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.

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.

The Summer of Math

Hey parents! Listen closely. Do you hear that? It’s the sound of school letting out for the summer!

You’ve got your summer camps planned, your squirt guns at the ready, and you’re all set to hit the library as many times as needed to keep your kids reading all summer long.

Now you need a plan to keep their math minds active.

At Talking Math with Your Kids, we’ve got you covered.

Announcing The Summer of Math.

A small sample of the fun to be had this summer!

Here’s how it works. You can head over to the Talking Math with Your Kids store, pay for a subscription to The Summer of Math, and all summer long we’ll ship you awesome, fun stuff that will keep you and your 5—10 year old busy playing and talking math.

You’ll color, count, make patterns, designs and shapes. You’ll read together, draw, and challenge yourselves. You’ll notice. You’ll wonder. You’ll play. And when school starts back up in the fall, your kids will remember this as the best, mathiest summer ever.

The details

Each month June—September, we’ll ship you a box that contains a bunch of great stuff—at least one book, at least one related set of mathy things to play with, and at least one special surprise. For example, in June you’ll get one beautiful math coloring book, one terrific activity book, all the supplies you need for both of these, a set of spiraling pentagons (so you can make your own awesome designs like those in the coloring book), and a little something extra we cannot yet divulge.

Plus a newsletter where we’ll share additional ideas, questions, cool math stuff we’ve been doing, and reports you send us of the mathy fun you’ve had this summer.

We’ll ship the first week of each month. One week before we ship it out, we’ll send you an email letting you know exactly what’s coming your way (except for the surprise—that’s always a surprise!) You can let us know if you need to add, delete, or swap anything out. We can easily credit you for things you already have (but it’s not likely you’ll already have much of what we’ve got planned), or substitute something new and awesome for it.

We’ll have a Facebook page where we’ll share our mathy adventures and encourage you to share yours.

What are you waiting for? Click on through and join us for The Summer of Math!

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.)

Talking Shapes with Kids

I have been spending time talking with kindergarteners, first and second graders in schools about my shapes book (coming from Stenhouse, Spring 2016). Many more school visits are ahead of me. I have written up some reflections for a more teacher-ish audience than this blog attracts. If you’re interested in the ways young children talk about shapes, and in what I hear in their ideas, hop on over to the sister-blog Overthinking My Teaching for the details.

You may be delighted to learn how much math there is in the simple collection of shapes below.

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.

Ten hundred Doras

There was a while when Tabitha (five and six years old at the time) would try to get away without wearing underpants when she dressed herself. Those days are pretty much over, but I still like to make sure she has done the complete job, so I ask her from time to time.

Tabitha (7 years old): I’m dressed!

Me: Are you wearing underpants?

T: Yup—Dora the Explorer.

Don’t worry. The child is not wearing these in the picture.

Me: Nice. How do you feel about your Dora the Explorer underpants?

T: I don’t really like Dora that much, but I have a thousand of them.

Me: That’s a lot.

T: I counted them once.

Me: All one thousand?

T: No. I don’t really have a thousand. I don’t even know how to count to a thousand. Just to ten hundred.

I pause for a moment. Does she mean one-hundred-ten? Can’t be. She must know that one-hundred-eleven comes next.

Me: Ten hundred. You mean like after nine hundred is ten hundred?

T: Yeah. That’s as high as I know how to count. I don’t even know how many a thousand is.

Me: A thousand is ten hundred.

T: Oh. Cool.

A few minutes later, I get an idea. I wonder how she would write ten hundred. She needs to get out the door for school so I make it quick. I ask her to read some numbers out loud as I write them.

• 900
• 832
• 110
• 1000.

For that last one, she says one thousand.

I ask how she would write ten hundred.

She writes, “1000”.

T: It’s the same.

Me: Because I just told you that. Right. How would you have written ten hundred before I told you it was the same as one thousand?

She shrugs her shoulders. Drat. Moment lost. We talk about hundreds for a moment. One hundred, two hundred, etc. up to ten hundred.

Then I have one more.

Me: OK. Last one, then off to school. How would you read this one?

I write 10,000.

She looks for a moment. And thinks.

T: Ten….

More thinking.

T: Ten thousand?

High five!

I zip up her sweatshirt and send her out the door to catch her bus.

So What Do We Learn?

A recent research article argued that children learn a lot about place value through everyday conversation, and that kindergarteners know a lot more about the structure of the number system than parents and kindergarten teachers (on average) think they do.

Here you can see that knowledge in action. Tabitha knows that 1000 is a big and important number. She knows the pattern that allows you to keep counting by hundreds. She has not put these two pieces together. A short conversation helped her put those two pieces together, and then to extend the pattern.

Starting the conversation

This didn’t start out as a math talk. It began as a clothing inspection. But the opportunity presented itself. Listen for those times your children use numbers, and ask follow up questions about them. You won’t get this much learning out of every such conversation, but if even 10% of those opportunities turn into a little bit of learning, the interest compounds.

I promise you that.

Making eight

I am writing a book. In the process of doing this, I come across homework assignments that parents find frustrating, and that they share on social media. These almost always get me thinking, and they frequently lead to math talks with my children.

This past weekend was one such instance.

Talking Math with Your Kids is not a place to hash out the details of whether this is a well written question, or whether this was an appropriate homework assignment for this child. We can discuss that on Twitter if you like, or through my About/Contact page.

Talking Math with Your Kids is about taking opportunities to have math conversations with our children. In that spirit, I share the conversation we had in our house.

Out of the blue, I asked Tabitha (7 years old) if I could ask her a math question. It was maybe Saturday afternoon. We had nothing special going on.

Me: Tabitha, can I ask you a math question?

Tabitha (7 years old): Yes.

Me: If I have eight things, and seven of them are in one hand, how many are in the other?

T: That’s not even a math question! That’s too easy!

Me: OK. But will you answer it anyway?

T: One.

Me: OK. What if I had five in one hand?

T: And you still had 8?

Me: Yeah.

She spent a few moments thinking.

T: Three.

I had a couple other questions, which I asked and she answered. The next day, I realized that I didn’t know how she knew that second one.

She was getting ready to brush her teeth on Sunday evening when I asked whether she remembered the previous day’s conversation. She did.

Me: How did you know it was three?

T: I counted.

Me: Like this? Five, then six, seven, eight?

T: Yeah. And that’s three. But actually, I kind of already had it memorized.

Me: Oh yeah? How did you memorize it?

T: Huh?

Me: Did you try to memorize it? When I want to memorize a phone number because someone told it to me and I don’t have a pen handy, I say it over and over to myself. Did you do that with 5 + 3?

T: No! I just have counted it out a lot of times.

Now, I should also mention that I asked Tabitha, If I had 8 things, and 8 of them were in one hand, how many would be in the other? She replied Zero without much hesitation. This If I have this many in one hand, how many are in the other formulation is probably less clumsy than the If this is one part, what is the other part? formulation on the original worksheet. But the intention is the same.

So What Do We Learn?

The kind of problem Tabitha and I were working with is called Part-Part-Whole. For young children, this is different from the standard “takeaway” problem because there is no “taking away”. I didn’t eat, lose, destroy or give away any of my eight things in these problems—I just have some in one hand and some in the other.

Because Part-Part-Whole involves a different way of thinking, it’s a good idea to practice some of these problems. It helps children to build a better understanding of addition and subtraction relationships if they see all the various ways these relationships appear in their worlds.

Tabitha herself pointed out an important principle of Talking Math with Your Kids: Many things that you hope to remember, you can remember by encountering them frequently. Tabitha has never sat down with flash cards to memorize her single-digit addition facts. Yet she is in second grade and is starting to feel confident with them.

She and I talked about familiarity—how maybe learning 5 + 3 is a little like learning the name of someone you see in your neighborhood. You don’t recognize the person as being the same person the first few times you see them. But eventually, if you see them frequently enough, you do recognize them, and you might introduce yourself. Pretty soon, you know their name. And if you just can’t seem to remember it? That’s when it’s time to drill yourself. That’s when you repeat the name over and over and over.

Starting the Conversation

Ask the questions I did. This is an easy conversation to have. If your child isn’t confident with addition and subtraction facts, ask about six in one hand instead of jumping to five in one hand.

More broadly, look for Part-Part-Whole opportunities to talk about. This is an important interpretation of subtraction, and one that is often neglected. Examples include apples (Our fruit bowl has 8 apples—5 are red, how many are green?), pets (There are 8 pets on our block—5 are cats, the rest are dogs. How many dogs?), et cetera.