# 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.

# Young children are more logical than you think they are

The kids (Griffin—12 years old—and Tabitha—10 years old) and I went on our annual summer camping trip recently.

This year it was Glacial Lakes State Park in western Minnesota.

While there, we visited the swimming beach where a family that included a 3-year-old girl was also enjoying the beautiful warm day.

There were minnows swimming in the shallow water, which the 3-year-old badly wanted to capture with her net. When she stood still, the fish would approach cautiously, but every time she moved, the fish quickly bolted.

Girl (three years old): I want to catch them, Mommy! Why do they swim away?

Mom: Maybe they’re scared.

[Thoughtful pause]

Girl: But I am not a monster! I am not a monster, Mommy.

Over the next several minutes, she repeated her monster claims a number of times. Eventually—as will happen with 3-year-olds—her attention shifted to other things and the swimming and splashing continued.

## So What Do We Learn?

Young children can think logically. This runs counter to some assumptions we make about three-year-olds, but it is true.

Here is the logical truth this girl understood:

The only thing to fear is monsters.

Fish fear me.

Therefore, I am a monster.

And deeper yet, she understood what logicians call the contrapositive.

The only thing to fear is monsters.

I am not a monster.

Therefore, the fish do not fear me (and so I can catch them).

This child was not expressing horror at being considered a monster. Rather she was a little frustrated that the fish were not behaving according to the logic she knew to be true. Or perhaps she was a little frustrated with the inadequate (and illogical) explanation her mother had provided.

In any case, her logic was perfect.

We don’t really expect this of three-year-olds but we should. Just as we don’t really expect rich place value ideas in kindergartners, but we should. If we keep our ears and eyes open, we’ll see it and hear it and be able to support its growth.

# 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.

Join the fun on Twitter.

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.

# On helping children to love math

Some version of the following comes through my email Inbox every so often.

My daughter does not like maths. How can I ignite the passion for maths? She’s 8 and I feel she’s got to learn the importance of maths but how can I do it?  A teacher told her Maths is not for everyone and she believes it. Help!

Here is a version of my standard response.

Your story strikes close to my heart.

You may well know that girls are much more likely to get these kinds of messages from teachers than boys are, and they are much more likely to internalize these messages, as their teachers are much more likely to be same-gender role models.

It is all heartbreaking.

And I’ve seen these forces first-hand this year with my 9-year-old daughter. Her teacher said to her in a parent-teacher conference, “Your mind is better with words than with numbers, isn’t it?”

This, despite extensive evidence that she is a super creative mathematical thinker. A significant fraction of that evidence is documented on my blog, Talking Math with Your Kids.

With my own children, I have taken the perspective that “loving math” or even “appreciating its importance” may not be reasonable goals. Instead, being able to see math in their lives, and becoming competent mathematicians is.

Of course I would love for my children to love math, just as I would love for them to love reading. But I can’t enforce those emotions. What I can do is infuse my children’s everyday world with shapes, patterns, and numbers just as I infuse their world with words and stories.

This blog is full of concrete examples of opportunities for this. The post about hot chocolate is probably the simplest and clearest example of how parents can make simple changes to support their kids’ developing mathematical minds.

I would also recommend spending some time reading the research posts. There’s a lot of useful and interesting research work going on in math education right now, especially as it pertains to elementary-aged children, parents, and math.

Please don’t hesitate to reach out if there is anything further I can do to support you and your daughter.

I wish you both the best!

Christopher

# Tessalation: A great new book

Tessalation is a terrific new picture book by Emily Grosvenor. The story involves a little girl whose mother needs a bit of peace and quiet, so sends her outside to play. While outside, Tessa (get it?) notices shapes fitting together without gaps everywhere she looks.

I helped sponsor Tessalation on Kickstarter this spring, and our hard copies arrived last week. Naturally Tabitha (9 years old) and I read it together right away.

Here are some of the things Tabitha, Griffin (11 years old) and I noticed and discussed while reading it, and afterwards:

• The turtles are delightful.
• While they are somewhat different turtles from the ones we’ve played with around the house for the last year, they have an important characteristic in common—two noses and two tails come together in both tessellations.
• There are tessellating leaves that look an awful lot like some shapes I’ve made and we’ve played with a number of times. We saw kites and hexagons and triangles in the leaves just as we have in the pink quadrilaterals below.
• We wondered whether this object counts as a tessellation. (It’s not from the book, but Tessa set a great example for us to notice and ask about tessellations in our world.)

All in all, Tessalation is perfectly aligned with the Talking Math with Your Kids spirit. It creates a richly structured and playful space for parents and children to notice things and to converse. The language is fun. The images are beautiful. Tabitha and I highly recommend it.

Quick notes: Tessalation will be a component of August’s Summer of Math box. It’s not too late to sign up! Also, we’ll soon have a Tessalation/Tiling Turtles combo pack available. You can order the book right now from Waldorf Books, and e-books from Amazon.

# Talking Math with Your Kids update

As spring approaches, it’s time to update readers on what’s going on behind the scenes at Talking Math with Your Kids.

# The blog

The pace of posting has slowed way down in recent months. Rest assured that we’re still talking math around the house, and that my dedication to helping others do the same remains strong. I have lots to write, but not much time to write it because…

# Math On-A-Stick

Two years ago, I began to wonder how to expand the work of this blog beyond the parents who have the time, technology, and inclination to read blogs.

One year ago, I pitched an idea for this to the Minnesota State Fair.

And last summer we inaugurated what is now an annual event: Math On-A-Stick. Planning is under way for year two, with help from the Minnesota Council of Teachers of Mathematics, The Math Forum, the National Council of Teachers of Mathematics, the Minnesota State Fair, and the Minnesota State Fair Foundation.

The number one question at the Fair was Where can we buy the turtles?

At the time, the answer was “Nowhere”. We had asked permission from their designers, Jos Leys and Kevin Lee, only to cut them for Math On-A-Stick. Soon afterwards, I got permission from Jos to make and sell these turtles. I also got permission from Kevin who adapted Jos’s design for laser cutting using his own software (which is a ton of fun, and which you can buy from him) Tesselmaniac.

# The store

The Talking Math with Your Kids Store, at talkingmathwithkids.squarespace.com, opened late last fall with tiling turtles as the main offering. It is now stocked with a number of things to support parents and children in math activities and conversations—Pattern Machines, Tiling Turtles, Spiraling Pentagons, a gorgeous coloring book, and more on the way soon.

Click on through and have a look if you haven’t done so yet.

# A book

I recently submitted the final manuscript for Which One Doesn’t Belong? A Better Shapes Book. There will be both a home/student edition, and a companion guide. It is being published by Stenhouse this summer.

# More

The big ideas continue to flow, and further collaborations are in the works. Keep an eye on this space. In the meantime, you can expect a few new posts in the coming weeks as my attention shifts from book-writing mode.

And don’t forget to follow the fun on Twitter at the #tmwyk hashtag, where people share young children’s beautiful ideas and questions on a daily basis.

# Let the children play

Talking Math with Your Kids has been on something of a summer hiatus as I’ve geared up for Math On-A-Stick at the Minnesota State Fair. It has been a wild ride.

I have spent the last four days playing and talking math with kids of all ages for eleven hours a day.

My number one message coming out of this work is Let the children play.

Have a peek at our flickr photo albums to see what’s been going on. Here’s a sample (Thanks to Kaytee Reid for sharing these beautiful images).

I have been paying close attention to how children behave in this space we’ve built. I’ll just write about the plastic eggs today, but they stand in as an example for all of our activities.

When children come to the egg table at Math On-A-Stick, they know right away what to do. There are plastic eggs, and there are large empty egg cartons. The eggs go in the cartons. No one needs to give them instructions. (This is by design, by the way.)

A typical three- or four-year old will fill the cartons haphazardly. She won’t be concerned with the order she fills it, nor with the colors she uses, nor anything else. She’ll just put eggs into the carton one at a time in a seemingly random order.

But when that kid plays a second or third time, emptying and filling her egg carton—without being told to do so—she usually begins to see new possibilities. After five or ten minutes of playing eggs, this child is filling the carton in rows or columns. Or she’s making patterns such as pink-yellow, pink-yellow… Or she’s counting the eggs as she puts them in the carton. Or she’s orienting all of the eggs so they are pointy-side up.

The longer the child plays, the richer the mathematical activity she engages in. This is because the materials themselves have math built into them. The rows and columns of the egg crate; the colors and shape of the eggs; the fact that the eggs can separate into halves—all of these are mathematical features that kids notice and begin to play with as they spend time at the table.

We have seen four-year-olds spend an hour playing with the eggs.

I have observed that the children who receive the least instruction from parents, volunteers, or me are the most likely to persist. These are the children who will spend 20 minutes or more exploring the possibilities in the eggs.

The children who receive instructions from adults are least likely to persist. When a parent or volunteer says, “Make a pattern,” kids are likely to do one of two things:

1. Make a pattern, quit, and move to something else
2. Stop playing without making a pattern

We adults have a responsibility to let the children play. We can be there to listen to their ideas as they do. We can play in parallel by getting our own egg cartons out and filling these cartons with our own ideas.

But when we tell kids to “make a pattern” or “use the colors”, we are asking the children to fill that carton with our ideas, rather than allowing them to explore their own.

Here are some ideas children have explored in the last few days. I look forward to the next week’s worth of wonder. (Photos all shared by visitor and volunteers through Twitter and Intagram—handles are in the image titles. Many thanks to all for your generous sharing.)

# A tale of two conversations

Here are two conversations about hot chocolate.

The first one didn’t happen. The second one did. Read them both, then I’ll tell you about their meaning.

Both conversations begin on a cold November night in Minnesota. Unseasonably cold. Fourteen degrees, to be precise (–10 Celsius).

Zero marshmallows for me on this cold night.

Tabitha (7 years old), Griffin (10 years old) and I get in the car to head for Tabitha’s basketball practice.

# What might have been

Me: Wow! It is cold!

Tabitha (7 years old): You know what you do when it’s cold? You make hot chocolate.

Me: Ooooo! Good idea! We can do that when we get back home after practice.

T: Does it count as dessert?

Me: If you have marshmallows in it, it does.

T: I won’t have any marshmallows, then. So I can have some Jell-O.

Griffin agrees that this is the way to go, and the conversation moves on to other things.

# What actually happened

Me: Wow! It is cold!

Tabitha (7 years old): You know what you do when it’s cold? You make hot chocolate.

Me: Ooooo! Good idea! We can do that when we get back home after practice.

T: Does it count as dessert?

Me: If you have two marshmallows in it, it does.

T: I’ll have zero marshmallows in mine, then, so I can have some Jell-O.

Griffin (10 years old): I’ll have one marshmallow, and a small serving of Jell-O. Wait, no! I know! I’ll cut a marshmallow in half!

I presume that this is in order to maximize his allowable Jell-O serving, while still retaining some marshmallow in his hot chocolate. It’s a scheme nearly as complicated as credit default swaps.

# So what do we learn?

One small difference changed the course of the conversation—my use of a number word. I could have said, “It counts as dessert if you have marshmallows in it.” But I did say, “It counts as dessert if you have two marshmallows in it.”

Using numbers—two marshmallows instead of just marshmallows—invited the children to talk about numbers. It invited them to use numbers to maximize their benefit. It invited them to think about numbers.

This invitation is important.

A few years back, researchers paid careful attention to the ways preschool teachers talked with their students. Those teachers who used more number words and concepts as they talked with children stimulated greater growth in math than those who used less math talk.

This was not a study about math instruction; it was a study about the math language that these teachers used when they weren’t teaching math. “Yes, you three may help me.” versus “Yes, you may help me.” is the sort of difference that matters.

Using number words and math concepts in everyday speech invites children to notice and to think about number. That’s what Talking Math with Your Kids is all about.

# A happy report from the field

Every once in a while, someone shares with me a lovely story of a conversation that they had with their kid that was inspired by the work on this blog. These stories are tremendously satisfying to me because they remind me that isn’t just me and my kids, and that it doesn’t just come naturally. Talking math with your kids is something you can learn.

Today’s report is from Zoe Ryder White, whom I have not met, but who heard about this site from a friend of the project, and who gave me permission to share it.

[I] used some tidbits already this morning – [My daughter] A. was making a giraffe and wanted each leg to be two wooden spools long. At first she wasn’t sure how many total she’d need, but when I asked how many a giraffe has, she quickly figured out the total was 8.

Before reading the talk math with our kids stuff I would’ve probably just said yep, you got it- but we ended up having a great conversation about all the different ways you could figure that problem out. SO FUN.

I am determined to raise a math-confident and math-curious kid. All the work you’re doing in your research is already making a concrete change! Thanks : )

That is the power of asking a follow up question. It is the power of asking, “How do you know that?”

# What makes a sandwich

The 3-year old daughter of fellow Minnesotan, fellow math teacher and friend Megan Schmidt made the following proclamation a couple weeks back.

This simple claim has led to lots of fun conversation. Let’s call the daughter veganmathpup (since she is the daughter of Twitter’s @Veganmathbeagle), or VMP for short.

All discussions with VMP are filtered through her mom via Twitter. All discussions with my own children are my best recollections of the recent silliness.

## Open faced sandwiches

Veganmathpup’s assertion boils down to this: A sandwich needs these things: (1) a slice of bread, (2) a filling, (3) another slice of bread. I wanted to know about open-faced sandwiches. Is an open-faced sandwich properly called a sandwich? VMP was silent on this matter. So I asked Tabitha.

Tabitha (7 years old): That counts as a half-sandwich…actually more than a half-sandwich.

So an open face sandwich is not actually a sandwich for Tabitha. This gave me a chance to introduce the term misnomer.

A week or so later, VMP claimed that “2, 3, 4 or 5 cookies can make a sandwich”. This was a clear violation of the earlier rule here. Two cookies, no filling? How can this be a sandwich when “It takes three things to make a sandwich”?

So I asked about Oreos. Does VMP think of an Oreo as 1 cookie? 2 cookies? Most importantly, Is an Oreo a sandwich? Megan related the following conversation.

Megan: [Handing VMP an Oreo]  VMP, I have a question.  Is this a sandwich?

VMP (3 years old):  [Examining carefully] Um, no.  It’s not.

Me:  Why isn’t this a sandwich?

VMP:  It doesn’t have things, like a burger.

Me: [Handing her two Oreos stacked on top of one another] Is this a sandwich?

VMP:  [Examining even closer this time] No.  it doesn’t have stuff in it. It needs lots of stuff inside like a burger to be a sandwich.  I want a burger.  Let’s get one [face full of oreos]. We won’t tell Daddy.

So many follow up questions I was unable to ask here. Does a Double Stuf Oreo have enough stuff inside to count as a sandwich? What about a Mega Stuf Oreo?

A Mega Stuf Oreo contains approximately 3.1 times the Stuf of a regular Oreo.

## Marshmallows

Then the plot thickened.

Megan went on to report that, even after opening the Oreo to demonstrate that there is a filling, VMP rejected the Oreo as a sandwich because the filling is white.

Allow me to summarize:

• Three things are required for a sandwich.
• Unless they are cookies, in which case you only need two.
• An Oreo is one cookie, so is not a sandwich.
• Even if you want to call the Oreo wafers cookies and the Stuf the filling an Oreo is still not a sandwich because the filling is white.
• The filling in a sandwich is properly referred to as a burger.

I saw a flaw in the logic, though.

Marshmallows are white.

I HAVE BEEN FOILED BY A THREE YEAR OLD!

## So what do we learn?

Children have ideas.

Children use their minds. They think about things.

We can contribute greatly to our children’s learning by probing those ideas.

Formulating precise definitions is an important part of doing mathematics. Sorting things into examples and non-examples is part of this process. It really doesn’t matter whether we are sorting shapes (square, not square) or food (sandwich, not sandwich). And when the child is three years old, it really doesn’t matter whether she is consistent in her sorting.

What matters is that she is thinking in this mathematical way.

## Starting the conversation

You can do as I did. Tell your child that another child says it takes three things to make a sandwich. Ask your child whether she agrees. Then ask about open face sandwiches and about Oreos.

But the bigger picture is important here too. There is a useful habit to develop as a parent—ask follow up questions when your child makes proclamations.

Other conversations we have had in this vein include Spirals, Circles and Armholes.