Season 4 | Episode 16 – Kristin Frang, Understanding the Roots of Fluency with Addition & Subtraction

Kristin Frang, Understanding the Roots of Fluency with Addition & Subtraction

ROUNDING UP: SEASON 4 | EPISODE 16

Research suggests that supporting students’ fluency with addition and subtraction hinges on understanding how children’s mathematical thinking develops. So what are the concepts and ideas that play a part in fluency with combinations to 10, 20, and beyond? 

Today, we’ll explore this question with Kristin Frang, director of instructional programs at Integrow Numeracy Solutions. 

BIOGRAPHY

Kristin Frang is the director of instructional programs for Integrow Numeracy Solutions. She designs resources and services that support states, districts, schools, and individuals in transforming numeracy education.

RESOURCES

”Understanding Units Coordination” Season 4, Episode 11 of the Rounding Up podcast

Integrow Numeracy Solutions

• website

• blog 

• email address

On Track to Numeracy book by Lucinda “Petey” MacCarty, Kurt Kinsey, David Ellemor-Collins, and Robert J. Wright

TRANSCRIPT

Mike Wallus: Welcome to the podcast, Kristin. It is so great to be talking with you today. 

Kristin Frang: It’s great to be here. I feel so honored to be on this podcast. 

Mike: Before we dive into a conversation about addition and subtraction, I'd like to do a bit of grounding. So you're currently the director of instructional programs for Integrow Numeracy Solutions. I wonder if briefly you could tell the listeners: What is Integrow Numeracy Solutions, and what's its mission? 

Kristin: Yeah. Integrow Numeracy Solutions’ mission is to transform numeracy education by connecting research with practice and empowering educators to advance student mathematical thinking and success. But I really want to bring that mission to life through a story, just a quick story, if I can. 

Prior to my role with Integrow, I was a K–12 mathematics consultant. And one of the things that I did was, when the Common Core [State Standards] were released, I worked with teachers to transition to the then-new standards. We studied many documents together, including progression documents that were included in the standards, and teachers were honestly fascinated by this idea of a progression and that they were embedded into the standard. But I remember an instance where we had been studying these progressions and a teacher came up and said to me, "I know where my students are at; I can see them in these progressions. But how do I get them to the next stage?" 

And I didn't have an answer (laughs) at that point. I was a former middle school and high school teacher. I was working with elementary teachers. I was studying, just like them, these progression documents, and I could only categorize the reasoning that was in front of us. And so that next step to say, "Oh, this is what I would do and bring into action in the classroom,” I didn't have an answer for.

And so that's really where I was introduced to Integrow—formerly [the] US Math Recovery Council, but now Integrow Numeracy Solutions. And at the heart of our mission to empower educators is to bring research to the classroom in accessible and practical ways that advance student reasoning. We do this in professional learning, we do it in supplemental resources, and we also hire and train educators to deliver high-dosage tutoring for students to accelerate their learning. 

Mike: I want to just linger on something you said, which was—and I really appreciate both the truth of the statement you made and also the vulnerability, which is to say—I think for many teachers, there's this experience of, “I can see my students in these progressions, but I'm not sure what to do when it comes to making moves to shift where they're at or help them move.” And I think that's a profound truth for so many teachers. And I think it's really important that folks like you, who are doing this work, acknowledge that that's a place you were in once as well because that's so true for so many of us. 

Kristin: Yeah. There's always a new thing where we're watching students, we're thinking about the next steps. And so often it boils down to categorizing the things that students are doing now, but not often figuring out: What are the true actions that we take with real children who are in front of us to get them to progress in their own reasoning? We can tell them the next step, but my belief system that is aligned with Integrow Numeracy Solutions is that the most powerful thing is to help students have those experiences and create that understanding themselves. And to do that, it's more complex than just knowing what the next benchmark is for them. 

Mike: I think that's a helpful introduction. And I also find it to be a good segue for all the questions that I wanted to explore today. So let me start here: It feels important to acknowledge that supporting students' addition and subtraction fluency actually hinges on understanding how children's mathematical thinking develops. So I wonder if you can talk about some of the concepts and the ideas that play a part in fluency when it comes to combinations of 10, combinations to 20, and even beyond. 

Kristin: Yeah. The words that we hear associated with fluency right now are “flexibility,” “efficiency,” “accuracy.” So we've moved on from just speed, which I think is a really positive place for us to be in education. But at the heart of flexibility, efficiency and accuracy is a quantitative understanding of arithmetic. I'm really glad that you had Amy Hackenberg on [the podcast] recently who discussed this concept of units coordination because throughout what we'll talk about, you'll see units coordination come out, but she's definitely the expert to explain it. Just a nod. Just listen to that episode [Season 4, Episode 11]. It was amazing. 

Thinking, though, specifically about fluency—fluency isn't just knowing all of these combinations. In the early stages of counting, students view a number simply as a count or result of a count of single items, and there's this critical shift in developing a unit as a fundamental tool of measurement. And that's the act of unitizing where a student conceives of a collection of items as one unit that's simultaneously made of smaller units. 

It is a common progression that once a student counts on, that then we would shift to building strategies to solve addition and subtraction within 20, and then of course with 100, and beyond, and then in other domains. But this is all happening in first and second grade for that addition and subtraction to 20 fluency. So attending to this numerical composite—understanding that when a child says “7” and sees that that represents counting from 1 to 7 without having to count—is a really big cognitive shift in their mathematical understanding and can be undermined with, “Oh, now that they're counting on, we're going to tell them these strategies.” And so we really do need to have some intentional instructional strategies to make sure that we're developing that first, that numerical composite, before we try to develop all these strategies for addition and subtraction to 20. Because that is the basis for children to move from a counting-based strategy to compose units. 

So when they can use a quantity like, "Oh, 8 plus 5, I can break apart this 5 into smaller parts and I can give some of those parts to the 8." So children at that point have to simultaneously hold 5 as a single unit while recognizing the 2 and the 3 make up the 5, but they can be moved to the 8 as well. That's really sophisticated. 

Mike: So I want to mark that because I think the notion that this is really sophisticated is important for folks to understand because I'll be vulnerable and honest: I didn't recognize the complexity of what children were grappling with when I started teaching, particularly as a person who was teaching kindergarten and first grade. I really saw my job as helping to build a set of rote procedures like counting and number sequence and memorizing combinations and the outcome of being able to count and the outcome of being able to quickly recall those. I think that's not in question, but understanding the mechanics and the evolution of kids' thinking that's going on, that's a big deal. This whole notion that you have a unit and the unit is composed of smaller units. And one of the things that you said that feels like a really big deal that could be lost is the idea that shifting from a counting-based strategy to a strategy that depends on this notion of units that have smaller units inside and that are also still a unit—that's such a big deal. In order to go from counting everything to counting on to being able to look at a number like 8 and say that it has a 5 and a 3 inside of it—all of that is connected to this notion of units inside of units. And I'm so glad you mentioned that. 

Kristin: Yeah. The mental actions that students are doing, making those visible, when we see children do it developmentally, we just assume it's easy. But the shifts that they're making in their understanding of units to move from that pre-numerical stage of “Everything is a 1 and I have to repeat it” to “Now this word can stand in for the count” to “Now I can embed units inside of other units.” There's so much happening, and they're so young at that age; we have to remember that too. 

Mike: So let's talk about some other important components of developing fluency. What else is an important primer for how people are thinking about this? 

Kristin: Yeah. Another important component is supporting students in developing the cognitive structures that allow students to anchor their understanding and quantitative meaning and develop that sophisticated reasoning. Many researchers, many authors have written in different ways and different names about these structures. So like a “mental structure,” “mental residue,” “mental tools,” “patterns of thought.” To name a few people, Zaretta Hammond, Betty [K.] Garner, Karen [S.] Karp are some people I've read and appreciate their thinking around that. 

So it's more than just allowing students to use manipulatives to solve problems. There's an intentionality in how we use tools and an explicit process used by educators to bring their mathematical world to life. So first, identifying key settings that emphasize mathematical structures. So the tool in front of them has a big role to play in the “math”—I put that in quotations—in the “math” that they see. 10-frames that highlight a quantity of 10, but also can show other quantities within 10, such as, like, a five or a double. It has an added layer of boxes that contain a number. Some contain a number or a counter and others are empty. So there's ways that kids are coming to understand quantity with the structure. 

Similarly, a bead rack can show a five structure, a double structure, depending on your representation. They can help kids think about exchanges and really kind of that movement of quantity in a real physical way. Using linking cubes, do you use them all in one color? Are you strategic about the color that you use to bring out mathematical structures for them? 

So once we think about the key setting and the structure that we're trying to help kids reason about, we want to pose intentional questions that orient students to those structures. So how do they see that 5 inside? How are we going to bring that out? It's obvious to us, but are they seeing that or are they seeing something different in the tool? Are they reasoning about something different? And so the intentionality behind how we question students during those activities also aids to building their cognitive structures. So it's not the tool itself that is the 8. It's that the child is seeing the 8 and they're seeing the 5 and the 3 in some empty boxes. 

And finally, I think the step that we miss a lot, especially in problem-based instruction or any kind of inquiry-based instruction, is this explicit time where we connect the symbols in formal mathematics directly to represent the child's thinking and the tool that they've been playing around with. So it's not just about knowing I can get an answer on the 10-frame, but it's [that] I'm abstracting that series of actions, and I'm then connecting it to this quantity that I've written in a symbol. And are there connections between those things? And if those things aren't happening—kids are doing all those parts and pieces, but really developing the cognitive structure that they can then themselves use and take with them, I think that's what's so powerful when we talk about fluency is they can take a cognitive structure with them and fill in the mathematics in the future [when] maybe they don't have an educator in front of them asking those questions. But if they've been through those processes, then they have that structure to fill in. 

Mike: There's a lot that you just said that I think is important and we could probably linger on a lot of it. But on the front end of this conversation, you said it's one thing to be able to see students in a progression, and it's another thing to think about, “What's my role or what are the tools that I have to help them shift?”

What I heard in that last part, particularly is this notion of almost like a translation between the physical materials kids are engaging with and the meaning that they're making of that, and then helping them to abstract that in a way where we have symbols that are representing either actions or quantities and the relationships that are happening. That part of the teacher's job and part of the moves that teachers have in their toolbox is this notion of translation—taking what I'm seeing kids doing and how what I'm hearing them say or do to make meaning of it, and then helping them make that abstraction is kind of one of the tools that's really important in a teacher's toolbox when they're thinking about helping kids make moves. 

In preparation for our interview, one of the things that stayed with me was you described how your own understanding of the meaning and the importance of fluency had shifted over time. And I'm wondering if you can talk about what you used to think and what is it that you think now about fluency. Could you talk about your own personal journey? 

Kristin: For sure. I used to think that knowing facts, just knowing them in a very static way—like I know the answer to 5 plus 3, I keep coming back to that fact—reduces the cognitive load when they were getting into higher grade levels. Well, they don't need to think about that problem, and they can think about what we're doing in seventh grade math or in algebra. 

But what I've come to understand is that the ways that students know their facts—more specifically how they're able to work with the units and the way they conceptualize the units that they are given, how they break them apart, how they put them back together—that's what matters as they go. So not just knowing the answer, but that these things can be taken apart and put back together. 

Anderson Norton is a researcher that I really love to listen to. And I listened to him at an Integrow conference once. And he talked about developing mathematics through repeatable mental actions. So this kind of relates back to those cognitive structures. One example of a group of mental actions is this idea of composable, reversible, and associative. So when I think about 8 plus 5, 5 is composed of a 2 and a 3, and I can reverse that to focus on the unit of 2, and then I can associate that quantity with the 8 to make a new unit while keeping intact the unit of 5. That's really complex, but that idea transcends the domains of mathematics. Now, I'm not an expert in units coordination research, so I hope I represented that correctly, but I've certainly experienced students struggling to keep track of different units as they work. So thinking about exponent rules, and they break apart these powers and they're writing them and they're learning all these patterns, but they're struggling to keep track of the units that they're working with. Factoring functions in algebra. We're asking them to break apart something and put it back together in these different forms, and they're losing track of these units. So these actions of composable, reversible, and associative have implications in many domains of mathematics. So the bottom line is we want to develop not the fact itself, but the mental action behind that fact. Anderson Norton, I hope I did that justice. 

Mike: I want to name something that I think is really important, particularly given the fact that your background is actually in secondary [education]. So what I take from this is this idea of working with units and the mental actions, that transcends arithmetic. It transcends whole numbers and even rational numbers. And it pays dividends and it keeps paying dividends in middle school and high school as kids are working in an algebra context. And I think that's worth saying out loud because it means that doing this work with elementary students to develop fluency is a bit of a twofer in the sense that you do get kids who end up with a bank of facts that they know, but they also have this underlying understanding of units and actions that pays dividends for them in the long run. Mathematics education, students' learning experience, is not a sprint or a series of handoffs. It's really a marathon. And those early experiences, they pay dividends and they keep paying dividends. I think that's really important because it reminds us, particularly as elementary educators, that we're part of a larger project. 

Kristin: Not only part of a project, but part of building a lifelong interest in mathematics as an actual body of research that's dynamic and not a set of things to memorize and learn so that mathematics does become applicable in these different fields because the way that I approach a problem as an expert mathematician is that I take things apart, I put them back together. That transcends many careers. It's not just about being a math teacher or a math professor. It's about coming to understand that I have autonomy and how I see relationships of things, whether they're numbers or shapes or maybe parts that I'm working on in some sort of creative field that I'm in, but that I can do all of these things and that I can be curious and repeat those actions and see how they play out in that particular study. 

Mike: That's well said. 

Well, let's talk about the what, the why, the how of combinations to 10 and 20. To begin, I want to note that we use the term “combinations,” and I'm wondering if you can say more about what you mean when you refer to combinations and why they matter. 

Kristin: Yeah. I mean combinations not to literally mean “addition,” but that combination is the idea of this relationship between parts and wholes. So that 2, 3, and 5 have this kind of additive relationship. I can put these parts together to make the whole; I can take a part out of the whole and be left with a part. I can have a part and wonder what part I need to make the whole. And so we sometimes talk about these in curriculums as “fact families,” but the emphasis should be on the relationship of the parts to the whole and not filling out that kind of mimicking of like, “I know the four sentences because I know this thing.” So, “If I know this, I also know this.” It feels really nuanced, but in action really quite specific. 

Mike: So I think that's really helpful and it really does lead me to my next question about how we help kids build their fluency with combinations to 10 and 20 and beyond. So given the why that you just articulated, it seems like the how is going to be substantially different from the ways that many, if not most, adults learn to build fluency. Can you talk about that, Kristin? 

Kristin: We start from key combinations first. We consider a set of combinations that would be really useful in a lot of contexts. And I think many listeners will be familiar with those key combinations: doubles. Combinations of 10, of course. 5 plus because I have five fingers and then I can add some more on it, and I'm showing some finger patterns. So those are things we normally work on with students anyways. But starting again, going back to my original statement from a quantitative perspective—so not the memorization of those facts, but that I really come to understand them as quantities that are useful to me. And then building from those key combinations—I also want to name before I build onto that, is that some kids just have other facts that are interesting to them that they bring. So it might be their age, it might be the combination of their siblings' ages. And so we don't want to ignore that we introduce key combinations to students, but that students also have combinations that are useful to them naturally. 

So once we have a set of those key combinations that we've come to think about and reason about, we can then build things that we don't know. We can transfer that. So 5 plus 3 can help me think about 4 plus 3. If I have a mental structure of a 10-frame or a bead rack that helps me think about, “Oh, there's just going to be one less counter on the top, and so I'm going to take that [counter] away.” So that idea of taking the 1 out of the number is a really important mental action of them disembedding that quantity. 

In addition, when we think about the 5 plus, the doubles, the partitions, we're thinking about combinations that will also transcend into multidigit combinations. So addition, subtraction—whether we're working with whole numbers or decimals, we can make tens, we can make hundreds, we can make wholes, we can make zeros. And those combinations of 10 are going to be really useful for us. 

Mike: I'm struck by the fact that the combinations and also the mental actions that accompany them, as you said, they really do scale up quite nicely. And it seems like they scale up in the sense that they can get used to understand and solve problems with larger whole numbers, but they can also scale in the sense that ideas will help kids, but they can also scale in the sense that the ideas can really help kids when they encounter fractions and decimals. I wonder if you could talk about that idea just a little bit. 

Kristin: Yeah. So thinking about a combination of 10 in this missing part. So 99 plus can help us when we're thinking about, that 99 is 1 away from 100. It can also help us think about 99 one-hundredths or 9 tenths as being one part or one unit away from a benchmark number that's really helpful for us. And so, it's just that the unit itself is different. So instead of just a whole, I'm one whole unit away from 100, I might be 1 tenth of a unit away from one whole, so the unit is just changing. 

The view of mathematics this way, again, is very dynamic. We're creating a world where children are thinking about units and units away across domains, across number systems. And if we come to regard units as things that we can act on, whether it's a single object or a group of objects or a shape—we can put them together, take them apart and reassociate them—I can think of a lot of my mathematical knowledge in this way and not as a static set of information that I learned. And so then I'm able to transfer that because I've done that mental action or I've thought about something being a unit away. 

Mike: That's fascinating because I'm going to go back to this whole notion of the relationship between 3 and 2 and 5. So 3 is 2 units away from a unit of 5 and three-fifths are 2 one-fifths away from a unit of five-fifths or one whole. This notion of units away from or units that combine to make other units, I really get now whether it's whole numbers or fractions, we're really talking about a unit that we've defined and then how many other units or how can we—how did you describe that? What was the language you used before about pulling a unit out? Was it “disembed”? 

Kristin: “Disembed,” yeah. 

Mike: That really plays regardless of the type of unit we're talking about. 

Kristin: Yeah. And remember back where we said this quantity had a meaning, so 7 stood for something. When we disembed, that unit still has meaning in the context of the original unit. So that's a really important point about disembedding is that it's not just that you take a part out, it's that part still has a relationship to the whole and you don't lose that relationship. 

Mike: As I hear you talking, there seem to be some themes that are jumping out. One is the importance of key fact combinations and the mental actions. Another is the role visual models play in learning those combinations. And I think finally, I hear you indicating that it's important for students to make connections between different representations of the same combination. Tell me what I understood properly. Tell me what you'd revise or add to the summary that I just offered. 

Kristin: Yes. I think we get a false sense that a student understands a concept when they're recognizing pattern, and that could be that they're recognizing pattern in a really intentional setting. Maybe they're using a 10-frame. But is that same relationship present in another setting? Success should not be measured by one instance of a child recognizing that pattern. And so one way of knowing that a child knows this is to see it in many contexts. And I think that's why it's so important for us to acknowledge the research around multiple representations in mathematics. And showing that knowledge in these multiple ways really does say that this is a connected set of knowledge that I can refer to as a child and not just be successful on this one day. That doesn't mean that that experience where they're recognizing the patterns is not important, but that can't be the measure of their success. 

So this also becomes challenging in our system that values assessment events so heavily and measuring against a set benchmark. And I just want to name that because that's a real challenge for teachers. And of course we want to develop this rich set of knowledge, and sometimes we have to say that this is the system that we live in. But the true measure of that knowledge is being able to take that knowledge and transfer it into these multiple representations or in these multiple spaces and be able to use that. And that's why we talk so much about fluency being flexible and not just about accuracy. 

Mike: You have me thinking more deeply than I have in a long time about the structure of some of the visual models and the physical materials that children use when they're engaged with the Bridges curriculum. I wonder if we could get specific and talk about a few of the visual models that support student learning. Are there features that make some models particularly valuable? 

Kristin: One I want to mention that we might not have talked about is just a child's fingers. I think sometimes we think child's fingers are not models for them because they're counting by 1 and we tend to want students to move to more efficient strategies. But these fingers actually become really efficient tools. We can exchange fingers, we can move them very easily. We have control, and they're always with us. And so the finger use itself, I think, is a really powerful tool for us to encourage students to use in very sophisticated ways. 

Mike: I mean, we literally have units of 1, units of 5, and a unit of 10 at our fingertips in front of us. I'm so glad you called that out because that's a tool that students can make use of, that teachers can make use of and that we can think of in a slightly different way than we had in the past when I just thought about fingers as a counting-by-1 resource, when actually fingers, [a hand], and hands, plural, are 1s, 5s, and 10s right there in front of you. 

Kristin: And they can stand in for other units if we're really sophisticated with sequences. So a 1 can be a 7 if we wanted it to be, and we can think really creatively about that. I mean, I think that depends on some other skills. But yeah, we have 1s, 5s, and 10s built right into our hands. 

Mike: That's exactly right. And you're making me think about the fact that when I skip-count or when I see students skip-count, oftentimes what's happening is I'm speaking the unit out loud and I'm holding up one finger to stand in for that unit on my hand to keep track of the number of units. So I totally hear what you're saying. 

Kristin: Yeah, very sophisticated. And then there's even more complex content, right? So thinking about hours and elapsed time, and we're crossing different kinds of numerical systems where you go from a 12 to a 1 is very complex, and then we can have these fingers as units as well to help us keep track of things. So of course, frames are a really powerful tool. Frames—specifically, 10-frames, 5-frames, 20-frames—provide an extra structure for students, especially when they're really thinking hard about some quantity pieces. So they might not be completely solid in that unit, but we don't have to say, "Oh, you have to count on first before we're going to try to explore some other patterns." Those things can be developing simultaneously. So frames provide this box that contains the unit for them and it becomes this really obvious count for them. They can see those individual discrete items, but they can also see what's missing really clearly because they're empty. 

Bead racks are a great support as well when you're thinking about that relational network that we want students to develop and not count by 1s. So we can exchange beads, and we can exchange quantities, and we don't have to exchange beads one by one. Sometimes frames, when we get to a space, it's inconvenient to have to move five counters at the same time where in a bead rack, you can just slide those five over or three over at the same time. 

I also want to mention linear bead racks. So taking that stacked bead rack and making it align really helps students think about a continuous model, which transfers to a number line and the idea of units being measurement. So we were talking about, “It's one away,” and so really conceptualizing that kind of next decade of numbers and one bead away. That's developing that idea of relative magnitude that's extremely helpful when we get to middle school and all of a sudden we're working in negative numbers. 

Mike: We're reaching the end of our time together. And before we go, I'm wondering if you could share contact information for Integrow Numeracy Solutions with our listeners. I'd really love to be able to offer that because we've just touched the surface of some of the ideas that you help educators explore in some of the training and the support that you all offer. 

Kristin: Yeah. If you'd like to find out more about us, a great place to go is our website, which is www.integrowmath.org, all one word. And we have a lot of different things you can explore from our events. There is actually, if you add a backslash “blog” to that [www.integrowmath.org/blog], you can go to our blog and read some of the ways that we think about our professional learning and some of the topics that I talked about today. If you want to reach out directly, feel free to email [e-mail] and someone will get you to the right place based on your question. 

Mike: And for listeners, we'll put a link to both of those in the show notes.

Before we leave, Kristin, I'll just ask one last question. Are there any recommendations that you have for folks interested in learning more about the ideas we've talked about today? It could be books, websites, articles, or even just a suggested practice for someone who wants to get started. 

Kristin: Yeah. For sure, take a look at the blogs on our website. They're little snippets of pieces of our trainings that you can take right with you into the classroom. Some ideas that I've talked about—help with bead racks, ideas around multiplication and division, and supporting students to think about those units. Our new publication, On Track to Numeracy from [Lucinda] “Petey” MacCarty, Kurt Kinsey, [David Ellemor-Colons, and Robert J. Wright], is designed to be an accessible, relatable and practical tool focused on supporting classroom teachers. It not only has the progressions that I started this podcast off talking about, but it has those teaching tests and progressions that help us answer the question of, “What do I do next now that I can understand where my students are?”

Mike: I think it's a great place to stop, Kristin. I want to thank you so much for joining us. It's really been a pleasure talking with you. 

Kristin: Thank you for having me. I've had a great time. 

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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