Kim Hartweg, Counting

Mike Wallus, Vice President for Educator Support


Counting is a process that involves a complex and interconnected set of concepts and skills. This means that for most children, the path to counting proficiency is not linear. Today we are talking with Dr. Kim Hartweg from Western Illinois University about the big ideas and skills that are a part of counting and ways educators can support their students on this important part of their math journey.

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Professor Kim Hartweg teaches mathematics education at Western Illinois University.


NCTM Classroom Conversations


Mike Wallus: Counting is a process that involves a complex and interconnected set of concepts and skills. This means that for most children, the path to counting proficiency is not a linear process. Today we're talking with Dr. Kim Hartweg from Western Illinois University about the big ideas and skills that are a part of counting and the ways educators can support their students on this important part of their math journey.

Well, hey, Kim, welcome to the podcast. We're excited to be talking with you about counting. 

Kim Hartweg: Ah, thanks for having me. I'm excited too. 

Mike: So I'm fascinated by all of the things that we're learning about how young kids count, or at least the way that they attend to quantities. 

Kim: Yeah, it's exciting what all is taking place, with the research and everything going on with early childhood education, especially in regards to number and number sense. And I think back to an article I read about a 6-month-old baby who's in a crib and there's three pictures in this crib. One of them has two dots on it, another one has one dot, and then a third one has three dots. And a drum sounds, and it goes, “boom, boom, boom.” And the 6-month-old baby turns their head and eyes and they look at the picture with three dots on it. And I just think that's exciting that even at that age they're recognizing that three dots [go] with three drum beats. So it's just exciting. 

Mike: So you're actually taking us to a place that I was hoping we could go to, which is, there are some ideas and some concepts that we associate with counting. And I'm wondering if we could start the podcast by naming and unpacking a few of the really important ones. 

Kim: OK, sure. I think of the fundamental counting principles, three different areas. And for me, the first one is that counting sequence, or just learning the language and that we count 1, 2, 3, 4, 5. However, in the English language, it's much more difficult [than] in other languages when we get beyond 10 because we have numbers like 11, 12, 13 that we never hear again. Like, you hear 21, 31, 41, but you don't hear 11 again. It's the only time it's ever mentioned. So I think it's harder for students to get that counting sequence for those who speak English. 

Mike: I appreciate you saying that because I remember reading at one point that in certain Asian languages, the number 11, the translation is essentially “10 and 1,” as opposed to for English speakers where it really is 11, which doesn't really follow the cadence of the number sequence that kids are learning: 1, 2, 3, 4, and so on. 

Kim: Exactly. Yes. 

Mike: It picks up again at 21, but this interim space where the teen numbers show up and we're first talking about a 10 and however many more, it's not a great thing about the English language that suddenly we decided to call those things that don't have that same cadence. 

Kim: Yeah, after you get past 20, yes. And if you think of kids when they hear the number 16, a lot of times they'll say, “A 1 and a 6 or a 6 and a 1?” Because they hear “sixteen,” so you hear the 6 first. But like you said, in other languages, it's “ten-six,” “ten-seven,” “ten-eight.” So, it kind of fits more naturally with the way we talk and the language. 

Mike: So there's the language of the counting sequence. Let's talk about a couple of the other things.

Kim: OK. One-to-one correspondence is a key idea, and I think of this when I was first starting to teach undergraduate students about early math education. I had kids at the same age, so at a restaurant or wherever we happened to be, I'd get out the sugar packets and I would have them count. And at first, when they're maybe 2 years old or so, and they're just learning the language, they may count those sugar packets as “1, 2, 3.” There may be two packets. There may be five packets. But everything is “1, 2, 3,” whether there's again, five packets or two packets. So once they get that idea that each time they say a number word that it counts for an actual object and they can match them up, that's that idea of one-to-one correspondence to where they say a number and they either point or move the object so you can tell they're matching those up. 

Mike: OK, let's talk about cardinality because this is one that I think when I first started teaching kindergarten, I took for granted how big of a leap this one is. 

Kim: Yeah, that's interesting. So once they can count out and you have five sugar packets and they count “1, 2, 3, 4, 5,” and you ask, “How many are there?”, they should be able to say, “five.” That's cardinality of number. If they have to count again—“1, 2, 3, 4, 5”—then they don't have cardinality of number, where whatever number they count last is how many is in that set. 

Mike: Which is kind of amazing actually. We're asking kids to decide that “I've figured out this idea that when I say a number name, I'm talking about an individual part of the count—until I say the last one, and then I'm actually talking about the entire set.” That's a pretty big leap for kids to start to make sense of. 

Kim: It is, and it's fun to watch because hear some of them say, “1, 2, 3, 4, 5. Five, there's five.” [laughs] So they kind of get that idea. But yeah, that cardinality of number is a key principle and leads into the conservation of numbers.

Mike: Let's talk about conservation of number. What I'm loving about this conversation is the way that you're using these concrete examples from your own children, from sugar packets, to help us make sense of something that we might be seeing, but we might not have a name for. 

Kim: Yeah, so the conservation of number, this is my favorite task when I have young kids around. I want to see if they can serve number or not. So I might first do the sugar [packet] thing or whatever and see if they can tell you how many there are. But the real fun is, do they conserve that number? 

So I think back to a friend of mine who brought her daughter over one time, and I had these toy matchbox cards on my table, and her name was Katie. And I said, “Katie, how many cars are there?” And she counted, “1, 2, 3, 4, 5. There's five toy cars.” And I moved them around and I said, “Now how many cars are there?” And she counted, “1, 2, 3, 4, 5. There's five toy cars.” So she has cardinality of number. However, I kept moving those cars into different positions, never adding or taking any away. That's all that were there the whole time. And after about seven or eight times, I said, “Now how many cars are there?” And her mom finally jumped in and said, “Katie, you've counted those already. There's five cars.” [laughs] And I said, “No, no, no. This is just whether she conserves number or not, it's a developmental-type thing.” 

But you know they conserve number when you ask them, “Well, now how many cars are there?” And they look at you and like, “Well, why would you ask that again? There's five.” [chuckles] So then they can conserve number. So It's real fun to do that with elementary students who are getting their number sense going and even before they enter school. However, there will be some that may not get that conservation of number until they're 5 or 6 years old. 

Mike: Let's talk about something you named earlier. I've heard people pronounce this as “soo-bi-ty-zing” or “su-bi-ty-zing” but in any case, it's really an important idea for people, especially if you're teaching young children to make sense of this. Can you talk about what that means? 

Kim: Yeah, so subitizing, I think that's interesting. You know, we work so hard getting kids to count and learn the language and have one-to-one correspondence, and then be able to eventually conserve number. But then we want them to just be able to recognize a set of numbers without counting. And that's when they're really starting to develop some number sense. 

I think of dice. And if you roll a single [die], we want students to just know that when there's an arrangement of four dice, they know it's four without having to count “1, 2, 3, 4.” So the subitizing idea, a lot of dice games, maybe some 10-frame cards, dot cards, lots of things like that can help students develop a little bit more of that subitizing, or recognizing a set of items without having to count those. 

Mike: So when I look at a set of three dots, I can just say, “That's three,” as opposed to an earlier point where a child might actually say, “1, 2, 3. That's three.”

Kim: Exactly. So that would be subitizing—just instantly knowing what's there without having to count. 

Mike: So I wonder if we could unpack two other counting behaviors that sometimes pop up with kids when they're combining or separating quantities. And what I'm thinking about is the difference between the child who counts everything and the child who either counts on from a number or counts back from a number. And I'm wondering if you can talk about what these two behaviors can tell us about how kids are thinking about the numbers that they're operating on. 

Kim: Yeah, it's so interesting when you have activities like a cup, and maybe you have eight counters and you put three under the cup and you say, “How many are here?” “Three.” And then you cover those up and you ask, “Well, how many are altogether?” There are some kids who don't have any trouble with counting on “4, 5, 6, 7, 8,” but there's other kids who have to lift up the cup and start again at 1. So they don't have that idea that there's three items under that cup whether you can see them or not. So it's difficult for them to be able to count on, and we shouldn't as teachers force that on them until they're ready to do it. So it's a hard concept for kids to get, but especially if they're not developmentally ready for it. 

Mike: I think that's a really nice caution because I think you could accidentally potentially get kids to mimic a practice that you're trying to show them, but without understanding there's some real danger that you're just causing confusion. 

Kim: Yeah, and we want to give kids the idea that counting collections and things, it's a fun thing to do. And I know there may be teachers that have seashells or rocks or different types of collections they might count, and we want students to count those and then discuss how they counted them, arrange them. 

And I'm thinking of this little girl that I saw on a video where she was counting eight bears, and she arranged them first by color, then counted how many there were. And the teacher then went on to use that and make a problem-solving task for her, such as, “Well, how many green bears do you have?” And she would count them. “Well, what if you gave me those green bears? Do you know what you would have left?” And she said, “Well, I don't know. Let's try it.” And I love that because I think that's the kind of idea we want students to have. They're counting, and “I don't know, let's try it.” They're excited about it. They're not afraid to take chances, and we don't want them to think that “Oh, this is difficult to do.” It's just, “Hey, let's try it. Give it a try here.” 

Mike: Well, I've heard people talking a lot about this idea of counting collections lately. It seems like we are almost rediscovering the value of a routine like that. I'm wondering if you could talk about the value that can come out of an experience of counting collections and help bring that idea to life for people. 

Kim: The idea here is that we want students to get good at counting. And the research is showing that students who maybe don't show one-to-one correspondence when they count out, maybe, eight counters, might show one-to-one correspondence when they count out 31 pennies, which seems like it shouldn't happen. But there's research out there that over 70 percent of them did better counting 31 pennies than they did with eight counters. 

So, I think what you count makes a big difference for kids—and to not hold them back, to not think that “OK, we've got to get one-to-one correspondence before we count this collection of 50 items.” I don't think that's the case. And the research is even showing that these ideas that we've talked about all develop concurrently. It's not a linear process. But this counting collections is kind of a big deal with that. And having students count, again, collections that they're interested in, writing number sentences about their collections, comparing what they counted with another partner, and then turning it into problem-solving questions where they're actually doing, “What happens if you lost five of yours?” or “What happens if you combined your collection with somebody else?” And turning it into where they're actually doing addition and subtraction, but not actually the formal process of that. 

Mike: The other thing that you made me think about is, I would imagine you could also have kids finish counting a collection and then you could ask them to represent it either on paper or in some other way. 

Kim: Exactly. And writing out those number sentences or even creating their own word problems so that they can ask a friend or a partner, it makes it fun. And it relates to what they've done. And let's face it, once you've taken that time to count those collections, you may as well get as much use out of it as you can. [chuckles]

Mike: Kim, you're making me think of something that I don't know that I had words for when I was teaching kindergarten, which is, when I look back now, I was looking to see that kids could do a particular thing like one-to-one correspondence or that they had cardinality before I would give them access to a task like counting collections. And I think what you're making me think is that those things shouldn't be a gatekeeper; that they actually develop by doing those things. Am I making sense to you? 

Kim: Yes. I always thought you had to have the language first. You had to be able to do one-to-one correspondence before you could get cardinality of number, and you needed cardinality of number before you could do conservation of number. But what the research is showing is, it develops concurrently with students; that it's not something that is a linear process by any means. So, when we have these activities, it's OK if they don't have one-to-one correspondence and you're doing problem-solving tasks with counters. We need to be planning these activities so they're getting all of this, and they will develop it as it fits in the schema of what they're working on and thinking of in their minds. 

Mike: So I want to bring up a set of manipulatives that are actually attached to our bodies, particularly when it comes to counting. I'm thinking about fingers. And part of what's on my mind is, again, to go back to my practice, there was a point in time where I was really hung up on whether kids should make use of their fingers when they're counting or when they're operating on numbers. And I'm wondering if you could just offer some guidance around that. 

Kim: Yes. I think again, it goes to that idea that we have these 10 fingers that are great manipulatives, that we shouldn't stop students from doing that. And I know there was a time when teachers would say, “Don't use your fingers,” “Don't count on your fingers.” And I get the idea that we want students to start to subitize eventually and make combinations and not have to count on their fingers, but to stop them from doing it when they need that would be very detrimental to them. 

And I actually have a story. When I was supervising student teachers, one teacher was telling a student, “Don't count with their fingers.” And I saw them nodding their head, and I went over and I said, “What are you doing?” He said, “Well, I can't count my fingers, so I'm using my tongue, and I'm counting my teeth.” [laughs] So coming up with a problem that way, still using a manipulative, but it wasn't their fingers. 

Mike: That's pretty creative. 

Kim: [laughs] Yeah. 

Mike: Well, part of what strikes me about it, too, is our entire number system is based on tens and ones, and we've got a set of them right in front of us, right? We're trying to get kids to make sense of shifting from units of one to units of ten or maybe units of five. So these tools that are attached to our bodies are actually pretty helpful because they're really the basis for our number system in a lot of ways. 

Kim: Yes, exactly. And being able to come up with even using your fingers to answer questions. I'm thinking, we want students to subitize. So even something to where there's a dot card that a teacher flashes for 3 seconds, and it's in the formation of maybe a five on a [die]. And you could have students, you know, hold up how many there are. And you could do that five or 10 times, with dot flashes. Or you could hold up one more than what you see on the [die]. So, they only see it for 5 seconds and the number's five, but they hold up six. So just uses of fingers to kind of make those connections can be very helpful. 

Mike: So before we go, you mentioned that you work with preservice teachers, folks who are getting ready to go into the field and work with elementary children in the area of mathematics. I'm wondering if there are any particular resources that really help your students, and perhaps teachers who are already in the field, just make sense of counting and number to really understand some of the ideas that we've been talking about today. Do you have anything in particular that you would recommend to teachers? 

Kim: Yeah, I'll just mention a few that we use a lot of. We do the two-color counters a lot where one side is yellow and one is red. But we do a lot of dot cards, where again, there are arrangements of dots on a card that you could just flash to a student, kind of like I've already explained. There's lots of resources on the National Council of Teachers of Mathematics website. That has 10-frame activities. 

And if you haven't used rekenreks before, I think those are pretty amazing as well—along with hundreds charts. And just being able to have students create some of their own manipulatives and their own numbers makes a huge difference for kids. 

Mike: I think that's a great place to close the conversation. Thank you so much for joining us, Kim. It's really been a pleasure chatting with you. 

Kim: Hey, thanks so much. It's been fun, Mike. Thanks for asking me. 

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.