On Belonging, and Not

I didn’t invent Which One Doesn’t Belong. I’ve told the story of how I came to the idea many times (notably in the teacher guide to my book), but the origins lie here:

Mine isn’t even the first book with this title.

Two editions of my book Which One Doesn't Belong? and a copy of a 1970's era Sesame Street book with the same title. This older book has a drawing of Cookie Monster in front of a plate with three chocolate chip cookies and a toy ball.

It does appear to be the first whose intention is to invite and welcome diverse perspectives—to take all true ideas about ways to categorize and distinguish, and to call them all “math”.

As with all creative endeavors, I built on the work of others. I learned from it, polished and adapted it, and I offered it up to my colleagues in the hopes that they would use these ideas to invite learners into the intellectual community of mathematics. I want Which One Doesn’t Belong? to widen the range of people who feel that they belong because their ideas matter. I want Which One Doesn’t Belong? to provide a frame for helping others to see the world from new mathematical perspectives, to ask new questions, and perhaps even to help the world see new mathematics.

In everyday life, the meaning of belonging is simple. Dirty socks don’t belong in the refrigerator. A basketball shoe doesn’t belong with a hiking boot. A ball doesn’t belong on a plate of cookies.

It is in this sense that mathematicians adapted the informal language of belonging to the mathematical register as a foundational concept in set theory. 0 either belongs to the set of natural numbers, or it does not (although which of these is true depends on whom you ask). \frac{1}{2} belongs to the set of rational numbers. A square belongs to the set of all polygons.

These two senses of the term belong are what make Which One Doesn’t Belong? a powerful tool for drawing learners into mathematical conversation.

Occasionally in my journeys, I have heard from someone concerned about another sense of belonging—that of students belonging in classrooms, schools, or even in mathematics itself.

I know some things about not belonging in this sense. I don’t belong in public schools—my sense of time and my priorities are very ill-suited to my long-term survival there. I am too interested in the complexity of learning simple ideas to belong in many college math departments, and too interested in the abstractions of mathematics to belong in most education departments. I am too intellectually impulsive and impatient to belong in higher education more generally. (Have I mentioned that I got the idea for and developed the first draft of Which One Doesn’t Belong? in a day? That within five minutes of first visiting the Alphabet Forest, I determined to build the math version of it?)

Perhaps in my Which One Doesn’t Belong? work, I have had some affinity not just for those mathematical outsiders whose ideas (such as the geometry of diamonds) are not valued in textbook mathematics. Perhaps I have also identified a bit with that triangle in the upper left, who has the wrong number of sides to belong on the opening pages of this book, but who does have the right orientation (for now….it is balanced rather precariously).

Four polygons in a 2 by 2 array.
Upper left: equilateral triangle balanced on a vertex
Upper right: square balanced on a vertex
Lower left: non-square, unshaded rhombus balanced on a vertex
Lower right: non-square rhombus resting on a side

It is in this third sense of belonging that people sometimes raise concerns. Are we inadvertently sending a message to students that, in the various ways they are different from their classmates, they might not belong? The most honest answer I have is that I don’t know.

We do know that children receive all kinds of implicit and explicit messages about whether they belong in this classroom, this school, doing this mathematics.

In skilled hands, Which One Doesn’t Belong? offers opportunities to unpack those messages—to question and reject them. Opportunities to find alternative ways of conceiving of self and community, as far more nuanced, complex, and important than the sorting of socks or of triangles.

I am also sure that in clumsy or malevolent hands, Which One Doesn’t Belong? could cause real harm. Is that harm alleviated by replacing the language of belonging with the language of differences or uniqueness? I don’t know.

When people have expressed concerns to me in the past, I have had two go-to messages. First is that we must never use children as the constituents of our Which One Doesn’t Belong? sets. Second is that you should absolutely adapt these ideas to work best in your context. Use the parts of this work that will build a welcoming and diverse community of mathematicians in your classroom; modify as necessary, and please report back to the wider community so that we may learn along with you.

I have learned a tremendous amount from the opportunities I’ve had to work with a wide range of audiences—to listen to children, adolescents, parents, and teachers tell me which one doesn’t belong and why. I have also learned a lot from teachers and others in talking about this work. I look forward to seeing how these ideas grow and change with the wisdom and experience of the community.

I look forward to seeing what Which One Doesn’t Belong becomes, and what it inspires.


What Makes a Pattern?

Amazing question on Twitter yesterday.

The reason this particular question came my way is that I’m working on the next book, which will be about patterns and have the same spirit as Which One Doesn’t Belong? and How Many?

In the coming weeks, I’ll work out some of the relevant ideas in this space. I invite you to play along in the comments, on Twitter, and by using anything I share here and reporting back (If you’d rather keep your ideas, critiques and wonderings private, hit me up on the Ab0ut/Contact page).

First up, I need to share a bit of research.

Young children are able to succeed on a more sophisticated pattern activity than they are frequently encouraged to do at home or at school.

This is an important conclusion of some recent research [paywall], and it matches the kinds of findings that are common when researchers look seriously at the mathematical minds of young children.

Here’s what that claim means. There are two types of tasks that young children frequently encounter when working on patterns:

  1. Make my pattern
  2. Extend my pattern

The researchers argue that you can successfully complete these tasks without actually engaging in any kind of mathematical thinking.

Make My pattern

If you put down a pattern of alternating red and blue tiles, then ask me to make your pattern using red and blue tiles, I don’t have to notice or analyze that pattern; I just have to place a red tile next to each red one, and a blue tile next to each blue one.

Extend my pattern

Similarly, I get into the rhythm of “blue, red, blue, red” and continue the pattern without ever explicitly dealing with the repetition of “blue red” as a distinguishing feature of the pattern.

That’s the less sophisticated patterning work that children encounter at home and school. So what are the more sophisticated tasks they can be successful with?

  1. Make my pattern in a new medium
  2. Make the smallest possible version of my pattern

Make my pattern in a new medium

I make a red-blue pattern just as before, but I give you things other than red and blue tiles. I ask you to make a pattern like mine. Maybe they are different colored tiles; maybe they are also entirely different objects. Either way, you cannot just match my pattern; you have to notice the structure of the pattern and reproduce that structure with new materials.

Make the smallest possible version of my pattern

I make a tower of red and blue cubes, and I ask you to make the shortest tower you can that still has my pattern. This task also forces you to notice the structure of the pattern rather than simply matching the original.

Before I read this research article a year ago, the book’s working title was What Comes Next? and I was struggling a bit to make this question have multiple correct and meaningful answers. The current working title is What Repeats? and I’m having quite a bit of success generating conversations with images such as the following.

What patterns do you see in each image? What repeats, and how?

(A careful reader may notice that I never answered the original question. That’s because I’m still thinking about it.)


Posters Posters Posters! (plus a prize drawing!)

I am working on my next book from Stenhouse, a counting titled How Many? I took a little time away from that this summer to work on a set of shapes posters. I designed them, Tracy helped me refine them, and the Stenhouse production team made them beautiful.

Poster images

Most of the shapes posters available to teachers are very very bad. (Seriously…go ahead and do a quick image search on shapes posters….I’ll wait.) Everything is in standard orientation; the triangles are mostly equilateral; the rectangles never are, and on and on.

The Which One Doesn’t Belong? shapes posters are mathematically correct, and they also give everyone in the room something to think about. Why is there a square on the rectangle poster? Is a heart a shape? What about a spiral? Does a curve have to be curvy? Does biggest count as a property? These are the kinds of questions I meant these posters to elicit, while still serving the noble purpose of being a visual reference for important geometry vocabulary.

I am very excited to see these posters out in the world. If you’ll help me spread the word that they exist (and where to find them), I’ll do two things: (1) Express my heartfelt gratitude, and (2) Put your name in a drawing for one of two classroom math play sets.

There are three ways to enter:

  1. Tweet a photograph of these posters in the wild to the #wodbposters hashtag.
  2. Design your own Which One Doesn’t Belong? poster and tweet a photo of it to the #wodbposters hashtag.
  3. Send me a note through the About/Contact page on this blog right here, and include this set of characters: #wodbposters . (This is essential as I’ll search my inbox for it when assembling the drawing.)

Here are the rules:

  • One entry per person
  • You may tweet as many times as you like on the hashtag (In fact, please do!), but together those will count as a single entry. #unitchat
  • You may have someone else tweet or write on your behalf. This is totally fair.
  • Entries end at 11:59 p.m. Central Time on November 1.
  • The drawing takes place using a spreadsheet and random.org on November
  • Winners will be notified by whatever means they entered (I’ll tweet you if you tweeted; email if you submitted through the contact page), and we’ll connect at that time to work out a shipping address.
  • Worldwide participation is welcome.
  • A classroom math play pack includes Tiling Turtles, Spiraling Pentagons, Curvy Truchet Tiles, 21st Century Pattern Blocks, and maybe something else, depending what prototypes I’m playing around with when I put the packs together.
  • If you have already tweeted a photo of the posters in the world, you probably didn’t use the #wodbposters hashtag, which is how I’ll search for them when the drawing comes to an end, so you’ll need to do that if you want to be entered.

Got questions? Hit me with them in the comments!


Good White People

Let me tell you a quick story.

A number of years ago, I was for the first time working with an immigrant population that was new to me. I had questions. I asked those questions of a colleague who had been working in this school for a number of years, hoping that he might have insights for me.

Instead, my every question was met with “Why are you asking this about this population when you might also see this behavior in the majority population?” with an insinuation that my even asking questions involving race and culture could be seen as indications of racism.

There is a certain brand of white liberalism that believes discussing race to be racist.

This is not healthy.

Here is a book recommendation: Good White People by Shannon Sullivan.

My major takeaway from this book is that if we don’t talk to our children about race, someone else will.

Do you know which white people can be counted on to talk about race to our children?


The folks who turned out with torches in Charlottesville last weekend will be more than happy to talk to our children about race.

It is important that white people with love in their hearts do so too. Just as we can talk about math with our children without always having the right answers, we can talk about race with our children without always having the right answers. Just as our children don’t have to get every math idea correct the first time they encounter it, they don’t have to nail every nuance the first time they encounter it either.

These are challenging times. This is an important book. White people need to read it.

Which Poster Doesn’t Belong?

(Cross-posted from Talking Math with Your Kids)

Two and a half years ago, I was developing Which One Doesn’t Belong? (before Stenhouse had signed on to publish it). I went on a tour of elementary classrooms to talk with K—5 students all around the Twin Cities about these collections of shapes. I learned a tremendous amount of course, and much of that learning went into the Teacher Guide (which Stenhouse convinced me needed to exist).

I learned a lot, and I also noticed something.

Most of those classrooms had some form of shapes posters on the walls. Triangles, rectangles, squares, and rhombi were proudly and prominently displayed so that students would be surrounded by correct geometry vocabulary. Most of those shapes posters had something important (and unfortunate) in common with the shapes books in the school library and in the children’s homes.

There were rarely squares on the rectangle poster. All of the triangles were oriented with one side parallel to the ground, and most of them were equilateral. Sometimes the shapes had smiley faces. You and I know that a triangle is still a triangle, no matter its orientation. I can assure you not all elementary school children know this. While the vocabulary is good on your standard shapes poster, the math is not. (I decided not to link to examples—you can do your own search and report back if you find my claims exaggerated.)

This summer, Stenhouse is helping all of us to fix this. You can now preorder Which One Doesn’t Belong? shapes posters.

Poster images

They come as a set of eight, with an insert in the spirit of the Which One Doesn’t Belong? Teacher Guide to help you facilitate student thinking and classroom conversation as they hang in your classroom.

1. Which SQUARE doesn’t belong?

2. Which RECTANGLE doesn’t belong?

3. Which RHOMBUS doesn’t belong?

4. Which HEXAGON doesn’t belong?

5. Which TRIANGLE doesn’t belong?

6. Which POLYGON doesn’t belong?

7. Which SHAPE doesn’t belong?

8. Which CURVE doesn’t belong?

These posters are filled with good mathematics. Consider the triangle poster on top of the pile. The triangle in the lower right is the only right triangle. The one in the upper right is the only equilateral triangle. The one in the upper left is the only isosceles triangle (or is it? do equilateral triangles count as isosceles?) The one in the lower left is the only one you can’t build out of the triangle in the lower right. Students will notice side lengths, angle measures, orientation, composition and decomposition, and more properties of triangles. Some will complain that not all of them are triangles (“too pointy” or “doesn’t have a bottom”). These posters let you and your students sit with—and play with—these ideas over a period of weeks or months.

So as you plan your back-to-school classroom organizing and decoration, I hope you’ll consider making space on your walls for these posters. And I definitely hope you’ll share your students’ ideas here and on Twitter.

Available for pre-order now. They’ll ship in early August.