Latest / Rounding Up / What Counts as Counting? with Dr. Christopher Danielson

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What Counts as Counting? with Dr. Christopher Danielson · Rounding Up

• 0:03
  What counts as counting? The question may sound simple,
• 0:07
  but take a moment to think about how you would answer. After all,
• 0:11
  we count all kinds of things, physical quantities, increments of time,
• 0:15
  lengths, money, as well as fractions and decimals.
• 0:20
  In this episode,
• 0:21
  we'll talk with Christopher Danielson about what counts as counting and how our
• 0:26
  definition might shape the way we engage with our students.
• 0:31
  Before we start today's episode,
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  I'd like to offer a bit of context to our listeners.
• 0:35
  This is the second half of a conversation that we originally had with
• 0:38
  Christopher Danielson back in the fall of 2025. At that time,
• 0:43
  we were talking about which one doesn't belong.
• 0:46
  This second half of the conversation focuses deeply on the question,
• 0:50
  what counts as counting. I hope you'll enjoy the conversation as much as I did.
• 0:55
  Well, welcome to the podcast, Christopher.
• 0:57
  I'm excited to be talking with you today.
• 1:00
  Thank you for the invitation. Delightful to be invited.
• 1:02
  So I'd like to talk a little bit about your recent work. The book,
• 1:05
  How Did Yo Count? In it,
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  you touch on what seems like a really important question, which is,
• 1:11
  what is counting?
• 1:13
  Would you care to share how your definition of counting has evolved over time?
• 1:17
  Yeah. So the previous book to How Did Yo Count was called How Many?
• 1:21
  And it was about units.
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  So the conversation that the book encourages would come from
• 1:28
  children and adults all looking at the same picture,
• 1:31
  but maybe counting different things.
• 1:33
  So how many was sort of an ill-formed question?
• 1:36
  You can't answer that until you've decided what to count. So for example,
• 1:40
  on the first page, first photograph is a pair of shoes,
• 1:43
  Doc Martin shoes sitting in a shoebox on a floor.
• 1:47
  And children will count the shoes. They'll count the number of pairs of shoes.
• 1:51
  They'll count the shoe laces.
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  They'll count the number of little silver holes that the shoelaces go through,
• 1:58
  which are called eyelets.
• 2:00
  And so the conversation there came from there being lots of different things to
• 2:03
  count. If you look at it, if I look at it,
• 2:05
  if we have a sufficiently large group of learners together having a
• 2:09
  conversation,
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  there's almost always going to be somebody who notices some new thing that they
• 2:14
  could count,
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  some new way of describing the thing that they're counting. One of the things
• 2:19
  that I noticed in those conversations with children,
• 2:23
  I noticed it again and again and again, was a particular kind of interaction.
• 2:27
  And so we're going to get now to what does it mean to count and how my view of
• 2:31
  that has changed. The eyelets,
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  there are five eyelets on each side of each shoe.
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  Two little flaps that come over, each has five of those little silver rings.
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  Super compelling for kids to count them. Most of the things on that page,
• 2:45
  there's not really an interesting answer to how did you count them. Shoelaces,
• 2:49
  they're either two or four. It's obvious how you counted them. But the eyelets,
• 2:52
  there's often an interesting conversation to be had there.
• 2:55
  So if a kid would say, "I counted 20 of those little silver holes," I would say,
• 2:59
  "Fabulous. How do you know there are 20?" And they would say,
• 3:01
  "I counted." In my mind,
• 3:04
  that was like an evasion. They felt like what they had been called on to do
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  by this strange man who's just come into our classroom and seems friendly
• 3:13
  enough, what they had been called on to do was say a number and a unit.
• 3:16
  And they said they had 20 silver things. We're done now.
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  And so by my asking them, "How do you know? " And they say,
• 3:22
  "I counted." It felt to me like an evasion because I counted as being one, two,
• 3:26
  three, four, five, all the way up to 20.
• 3:29
  And they didn't really want to tell me about anything more complicated than
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  that. It was just sort of an obvious I counted.
• 3:34
  So in order to counter what I felt like was an evasion, I would say, "Oh,
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  so you said to yourself, one, two, three, and then blah, blah, blah, 18, 19,
• 3:43
  20." And they'd be like, "No, there were 10 on each shoe." Or, "No,
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  there's five on each side." Or rarely there would be the kid who would see there
• 3:50
  were four bottom eyelets across the four flaps on the two shoes
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  and then another row and another row. Some kids would say there's five rows of
• 3:59
  four of them, which are all fabulous answers.
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  But I thought initially that that didn't count as counting.
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  After hearing it enough times, I started to wonder,
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  is it possible that kids think five rows of four,
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  four groups of five, two groups of 10, counted by twos and one, two, three,
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  four, all the way up to 19 and 20?
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  Is it possible that kids conceive of all of those things as ways of counting,
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  that all of those are encapsulated under counting?
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  And so I began because of the ways children were responding to me to
• 4:35
  think differently about what it means to count.
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  So when I first started working on this next book, how did you count?
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  I wanted it to be focused on that.
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  The focus was deliberately going to be on the ways that you count.
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  We're all going to agree that we're counting tangerines,
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  we're all going to agree that we're counting eggs,
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  but the conversation is going to become because there are rich ways that these
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  things are arranged,
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  rich relationships that are embedded inside of the photographs. And what I found
• 5:02
  was when I would go on Twitter and throw out a picture of some tangerines and
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  ask how people counted,
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  and I would get back the kind of thing that was how I had previously seen
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  counting. So I would get back from some people, "There are 12." I'd ask,
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  "How did you count?" And they'd say, "I didn't. I multiplied three times four.
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  I didn't.
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  I multiplied two times six." But then on reflection through my own
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  mathematical training,
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  I know that there's a whole field of mathematics called combinatorics,
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  which if you asked a mathematician,
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  "What is combinatorics?" Nine times out of 10, the answer is going to be,
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  it's the mathematics of counting.
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  And it's not mathematicians sitting around going one, two, three, four, or two,
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  four, six, eight.
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  It's looking for structures and ways to count the number of possibilities there
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  are, the number of ...
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  If we're thinking about calculating probabilities of winning the lottery,
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  somebody's got to know what the probabilities are of choosing winning numbers,
• 5:59
  of choosing five out of six winning numbers. And the field of cognitorics is
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  what does that. It counts possibilities.
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  So I know that mathematicians and kindergartners,
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  this is what I've learned in both my graduate education and in my postgraduate
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  education working with kindergartners,
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  is that they both think about counting in this rich way.
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  It's any work that you do to know how many there are.
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  And that might be one by one. It might be skip counting.
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  It might be multiplication. It might be using some other kind of structure.
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  I think that's really interesting because there was a point in time where I saw
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  counting as a fairly rote process, right?
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  Where I didn't understand that there were all of these elements of counting,
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  meaning one-to-one correspondence and quantity versus being able to just say
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  the wrote, count out loud.
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  And so one way that I think counting and its meaning have expanded for me
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  is to kind of understand some of those pieces.
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  But the thing that occurs to me as I hear you talk is that I think one of the
• 6:55
  things that I've done at different points, and I wonder if people do, is say,
• 6:59
  that's all fine and good, but counting is counting.
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  And then we've suddenly shifted and we're doing something called addition or
• 7:07
  multiplication.
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  And this is really interesting because it feels like you're drawing a much
• 7:13
  clearer connection between those critical,
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  emergent ideas around counting and these other things we do to try to figure out
• 7:20
  the answer to how many or how did you count. Tell me what you think about that.
• 7:25
  Yeah. So this for me is the project, right?
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  This book is an instantiation of this larger project way of viewing the world
• 7:33
  of mathematics through the lens of what it means to learn it.
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  And I would describe that larger project through some imagery and appealing to
• 7:44
  teachers' ideas about what it means to have a classroom conversation. For me,
• 7:49
  learning is characterized by increasing
• 7:54
  sophistication, increasing expertise with whatever it is that I'm studying.
• 7:58
  And so when I put several different triangular
• 8:03
  arrangements of things, in the book,
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  there's a triangular arrangement of bowling pins,
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  which lots of kids know from having bold in their lives and other kids don't
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  have any experiences with them,
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  but the image is rich and vivid and they're able to do that counting.
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  And then later on,
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  there's a triangular arrangement of what turned out to be very bland, gooey,
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  and nasty, but beautiful to photograph, pink pudding cups. Later on,
• 8:26
  there are two triangles of eggs. And so what I'm asking of kids,
• 8:31
  I'm always imagining a child and a parent sitting on a couch reading
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  these books together, but also building them for classrooms.
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  Any of this could be like a thing that happens at home,
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  a thing that happens for a kid individually or a classroom full of children led
• 8:46
  by a teacher. Thinking about the second picture of the pudding cups,
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  my hope and expectation is that at least some children will
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  say, "Okay,
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  there are six rows in this triangle and there were four rows previously.
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  So I already know these first four are 10.
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  I don't have to do any more work and then five plus six, right?
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  And then that demonstrates some learning.
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  They're more expert with this triangle than they would have been previously."
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  I'm also expecting that there's going to be some kid who's counting them one by
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  one, and I'm expecting that there are going to be some kids who are like,
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  "You know what? That six up top and the one makes seven and the five and
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  the two make seven and the four and the three." So it's three sevens,
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  there's 21.
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  I'm expecting that we're going to have in a reasonably large population of
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  third, fourth, fifth graders, sort of the target audience for this book,
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  we're going to have some kids who are doing each of these. And for me,
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  getting back to this larger project,
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  that is a rich task which can be approached in a bunch of different ways and all
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  of those children are doing the same sort of task.
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  They're all counting at various levels of sophistication representing various
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  opportunities to learn previously,
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  various ways of applying their new learning as they're having conversations,
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  looking at new images, hearing other people's ideas,
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  but that larger project of building something that is rich enough for everybody
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  to be able to find something new in,
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  but simple enough for everybody to have access to, yeah,
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  that's the larger project.
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  So one of the things that I found myself thinking about when I was thinking
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  about my own experiences with dot talks or some of the supertizing images that
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  I've used and the book that you have is there's something about the way that
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  a set of items can be arranged.
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  And I think what's interesting about that is I've heard you say that that
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  arrangement can both reveal structure in terms of number,
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  but it can also make connections to ideas in geometry.
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  And I wonder if you could talk a little bit about that.
• 10:44
  Yeah.
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  I'll draw a quick distinction that I think will be helpful.
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  If you've ever seen bowling pins, right? It's four, three, two, one,
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  the one is at the front, four is at the back,
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  arranged so that the three fit into the spaces between the four as you're
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  looking at it from the front. Very iconic arrangement.
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  And you can quickly tell that it's a symmetric triangle and the longest row is
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  four. You might just know that that's 10.
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  But if you take those same bowling pins and just toss them around inside of a
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  classroom or inside of a closet and they're just lying on the floor,
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  so they're all in your field of vision,
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  you don't know that there's 10 right away.
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  You have to do a different kind of work in order to know that there are 10 of
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  them. In that sense,
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  the structure of the triangle with the longest row of four is a thing
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  that you can start to recognize as you learn about triangles and ultimately what
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  mathematicians refer to as triangular numbers.
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  That's a thing you can learn to recognize,
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  but learning to recognize 10 in that arrangement doesn't afford you anything
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  when it's 10 scattered around on the floor. Unless you do a little abstraction,
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  there's a story in the book about a lovely sixth grader who proceeded to tell me
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  about how the bowling pin arrangement matches a way that she thinks about
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  things. Because if she's ever going about her life, I don't know,
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  making a bracelet or buying groceries,
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  collecting pencils for first day of school or whatever.
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  If she wants to count them,
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  and it looks like there's probably fewer than a hundred, but more than five,
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  she will grab a set of four, a set of three, a set of two, a set of one,
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  and she'll know that's 10. Unprompted by me,
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  except that we had this bowling pin arrangement.
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  So there are ways to abstract from that. You can use these structures that
• 12:22
  you've noticed in order to do something that isn't structured that way,
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  but the four, three, two, one thing probably came from recognizing that four,
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  three, two, one made this nice little geometric arrangement. So our eyes,
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  our brains are tuned to symmetry and to beauty and elegance,
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  and there is something much more lovely about a nice arrangement of four, three,
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  two, one than there is about a bunch of scattered things.
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  And so a lot of those things are things that have been captured by
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  mathematicians. So we have words for square numbers,
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  three times three is nine because you can make three rows of three and you make
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  something that looks nice that way. Triangular numbers,
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  there are other figurine numbers like hexagonal numbers,
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  but yet innate in our minds, there is an appeal to symmetry.
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  And so if we start arranging things in symmetric patterned ways that will be
• 13:09
  appealing to our brains and to our eyes and to our mathematical minds,
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  and my goal is to try to tap into that in order to help kids become
• 13:18
  more powerful mathematicians.
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  So I want to go back to something you said earlier,
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  and I think it's an important distinction before I ask this next question.
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  One of the things that's fascinating is that a child could engage with this kind
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  of image,
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  and there doesn't necessarily have to be an adult in the room or a teacher who's
• 13:36
  guiding them.
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  But what I was thinking about is if there is a student or a pair of students or
• 13:43
  a classroom of students and you're an educator and you're engaging them with one
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  of these images, how do you think about the educator's role in that space?
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  What are they trying to do? How should they think about their purpose?
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  And then I'm going to ask a sub question.
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  To what extent do you feel like annotation is a part of what an educator
• 14:03
  might do?
• 14:04
  Yes.
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  One thing that teachers are generally more expert at than young
• 14:10
  children is being able to state something simply,
• 14:15
  clearly, concisely in a way that lots of other people can understand.
• 14:19
  If you listen to children thinking aloud,
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  it is often hesitant and halting and it goes in different directions
• 14:26
  and units get left off.
• 14:28
  So they'll say three and then four more is eight and they've left off the fact
• 14:32
  that the four were ... I mean, you could just easily get lost.
• 14:36
  And so one of the roles that a teacher plays can certainly be to help make clear
• 14:40
  to other students the ideas that a particular student is expressing and
• 14:46
  at the same time, often helping make more clear for that student, right?
• 14:50
  Often a restating or a question or an introduction of a vocabulary word
• 14:55
  that seems like it's going to be helpful right now will not just be helpful to
• 14:58
  other people to understand it for the whole class,
• 15:00
  but will be helpful for the student in clarifying their own ideas and their own
• 15:03
  thinking, solidifying it in some kind of way. So that's one of the roles.
• 15:07
  I know that there are also roles that involve,
• 15:09
  and I think about this a lot whenever I'm working with learners, status, right?
• 15:13
  Making sure that children that have different perceived status in the classroom
• 15:18
  are able to be lifted up,
• 15:19
  that we're not just hearing from the kid who's been identified as the math kid.
• 15:24
  So I think intellectual status, social status, those are going to be balances,
• 15:27
  right?
• 15:28
  I also understand that teachers have a role in making sure that children are
• 15:33
  listening to each other. If I'm working with learners,
• 15:35
  I can't always be the one to do the restating.
• 15:38
  I've got to make sure there are times where kids are required to try to
• 15:41
  understand each other's thinking and not just the teacher's restatement of that
• 15:45
  thinking. There are just so many balances.
• 15:47
  But I would say that some top ones for me,
• 15:49
  if I'm thinking about how to make choices,
• 15:52
  thinking about raising up the status of all learners as intellectual resources,
• 15:58
  making good on a promise that I make to children,
• 16:00
  which is that any way of counting these things is valid and not telling a kid,
• 16:05
  "Oh no, no, no, we're not counting one by one today." Or, "Oh no, no, no,
• 16:09
  that's too sophisticated. That's too advanced of a ...
• 16:12
  We can't share that because nobody will understand it.
• 16:14
  " So making good on that promise that I make at the beginning, which is,
• 16:17
  "I really want to know how you counted." Making sure that learners are able to
• 16:21
  get better at expressing the ideas that are in their heads using language and
• 16:24
  gesture and making sure that learners are communicating with each other and not
• 16:28
  just with me as a teacher.
• 16:29
  Those seem like four important tensions and a talented and experienced
• 16:33
  elementary teacher could probably name like 10 other tensions that they're
• 16:37
  keeping in mind all at the same time. Behavior, classroom management,
• 16:40
  but also some ideas around multilingual learners. Yeah,
• 16:44
  a lot of respect for the kind of balances that teachers have to maintain and the
• 16:48
  kinds of tensions that they have to choose when to use and when to
• 16:53
  gloss over or not worry about for right now. So you ask about annotation
• 16:58
  and absolutely I think about multiple representations of mathematical
• 17:03
  ideas. And so far I've only focused on the role of the teacher in a classroom
• 17:07
  discussion and thinking about gesture,
• 17:09
  thinking about words and other language forms,
• 17:13
  but I haven't focused on writing and annotation is absolutely a role that
• 17:15
  teachers can play. For me,
• 17:18
  the thing that I want to have happen is I want children to see their
• 17:23
  ideas represented in multiple ways.
• 17:26
  So if they've described for the class something in words and
• 17:30
  gestures,
• 17:31
  then there are sort of two natural easy annotations for a teacher to do or a
• 17:35
  teacher to have students do, which is one,
• 17:38
  make those gestures and words explicit in the image.
• 17:42
  And that's where something like a smart board or projecting onto a whiteboard,
• 17:47
  lots of technologies that teachers use for this kind of stuff,
• 17:50
  but where we can write directly on the image. So if you said you put the one and
• 17:52
  the four together in the bowling pins and then the three and the two,
• 17:55
  then I might make a loopy thing that goes around the four and the one,
• 17:59
  and I might circle the three and the two, right?
• 18:01
  And so that adds both some clarity for students looking,
• 18:04
  but also is a model for here's how we can start to annotate our images.
• 18:09
  But then I'm also probably going to want to write four plus one,
• 18:12
  maybe in parentheses, plus three plus two in parentheses,
• 18:16
  so that we can connect the four to the four that are circled,
• 18:20
  the one to the one that circled the four plus one in parentheses,
• 18:22
  identifying that as a group, like a thing that has a mathematical purpose.
• 18:26
  It's communicating part of an idea and that that connects back.
• 18:30
  Teachers are super skilled at using color to do that,
• 18:32
  right? So four plus one might be written in red to match the red circle that
• 18:37
  goes around here using not green because of color blindness.
• 18:40
  They're using blue to do three plus two in parentheses over here and teachers
• 18:44
  might make other choices, right?
• 18:45
  We might sometimes use color to annotate in the image,
• 18:49
  but then just black here so that we aren't doing all of that work of
• 18:53
  corresponding for kids and are asking kids to try to do some of that
• 18:56
  corresponding work and we might do it the other way around as well.
• 18:59
  So annotation as a way of adding, I think,
• 19:03
  a couple of dimensions to the conversation.
• 19:06
  And I have to shout out a fabulous teacher who I know through math Twitter,
• 19:09
  Simon Gregg is a teacher in an international school in
• 19:14
  Tulus, France,
• 19:15
  and he has done amazing work with using and producing his
• 19:20
  own, which one doesn't belongs and annotating them and having kids do them,
• 19:24
  how many,
• 19:25
  and then there are a few examples of his work with kids in the teacher guide for
• 19:29
  How Did Yo Count. Yeah, he's just a true master at annotation.
• 19:32
  So go find Simon Greg on the social media if you want to learn some beautiful
• 19:37
  things about representing kids' ideas in writing.
• 19:41
  Love it.
• 19:42
  So the question that I typically will ask any guest before the close of the
• 19:46
  interview is, what are some resources that educators might grab onto,
• 19:51
  be they yours or other work in the field that you think is really powerful that
• 19:54
  supports the kind of work that we've been talking about?
• 19:57
  What would you offer to someone who's interested in continuing to learn and
• 20:00
  maybe to try this out?
• 20:02
  In the teacher guide of How Did You Count,
• 20:04
  I make mention of which of the Number Talks books was
• 20:09
  most powerful for me.
• 20:11
  But if you want to take a look at that page in the teacher book and then throw a
• 20:14
  link in and a shout out to the folks who wrote it.
• 20:17
  Joe Bowler and Kathy Humphreys wrote a book called Connecting Mathematical
• 20:21
  Ideas. It's old enough that there are some CD ROMs in it.
• 20:25
  I don't know if there's a new edition.
• 20:27
  I'm sure used ones are available on all the places you buy used books,
• 20:31
  but the expert work that the teacher Kathy Humphrey does,
• 20:34
  as described in the book, even if you can't use the CD Roms in your computer,
• 20:38
  expert work at drawing out students' ideas and then the two collaborating to
• 20:42
  reflect on that lesson, the connections they were drawing.
• 20:45
  It's been a while since I read it,
• 20:46
  but I imagine the annotations have got to come up,
• 20:49
  fabulous resources for thinking about how these ideas pertain to middle school
• 20:52
  classrooms,
• 20:53
  but absolutely stuff that we can learn as college teachers or as elementary
• 20:57
  teachers on either side of that bridge from arithmetic to algebra.
• 21:01
  So for listeners, just so you know,
• 21:03
  we're going to add links to the resources that Christopher referred to in all of
• 21:06
  our show notes for folks' convenience. Christopher,
• 21:09
  I think this is probably a good place to stop. Thank you so much for joining us.
• 21:12
  It's absolutely been a pleasure chatting with you.
• 21:14
  Yeah. Thank you for the invitation for your thoughtful prep work and
• 21:19
  support of both the small and the larger projects along the way.
• 21:23
  I appreciate that.
• 21:24
  I appreciate all of you at Bridges and the Math Learning Center.
• 21:27
  You do fabulous work.
• 21:32
  This podcast is brought to you by the Math Learning Center and the Meyer Math
• 21:35
  Foundation,
• 21:36
  dedicated to inspiring and enabling all individuals to discover and develop
• 21:41
  their mathematical confidence and ability.