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 ...