Season 4 | Episode 10 – What Counts as Counting? Guest: Dr. Christopher Danielson, Part 2

What Counts as Counting? with Dr. Christopher Danielson

ROUNDING UP: SEASON 4 | EPISODE 10

What counts as counting? The question may sound simple, but take a moment to think about how you would answer. After all, we count all kinds of things: physical quantities, increments of time, lengths, money, as well as fractions and decimals. 

In this episode, we'll talk with Christopher Danielson about what counts as counting and how our definition might shape the way we engage with our students.

BIOGRAPHY

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.

RESOURCES

How Did You Count? A Picture Book by Christopher Danielson

How Many?: A Counting Book by Christopher Danielson

Following Learning blog by Simon Gregg

Connecting Mathematical Ideas by Jo Boaler and Cathleen Humphreys 

TRANSCRIPT

Mike Wallus: Before we start today's episode, I'd like to offer a bit of context to our listeners. This is the second half of a conversation that we originally had with Christopher Danielson back in the fall of 2025. At that time, we were talking about [the instructional routine] Which one doesn't belong? This second half of the conversation focuses deeply on the question “What counts as counting?” I hope you'll enjoy the conversation as much as I did. 

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: So I'd like to talk a little bit about your recent work, the book How Did You Count?[: A Picture Book] In it, you touch on what seems like a really important question, which is: “What is counting?” Would you care to share how your definition of counting has evolved over time?

Christopher: Yeah. So the previous book to How Did You Count? was called How Many?[: A Counting Book], and it was about units. So the conversation that the book encourages would come from children and adults all looking at the same picture, but maybe counting different things. So “how many?” was sort of an ill-formed question; you can't answer that until you've decided what to count. 

So for example, on the first page, the first photograph is a pair of shoes, Doc Marten shoes, sitting in a shoebox on a floor. And children will count the shoes. They'll count the number of pairs of shoes. They'll count the shoelaces. They'll count the number of little silver holes that the shoelaces go through, which are called eyelets. And so the conversation there came from there being lots of different things to count. If you look at it, if I look at it, if we have a sufficiently large group of learners together having a conversation, there's almost always going to be somebody who notices some new thing that they could count, some new way of describing the thing that they're counting.

One of the things that I noticed in those conversations with children—I noticed it again and again and again—was a particular kind of interaction. And so we're going to get now to “What does it mean to count?” and how my view of that has changed. The eyelets, there are five eyelets on each side of each shoe. Two little flaps that come over, each has five of those little silver rings. Super compelling for kids to count them. Most of the things on that page, there's not really an interesting answer to “How did you count them?” Shoelaces, they're either two or four; it's obvious how you counted them. But the eyelets, there's often an interesting conversation to be had there. So if a kid would say, "I counted 20 of those little silver holes," I would say, "Fabulous. How do you know there are 20?" And they would say, "I counted." In my mind, that was like an evasion. They felt like what they had been called on to do by this strange man who's just come into our classroom and seems friendly enough, what they had been called on to do was say a number and a unit. And they said they had 20 silver things. We're done now. And so by my asking them, "How do you know? " And they say, "I counted." It felt to me like an evasion because I counted as being 1, 2, 3, 4, 5, all the way up to 20. And they didn't really want to tell me about anything more complicated than that. It was just sort of an obvious “I counted.” So in order to counter what I felt like was an evasion, I would say, "Oh, so you said to yourself, 1, 2, 3, and then blah, blah, blah, 18, 19, 20." And they'd be like, "No, there were 10 on each shoe." Or, "No, there's 5 on each side." Or rarely there would be the kid who would see there were 4 bottom eyelets across the 4 flaps on the 2 shoes and then another row and another row. Some kids would say there's 5 rows of 4 of them, which are all fabulous answers. But I thought, initially, that that didn't count as counting. After hearing it enough times, I started to wonder, “Is it possible that kids think 5 rows of 4, 4 groups of 5, 2 groups of 10, counted by 2s and 1, 2, 3, 4, all the way up to 19 and 20—is it possible that kids conceive of all of those things as ways of counting, that all of those are encapsulated under counting?” And so I began because of the ways children were responding to me to think differently about what it means to count. 

So when I first started working on this next book, How Did You Count?, I wanted it to be focused on that. The focus was deliberately going to be on the ways that you count. We're all going to agree that we're counting tangerines; we're all going to agree that we're counting eggs, but the conversation is going to come because there are rich ways that these things are arranged, rich relationships that are embedded inside of the photographs.

And what I found was, when I would go on Twitter and throw out a picture of some tangerines and ask how people counted, and I would get back the kind of thing that was how I had previously seen counting. So I would get back from some people, "There are 12." I'd ask, "How did you count?" And they'd say, "I didn't. I multiplied 3 times 4.” “I didn't. I multiplied 2 times 6." 

But then, on reflection through my own mathematical training, I know that there's a whole field of mathematics called combinatorics. Which if you asked a mathematician, "What is combinatorics?," 9 times out of 10, the answer is going to be, “It's the mathematics of counting.” And it's not mathematicians sitting around going “1, 2, 3, 4” or “2, 4, 6, 8.” It's looking for structures and ways to count the number of possibilities there are, the number of—if we're thinking about calculating probabilities of winning the lottery, somebody's got to know what the probabilities are of choosing winning numbers, of choosing five out of six winning numbers. And the field of combinatorics is what does that. It counts possibilities. 

So I know that mathematicians and kindergartners—this is what I've learned in both my graduate education and in my postgraduate education working with kindergartners—is that they both think about counting in this rich way. It's any work that you do to know how many there are. And that might be one by one; it might be skip-counting; it might be multiplication; it might be using some other kind of structure.

Mike: I think that's really interesting because there was a point in time where I saw counting as a fairly rote process, right? Where I didn't understand that there were all of these elements of counting, meaning one-to-one correspondence and quantity versus being able to just say the rote count out loud. And so one way that I think counting and its meaning have expanded for me is to kind of understand some of those pieces. But the thing that occurs to me as I hear you talk is that I think one of the things that I've done at different points, and I wonder if people do, is say, “That's all fine and good, but counting is counting.” And then we've suddenly shifted and we're doing something called addition or multiplication. And this is really interesting because it feels like you're drawing a much clearer connection between those critical, emergent ideas around counting and these other things we do to try to figure out the answer to how many or how did you count. Tell me what you think about that.

Christopher: Yeah. So this for me is the project, right? This book is an instantiation of this larger project, a way of viewing the world of mathematics through the lens of what it means to learn it. And I would describe that larger project through some imagery and appealing to teachers' ideas about what it means to have a classroom conversation. 

For me, learning is characterized by increasing sophistication, increasing expertise with whatever it is that I'm studying. And so when I put several different triangular arrangements of things—in the book, there's a triangular arrangement of bowling pins, which lots of kids know from having bowled in their lives and other kids don't have any experiences with them, but the image is rich and vivid and they're able to do that counting. And then later on, there's a triangular arrangement of what turned out to be very bland, gooey, and nasty, but beautiful to photograph: pink pudding cups. Later on, there are two triangles of eggs. And so what I'm asking of kids—I'm always imagining a child and a parent sitting on a couch reading these books together, but also building them for classrooms. Any of this could be like a thing that happens at home, a thing that happens for a kid individually or a classroom full of children led by a teacher. Thinking about the second picture of the pudding cups, my hope and expectation is that at least some children will say, "OK, there are 6 rows in this triangle and there were 4 rows previously. So I already know these first four are 10. I don't have to do any more work, and then 5 plus 6, right?” And then that demonstrates some learning. They're more expert with this triangle than they would have been previously. 

I'm also expecting that there's going to be some kid who's counting them 1 by 1, and I'm expecting that there are going to be some kids who are like, "You know what? That 6 up top and the 1 makes 7 and the 5 and the 2 make 7, and the 4 and the 3. So it's 3 sevens. There's 21.” I'm expecting that we're going to have—in a reasonably large population of third, fourth, fifth graders, sort of the target audience for this book—we're going to have some kids who are doing each of these. And for me, getting back to this larger project, that is a rich task, which can be approached in a bunch of different ways, and all of those children are doing the same sort of task. They're all counting at various levels of sophistication representing various opportunities to learn previously, various ways of applying their new learning as they're having conversations, looking at new images, hearing other people's ideas, but that larger project of building something that is rich enough for everybody to be able to find something new in, but simple enough for everybody to have access to—yeah, that's the larger project.

Mike: So one of the things that I found myself thinking about when I was thinking about my own experiences with dot talks or some of the subitizing images that I've used and the book that you have, is: There's something about the way that a set of items can be arranged. And I think what's interesting about that is I've heard you say that that arrangement can both reveal structure, in terms of number, but it can also make connections to ideas in geometry. And I wonder if you could talk a little bit about that.

Christopher: Yeah. I'll draw a quick distinction that I think will be helpful. If you've ever seen bowling pins, right? It's four, three, two, one. The one [pin] is at the front; the [row of] four is at the back. Arranged so that the three fit into the spaces between the four as you're looking at it from the front. Very iconic arrangement. And you can quickly tell that it's a symmetric triangle and the longest row is four. You might just know that that's 10. But if you take those same bowling pins and just toss them around inside of a classroom or inside of a closet and they're just lying on the floor, so they're all in your field of vision, you don't know that there's 10 right away. You have to do a different kind of work in order to know that there are 10 of them. In that sense, the structure of the triangle with the longest row of four is a thing that you can start to recognize as you learn about triangles and ultimately what mathematicians refer to as triangular numbers.

That's a thing you can learn to recognize, but learning to recognize 10 in that arrangement doesn't afford you anything when it's 10 [pins] scattered around on the floor. Unless you do a little abstraction. There's a story in the book about a lovely sixth grader who proceeded to tell me about how the bowling pin arrangement matches a way that she thinks about things. Because if she's ever going about her life, I don't know, making a bracelet or buying groceries, collecting pencils for the first day of school or whatever. If she wants to count them, and it looks like there's probably fewer than 100 but more than 5, she will grab a set of 4, a set of 3, a set of 2, a set of 1, and she'll know that's 10. Unprompted by me, except that we had this bowling pin arrangement. 

So there are ways to abstract from that. You can use these structures that you've noticed in order to do something that isn't structured that way, but the 4, 3, 2, 1 thing probably came from recognizing that 4, 3, 2, 1 made this nice little geometric arrangement. So our eyes, our brains, are tuned to symmetry and to beauty and elegance, and there is something much more lovely about a nice arrangement of 4, 3, 2, 1 than there is about a bunch of scattered things. And so a lot of those things are things that have been captured by mathematicians. So we have words for square numbers—3 times 3 is 9 because you can make 3 rows of 3 and you make something that looks nice that way. Triangular numbers, there are other figurate numbers like hexagonal numbers, but yet innate in our minds, there is an appeal to symmetry. And so if we start arranging things in symmetric patterned ways that will be appealing to our brains and to our eyes and to our mathematical minds, and my goal is to try to tap into that in order to help kids become more powerful mathematicians.

Mike: So I want to go back to something you said earlier, and I think it's an important distinction before I ask this next question. One of the things that's fascinating is that a child could engage with this kind of image, and there doesn't necessarily have to be an adult in the room or a teacher who's guiding them. But what I was thinking about is: If there is a student or a pair of students or a classroom of students, and you're an educator and you're engaging them with one of these images, how do you think about the educator's role in that space? What are they trying to do? How should they think about their purpose? And then I'm going to ask a sub-question: To what extent do you feel like annotation is a part of what an educator might do?

Christopher: Yes. One thing that teachers are generally more expert at than young children is being able to state something simply, clearly, concisely in a way that lots of other people can understand. If you listen to children thinking aloud, it is often hesitant and halting and it goes in different directions and units get left off. So they'll say, “3 and then 4 more is 8” and they've left off the fact that the 4 were—I mean, you could just easily get lost. And so one of the roles that a teacher plays can certainly be to help make clear to other students the ideas that a particular student is expressing and at the same time, often helping make it more clear for that student, right? Often a restating or a question or an introduction of a vocabulary word that seems like it's going to be helpful right now will not just be helpful to other people to understand it for the whole class, but will be helpful for the student in clarifying their own ideas and their own thinking, solidifying it in some kind of way.

So that's one of the roles. I know that there are also roles that involve—and I think about this a lot whenever I'm working with learners—status, right? Making sure that children that have different perceived status in the classroom are able to be lifted up. That we're not just hearing from the kid who's been identified as “the math kid.” So I think intellectual status, social status, those are going to be balances, right? 

I also understand that teachers have a role in making sure that children are listening to each other. If I'm working with learners, I can't always be the one to do the restating. I've got to make sure there are times where kids are required to try to understand each other's thinking and not just the teacher's restatement of that thinking. There are just so many balances. But I would say that some top ones for me, if I'm thinking about how to make choices, thinking about raising up the status of all learners as intellectual resources, making good on a promise that I make to children, which is that any way of counting these things is valid and not telling a kid, "Oh no, no, no, we're not counting 1 by 1 today" or, "Oh no, no, no, that's too sophisticated. That's too advanced of a—We can't share that because nobody will understand it." 

So making good on that promise that I make at the beginning, which is, "I really want to know how you counted." Making sure that learners are able to get better at expressing the ideas that are in their heads using language and gesture and making sure that learners are communicating with each other and not just with me as a teacher. Those seem like four important tensions, and a talented and experienced elementary teacher could probably name like 10 other tensions that they're keeping in mind all at the same time: behavior, classroom management, but also some ideas around multilingual learners. Yeah, a lot of respect for the kind of balances that teachers have to maintain and the kinds of tensions that they have to choose when to use and when to gloss over or not worry about for right now.

So you ask about annotation and, absolutely, I think about multiple representations of mathematical ideas. And so far I've only focused on the role of the teacher in a classroom discussion and thinking about gesture, thinking about words and other language forms, but I haven't focused on writing and annotation is absolutely a role that teachers can play. For me, the thing that I want to have happen is I want children to see their ideas represented in multiple ways. So if they've described for the class something in words and gestures, then there are sort of two natural easy annotations for a teacher to do or a teacher to have students do, which is, one, make those gestures and words explicit in the image. And that's where something like a smartboard or projecting onto a whiteboard—lots of technologies that teachers use for this kind of stuff—but where we can write directly on the image. So if you said you put the 1 and the 4 together in the bowling pins and then the 3 and the 2, then I might make a loopy thing that goes around the 4 and the 1, and I might circle the 3 and the 2, right? And so that adds both some clarity for students looking, but also is a model for: Here's how we can start to annotate our images. 

But then I'm also probably going to want to write 4 plus 1, maybe in parentheses, plus 3 plus 2 in parentheses, so that we can connect the 4 to the four [items] that are circled, the 1 to the one that is circled, the 4 plus 1 in parentheses, identifying that as a group, like a thing that has a mathematical purpose. It's communicating part of an idea and that that connects back. Teachers are super skilled at using color to do that, right? So 4 plus 1 might be written in red to match the red circle that goes around here, using not green because of color blindness. They're using blue to do 3 plus 2 in parentheses over here. And teachers might make other choices, right? We might sometimes use color to annotate in the image, but then just black here so that we aren't doing all of that work of corresponding for kids and are asking kids to try to do some of that corresponding work. And we might do it the other way around as well. 

So annotation as a way of adding, I think, a couple of dimensions to the conversation. And I have to shout out a fabulous teacher who I know through math Twitter. Simon Gregg is a teacher in an international school in Toulouse, France. And he has done amazing work with using and producing his own Which one doesn't belong?s, and annotating them and having kids do them; how many?; and then there are a few examples of his work with kids in the teacher guide for How Did You Count? Yeah, he's just a true master at annotation. So go find Simon Gregg on social media if you want to learn some beautiful things about representing kids' ideas in writing.

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: In the teacher guide of How Did You Count?, I make mention of which of the number talks books was most powerful for me. But if you want to take a look at that page in the teacher book and then throw a link in and a shout out to the folks who wrote it. Jo Boaler and Cathleen Humphreys wrote a book called Connecting Mathematical Ideas. It's old enough that there are some CD-ROMs in it. I don't know if there's a new edition; I'm sure used ones are available on all the places you buy used books. But the expert work that the teacher Cathy Humphreys does, as described in the book—even if you can't use the CD-ROMS in your computer—expert work at drawing out students' ideas, and then the two collaborating to reflect on that lesson, the connections they were drawing. It's been a while since I read it, but I imagine the annotations have got to come up. Fabulous resources for thinking about how these ideas pertain to middle school classrooms, but absolutely stuff that we can learn as college teachers or as elementary teachers on either side of that bridge from arithmetic to algebra.

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 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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