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

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The idea of comparing items and looking for similarities and differences has
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been explored by many math educators that said,
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Christopher Danielson has taken this idea to new heights inspired by Sesame
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Streets. One of these things is not like the others. Christopher wrote the book,
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which one Doesn't Belong? In this episode,
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we'll ask Christopher about the which one doesn't belong routine and the
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features that make it such a powerful learning experience for students.
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Welcome to the podcast, Christopher. I'm excited to be talking with you today.
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Thank you for the invitation. Delightful to be invited.
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I would love to chat a little bit about the routine, which one doesn't belong.
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So I'll ask a question that I often will ask folks, which is,
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if I'm a listener and I don't have prior knowledge of that routine,
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how would you describe it for someone?
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Yeah.
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Sesame Street back in the day had a routine called Which one Doesn't Belong.
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There was a little song that went along with it, and for me,
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the iconic Sesame Street image is Grover is on the
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stairs up to the brownstone on the Sesame Street set,
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and there are four circles drawn in a two by two grid in chalk on the wall.
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And there are a few of the adults and a couple of the puppets sitting around and
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they're asking Grover and singing the song, which one of them doesn't belong.
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There are four circles, three of them are large and one is small,
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or maybe it's the other way around. I don't remember.
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So there's one right answer, and Grover is thinking really hard,
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think real hard as part of the song they're singing to him,
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he's under a lot of pressure to come up with which one doesn't belong in.
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Fortunately, Grover Succeeds, Grover's a hero,
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but what we're wanting kids to attend to, there is size.
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There are three things that are the same size. All of them are the same shape,
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three that are the same size, one that has a different size.
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They're wanting to attend to size. Lovely.
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This one doesn't belong because it is a different size,
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just like my underwear doesn't belong in my socks drawer
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because it has a different function. I mean, it's not for me. There is,
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we could talk a little bit about this in a moment.
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The belonging is in that mathematical and everyday sense of objects and whether
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they belong.
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So that's the Sesame Street version through a long chain of math
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educators.
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I came across a sort of tradition that had been flying
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along under the radar of rethinking that with the idea being that
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instead of there being one property to attend to,
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we're going to have a rich set of shapes that have rich and interesting
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relationships with each other. And so which one doesn't belong?
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Depends on which property you're attending to.
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So the first page of the book that I published called, which one Doesn't Belong,
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has four shapes on it. One is an equilateral triangle standing on a vertex.
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One is a square standing on a vertex. One is abus,
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a non square rhombus standing on its vertex and it's not colored in.
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All the other shapes are colored in.
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And then there is the same non square thrombus colored in, resting on a side,
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all sort of simple shapes that offer
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simple introductory properties,
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but different people are going to notice different things.
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Some kids will hone in on that.
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The one in the lower left doesn't belong because it's not colored in.
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Other kids will say, well,
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I'm counting the number of sides or the number of corners.
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And so the triangle doesn't belong because all the others have four and it has
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three. Others will think about angle measure, they'll choose a square.
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Others will think about orientation.
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I've been taken a task by a couple of people about this.
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Kindergartners are still thinking about orientation as one of the properties.
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So the shape that is in the lower right on that first page is a rhombus resting
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on a side instead of on a vertex.
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And kids will describe it as the one that feels like it's leaning over or that
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has a flat bottom or it's pointing up into the right and all the others are
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pointing straight up and down. So that's the routine.
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And then things as with how did you count as with how many?
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As you page your way through the book, things get more sophisticated.
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And for me, the entry was a geometry book because when my kids were small,
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we had sort of these simplistic shapes books,
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but really rich narrative stories in picture books that we could read. And it
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was always a bummer to me that we'd read these rich stories about characters
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interacting. We'd see how their interactions,
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their conflicts relate to our own lives,
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and then we'd get to the math books and it would be like triangle,
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always equilateral, always on a side square,
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never a square on the rectangle page.
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Rectangle gets a different page from square.
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And so we understand culturally that children can deal with and
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are interested in and find fascinating and imaginative rich narratives,
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but we don't understand as a culture that children also have rich math minds.
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So for a long time I wanted there to be a Better Shapes book and there are some
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better shapes books. They're not all like that,
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but they're almost all like that.
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And so I had this idea after watching one of my colleagues here in Minnesota,
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Terry Weiberg, this routine, he was doing it with fractions,
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but about a week later I thought to myself, Hey, wait a minute,
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what if I took Terry's idea about there not being one right answer,
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but any of the four could be and combine that with my Wish for a Better Shapes
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book. And along came which one doesn't belong as a Shapes book.
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So there's a square and a rectangle on the same page.
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There are shapes with curvy sides and shapes with straight sides on the same
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page and kids have to wrestle with or often do wrestle with what does it mean to
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be a vertex or a corner?
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A lot of really rich ideas can come out of some well chosen simple examples.
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I chose to do it in the field of geometry,
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but there are lots of other mathematical objects as well as non mathematical
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objects. You could apply the same mathematical thinking too.
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So I think you have implicitly answered the question that I'm going to ask.
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If you were to say at the broadest level,
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regardless of whether you're using shapes, numbers,
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images,
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whatever the content is that an educator selects to put into the two by
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two, that is structurally the way that which one doesn't belong is set up,
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what's it good for?
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What should a teacher think about in terms of this will help me or will help my
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students? Fill in the blank.
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How do you think about the value that comes out of this?
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Which one doesn't belong structure and experience.
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Multidimensional for me,
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I don't know if I'll remember to say all of the dimensions,
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so I'll just try to mention a couple that I think are important.
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One is that I'm going to make you a promise that whatever mathematical ideas you
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bring to this classroom during this routine are going to be valued.
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The measure of what's right,
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what counts as a right answer here is going to be what's true,
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not what I thought of when I was setting up this set.
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I think there is a lot of power in making that promise and then in
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holding that promise, it is really, really easy.
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All of us have been there as teachers that we make an instructional promise to
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kids that then there comes a time where it either inadvertently or we make a
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decision to break that promise.
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I think there's a lot of costs to that. I know from my own experience as a
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learner,
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from my own experiences as a teacher that there can be a high cost to that.
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So valuing ideas, I think this is a space I love having.
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Which one doesn't belong as a time that we can set aside for the measure of
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what's right is what's true.
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So when children are making claims about this one in the upper right doesn't
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belong, I want you to for a moment try to think like that person,
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even if you disagree, that that's important.
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And so teachers have to play that role. Also. Where that comes up a lot is in,
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especially when I'm talking with adults,
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if I'm talking to parents about which one doesn't belong,
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often parents who don't identify as math people or who explicitly identify as
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non-math people will say that one in the lower left,
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it's not colored in. But I don't think that really counts in that moment.
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Kids are less likely to make that apology,
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but adults will make that apology all the time.
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And in that moment I have to both bring the adult in as a mathematical thinker,
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but also model for them what does it look like when their kid chooses something
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that the parent doesn't think counts. So for me,
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the real thing that which one doesn't belong is doing is teaching children,
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giving children practice and expertise,
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therefore learning about a particular mathematical practice,
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which is abstraction that when we look at these sets of shapes,
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there are lots of properties. And so we have to for a moment,
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just think about number of sides, and if we do that,
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then the triangle doesn't belong because of the other four.
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But as soon as we shift the property and say, well,
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let's think about angle measures,
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then the ways that we're going to sort those shapes the relationships that they
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have with each other changes. And that's true with all mathematical objects.
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And you can do that kind of mathematical thinking with non mathematical objects.
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One of my favorite, which one doesn't belong sets,
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is there's a donut chocolate donut. There's a coffee cup,
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one of those speckled blue camping metal coffee cups.
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There's half a hamburger bun with a bunch of seeds on top and then there is a
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square everything bagel. And so as kids start thinking about that, they're like,
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well, if we're thinking about holes, the hamburger bun doesn't have a hole.
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If we're thinking about speckling, the chocolate donut isn't speckled.
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If we're thinking about whether it's an edible substance,
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the coffee cup is not edible. And so that's that same abstraction.
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If we pay attention to just this one property that forces a sort,
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if we pay attention to a different property,
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we're going to get a different sort. And that's one of the practices of
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mathematicians on a regular basis.
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So regular that often when we're doing mathematics,
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we don't even notice that we're doing it.
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We don't notice that we're asking kids to ignore all the other properties of the
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number two except for its evenness right now. If you do that,
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then two and four are like each other.
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But if we're supposed to be paying attention to primally as to a prime number,
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then two and four are not like each other. All mathematical objects,
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all mathematicians have to do that kind of sort on the objects that they're
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working with.
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I had a college algebra class at the community college while I was working on
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which one doesn't belong.
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And so I was test driving this with graphs and my students.
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I could still see Rosalie in the middle of the room room full of 45 adults
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ranging from 17 to 52. And I'm this 45-year-old college instructor,
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and we have three parabola and one absolute value function.
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So parabola y equals X squared.
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It's that nice curving swooping thing that goes up at one end down to a nice
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bowl and then up again, there's one that's upside down.
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I think there's one pointing sideways.
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And then an absolute value function is the same idea,
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except it's two lines coming together to make a bowl sort of a very sharp bowl
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instead of being curves. And we got this lovely, which one doesn't belong,
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right?
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So we've got this lovely collection of them and Rosalie like her eyebrows are
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getting more and more knitted as this conversation goes on.
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So finally she raised her hand, I call on her and she says, Mr. Daniels,
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and I get that all of these things are true about these,
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but which ones matter,
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which is a fabulous question that within itself holds a lot of tensions
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that Rosalie is used to being in math class and being told what
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things she's supposed to pay attention to. And so in some ways it's sort of
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disturbing to have me up there and I get that up there front of the classroom
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valuing all these different ways of viewing these graphs because she's like,
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which one is going to matter?
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When you ask me this question about something on an exam,
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which ones matter? But truly,
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the only intellectually honest answer to her question is, well, it depends.
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Are we paying attention to direction of concavity?
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Then the one that's pointing sideways doesn't count any one of these is it
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depends on whether you're studying algebra or whether you're studying geometry
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or topology. And I did give her, I think,
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I hope what was a satisfying answer after giving her the true but not very
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satisfying answer of it depends, which is something like, well,
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in the work we're about to do with absolute value functions,
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the direction that they open up and how steeply they open up are going to be the
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things that we're really attending to and we're not going to be attending as
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much to how they are or not like parais,
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but seeing how they have some properties in common with these parais is probably
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going to be really useful for us.
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That actually makes me think of one,
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a statement of what I think is really powerful about this. And then two,
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a pair of questions that I think are related.
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It really struck me Rosalie's question how different the
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experience of engaging with a,
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which one doesn't belong is from what people have traditionally considered math
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task where there is in fact an answer, right?
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There's something that the teacher's like, yep, that's the thing.
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Even if it's perhaps obscured by the task at first,
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ultimately oftentimes there is a thing and a which one doesn't belong is a very,
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very different type of experience.
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So that really does lead me to two questions.
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One is what is important to think about when you're facilitating A,
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which one doesn't belong experience?
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And then maybe even the better question to start with is what's important to
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think about when you're planning for that experience?
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Facilitating is going to be about making a promise to kids.
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That measure of what's right is what's true.
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I'm interested in the various ways that you're thinking and doing all the kind
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of work that we discussed, but now in this context of geometry,
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or in my case in the college algebra classroom,
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in the context of algebraic representations planning,
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I have been so deeply influenced by the work of
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Peg Smith and her colleagues and the five practices for facilitating
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mathematical conversations. And in particular,
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I think in planning for these conversations, planning a set,
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when I'm deciding what shapes are going to go in the set or how I'm going to
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arrange the eggs and the egg carton,
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or how many half avocados am I going to put on the cutting board?
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I'm anticipating one of those practices.
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What is it that kids are likely to do with this? And if I can't
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anticipate anything interesting that they're going to do with it,
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then either my imagination isn't good enough and I better go try it out with
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kids,
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or my imagination is absolutely good enough and it's just kind of a junkie thing
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that's not going to take me anywhere and I should abandon it. So over time,
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I've gotten so much better at that anticipating work because I have learned,
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I've become much more expert at what kids are likely to see.
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But I also always get surprised in a sufficiently large group of kids,
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somebody will notice something or have some way of articulating differences
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among the shapes.
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Even these simple shapes on the first page that I haven't encountered before,
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and I get to file that away again for next time.
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That's learning that gets fed back into the machine,
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both for the next time I'm going to work with a group of kids,
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but also for the next time I'm sitting down to design an experience.
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You have me thinking about something else,
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which is what closure might look like in an experience like this.
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Because I'm struck by the fact that there might be some really intentional
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choices of the items in the which one doesn't belong.
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So the four items that end up being there,
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it may be designed to drive a conversation around a set of properties or a set
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of relationships,
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and yet at the same time be open enough to allow lots of kids to be right in the
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things that they're noticing. And so if I've got a which one doesn't belong,
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that kind of is intended to draw out some ideas or have kids notice some of
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those ideas and articulate them, what does closure look like?
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Because I could imagine you don't know what you're going to get necessarily from
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kids when you put a, which one doesn't belong in front of them.
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So how do you think about different ways that a routine or experience like this
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might close for a teacher and for students?
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Yeah,
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I think one of the best roles that a teacher can play at the end of a which one
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doesn't belong conversation is going back and summarizing the
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various properties that kids attended to.
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Because as they're being presented and maybe annotated,
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we're noticing them sort of one by one,
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and we might not have a moment to set them aside.
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It might take a minute for a kid to draw out their ideas about the orientation
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of this shape,
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and it might take a little bit and some clarification with another kid about how
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they were counting sides.
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They might not have great words for sides or corners and it's gestures and we're
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all trying to figure things out. And so by the time we figured that out,
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we've forgotten about the orientation answer that we had before.
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So I think a really powerful move,
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one of many that are in teachers' toolkits is to come back and say, all right,
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so we looked at these four shapes and what we noticed is that if you're paying
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attention to how this thing is sitting on the page to its orientation,
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which direction it's pointing,
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then this one didn't belong and Susie gave us that answer. And then another
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thing you might pay attention to, another property could be the number of sides.
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If you're paying attention to the number of sides the triangle doesn't belong,
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and we got that one from Brent, right?
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And so run through some of the various properties.
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Also noticing along the way that there were two reasons to pick the triangle as
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the one that doesn't belong. It might be the sides,
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and it might be you might have some other reason for picking it.
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That isn't the number of sides for kindergartners,
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the number of corners or vertices,
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and the number of sides are not yet obviously the same as each other.
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So for a lot of kindergartners that feels like two answers rather than one.
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Older audiences are more likely to know that that's going to be the same.
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So yeah,
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I think the being able to come back and state succinctly after we've had this
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conversation, valuing each of the contributions that came along,
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but also being able to compare them. Maybe we're writing them down as part of
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our annotation. There might be other ways that we do that.
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But I think summarizing so that we can look at this set of ideas that's been
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brought out altogether, I think is a really powerful way.
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One other quick thing about designing, which is I hear this a lot from teachers.
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They're saying, okay, so we're studying quadrilaterals. So I made a,
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which one doesn't belong with four quadrilaterals?
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And nobody noticed that they were all quadrilaterals,
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to which I say they didn't notice because you didn't contrast that property.
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So if there's a property you want to bring out,
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you better make sure I think that you have three things that have it and one
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that doesn't, or vice versa, three that don't, and one that does,
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because then that's a thing for kids to notice.
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They're not going to notice what they all have in common.
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That's not the task we're asking them.
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So if you want to make one about Quadrilaterals, throw a Pentagon in there.
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Love it.
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So the question that I typically will ask any guest before the close of the
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interview is,
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what are some resources that might grab onto be they yours
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or other work in the field that you think is really powerful that supports the
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kind of work that we've been talking about?
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What would you offer to someone who's interested in continuing to learn and
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maybe to try this out?
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So we've referred to number talks. That talks and number talks.
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Those are both phrases that can be Googled. There are three books,
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which one Doesn't Belong? How Did You Count? All published by Stenhouse,
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all available as a hardcover book,
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hardcover student book or home picture book.
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So for listeners, just so you know,
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we're going to add links to the resources that Christopher referred to in all of
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our show notes for folks Convenience. Christopher,
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I think this is probably a good place to stop. Thank you so much for joining us.
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It's absolutely been a pleasure chatting with you.
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Yeah, thank you for the invitation for your thoughtful prep work and
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support of both the small and the larger projects along the way.
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I appreciate that.
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I appreciate all of you at Bridges and the Math Learning Center.
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You do fabulous work.
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This concludes part one of our discussion with Christopher Danielson.
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Christopher is going to join us again later this season where we'll have a
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conversation about the nature of counting and how an expanded definition of
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counting might help support students later in their mathematical journey.
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I hope that you'll join us for this conversation.
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This podcast is brought to you by the Math Learning Center and the Meyer Math
19:11
Foundation dedicated to inspiring and enabling all individuals to
19:16
discover and develop their mathematical confidence and ability.