What Could This Be? A Partial Shapes Book

We had a fun conversation on Twitter recently in which Adina R (@adinam225) asked,

What language might children who haven’t learned about right angles use to describe the angles in a rectangle and square?

Tracy Zager had some lovely things to say about how children will use examples and gestures to communicate their meaning.

Then I got to thinking about how young children may not have had many opportunities to think about the size of each angle in a rectangle.

Informally, kids will have heard a lot about how a rectangle has “four angles the same size” but if you know that, you don’t necessarily know that any particular size is necessary. Could it be four small angles, or four large angles?

I wanted to design something that would allow children to explore their ideas about that. So I started sketching out some ideas, and the result is a very early draft of a new book.

Imagine you are looking through a circular hole, and you see the blue region that is printed on the next page. What could that blue shape be? Could it be a rectangle? Does it have to be?

In order to answer those questions, it’s going to be useful to think about the angles and the side lengths of shapes you see in your head.

Ideally, a published version of this book will have a whiteboard page with a hole and come with a dry-erase marker so you can sketch your ideas before you turn the page and see a collection of shapes it could be.

For now though, you’ll just want to print the book out, sit down with a child and a pencil and have some fun talking shapes. As always, feedback welcome. Find me on Twitter, or send a note through the Contact page on this blog.

Here’s the download link.

If you like books in development, you should know about Is It or Not? Vehicles, a joyful adventure in mathematical definitions—fun for all ages!

Update, and a newsletter

It is undeniable that the pace of posting on this blog has slowed tremendously as my children have aged out of the 4-to-10 year old sweet spot of this work. I have ideas for reformatting and repurposing existing content, and for moving the blog writing forward, and I am hopeful that time will surface for this in the near future.

In the meantime, I wanted to let folks know that I started a newsletter. The first couple of issues went out before we all got sequestered in our homes, and you can read those in the archive. The next one is chambered and ready to go out soon.

So if you’re interested in semi-regular ideas and opportunities related to kids and math, mostly outside of school, then head on over to tinyletter and sign up.

I’ll also put in a plugs here in case you’re looking for something to amuse yourself and/or family with right now. Truchet Blocks are new and available at the Talking Math with Your Kids Store. They are wooden blocks, and so are fun for all ages. They also have some sophisticated geometry built in that allows for beautiful patterns and images.

Vehicle Chat [Invitation]

Note: A link and instructions for downloading and printing a draft of this book are at the end of this post.

It all started with a disagreement between Kassia Wedekind and her daughter Lulu.

Pretty soon, it wasn’t just elevators we were arguing about, and it wasn’t just Kassia and Lulu weighing in. #vehiclechat was born.

With Math On-A-Stick on the horizon, I put together a prototype concept book, and now I am here to report to you what I have learned from many dozens of conversations about the nature of vehicles.

The book begins with some easy, quick decisions. Dump truck? Yes. Salad? No. (Usually. More on this below.) Airplane? Of course! Next up are increasingly controversial cases, including elevator, horse, and broken-down bus with no wheels.

image credit: Don O’Brien on flickr.com, licensed CC-BY

There are about a dozen subconcepts pertaining to vehicles that people carry around in their heads, but which most people have never spoken out loud. Some of these subconcepts are in conflict with each other; some are independent of each other. Some people cling tightly to one or two of them; others loosely hold four or five.

Here are some examples, in no particular order:

  1. Vehicles carry people (not just things).
  2. Vehicles must have wheels.
  3. No living things are vehicles.
  4. Vehicles take people or things from Point A to Point B. (And then there is 4a. Point A and Point B must be different points from each other.)
  5. Vehicles must have an engine, motor, or some other dedicated power source.
  6. A vehicle must have—as its designed and primary purpose—transportation across the surface of the Earth.
  7. Once a vehicle; always a vehicle.
  8. If it could be a vehicle, with a little work or repair, then it is now a vehicle.

One thing we know about math learning is that we nearly always work from examples and mental images of things rather than from logical definitions of things. This is why you can have a second grader tell you in all earnestness that a triangle is a shape with three sides, and also tell you that the image below is not a triangle because it doesn’t look like one.

Knowing a definition (three-sided shape) doesn’t guarantee that you’ll use that definition for naming things. A classic paper by Vinner and Tall details this disconnect as being about concept image and concept definition, and it’s related to Kahneman’s Thinking Fast and Slow. Concept images are the tools of the fast judging brain, while concept definitions are the tools of slow analytical mind.

Somewhere into those controversial cases, there is one that makes everybody think. It’s not the same example for everyone, but each person has at least one. My vehicular conversation partner and I will cruise past airplane and tricycle, and then they will pause and smile and look off into the middle distance. That moment is the one I seek—when you notice that your quick reactions aren’t good enough and now it’s time to think—even if only for a brief moment.

Also I have learned that metaphorical examples are more comfortable for many people than close, but not-quite-right examples. This means that I’ll often hear a quick “Yes” to salad (a vehicle for nutrients into my body), but a slow “Yes” or even a “No” to elevator (because it doesn’t have obvious wheels, or has a limited range of travel, or for some other reason). 

Most of all, I have learned that these conversations are really a lot of fun. People smile. They laugh. They think of additional challenging examples. They ask “What about?” and “What if?” They imagine what other people might say, and why particular examples might be controversial.

This is a social game of negotiating meaning, and of noticing that language which seems so precise is often not that at all. Play along with us! Offer up some examples in the comments, or some additional vehicle subconcepts not listed above. Or join the conversation on Twitter.

Here’s a link to download a copy of the book. You’ll want to print two-sided, FLIPPING ON THE SHORT SIDE, and then trim top and bottom with scissors or a paper cutter. The book then assembles tidily as a 5.5 inch square.

And if you like books in development, check out a fun new book about shapes: What Could It Be?

Book Launch Party Tuesday!

A new version of How Many? is coming out on Tuesday, from Charlesbridge. It has an all new design, some new words, and mostly the same images, but is published for the home market. (Whereas Stenhouse publishes primarily for the teacher professional development market, and so is still the place to pick up the teacher guide.)

Charlesbridge did a lovely job with Which One Doesn’t Belong? earlier this year, and this one has come out beautifully as well. Both have been well reviewed in Kirkus Reviews.

And we’re celebrating with a launch party at the Red Balloon here in Saint Paul, MN. Tuesday, September 10 at 6:30 p.m. (The Red Balloon is on Grand Avenue, just west of Victoria.)

We’d love to have you join us! We’ll count and play with math toys and maybe even chat about vehicles!

Talking Patterns with Kids

I am working on another book; this one will be about patterns.

That means I need to learn about how people learn about patterns, and that means going to school.

So yesterday I packed a collection of images and asked 3rd, 4th, and 5th graders what patterns they saw.

Turtles are based on a Jos Leys and Kevin Lee image. The basket is on display at the Museum of Fine Arts in Boston.

Of course they blew me away. Perhaps a favorite moment was when a fifth grader told me that each turtle made space for another turtle, and that also each turtle needed the other turtles to make space for it.

Across the images, these students saw repeating and growing patterns. They saw patterns involving shapes, colors, numbers, and combinations of all of these.

I encourage you, the reader, to take some time with some children and ask each other these questions:

  • What patterns do you see?
  • So what repeats?
  • How does it repeat?

Then report back. Let’s learn together.

In the meantime, you should know that a new version of Which One Doesn’t Belong? came out this week. Same shapes as the original, new book design, new colors, and a few new words. Importantly, this is the first in a series of collaborations between Charlesbridge and Stenhouse by which a home version of each Stenhouse book is issued by Charlesbridge about a year after first publication.

Look for it (or even better…ASK for it) at your favorite bookseller.

Public Math

This is Molly.

This is Chris.

Molly is a math leader who lives in Vancouver, WA and started a project called Math Anywhere. Chris lives in Chicago where he supervises secondary math for the Chicago Public Schools and leaves math in places for others to find.

I met each of them through the work of this blog, and through Math On-A-Stick. We agreed that it would be fun to have some together this summer to brainstorm and prototype new forms of creative, playful math engagement for people in unexpected places.

One thing to know about me if we ever have the chance to work together is that I don’t work small or slow. One year I got the idea for Math On-A-Stick; the next year it was up and running at full scale at the nation’s second-largest state fair.

That tendency of mine turned an afternoon with Molly and Chris into a 48-hour extravaganza we’re calling the Public Math Gathering. With the generous support of Desmos and CPM, we added seven more amazing folks to our guest list and got started planning.

This is Morgan.

Morgan teaches at South High School in Minneapolis and runs a math circle at the Rondo branch of the St Paul Public Library each summer. We brought Morgan on board to help us plan and execute this work.

This is the whole team, including the four named above and our friends Aeriale Johnson, Manju Connolly, Lamia Abukhadra, Andrés Lemus-Spont, Amy Nolte, and Lara Jasien.

Together, this team includes designers, artists, elementary and secondary teachers, researchers, writers, and teacher educators.

Here’s how we have spent our time.

On Friday evening, we walked the boundaries of the Midway, Macalester-Groveland, and Frogtown neighborhoods of Saint Paul looking for opportunities for mathematical provocations. Here are some things we noticed and talked about.


On Saturday, we volunteered as a group at Math On-A-Stick, and then went off-site to do two pieces of work:

(1) Break down and critique our experiences at Math On-A-Stick, and

(2) Design new math provocations for new contexts.

Our reflections on Math On-A-Stick involved these questions:

  • Who was there?
  • Who was not there?
  • How did you notice those who there engaging with the materials and the space?
  • What are the affordances of the Minnesota State Fair as a place for designing math interactions?
  • What are the limitations?

Our design work followed from this invitation:

Let’s design place-based opportunities for [X] to engage in math provocations that are [Y]. 

We filled in many possibilities for X and for Y. For some of us, X might be children or teenagers or families or the math anxious. And Y might be delightful or memorable or identity-changing or surprising.

We moved on to prototyping, and we’ll test drive some new ideas at the Fair on Sunday morning.

In Defense of the Diamond

I had an opportunity to play math with children and parents at a local elementary school last night. (Shoutout to Oak Ridge Elementary! Thanks for hosting!) The Pattern Machines were a big hit.

One seven-year old spent some time deciding what to make and a couple of minutes idly punching buttons before she got down to work. In a short time, she declared she had made a square and presented it for my approval.

I noticed her square, as this seemed important to her, and I told her it gave me an idea. On my own Pattern Machine (leaving hers untouched—an important tenet of this work is never to take the pencil out of the child’s hand!), I made the thing below and asked her whether I too had made a square.

“No. That’s a diamond,” she declared.

I rotated my Pattern Machine 45°.

“Now?” I asked.

Now it’s a square.” she said with a knowing look.

Diamond also came up when I talked with children and adults about a page in Which One Doesn’t Belong?

I used to think that diamond was a lazy term for rhombus, but it is not. Diamond has a stable and robust meaning that is different from rhombus. On that Which One Doesn’t Belong? page, the upper right and the lower left shapes are diamonds. The lower right is not, yet each of these three is a rhombus.

Also sometimes a pentagon is a diamond.

So I propose we treat diamonds as we do other mathematical objects.

Let’s build rock solid definitions of them—definitions that we can take as shared and use to sort diamonds from not-diamonds.

Let’s investigate the consequences of those definitions.

Let’s investigate conjectures and prove theorems.

Together, let’s build a rich field of mathematical inquiry.

I’ll start us off. Some diamonds can be cut into smaller diamonds, as in the example below. Can ALL diamonds be cut into smaller diamonds? If not, which ones can and which ones cannot?

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.


Cocoa Puff or Cocoa Puffs: The language of nothing

In honor of Tabitha turning 11 this week, here’s a conversation from 6 years ago. 

We have a little family tradition. When we go grocery shopping the weekend before your birthday, you can choose one box of any cereal you want-no restrictions. In the weeks and months leading up to the grand event, much time is spent in the cereal aisle weighing the advantages of the various sugar-laden options.

The week before turning five, Tabitha nearly dropped the ball. She just grabbed the first box of anything at hand. I don’t remember what it was, but it seemed out of character for her. I reminded her of the cereals she had been coveting as recently as the previous week.

She went for the generic Cocoa Puffs.

I steered her towards the real deal. If you’re only gonna eat ’em once a year, you might as well have the sugar-addled bird bouncing off the box in front of you, right?


A sugar-addled bird

One morning shortly afterwards, we had this conversation:


Tabitha: Do I have Cocoa Puffs or Cocoa Puff in my hand?

Me: Well, you have four Cocoa Puffs.

T: [with only one in her hand now] Do I have Cocoa Puffs or Cocoa Puff?

Me: You have Cocoa Puff.

T: [huge smile] Right!

Me: [with empty hand displayed] Do I have Cocoa Puffs or Cocoa Puff in my hand?

T: [silent but smirking]

Me: Well…Is it Cocoa Puff or Cocoa Puffs?

T: [continued silence]

Me: I have zero…

T: [bigger smile]

Me: …Cocoa…

T: Puffs!

Me: Yeah. Isn’t that weird? If you have one, it’s Puff; if you have none it’s Puffs.

T: I knew that.

Me: Of course you did.

T: No! I knew that; I was showing you that [you had zero] by not saying anything-zero words!

So what do we learn?

Children listen carefully to language patterns. They do not learn a native language like a second language in school. The rules are not carefully explained to them one at a time.

Instead they listen, speak, get corrected, and try again. All of this can be tremendously fun for child and parent alike.

It is an odd quirk of English that zero is plural, grammatically speaking. We talk about having one child, but zero children. More commonly, we use no instead of zero, as in My neighbors have no children. The grammar is the same either way; saying My neighbors have no child sounds funny to our ears.

Starting the conversation

In discussing place value, zero is sometimes called a place holder. To understand that, children need to understand zero as a number. They need to understand that zero can legitimately answer the question, How many are there?

We talk a lot about zero in our house. You can too. Ask your children, “Would you rather have one cookie, two cookies or zero cookies?” Ask who has more of something, even when one of the people has none.

The Mad Hatter in Alice in Wonderland gives an example of this. The March Hare offers Alice “some more tea”. When Alice says she can’t possibly have more, since she hasn’t had any yet, the March Hare replies, “It’s very easy to take more than nothing.”

Another silly language game we play in our house is this. If you look in the pantry and see that there are three cookies left, you can report this in the following two ways: (1) “I checked the cookies; there are three left,” and (2) “There are three cookies.” If, however, there are no cookies in the pantry, these two ways of reporting the sad fact become: (1) “I checked the cookies; there are none left,” and (2) “There are none cookies.” We like to treat none as a number. There is no good reason for this; it is for personal amusement purposes only.


Tabitha again chose Cocoa Puffs on this, the week of her eleventh birthday. She is enjoying them, but she has also stated the obvious—they look like rabbit poop.

Things that give me hope

I am excited to see more and more people working hard to connect students’ informal mathematical thinking to the more formal work of schooling.

The emphasis in the school-home relationship used to be on helping kids do homework (as parodied in these five seconds of the newest Incredibles trailer).

No more! These days there are plenty of projects that seek to stimulate children’s math minds in ways that parallel what we do with literacy.

I’m thinking of the beautiful work of The Museum of Mathematics in New York City, and of Dan Finkel’s Prime Climb and Tiny Polka Dots games. I’m thinking of Malke Rosenfeld’s work, and of Bedtime Math and their associated research at the University of Chicago. and I’m thinking of Table Talk Math.

I’m thinking also of Eugenia Cheng, whose How to Bake Pi does for adults what I want parents to do for kids—show how their natural ways of thinking about their everyday worlds are deeply mathematical.

Some of the momentum for these projects can be traced back to the Cognitively Guided Instruction (CGI) research at University of Wisconsin, which demonstrated that when teachers know the informal ideas about numbers and operations that kids bring to school, those teachers are more effective at helping students learn the formal mathematics of school. The copyright on the first CGI book—titled Children’s Mathematics—is 1999, and it documents research that had been going on for some time before that.

Many of us doing this work now are deeply influenced by this work. Progress on this sort of thing is slow. Time spans are measured in decades, not months or years. But it’s a vibrant space that’s growing. I am optimistic.

Now for the point of today’s post. I want to recommend a delightful new book, Funville Adventures by A.O. Fradkin and A.B. Bishop, and published by Natural Math.

Funville Adventures involves a series of characters in a fanstatical land. Each has a magical power; these powers interact. You think you’re just following some fun and silly adventures on the playground; really, you’re thinking about one of the most important ideas of higher mathematics—functions.

Yet true to the nature of most of the projects I discussed above (and to the nature of this blog), it doesn’t matter whether you know about the relationship between the story and the mathematics. If you do, that’s great. If you don’t but are curious, there’s an addendum for that, and if you just want to stay at the level of the story, you’ll exercise your math mind thinking about the relationship between growing and shrinking, the relationship between doubling and halving, and why flipping upside down has no sibling.

Funville Adventures should be in every Talking Math with Your Kids-friendly library. I supported it as a Kickstarter; I’m a big fan of A.O. Fradkin’s blog. The book is on sale right now. More info and reviews here.