these gifts of candy he gives me.

Here’s how my after-work greeting went this evening.

Griffin (12 years old): Happy birthday, Dad.

Me: Thanks.

G: One thing I’ve noticed about you is that you eat the candy I give you incredibly slowly.

Me: I know. But actually I ate half of one today.

G: Half of a bar, or half of all the bars?

Me: Half of one bar. And then maybe I’ll have another half tomorrow.

G: Oh brother.

Me: And since I know 3 divided by 1/2 is 6…

G: You ate one-sixth of it.

Me: And it’ll last me 6 days.

Having arrived home a bit chilly and damp from the bike ride in the 45° rain, I went downstairs for a shower and he returned to his iPod.

## So What Do We Learn?

I haven’t written a lot about this boy recently because he is in a phase of rejecting everything the adults around him care about. All adolescents go through some form of this. He is doing it with gusto.

In any case, the groundwork we’ve laid in the early years has paid off. When math is useful for his purposes, he will use it. Here, he wanted to prove his point that I am a painfully slow candy consumer. That made it important to clarify that I had not eaten half of my candy, but only half of one bar of candy.

We play around with units like this frequently. It has contributed to both children’s place value understanding, as well as their fraction work.

## Starting the Conversation

Ask frequently about the units that are attached to the numbers in your lives. When you’re cooking, ask, Should we use 3 eggs or 3 dozen eggs? Ask about how many pieces of candy a pack of Whoppers is at Halloween.

Look at these pictures—one at a time—and ask How many? Challenge yourselves to find different numbers, and different units. (For example, there are 15 avocado halves, 7.5 avocados, 8 pits, 7 holes, and 1 cutting board).

# Counting grapes

I am pretty sure I have mentioned this before, but one of my proudest achievements has been watching a “Talking Math with Your Kids” hashtag (#tmwyk) blossom on Twitter in the past few months. Now, on a nearly daily basis I (and you, if you join us over there) get to see conversational gems such as Kindergarten kids talking about Spirals and cool math prompts such as Counting Grapes.

Michael Fenton—a father and math teacher—sent this photograph into the #tmwyk world recently. Naturally, I had to talk with Tabitha and Griffin about it.

The conversation with Tabitha (7 years old), I captured on video.

Here’s the transcript:

Me: Which one of these bowls has more grapes?

Tabitha: (7 years old): [points to a bowl, probably the one on the right but hard to tell] Obviously!

Me: What do you mean, ‘obviously’?

T: I mean look at this! One, two, three, four, do you mean halfs?

There is a thoughtful pause.

T: Actually…

She points to the bowl on the left.

T: Cause these are halves

Me: But how do you know that there’s more here than here?

T: Cause look.

She uses her thumb and finger to indicate that halves of grapes are getting put into pairs to make whole grapes.

T: One, two, three, four

Now she shifts to the bowl on the left and counts the whole grapes individually.

T: One, two, three, four, five.

## So what do we learn?

The key moment is right here: I mean look at this! One, two, three, four, do you mean halfs? (This occurs 8 seconds into the video.)

That is when she notices—on her own—that half grapes are not worth the same as whole grapes. It is where she shifts her attention from items (of which there are 5 on the left and 8 on the right) to whole grapes (5 on the left, but only 4 on the right).

The rest is tidying up details. The learning happens in that one brief moment of insight.

## Starting the conversation

Ask your own child this question when you have a spare moment. Don’t correct or interrupt. Just listen. Object if their explanations are incomplete, but otherwise just listen.

## Technical notes (and acknowledgements and thanks)

This was our first video using Google Glass.

There will be many more, I am sure. I’ll write more about this in the future, and I am happy to discuss with any interested parties. (You can hit me through the About/Contact link here on the blog.)

In the meantime, I want to thank Go Kart Labs for their sponsorship and financial support. They funded most of the cost of my Google Glass through a generous donation. These folks are smart, kind and interested in the overall goal of the Talking Math with Your Kids project, which is developing a world full of intelligent, creative and curious citizens. Upstanding people who do beautiful web-design work here in Minnesota.

# Twister

Tabitha received a Twister game for her recent birthday (7 years old!) She enjoys a version of the game in which one person spins and the other follows instructions until, as Tabitha puts it with much delight, the cookie crumbles.

The players switch roles for the next round. No score is kept.

She wants to play a round one recent Sunday evening. I have been writing, so I have her set it up in the kitchen while I finish up.

She comes back to me with questions.

Tabitha (7 years old): Daddy! What’s six plus six plus six plus six?

Me: Wait. How many sixes?

T: Four.

Me: Twenty-four.

T: Yes! I counted them right!

Me: Huh?

She takes me into the kitchen to show me the Twister board.

T: See? One, two, three, four, five, six.

She is counting the green dots in one row.

T: Then one, two, three, four

She is counting the rows.

Me: So four sixes is 24. Nice. Can I show you something cool? It’s also six fours. See? One, two, three, four.

I am counting the dots in one column—each a different color.

Me: Then one, two, three, four, five, six.

I am counting the columns.

Me: So four sixes and six fours are the same.

T: Like the dominoes.

She is referring to a recent homework assignment in which dominoes were used to demonstrate that 6+4 is the same as 4+6, and that this is true as a general principle about addition.

## So what do we learn?

Rows and columns are fun, fun, fun.

Malke Rosenfeld of Math in Your Feet reminds me regularly that children love to play in structured space. She uses blue tape on the floor for her math/dance lessons and has noticed that children love to play freely in and around the spaces created by the tape (seriously: click that link, have a read and then go buy some painter’s tape!). The same thing is true for the Twister board. It creates a structured space for Tabitha to explore at a scale that allows her to use her whole body. That’s a good time for a seven-year-old.

But children don’t always notice the rows and columns in an arrangement like the Twister board. They need to learn to notice it. This is an important step on the path to learning multiplication. The fact that our conversation began with “What is 6+6+6+6 ?” tells me that Tabitha notices the rows and the columns. She knows that the answer to 6+6+6+6 should be the same as her count. By introducing the language of “four sixes” and “six fours”, I am trying to help her notice the multiplication structure underlying her ideas.

## Starting the conversation

Arrange things in rows and columns. When you do, the whole thing is called an array.

Point out arrays in the world. Count the number in each row together, and count the number of rows. Notice together whether the numbers switch if you count the number in each column and count the columns. Does eight rows of six become six columns of eight? Does this happen for all numbers?

Here are some of my favorite arrays.

# 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.

# 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.

# 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.

# The biggest number

I do not recall the beginning of this conversation, but I do recall that we were eating pizza at the dinner table when Tabitha anticipated my turn in the dicussion.

Tabitha (6 years old): I know what you’re going to say, Daddy. “Counting never ends.”

Me: I suppose that sounds like something I would say, yes.

T: What’s the biggest number, though? Googolplex?

Quick tutorial. A “googol”—spelled that way—refers to this number: $10^{100}$, or “a one followed by a hundred zeroes”.

$10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000$

It is, of course, a very big number. Far too big to be practical in any meaningful sense. The very idea of such a large number having a name is fascinating to children. Most children (in my experience) encounter one googol in their social interactions with other children. The googol does not appear in the Common Core State Standards.

A “googolplex” is $10^{10^{100}}$, or $10^{googol}$ or “a one followed by a googol zeroes”. You cannot write this number out in standard form.

You may Google googol for lots of interesting characterizations of how extremely silly this very large number is.

For example, you will not live for one googol seconds.

Indeed, the universe has not existed for one googol seconds (not even by the greatest estimates of its age—not even close).

You get the idea.

Me: Well, like you said I would say, counting never ends, so no googolplex is not the biggest number.

T: If you counted by 10,000 could you ever get to googolplex in your life?

Me: No.

T: If you counted by 11,000?

Me: No.

T: 12,000? 13,000?

Me: No. Even if you counted by googol, you couldn’t get to googolplex in your lifetime.

T: Well, what if you counted by googolplex?

Me: Well sure. It would the start of your count, wouldn’t it?

She decides to demonstrate this (Side note, we have been counting by various numbers of late).

T: Googolplex.

She smiles broadly, congratulating herself for successfully counting to what she has perceived to be the largest number.

We discuss further the existence of a largest number. Then Tabitha makes a claim that takes us in a different direction.

T: Eventually, numbers just go back to the beginning.

Me: So if you keep counting, you get to zero?

T: No.

Me: One?

T: No, Daddy! Don’t you remember there are numbers before zero?

# So what do we learn?

Big numbers are fun. Boy howdy are big numbers fun. Children love to talk about the biggest number, and whether one exists. There is all kinds of lovely thinking going on when they ask these kinds of questions.

Talking about big numbers often leads to talking about infinity. If there is no biggest number, it is because numbers go on forever. The only thing Tabitha has experience with that goes on forever is a loop. She drew on that loop metaphor in imagining that numbers go back to the beginning eventually.

# Starting the conversation

Listen for the biggest number talk. It often surfaces when children are comparing their athletic prowess (I can jump 2 sidewalk squares! I can jump 100 sidewalk squares! Pretty soon, someone is claiming to be able to jump googol or infinity sidewalk squares.)

When it surfaces, support it. Play and explore with your child. Answer questions. Ask questions. Talk about it and have fun. Look stuff up together when the questions go past your own knowledge. Shoot me a question here at Talking Math with Your Kids if I can answer any of those for you.