Season 3 | Episode 02 - Principles for Responsive Curriculum Use - Guest: Dr. Corey Drake

Rounding Up

Rounding Up

Season 3 | Episode 2 – Responsive Curriculum

Guest: Dr. Corey Drake

Mike Wallus: When it comes to curriculum, educators are often told to implement with “fidelity.” But what does fidelity mean? And where does that leave educators who want to be responsive to students in their classrooms? Today we're talking with Dr. Corey Drake about principles for responsive curriculum use that invite educators to respond to the students in their classrooms while still implementing curriculum with integrity.

Mike: One of the age-old questions that educators grapple with is how to implement a curriculum in ways that are responsive to the students in their classroom. It's a question I thought a lot about during my years as a classroom teacher, and it's one that I continue to discuss with my colleague at MLC, Dr. Corey Drake. As a former classroom teacher and a former teacher educator who only recently began working for an organization that publishes curriculum, Corey and I have been trying to carve out a set of recommendations that we hope will help teachers navigate this question. Today on the podcast, we'll talk about this question of responsive curriculum use and offer some recommendations to support teachers in the field.

Mike: Welcome back to the podcast, Corey. I'm excited to have you with us again.

Corey Drake: It's great to be with you again.

Mike: So, I've been excited about this conversation for a while because this question of, “What does it mean to be responsive to students and use a curriculum?” is something that teachers have been grappling with for so long, and you and I often hear phrases like “implementation with fidelity” used when folks are trying to describe their expectations when a curriculum's adopted.

Corey: Yeah, I mean, I think this is a question teachers grapple with. It's a question I've been grappling with for my whole career, from different points of view from when I was a classroom teacher and a teacher educator and now working at The Math Learning Center. But I think this is the fundamental tension: “How do you use a set of published curriculum materials while also being responsive to your students?” And I think ideas like implementation with fidelity didn't really account for the responsive-to-your-students piece. Fidelity has often been taken up as meaning following curriculum materials, page by page, word for word, task for task. We know that's not actually possible. You have to make decisions, you have to make adaptations as you move from a written page to an enacted curriculum. But still the idea of fidelity was to be as close as possible to the written page. Whereas ideas like implementation with integrity or responsive curriculum use are starting with what's written on the page, staying consistent with the key ideas of what's on the page, but doing it in a way that's responsive to the students who are sitting in front of you. And that's really kind of the art and science of curriculum use.

Mike: Yeah, I think one of the things that I used to think was that it was really a binary choice between something like fidelity, where you were following things in what I would've described as a lockstep fashion. Or the alternative, which would be, “I'm going to make everything up.” And you've helped me think, first of all, about what might be some baseline expectations from a large-scale curriculum. What are we actually expecting from curriculum around design, around the audience that it's written for? I wonder if you could share with the audience some of the things that we've talked about when it comes to the assets and also the limitations of a large-scale curriculum.

Corey: Yeah, absolutely. And I will say, when you and I were first teachers probably, and definitely when we were students, the conversation was very different. We had different curriculum materials available. There was a very common idea that good teachers were teachers who made up their own curriculum materials, who developed all of their own materials. But there weren't the kinds of materials out there that we have now. And now we have materials that do provide a lot of assets, can be rich tools for teachers, particularly if we release this expectation of fidelity and instead think about integrity. So, some of the assets that a high-quality curriculum can bring are the progression of ideas, the sequence of ideas and tasks that underlies almost any set of curriculum materials; that really looks at, “How does student thinking develop across the course of a school year?” And what kinds of tasks, in what order, can support that development of that thinking.

Corey: That's a really important thing that individual teachers or even teams of teachers working on their own, that would be very hard for them to put together in that kind of coherent, sequential way. So, that's really important. A lot of curriculum materials also bring in many ideas that we've learned over the last decades about how children learn mathematics: the kinds of strategies children use, the different ways of thinking that children bring. And so, there's a lot that both teachers and students can learn from using curriculum materials. At the same time, any published set of large-scale curriculum materials are, by definition, designed for a generic group of students, a generic teacher in a generic classroom, in a generic community. That's what it means to be large scale. That's what it means to be published ahead of time. So, those materials are not written for any specific student or teacher or classroom or community.

Corey: And so, that's the real limitation. It doesn't mean that the materials are bad. The materials are very good. But they can't be written for those specific children in that specific classroom and community. That's where this idea that responsive curriculum use and equitable instruction always have to happen in the interactions between materials, teachers, and students. Materials by themselves cannot be responsive. Teachers by themselves cannot responsibly develop the kinds of ideas in the ways that curriculum can, the ways they can when using curriculum as a tool. And, of course, students are a key part of that interaction. And so, it's really thinking about those interactions among teachers, students, and materials and thinking about, “What are the strengths the materials bring? What are the strengths the teacher brings?” The teacher brings their knowledge of the students. The teacher brings their knowledge of the context. And the students bring, of course, their engagement and their interaction with those materials. And so, it's thinking about the strengths they each bring to that interaction, and it's in those interactions that equitable and responsive curriculum use happens.

Mike: One of the things that jumps out from what you said is this notion that we're not actually attempting to fix “bad curriculum.” We're taking the position that curriculum has a set of assets, but it also has a set of limitations, and that's true regardless of the curriculum materials that you're using.

Corey: Absolutely. This is not at all about curriculum being bad or not doing what it's supposed to do. This assumes that you're using a high-quality curriculum that does the things we just talked about that has that progression of learning, those sequences of tasks that brings ideas about how children learn and how we learn and teach mathematics. And then, to use that well and responsibly, the teacher then needs to work in ways, make decisions to enact that responsibly. It's not about fixing the curriculum. It's about using the curriculum in the most productive and responsive ways possible.

Mike: I think that's good context, and I also think it's a good segue to talk about the three recommendations that we want educators to consider when they're thinking about, “What does it mean to be responsive when you're using curriculum?” So, just to begin with, why don't we just lay them out? Could you unpack them, Corey?

Corey: Yeah, absolutely. But I will say that this is work you and I have developed together and looking at the work of others in the field. And we've really come up with, I think, three key criteria for thinking about responsive curriculum use. One is that it maintains the goals of the curriculum. So again, recognizing that one of the strengths of curriculum is that it's built on this progression of ideas and that it moves in a sequential way from the beginning of the year to the end of the year. We want teachers to be aware of, to understand what the goals are of any particular session or unit or year, and to stay true to those goals, to stay aligned with those goals. But at the same time, doing that in ways that open up opportunities for voice and choice and sensemaking for the specific students who are in front of them in that classroom. And then the last is, we're really concerned with and interested in supporting equitable practice. And so, we think about responsive curriculum use as curriculum use that reflects the equity-based practices that were developed by Julia Aguirre and her colleagues.

Mike: I think for me, one of the things that hit home was thinking about this idea that there's a mathematical goal and that goal is actually part of a larger trajectory that the curriculum's designed around. And when I've thought about differentiation in the past, what I was really thinking about was replacement that fundamentally altered the instructional goal. And I think the challenge in this work is to say, “Am I clear on the instructional goal? And do the things that I'm considering actually maintain that for kids or are they really replacing them or changing them in a way that will alter or impact the trajectory?”

Corey: I think that's such a critical point. And it's not easy work. It's not always clear even in materials that have a stated learning goal or learning target for a session. There's still work to do for the teacher to say, “What is the mathematical goal? Not the activity, not the task, but what is the goal? What is the understanding I'm trying to support for my students as they engage in this activity?” And so, you're right. I think the first thing is, teachers have to be super clear about that because all the rest of the decisions flow from understanding, “What is the goal of this activity, what are the understandings that I am trying to develop and support with this session? And then I can make decisions that are enhancing and providing access to that goal, but not replacing it. I'm not changing the goal for any of my students. I'm not changing the goal for my whole group of students. Instead, I'm recognizing that students will need different ways into that mathematics. Students will need different kinds of supports along the way. But all of them are reaching toward or moving toward that mathematical goal.”

Mike: Yeah. When I think about some of the options, like potentially, number choice; if I'm going to try to provide different options in terms of number choice, is that actually maintaining a connection to the mathematical goal, or have I done something that altered it? Another thing that occurs to me is the resources that we share with kids for representation, be it manipulatives or paper, pencil, even having them talk about it—any of those kinds of choices. To what extent do they support the mathematical goal, or do they veer away from it?

Corey: Yeah, absolutely. And there are times when different numbers or different tools or different models will alter the mathematical goal because part of the mathematical goal is to learn about a particular tool or a particular representation. And there are other times when having a different set of numbers or a different set of tools or models will only enhance students' access to that mathematical goal because maybe the goal is understanding something like two-digit addition and developing strategies for two-digit addition. Well then, students could reach that goal in a lot of different ways. And some students will be working just with decade numbers, and some students will be working with decades and ones, and some students will need number pieces, and others will do it mentally. But if the goal is developing strategies, developing your understanding of two-digit addition, then all of those choices make sense, all of those choices stay aligned with the goal.

Corey: But if the goal is to understand how base ten pieces work, then providing a different model or telling students they don't need to use that model would, of course, fundamentally alter the goal. So, this is why it's so critically important that we support teachers in understanding, making sense of the goal, figuring out how do they figure that out. How do you open a set of curriculum materials, look at a particular lesson, and understand what the mathematical goal of that lesson is? And it's not as simple as just looking for the statement of the learning goal and the learning target. But it's really about, “What are the understandings that I think will develop or are intended to develop through this session?”

Mike: I feel like we should talk a little bit about context, because context is such a powerful tool, right? If you alter the context, it might help kids surface some prior knowledge that they have. What I'm thinking about is this task that exists in Bridges where we're having kids look at a pet store where there are arrays of different sorts and kinds of dog foods or dog toys or cat toys. And I remember an educator saying to me, “I wonder if I could shift the context.” And the question that I asked her is, “If you look at this image that we're using to launch the task, what are the particular parts of that image that are critical to maintain if you're going to replace it with something that's more connected to your students?”

Corey: Connecting to your students, using context to help students access the mathematics, is so important and such an empowering thing for teachers and students. But you're asking exactly the right question. And of course, that all relates to, “What's the mathematical goal?” Again. Because if I know that, then I can look for the features of the context that's in the textbook and see the ways in which that context was designed to support students in reaching that mathematical goal. But I can also look at a different context that might be more relevant to my students, that might provide them better access to the mathematics. And I can look at that context through the lens of that mathematical goal and see, “Does this context also present the kinds of features that will help my students understand and make sense of the mathematical goal?” And if the answer is yes, and if that context is also then more relevant to my students or more connected to their lives, then great. That's a wonderful adaptation. That's a great example of responsive curriculum use. If now I'm in a context that's distracting or leading me away from the mathematical goal, that's where we run into adaptations that are less responsive and less productive.

Mike: Well, and to finish the example, the conversation that this led to with this educator was she was talking about looking for bodegas in her neighborhood that her children were familiar with, and we end up talking quite a bit about the extent to which she could find images from the local bodega that had different kinds of arrays. She was really excited. She actually did end up finding an image, and she came back, and she shared that this really had an impact on her kids. They felt connected to it, and the mathematical goal was still preserved.

Corey: I love that. I think that's a great example. And I think the other thing that comes up sometimes when we present these ideas, is maybe you want to find a different context that is more relevant to your students that they know more about. Sometimes you might look at a context that's presented in the textbook and say, “I really love the mathematical features here. I really see how knowing something about this context could help my students reach the mathematical goal, but I'm going to have to do some work ahead of time to help my students understand the context, to provide them some access to that, to provide them some entry points.” So, in your example, maybe we're going to go visit a pet store. Maybe we're going to look at images from different kinds of stores and notice how things are arranged on shelves, and in arrays, and in different combinations. So, I think there are always a couple of choices. One is to change the context. One is to do some work upfront to help your students access the context so that they can then use that context to access the mathematics. But I think in both cases, it's about understanding the goal of the lesson and then understanding how the features of the context relate to that goal.

Mike: Let's shift and just talk about the second notion, this idea of opening up space for students' voice or for sensemaking when you're using curriculum. For me at least, I often try to project ideas for practice into a mental movie of myself in a classroom. And I wonder if we could work to help people imagine what this idea of opening space for voice or sensemaking might look like.

Corey: I think a lot of times those opportunities for opening up voice and choice and sensemaking are not in the direct, action steps or the direct instructions to teachers within the lesson, but they're kind of in the in-between. So, “I know I need to introduce this idea to my students, but how am I going to do that? What is that going to look like? What is that going to sound like? What are students going to be experiencing?” And so, asking yourself that question as the lesson plays out is, I think, where you find those opportunities to open up that space for student voice and choice. It's often about looking at that and saying, “Am I going to tell students this idea? Or am I going to ask them? Are students going to develop their strategy and share it with me or turn it in on a piece of paper? Or are they going to turn and talk to a partner? Are they going to share those ideas with a small group, with a whole group? What are they going to listen for in each other's strategies? How am I going to ask them to make connections across those strategies? What kinds of tools am I going to make available to them? What kinds of choices are they going to have throughout that process?”

Corey: And so, I think it's having that mental movie play through as you read through the lesson and thinking about those questions all the way through. “Where are my students going to have voice? How are they going to have choice? How are students going to be sensemaking?” And often thinking about, “Where can I step back, as the teacher, to open up that space for student voice or student choice?”

Mike: You're making me think about a couple things. The first one that really jumped out was this idea that part of voice is not necessarily always having the conversation flow from teacher to student, but having a turn and talk, or having kids listen to and engage with the ideas that their partners are sharing is a part of that idea that we're creating space for kids to share their ideas, to share their voice, to build their own confidence around the mathematics.

Corey: Absolutely. I think that, to me, is the biggest difference I see when I go into different classrooms. “Whose voice am I hearing most often? And who's thinking do I know about when I've spent 20 minutes in a classroom?” And there are some classrooms where I know a lot about what the teacher's thinking. I don't know a lot about what the students are thinking. And there are other classrooms where I can tell you something about the thinking of every one of the students in that room after 10 minutes in that classroom because they're constantly turning and talking and sharing their ideas. Student voice isn't always out loud either, right? Students might be sharing their ideas in writing, they might be sharing their ideas through gestures or through manipulating models, but the ideas are communicating their mathematical thinking. Really, student communication might be an even better way to talk about that because there are so many different ways in which students can express their ideas.

Mike: Part of what jumped out is this notion of, “What do you notice? What do you wonder?” Every student can notice, every student can wonder. So, if you share a context before you dive right into telling kids what's going to happen, give them some space to actually notice and wonder about what's going on, generate questions, that really feels like something that's actionable for folks.

Corey: I think you could start every activity you did with a, “What do you notice? What do you wonder?” Students always have ideas. Students are always bringing resources and experiences and ideas to any context, to any task, to any situation. And so, we can always begin by accessing those ideas and then figuring out as teachers how we might build on those ideas, where we might go from there. I think even more fundamentally is just this idea that all students are sensemakers. All students bring brilliance to the classroom. And so ,what we need to do is just give them the opportunities to use those ideas to share those ideas, and then we as teachers can build on those ideas.

Mike: Before we close this conversation, I want to spend time talking about responsive curriculum use being a vehicle for opening up space for equity-based practice. Personally, this is something that you've helped me find words for. There were some ideas that I had an intuitive understanding of. But I think helping people name what we mean when we're talking about opening space for equity-based practice is something that we might be able to share with folks right now. Can you share how a teacher might take up this idea of creating space for equity-based practice as they're looking at lessons or even a series of lessons?

Corey: Yeah, absolutely. And I think student voice and choice are maybe outcomes of equity-based practices. And so, in a similar way, I think teachers can begin by looking at a lesson or a series of lessons and thinking about those spaces and those decisions in between the action steps. And again, asking a series of questions. The equity-based practices aren't a series of steps or rules, but really like a lens or a series of questions that as a teacher, you might ask yourself as you prepare for a lesson. So, “Who is being positioned as mathematically capable? Who's being positioned as having mathematically important ideas? Are all of my students being positioned in that way? Are some of my students being marginalized? And if some of my students are being marginalized, then what can I do about that? How could I physically move students around so that they're not marginalized? How can I call attention to or highlight a certain student's ideas without saying that those ideas are the best or only ideas? But saying, ‘Look, this student, who we might not have recognized before as mathematically capable and brilliant, has a really cool idea right now.’”

Corey: You and I have both seen video from classrooms where that's done brilliantly by these small moves that teachers can make to position students as mathematically brilliant, as having important or cool or worthwhile ideas, valuable ideas to contribute. So, I think it's those kinds of decisions that make such a difference. Those decisions to affirm learners' identities. Those aren't big changes in how you teach. Those are how you approach each of those interactions minute by minute in the classroom. How do you help students recognize that they are mathematicians, that they each bring valuable ideas to the classroom? And so, it's more about those in-between moments and those moments of interaction with students where these equity-based practices come to life.

Mike: You said a couple things that I'm glad that you brought out, Corey. One of them is this notion of positioning. And the other one that I think is deeply connected is this idea of challenging places where kids might be marginalized. And I think one of the things that I've been grappling with lately is that there's a set of stories or ideas and labels that often follow kids. There are labels that we affix to kids within the school system. There are stories that exist around the communities that kids come from, their families. And then there are also the stories that kids make up about one another, the ideas that carry about, “Who's good at math? Who's not? Who has ideas to share? Who might I listen to, and who might I not?” And positioning, to me, has so much opportunity as a practice to help press back against those stories that might be marginalizing kids.

Corey: I think that's such an important point. And I think, along with that is the recognition that this doesn't mean that you, as the individual teacher, created those stories or believed stories or did anything to perpetuate those stories—except if you didn't act to disrupt them. Because those stories come from all around us. We hear Pam Seda and Julia Aguirre and people like that saying, “They're the air we breathe. They're the smog we live in. Those stories are everywhere. They're in our society, they're in our schools, they're in the stories students tell and make up about each other.” And so, the key to challenging marginality is not to say, “Well, I didn't tell that story, I don't believe that story. But those stories exist, and they affect the children in my classroom, so what am I going to do to disrupt them? What am I going to do? Because I know the stories that are told about certain students, even if I'm not the one telling them, I know what those stories are. So how am I going to disrupt them to show that the student who the story or the labels about that student are, that they are not as capable, or they are behind or struggling or ‘low students.’ What am I going to do to disrupt that and help everyone in our classroom community see the brilliance of that child, understand that that child has as much to contribute as anybody else in the math classroom?” And that's what it means to enact equity-based practices.

Mike: You're making me think about an interview we did earlier this year with Peter Liljedahl, and he talked about this idea. He was talking about it in the context of grouping, but essentially what he was saying is that kids recognize the stories that are being told in a classroom about who's competent and who's not. And so, positioning, in my mind, is really thinking about—and I've heard Julia Aguirre say it this way—“Who needs to shine? Whose ideas can we bring to the center?” Because what I've come to really have a better understanding of, is that the way I feel about myself as a mathematician and the opportunities that exist within a classroom for me to make sense of math, those are really deeply intertwined.

Corey: Yes, yes, absolutely. We are not focusing on marginality or identity just because it makes people feel good, or even just because it's the right thing to do. But actually in the math classroom, your identity and the expectations and the way you're positioned in that classroom fundamentally affect what you have opportunities to learn and the kinds of math you have access to. And so, we will do this because it's the right thing to do and because it supports math learning for all students. And understanding the role of identity and marginality and positioning in student learning is critically important.

Mike: You're making me think about a classroom that we visited earlier this year, and it was a really dynamic math discussion. There was a young man, I'll call him David, and he was in a multilingual classroom. And I'm thinking back on what you said. At one point you said, “I can go into a classroom, and I can have a really clear idea of what the teacher understands, and perhaps less so with the kids.” In this case, I remember leaving thinking, “I really clearly understand that David has a deep conceptual understanding of the mathematics.” And the reason for that was, he generally volunteered to answer every single question. And it was interesting. It's not because the educator in the classroom was directing all of the questions to him, but I really got the sense that the kids, when the question was answered, were to almost turn their bodies because they knew he was going to say something. And it makes me think David is a kid who, over time, not necessarily through intention, but through the way that status works in classrooms, he was positioned as someone who really had some ideas to share, and the kids were listening. The challenge was, not many of them were talking. And so, the question is, “How do we change that? Not because anyone has any ill intent toward those other children, but because we want them to see themselves as mathematicians as well.”

Corey: Yeah, absolutely. And that is part of what's tricky about this is that that's so important is that I think for many years we've talked about opening up the classroom for student talk and student discourse. And we do turn and talks, and we do think pair shares. And we've seen a lot of progress, I think, in seeing those kinds of things in math classrooms. And I think the next step to that is to do those with the kind of intentionality and awareness that you were just demonstrating there; which is to say, “Well, who's talking and how often are they talking? And what sense are people making of the fact that David is talking so much? What sense are they making? What stories are they telling about who David is as a mathematician? But also who they are as mathematicians. And what does it mean to them that even though there are lots of opportunity for students talk in that classroom, it’s dominated by one or maybe two students. And so, we have opened it up for student discourse, but we have more work to do. We have more work to say, “Who's talking, and what sense are they making, and what does that look like over time? And how is mathematical authority distributed? How is participation distributed across the class? And, in particular, with intentionality toward disrupting some of those narratives that have become entrenched in classrooms and schools.”

Mike: I think that's a great place for us to stop. I want to thank you again for joining us, Corey. It was lovely to have you back on the podcast.

Corey: Thanks. It was great to be with you.

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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