Kendra Lomax, Cognitively Guided Instruction: Turning Big Ideas into Practice

Mike Wallus, Vice President for Educator Support

Rounding Up: Season 1 | Episode 7

Today on the podcast, we’re talking with Dr. Kendra Lomax from the University of Washington about a body of research called Cognitively Guided Instruction and the promise it holds for elementary educators and students.

More Episodes


If you’re interested in more on this topic, consider the following article for further reading:

Cognitively Guided Instruction


Mike Wallus: Have you ever had an experience during your teaching career that fundamentally changed how you thought about your students and the role that you play as an educator? For me, that shift occurred during a sweltering week in July of 2007, when I attended a course on cognitively guided instruction. Cognitively guided instruction, or CGI, is a body of research that has had a massive impact on elementary mathematics over the past 20 years. Today on the podcast, we're talking with Dr. Kendra Lomax, from the University of Washington, about CGI and the promise it holds for elementary educators and students. Well, Kendra, welcome to the podcast. It's so great to have you on. 

Kendra Lomax: Well, thanks for having me. 

Mike: Absolutely. I'm wondering if we can start today with a little bit of background; part history lesson, part primer to help listeners understand what CGI is. So, can you just offer a brief summary of what CGI is and the questions that it's attempted to shed some light on? 

Kendra: Sure, I'll give it my best try. So, CGI is short for cognitively guided instruction, and it's a body of research that began some 30 years ago with Tom Carpenter and Elizabeth Fennema. And there's lots of other scholars that since then have kind of built upon that body of research. They really tried to think about and understand how children develop mathematical ideas over time. So, they interviewed and studied and watched really carefully what young children did as they solve whole-number problems. So, you may have heard about the book Children's Mathematics, and that's where you can read a lot about cognitively guided instruction and [it] summarizes some of that research. And they really started with whole-number computation and then have kind of expanded into areas like fractions and decimals, learning about how kids develop ideas about algebraic thinking, as well as early ideas around counting and quantity. 

Mike: Uh-hm. 

Kendra: So, there's a couple of books that are kind of in the CGI family. Young Children's Mathematics includes those original authors, as well as Nick Johnson and Megan Franke, Angela Turrou, and Anita Wager. That fractions and decimals work was really led by Susan Empson and Linda Levi. And then, like I mentioned, Thinking Mathematically is the text by the original authors that kind of talks about algebra. So, in all of those texts that summarize this research, basically, we're trying to understand how do children develop ideas over time? And Tom and Liz really set an example for all of us to follow in how they thought about sharing this research. They had a deep respect for the wisdom of teachers and the work that they do with young children. So, you won't find any sort of prescription in the CGI research about how to teach, exactly, or a curriculum. Because their approach was to share with teachers the research that they had done when they interviewed and listened to all of these many children solving problems, and then learn from the teachers themselves. What is it that makes sense to do in response to what we now know about how children develop mathematical ideas? 

Mike: I mean, it's kind of a foundational shift in some ways, right? It reframes how to even think about instruction, at least compared to the traditional paradigm, right?

Kendra: Yeah, it's less a study of how best to teach children and really a study and a curiosity about how children bring the ideas that they already have to their work in the math classroom, and how they build on those ideas over time. 

Mike: Definitely. It's funny, because when I think about my first exposure I think that was the big aha, is that my job was to listen rather than to impose or tell or perfectly describe how to do something. And it's just such a sea change when you rethink the work of education. 

Kendra: Definitely. And it feels really joyful, too, right? You get to be a student of your students and learn about their own thinking and be really responsive to them in the moment, which certainly provides lots of challenges for teachers. But also, I think, just a sense of genuine relationship with children and curiosity and a little bit of joy. 

Mike: Definitely. So, I'm wondering if we could dig into a little bit of the whole-number work, because I think there's a bit that we were talking about with CGI, which is really the way in which you approach students, right? And the way that you listen to students for cues on what they're thinking is. But the research did reveal some ways to construct a framework for some of the things you see when children are thinking. 

Kendra: So, if you read the book Children's Mathematics, you might notice or recognize some of those ideas, because CGI is one of the research bases for the Common Core state math standards. So, when you're looking through your grade-level standards and you see that they're suggesting particular problem types, number sizes, or strategies that children might use, much of that is based on the work of cognitively guided instruction, as well as other bodies of research. So, it might sound familiar when you read through the book yourself. And what CGI helps reveal is that there's a somewhat predictable sequence: That young children develop strategies for whole-number operation for working with whole number computational problems. 

Mike: Yeah. Can you talk about that, Kendra? 

Kendra: Yeah. So, young children are going to start out with what we call direct modeling, where they are going to directly model the context of the problem. So, if we give them a story problem, they'll act out or model or show or gesture, to show the action of the problem. So, if it describes eating something (makes eating sounds), you can imagine, right, the action that goes along with eating? And we're all very familiar with it. So, they're going to show maybe, the cookies, and then cross out the ones that get eaten… 

Mike: Uh-hm. 

Kendra: …right? So, they're really going to directly model the action or relationship described in the problem. And they're going to also represent all the quantities in the problem, which is different. What they learn over time is to count on or count back. So, some of the counting strategies where they learn, “Gosh, I don't want to make all the quantities in this problem.” It becomes too difficult, too cumbersome. And they learn that they could count on from one of the quantities or count back. So, in that cookie example, maybe there are seven cookies on a plate, and I have two of them for dessert, right? (makes eating sounds) They go away. So, in direct modeling, they're going to show the seven cookies. They're going to remove those two cookies that get eaten, and then count how many are left. Where in counting on—so they have had lots of experiences of direct modeling—they can say, “Gosh, I don't really want to draw that seven. I'm going to imagine the seven…” 

Mike: Uh-hm. 

Kendra: “ …And I can maybe count backwards from there.”

Mike: So, like, 7, 6, 5. 

Kendra: Yeah. Right. So, I don't have to make the seven. I can just imagine it. And I keep track of those two that I'm counting back. 

Mike: That totally makes sense. And as a former kindergarten and first-grade teacher, it's an amazing thing to actually see that shift happen. 

Kendra: Right? And it's really specialized knowledge that teachers develop to pay attention to that shift. It's easy for everybody else to kind of miss it. But for teachers, it's a really important shift to pay attention to. 

Mike: I used to say to parents, when I would try to describe this, it's something that we almost aren't conscious of being able to do. But it's a gigantic step to go from imagining a quantity as a set of ones to imagining a quantity that is a number that you can count back from or count forward from. It's a gigantic leap. Even though to us, we've forgotten what big of a leap that was because it's been so long since we took it. 

Kendra: Yeah. That's one thing I love about studying children's mathematics, is, like, you get to experience that wonderment all over again… 

Mike: Uh-hm. 

Kendra: …in the things that we kind of, as adults, take for granted in how we think about the world. 

Mike: Yeah. I think you really clearly articulated the shift that kids make when they move from direct modeling, the action and the quantities, to that kind of shift in their thinking and also their efficiency of being able to count on or count back. Is there more to, kind of, the trajectory that kids are on from there? 

Kendra: There is, yeah. So, after children have had lots of experiences to direct model, and then learn to become more efficient with that, and counting on or counting back, then they might start inventing. We call them invented algorithms, which is a fancy way to say that they think about the relationship between quantities and start putting them together and taking them apart in more efficient ways. So, they might use their understanding of groups of 10, right? So, in that example, with the cookies—seven cookies and eating two of them—I might know something about the relationship with fives… 

Mike: Uh-hm. 

Kendra: …Five and two make a seven. So, they start to develop some sense of how numbers go together, and how the operations really behave. So, in addition, I can kind of add them in any order that I want to, right? So, we see these called in the Common Core standards, Strategies Based on Place Value, Properties of Operation, and the relationship between addition, subtraction, or multiplication, division. 

Mike: That's super helpful to actually connect that language in Common Core to what you might see, and how that translates into, kind of, what one might read about in some of the CGI research. 

Kendra: Right. It'd be lovely if we all had the exact same (laughs) names, wouldn't it? 

Mike: Definitely. One of the questions that I suspect people who might be new to this conversation are asking is, what are the conditions that I can put in place? Or what are the things that I might, as a teacher, be able to influence that would help kids move and make some of these shifts. Knowing that the answer isn't direct instruction. I could get a kid to mimic counting on, but if they're still really thinking about numbers in the sense of a direct modeler, they haven't really shifted, right? So, my wondering is, how would you describe some of the ways that teachers can help nudge children, or kind of set up situations that are there to help kids make the shift without telling, or… 

Kendra: (chuckles) 

Mike: …like, giving away the game? 

Kendra: Totally. Yeah. That's one takeaway that I'm always on the lookout for when people hear about CGI and this trajectory that's somewhat predictable. 

Mike: Uh-hm. 

Kendra: Let's just teach them the next strategy then, right? 

Mike: Right. 

Kendra: And what's important to remember is that these are called invented algorithms for a reason. Mike: Uh-hm. 

Kendra: Because children are actually inventing mathematics. It's amazing. Kindergartners are inventing mathematics. And so, our role is really to create the right opportunities for them to do that important work. And like you're saying, when they're ready for the next ideas that they're building on their existing knowledge, rather than us kind of coming in and trying to create that artificially. 

Mike: Uh-hm. 

Kendra: So, again, like, Liz and Tom really kind of taught us to be students of our students as well as students of teachers. 

Mike: Uh-hm. 

Kendra: So, what we've learned over time…some of the things that teachers have found really productive for supporting students to kind of move through this trajectory, to create increasingly efficient strategies, is really about thinking about carefully choosing the problems that we've put in front of students. 

Mike: Uh-hm. 

Kendra: So, paying attention to the context. Is it familiar to them? Is it reasonable for the real world? Are we helping kids see that mathematics is all around them. 

Mike: Uh-hm. 

Kendra: Paying attention to the quantities that we select. So, if we want them to start thinking about those relationships with five and 10, or as they get older with hundreds and thousands, that we’re intentional about the quantities that we choose for those problems. 

Mike: Right. 

Kendra: And then, of course we know that students learn a lot from not just us, but their relationships and their discussions with their classmates. So, really orchestrating classroom discussions, thinking about choosing students to work together so that they can both learn from one another, and really just finding ways to help students connect their current thinking with the new ideas that we know are on the horizon for them. 

Mike: I would love for you to say a little bit more about number choice. That is such a powerful strategy that I think is underutilized. So, I'm wondering if you could just talk about being strategic around the number choices that you offer to kids. Can you say more about that? 

Kendra: Sure! It's going to depend on grade level, of course, right? 

Mike: Uh-hm. 

Kendra: Because they're going to be working with very different quantities early in elementary and then later on… One thing I would say, across all of the grade levels, is to not limit students whenever possible. So, sometimes we want to give problems that kids are really comfortable with, and we know they're going to be successful. But if I'm thinking of how they develop more efficient strategies, sometimes the growth comes in making it a little tricky. So, giving quantities that are just a little bit beyond where they're counting as young children, so they develop the need to learn that counting sequence. Or, as we're working with older students, if we know that particular multiplication facts are less familiar to students. Giving them that nudge by creating story context, where they can really make sense of the action of the relationship that's happening in it, but maybe choosing that times seven that we know has been tricky for kids, right? 

Mike: Yep. 

Kendra: So, I would just encourage people to not shy away from problems that we know pose some challenge to students. That's actually where a lot of the meat and the rigor happens. And, but then we also want to provide support inside of those, right? So, working with a partner. 

Mike: Definitely.

Kendra: Or making sure they have access to those counting charts. That's one thing I would say across grade levels. 

Mike: Yeah. So, you made me think of something else. It's fascinating to have this conversation, Kendra, ’cause it reminds me of all the things that I had to learn over time. And I think one of the things that I'm wondering if you could talk a little bit more about is, the types of problems and how the problem that you choose for a given group of students might influence whether they're direct modeling or they're counting on or whether they're using invented algorithms. Because I think, for me, one of the things that it took a while to make sense, is that the progression isn't necessarily linear, right? Like, if I'm counting on in a certain context, that doesn't mean I'm counting on in all contexts or direct modeling or what have you. So, I'm curious if you could talk a little bit about problem types and now how those influence what things students sometimes show us. 

Kendra: Yeah. I'm glad you brought that up. When we describe, kind of, that trajectory of strategies, it sounds really nice and tidy and organized and like it is predictable in some ways. But like you're saying, it also depends on the kind of problem and the number size that we're putting in front of children. So that trajectory kind of iterates again and again throughout elementary school. So, as we pose more complex problem types…so, for example, the cookies problem where I have seven cookies, I eat two of them and the result is what's at the end of the story, right? The cookies left over. 

Mike: Uh-hm. 

Kendra: If I now make that problem, I have some cookies on a plate. I ate two of them, and I have five left over. All the kindergarten teachers, actually all the elementary school teachers… 

Mike: (laughs) Yes. 

Kendra: …can automatically recognize that's going to be a more tricky problem, right? Mike: Uh-hm. 

Kendra: Where do I start!? Especially if I'm direct modeling, right? We know they start and follow the exact action of the story. 

Mike: Absolutely (chuckles). 

Kendra: So, as we pose more complex problem types, you're right. You're going to see that they might use less efficient strategies because they're really making sense. They're like, “Wait, what's the relationship that's happening in this story? Where do I begin? Where are the cookies at the beginning, middle, and end of this story?” So, we see that happen throughout elementary school. So, it's not that direct modeling is for kindergartners. And that invented algorithms are for fifth grade. It's that as new ideas get introduced, as we make problems more complex, maybe increasing the number size or now we're working with fractions and decimals… 

Mike: Uh-hm.

Kendra: We see this happen all over again. Kids begin with direct modeling to make sense of the situation. Then they build on that and get a little bit more efficient with some counting kinds of strategies. And then over time with lots of practice with that new problem type, those new numbers, um, they develop those invented algorithms again. 

Mike: So, this makes me think of something else, Kendra. How would you describe the role of representation in this process? That could mean manipulatives that students choose to use. It could mean things that they choose to draw, visual models. How does representation play in the process? 

Kendra: Yeah. So, oftentimes I hear people say, “This student used cubes. That was their strategy.” Or “This student used a drawing. That was their strategy.” And that's really not enough information to know the mathematical work that that child is doing. Did they use cubes as a way to count on? 

Mike: Uh-hm. 

Kendra: Are they keeping track of only one of the quantities but using cubes to do so? Are they doing a drawing that actually represents groups of 10? And they're using ideas about place value inside of it, which is different than if they're just drawing by ones, right? So, there's lots of detail inside of those representations that's important to pay attention to. 

Mike: Yeah. I’m thinking about one of my former kindergartners. I remember that I had some work that she had done in the fall, and then I had another bit of work that she had done in the spring. And the fall was this (chuckles) very detailed drawing of, like, a hundred circles. And then in the spring, she was unitizing, right? She had a bunch of circles and then within [them] had labeled that each of those were 10. And it just struck me, like, “Wow, that is a really tangible vision of how she was drawing in both cases.” But her representation told a really different story about what she understood about math, about numbers, about the base 10 system. 

Kendra: Right. And those might be very different starting points. As you, the teacher, you walk over and you see those two different kinds of drawings… 

Mike: Uh-hm. 

Kendra: …your conversation or your prompt for them…your next step for them might be pretty different. 

Mike: Absolutely. 

Kendra: Even though they're both drawing. 

Mike: Yeah. Well, let me ask you this, ’cause I think I struggled with this a little bit when I first started really thinking about CGI. I had gone to a training and left incredibly inspired and was excited. And one of the things that I was trying to reconcile at that time is, like, I do have a curriculum resource that I'm using, and I wonder how many teachers sometimes struggle with that? I've learned these ideas about how children think, how to listen…what are some of the teacher moves I can make? And I'm also trying to integrate that with a tool that I'm using as a part of my school or my district. So, what are your thoughts about that? 

Kendra: That makes a lot of sense. And I think that happens a lot of time in professional learning, where we learn a new set of ideas and then we're wrestling with how do they connect with the things I'm already doing? How do I use them in my own classroom? So, I really appreciate that challenge. I guess one way I like to think about it is that the trajectory that CGI helps us know about how children develop ideas over time is a little bit like a roadmap that I can use regardless of the curricular materials that I have in front of me. And it helps me understand what is on the horizon for that child. What's next for them and their learning? Depending on the kind of strategies that they're using and the kinds of problems that we're hoping to be giving them access to in that grade level, I can look at my curricular materials in front of me and use that roadmap to help me navigate it. So, we were talking about number selection. So, I might take that lens as I look at the curriculum in front of me and think about, “Are these the right numbers to be using? What will my students do with the problem… 

Mike: Uh-hm. 

Kendra: …that is suggested in my curricular materials?” To anticipate how my discussion is going to go and what kinds of strategies I might want to highlight in my discussion. So, I really like to think of it as the professional knowledge that teachers need in order to make sense of their curriculum materials and make informed decisions about how to use those really purposefully. 

Mike: Yeah. The other thing that strikes me, that I'm connecting to what you said earlier, is that I could also look at the problem and think about, “Does the context actually connect with what I know about my children? Can I somehow shift the context in a way that makes it more accessible to them while still maintaining the structure, the problem, the mathematics, and such?”

Kendra: Right. Yeah. Are there small revisions I can make? Because, uh, I don't envy curriculum writers (chuckles) at all because there's no way you can write the exact right problem for every day, for every child across the country. So, as teachers, we have to make really smart decisions and make those really manageable. Because teachers are very busy people. 

Mike: Sure. 

Kendra: But those manageable, kind of, tweaks or revisions to make it really connected to our students’ lives. 

Mike: Yeah. I think the other thing that's hitting me is that, when you've started to make sense of the progression that children go through, it's a little bit like putting on a pair of glasses that allow you to see things slightly differently and understand that skill of noticing. That's universal. It doesn't necessarily come and go with a curriculum. It's something that's important. Knowing your students is always going to be something that's important for teachers, regardless of the curriculum materials they've got. 

Kendra: Yep. That's right. 

Mike: So, here's my, I think my last question. And it's really, it's a resource one. So, if I'm a listener who's interested in learning more about CGI, if this is really my first go at understanding the ideas, what would you recommend for someone who's just getting started thinking about this and maybe is walking away thinking, “Gosh, I'd like to learn more.”

Kendra: Sure. Well, I mentioned the whole laundry list of great texts that you can dig into more. So, Children's Mathematics being the one on whole-number operation across grade levels. I find that, like, preschool through first- or second-grade teachers have found Young Children's Mathematics incredibly impactful. It helps connect ideas about counting in quantity with these ideas about problem-solving and operation. And then kind of connects them and helps us think about how to support students to develop those really important early ideas. 

Mike: Uh-hm. 

Kendra: Anybody who I have talked to that has read Extending Children's Mathematics: Fractions and Decimals has found it incredibly impactful. 

Mike: I will add myself to that list, Kendra. It blew my mind. 

Kendra: Yeah, us, too! Everybody who read it was like, “Ohhh, I see now.” It points out a lot of really practical ways for us to pay attention. It offers a trajectory much like whole-number about how children develop ideas and also kind of suggests some problems that will help us support students as they're developing those ideas. So, [I] definitely recommend those. And then, Thinking Mathematically is another great text that helps us connect arithmetic and algebra, as we're thinking about how to make sure that students are set up for success as they start thinking more algebraically. And [it] digs into a little bit of—I talked about young children inventing mathematics—I think even further describes the ways that they invent important properties of operation that can be really interesting to read about. 

Mike: That's fantastic. Kendra, thank you so much for joining us. It's really been a pleasure talking to you today. 

Kendra: Thanks for having me. 

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.