Which One Doesn't Belong Routine with Dr. Christopher Danielson

The idea of comparing items and looking for similarities and differences has been explored by many math educators. Christopher Danielson has taken this idea to new heights. Inspired by the Sesame Street song "One of These Things (Is Not Like the Others)," Christopher wrote the book Which One Doesn't Belong? In this episode, we'll ask Christopher about the routine of the same name and the features that make it such a powerful learning experience for students.

Christopher Danielson started teaching in 1994 in the Saint Paul (MN) Public Schools. He earned his PhD in mathematics education from Michigan State University in 2005 and taught at the college level for 10 years after that. Christopher is the author of Which One Doesn't Belong?, How Many?, and How Did You Count? Christopher also founded Math On-A-Stick, a large-scale family math playspace at the Minnesota State Fair.

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Mike Wallus: The idea of comparing items and looking for similarities and differences has been explored by many math educators. That said, Christopher Danielson has taken this idea to new heights. Inspired by Sesame Street's [song] "One of These Things (Is Not Like the Others)," Christopher wrote the book Which One Doesn't Belong? In this episode, we'll ask Christopher about the Which one doesn't belong? routine and the features that make it such a powerful learning experience for students.

Well, welcome to the podcast, Christopher. I'm excited to be talking with you today.

Christopher Danielson: Thank you for the invitation. Delightful to be invited.

Mike: I would love to chat a little bit about the routine Which one doesn't belong? So, I'll ask a question that I often will ask folks, which is: If I'm a listener, and I don't have prior knowledge of that routine, how would you describe it for someone?

Christopher: Yeah. Sesame Street, back in the day, had a routine called Which one doesn't belong? There was a little song that went along with it. And for me, the iconic Sesame Street image is [this:] Grover is on the stairs up to the brownstone on the Sesame Street set, and there are four circles drawn in a 2-by-2 grid in chalk on the wall. And there are a few of the adults and a couple of the puppets sitting around, and they're asking Grover and singing the song, "Which One of Them Doesn't Belong?" There are four circles. Three of them are large and one is small—or maybe it's the other way around, I don't remember. So, there's one right answer, and Grover is thinking really hard—"think real hard" is part of the song. They're singing to him. He's under kind of a lot of pressure to come up with which one doesn't belong and fortunately, Grover succeeds. Grover's a hero.

But what we're wanting kids to attend to there is size. There are three things that are the same size. All of them are the same shape, three that are the same size, one that has a different size. They're wanting to attend to size. Lovely. This one doesn't belong because it is a different size, just like my underwear doesn't belong in my socks drawer because it has a different function. I mean, it's not—for me there is, we could talk a little bit about this in a moment. The belonging is in that mathematical and everyday sense of objects and whether they belong.

So, that's the Sesame Street version. Through a long chain of math educators, I came across a sort of tradition that had been flying along under the radar of rethinking that, with the idea being that instead of there being one property to attend to, we're going to have a rich set of shapes that have rich and interesting relationships with each other. And so Which one doesn't belong? depends on which property you're attending to.

So, the first page of the book that I published, called Which One Doesn't Belong?, has four shapes on it. One is an equilateral triangle standing on a vertex. One is a square standing on a vertex. One is a rhombus, a nonsquare rhombus standing on its vertex, and it's not colored in. All the other shapes are colored in. And then there is the same nonsquare thrombus colored in, resting on a side. So, all sort of simple shapes that offer simple introductory properties, but different people are going to notice different things. Some kids will hone in on that. The one in the lower left doesn't belong because it's not colored in. Other kids will say, "Well, I'm counting the number of sides or the number of corners. And so, the triangle doesn't belong because all the others have four and it has three." Others will think about angle measure, they'll choose a square. Others will think about orientation. I've been taken to task by a couple of people about this. Kindergartners are still thinking about orientation as one of the properties. So, the shape that is in the lower right on that first page is a rhombus resting on a side instead of on a vertex. And kids will describe it as "the one that feels like it's leaning over" or that "has a flat bottom" or "it's pointing up and to the right" and all the others are pointing straight up and down. So that's the routine. And then things, as with "How Did You Count?" as with "How Many?" As you page your way through the book, things get more sophisticated. And for me, the entry was a geometry book because when my kids were small, we had sort of these simplistic shapes books, but really rich narrative stories in picture books that we could read. And it was always a bummer to me that we'd read these rich stories about characters interacting. We'd see how their interactions, their conflicts relate to our own lives, and then we'd get to the math books, and it would be like, "triangle: always equilateral, always on a side." "Square: never a square on the rectangle page." Rectangle gets a different page from square. And so, we understand culturally that children can deal with and are interested in and find fascinating and imaginative rich narratives, but we don't understand as a culture that children also have rich math minds.

So, for a long time I wanted there to be a better shapes book, and there are some better shapes books. They're not all like that, but they're almost all like that. And so, I had this idea after watching one of my colleagues here in Minnesota, Terry Wyberg. This routine, he was doing it with fractions, but about a week later I thought to myself, "Hey, wait a minute, what if I took Terry's idea about there not being one right answer, but any of the four could be, and combine that with my wish for a better shapes book?" And along came Which One Doesn't Belong? as a shapes book. So, there's a square and a rectangle on the same page. There are shapes with curvy sides and shapes with straight sides on the same page, and kids have to wrestle with or often do wrestle with: What does it mean to be a vertex or a corner? A lot of really rich ideas can come out of some well-chosen, simple examples. I chose to do it in the field of geometry, but there are lots of other mathematical objects as well as nonmathematical objects you could apply the same mathematical thinking to.

Mike: So, I think you have implicitly answered the question that I'm going to ask. If you were to say at the broadest level, regardless of whether you're using shapes, numbers, images—whatever the content is that an educator selects to put into the 2-by-2, that is structurally the way that Which one doesn't belong? is set up—what's it good for? What should a teacher think about in terms of "This will help me or will help my students…," fill in the blank. How do you think about the value that comes out of this Which one doesn't belong? structure and experience?

Christopher: Multidimensional for me. I don't know if I'll remember to say all of the dimensions, so I'll just try to mention a couple that I think are important.

One is that I'm going to make you a promise that whatever mathematical ideas you bring to this classroom during this routine are going to be valued. The measure of what's right, what counts as a right answer here, is going to be what's true—not what I thought of when I was setting up this set. I think there is a lot of power in making that promise and then in holding that promise. It is really, really easy—all of us have been there as teachers—[to] make an instructional promise to kids, [but] then there comes a time where it either inadvertently or we make a decision to break that promise. I think there's a lot of costs to that. I know from my own experience as a learner, from my own experiences as a teacher, that there can be a high cost to that. So valuing ideas, I think this is a space. I love having Which one doesn't belong? as a time that we can set aside for the measure of "what's right is what's true." So, when children are making claims about this one in the upper right doesn't belong, I want you to for a moment try to think like that person, even if you disagree that that's important. And so, teachers have to play that role also.

Where that comes up a lot is in, especially when I'm talking with adults, if I'm talking to parents about Which one doesn't belong?, often parents who don't identify as math people or who explicitly identify as nonmath people, will say, "That one in the lower left, it's not colored in. But I don't think that really counts." In that moment, kids are less likely to make that apology, but adults will make that apology all the time. And in that moment, I have to both bring the adult in as a mathematical thinker but also model for them: What does it look like when their kid chooses something that the parent doesn't think counts? So, for me, the real thing that Which one doesn't belong? is doing is teaching children, giving children practice and expertise—therefore learning—about a particular mathematical practice, which is abstraction. That when we look at these sets of shapes, there are lots of properties. And so, we have to for a moment, just think about number of sides. And if we do that, then the triangle doesn't belong because of the other four. But as soon as we shift the property and say, "Well, let's think about angle measures," then the ways that we're going to sort those shapes, the relationships that they have with each other, changes. And that's true with all mathematical objects.

And you can do that kind of mathematical thinking with non-mathematical objects. One of my favorite Which one doesn't belong? sets is: There's a doughnut, a chocolate doughnut; there's a coffee cup, one of those speckled blue camping metal coffee cups; there's half a hamburger bun with a bunch of seeds on top; and then there is a square everything bagel. And so, as kids start thinking about that, they're like, "Well, if we're thinking about holes, the hamburger bun doesn't have a hole. If we're thinking about speckling, the chocolate doughnut isn't speckled. If we're thinking about whether it's an edible substance, the coffee cup is not edible." And so that's that same abstraction. If we pay attention to just this one property, that forces a sort. If we pay attention to a different property, we're going to get a different sort. And that's one of the practices of mathematicians on a regular basis. So regular that often when we're doing mathematics, we don't even notice that we're doing it. We don't notice that we're asking kids to ignore all the other properties of the number 2 except for its evenness right now. If you do that, then 2 and 4 are like each other. But if we're supposed to be paying attention to primality as to a prime number, then 2 and 4 are not like each other. All mathematical objects, all mathematicians have to do that kind of sort on the objects that they're working with.

I had a college algebra class at the community college while I was working on Which One Doesn't Belong?, and so, I was test-driving this with graphs and my students. I can still see Rosalie in the middle of the room—a room full of 45 adults ranging from 17 to 52, and I'm this 45-year-old college instructor—and we have three parabolas and one absolute value function. So, a parabola is "y equals x squared." It's that nice curving swooping thing that goes up at one end down to a nice bowl and then up again. There was one that's upside down. I think there was one pointing sideways. And then an absolute value function is the same idea, except it's two lines coming together to make a bowl, sort of a very sharp bowl, instead of being curved. And we got this lovely Which one doesn't belong?, right? So, we've got this lovely collection of them. And Rosalie, her eyebrows are getting more and more knitted as this conversation goes on. So finally, she raises her hand. I call on her, and she says, "Mr. Danielson, I get that all of these things are true about these, but which ones matter?" Which is a fabulous question that within itself holds a lot of tensions that Rosalie is used to being in math class and being told what things she's supposed to pay attention to.

And so, in some ways it's sort of disturbing to have me up there, and I get that, up there in front of the classroom valuing all these different ways of viewing these graphs because she's like, "Which one is going to matter when you ask me this question about something on an exam? Which ones matter?"

But truly, the only intellectually honest answer to her question is, "Well, it depends. Are we paying attention to direction of concavity? Then the one that's pointing sideways doesn't count." Any one of these is, it depends on whether you're studying algebra or whether you're studying geometry or topology. And I did give her, I think—I hope—what was a satisfying answer after giving her the true but not very satisfying answer of "It depends," which is something like, "Well, in the work we're about to do with absolute value functions, the direction that they open up and how steeply they open up are going to be the things that we're really attending to, and we're not going to be attending as much to how they are or are not like parabolas. But seeing how they have some properties in common with these parabolas is probably going to be really useful for us.

Mike: That actually makes me think of, one, a statement of what I think is really powerful about this. And then, two, a pair of questions that I think are related.

It really struck me—Rosalie's question—how different the experience of engaging with a Which one doesn't belong? is from what people have traditionally considered math tasks where there is in fact an answer, right? There's something that the teacher's like, "Yep, that's the thing." Even if it's perhaps obscured by the task at first, ultimately, oftentimes there is a thing and a Which one doesn't belong? is a very, very different type of experience.

So that really does lead me to two questions. One is: What is important to think about when you're facilitating a Which one doesn't belong? experience? And then, maybe even the better question to start with is: What's important to think about when you're planning for that experience?

Christopher: Facilitating is going to be about making a promise to kids. That measure of "what's right is what's true." I'm interested in the various ways that you're thinking and doing all the kind of work that we discussed but now in this context of geometry, or in my case in the college algebra classroom, in the context of algebraic representations.

Planning. I have been so deeply influenced by the work of Peg Smith and her colleagues and the five practices for facilitating mathematical conversations. And in particular, I think in planning for these conversations, planning a set—when I'm deciding what shapes are going to go in the set, or how I'm going to arrange the eggs in the egg carton, or how many half avocados am I going to put on the cutting board—I'm anticipating one of those practices: What is it that kids are likely to do with this? And if I can't anticipate anything interesting that they're going to do with it, then either my imagination isn't good enough, and I better go try it out with kids or my imagination is absolutely good enough and it's just kind of a junky thing that's not going to take me anywhere, and I should abandon it. So over time, I've gotten so much better at that anticipating work because I have learned, I've become much more expert at what kids are likely to see. But I also always get surprised. In a sufficiently large group of kids, somebody will notice something or have some way of articulating differences among the shapes, even these simple shapes on the first page, that I haven't encountered before. And I get to file that away again for next time. That's learning that gets fed back into the machine, both for the next time I'm going to work with a group of kids, but also for the next time I'm sitting down to design an experience.

Mike: You have me thinking about something else, which is what closure might look like in an experience like this. Because I'm struck by the fact that there might be some really intentional choices of the items in the Which one doesn't belong? So, the four items that end up being there, [they] may be designed to drive a conversation around a set of properties or a set of relationships—and yet at the same time be open enough to allow lots of kids to be right in the things that they're noticing.

And so, if I've got a Which one doesn't belong? that kind of is intended to draw out some ideas or have kids notice some of those ideas and articulate them, what does closure look like? Because I could imagine you don't know what you're going to get necessarily from kids when you put a Which one doesn't belong? in front of them. So, how do you think about different ways that a routine or experience like this might close for a teacher and for students?

Christopher: Yeah, I think one of the best roles that a teacher can play at the end of a Which one doesn't belong? conversation is going back and summarizing the various properties that kids attended to. Because as they're being presented and maybe annotated, we're noticing them sort of one by one. And we might not have a moment to set them aside. It might take a minute for a kid to draw out their ideas about the orientation of this shape. And it might take a little bit and some clarification with another kid about how they were counting sides. They might not have great words for "sides" or "corners," and [instead they use] gestures, and we're all trying to figure things out. And so, by the time we figured that out, we've forgotten about the orientation answer that we had before.

So I think a really powerful move, one of many that are in teachers' toolkits, is to come back and say, "All right, so we looked at these four shapes, and what we noticed is that if you're paying attention to how this thing is sitting on the page, to its orientation, which direction it's pointing, then this one didn't belong, and Susie gave us that answer. And then another thing you might pay attention to, another property could be the number of sides. If you're paying attention to the number of sides the triangle doesn't belong, and we got that one from Brent, right?" And so run through some of the various properties.

Also, noticing along the way that there were two reasons to pick the triangle as the one that doesn't belong. It might be the sides, and it might be, you might have some other reason for picking it that isn't the number of sides. For kindergartners, the number of corners, or vertices, and the number of sides are not yet obviously the same as each other. So, for a lot of kindergartners that feels like two answers rather than one. Older audiences are more likely to know that that's going to be the same.

So yeah, I think that being able to come back and state succinctly after we've had this conversation—valuing each of the contributions that came along, but also being able to compare them, maybe we're writing them down as part of our annotation. There might be other ways that we do that. But I think summarizing so that we can look at this set of ideas that's been brought out altogether, I think is a really powerful way.

One other quick thing about designing, which is—I hear this a lot from teachers, they're saying, "OK, so we're studying quadrilaterals. So, I made a Which one doesn't belong? with four quadrilaterals. And nobody noticed that they were all quadrilaterals." To which I say, "They didn't notice because you didn't contrast that property." So, if there's a property you want to bring out, you better make sure, I think, that you have three things that have it and one that doesn't. Or vice versa—three that don't, and one that does—because then that's a thing for kids to notice. They're not going to notice what they all have in common because that's not the task we're asking them. So, if you want to make one about quadrilaterals, throw a pentagon in there.

Mike: Love it.

So, the question that I typically will ask any guest before the close of the interview is, what are some resources that educators might grab onto, be they yours or other work in the field that you think is really powerful, that supports the kind of work that we've been talking about? What would you offer to someone who's interested in continuing to learn and maybe to try this out?

Christopher: So, we've referred to number talks. "Dot talks" and "number talks," those are both phrases that can be googled. There are three books, Which One Doesn't Belong?, How Many?, How Did You Count?—all published by Stenhouse, all available as a hardcover book, hardcover student book, or home picture book.

Mike: So, for listeners, just so you know, we're going to add links to the resources that Christopher referred to in all of our show notes for folks' convenience.

Christopher, I think this is probably a good place to stop. Thank you so much for joining us. It's absolutely been a pleasure chatting with you.

Christopher: Yeah, thank you for the invitation, for your thoughtful prep work, and support of both the small and the larger projects along the way. I appreciate that. I appreciate all of you at Bridges and The Math Learning Center. You do fabulous work.

Mike: This concludes part one of our discussion with Christopher Danielson. Christopher is going to join us again later this season, where we'll have a conversation about the nature of counting and how an expanded definition of counting might help support students later in their mathematical journey. I hope that you'll join us for this conversation.

This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.

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