WestEd’s Leading Together Webinar Series: Collaborative Lesson Inquiry Cycles: Challenging and Supporting English Learners in Secondary Mathematics

Hello, everyone, and welcome to the eighth session of our Leading Together series. In these 30-minute learning webinars, WestEd experts are sharing research and evidence-based practices that help bridge opportunity gaps, support positive outcomes for children and adults, and help build thriving communities. Today’s topic, Collaborative Lesson Inquiry Cycles: Challenging and Supporting English Learners in Secondary Mathematics. Our featured speakers today are Haiwen Chu, research director for the English Learner and Migrant Education team at WestEd, and Monique Evans. Monique is a program associate for the English Learner and Migrant Education team at WestEd. Thank you all very much for joining us. My name is Danny Torres. I’m associate director of events and digital media for WestEd. I’ll be your host.

Now, before we move into the contents of today’s webinar, I’d like to take a brief moment to introduce WestEd. As a nonpartisan research, development, and service agency, WestEd works to promote excellence, improve learning, and increase opportunity for children, youth, and adults. Our staff partner with policymakers, district leaders, school leaders, communities, and others, providing a broad range of tailored services, including research and evaluation, professional learning, technical assistance, and policy guidance. We work to generate knowledge and apply evidence and expertise to improve policies, systems, and practices. Now I’d like to pass the mic over to Haiwen. Haiwen, take it away.

Haiwen Chu:

Great. Thank you, Danny, for that kind introduction. Monique and I will now briefly introduce ourselves before we dive right into the content of today’s webinar. We thank you for joining us. My name is Haiwen Chu, and I have been working at WestEd for over a decade at the intersection of theory, research, and practice, but my heart is still in the secondary math classroom, and that’s why I’m really happy to share with you what we’re gonna be doing today, which is in collaboration with secondary mathematics educators of students who are bureaucratically designated as English learners, looking at ways to connect what we know from research and from theory directly into practice. And so I’ll turn it over to Monique to give a brief introduction of herself before we dive right in.

Monique Evans:

Thanks, Haiwen. My name’s Monique Evans. I was a secondary math classroom educator for about 19 years, and within part of that time, I became a math instructional coach for about 11 years. And I really enjoy working with secondary math educators to support our English learners.

Haiwen Chu:

Thank you, Monique. We’re now gonna dive right into the presentation. And so I wanted to give you a broad outline of how our time is going to be allocated in terms of these collaborative lesson inquiry cycles.

[NARRATOR]:

A presentation slide appears on screen, Collaborative Lesson Inquiry Cycles: Developing Teacher Expertise. Ambitious Learning for Secondary English Learners.

Haiwen Chu:

First, in our agenda, as you’ll see, I do want to be very explicit about our theory of teacher expertise, the domains of teacher expertise, because how we conceptualize teacher expertise is critical to how we then create opportunities, invite them into opportunities to develop that expertise. We’ll then spend the bulk of today’s presentation on one out of many ways that we can expand teacher expertise, which is the famous collaborative lesson inquiry cycles. And we will provide you with concrete examples taken from classroom practice of what that actually looks like for students in terms of talk and in terms of thinking all around learning important mathematics with students who are bureaucratically designated as English learners.

And so let’s dive right in then to this question of, what is teacher expertise? What are the domains of teacher expertise? And how may they be connected in terms of professional learning? And so as you’ll see on the next slide, there are multiple domains of teacher expertise, first identified by Lee Shulman and later refined by Aida Walqui.

[NARRATOR]:

A diagram is shown under the title Domains of Teacher Expertise. Five connected diamonds form the diagram and together, the diamonds are arranged to create a larger interconnected diamond shape. At the top is vision, on the right is knowledge, at the bottom is practice, and on the left is motivation and in the center is reflection. Green arrows link the diamonds in multiple directions all within the surrounding influence of context.

Haiwen Chu:

And we’re gonna begin in the top of this diagram with this notion of vision. Vision encompasses teachers’ beliefs, including how they see their students, not just who their students are today, but who their students may yet become through fruitful opportunities to learn. And so that vision of learning of who students are and who they may be and of what teaching therefore is, is connected intimately to the domain you’ll see on the left of the screen, which is teachers’ motivation for entering and staying in the profession of teaching, given all of the challenges we’re aware of, in terms of how they view their role as a teacher and how they stay in the game with all of the challenges that come with the job that are both surrounded by and supported by the contexts in which teachers work.

So these questions of vision and motivation are critical to secondary mathematics teaching because the beliefs that we have about the learning of mathematics and the students who are doing that learning shape the opportunities that we then offer. So that is vision and motivation as related by context. What we’re gonna focus on, as you’ll see on the next slide, is this question of, what can we focus on in collaborative lesson inquiry cycles? And so the purpose of these cycles is to work at the nexus of knowledge, practice, and reflection. Reflection is going to be the main vehicle through which teacher learning happens. It happens before they teach, as they plan. It happens as they teach, as they adjust. It happens after they teach, as they look back and begin to understand what worked for whom and why and under what conditions.

All of this serves to expand teachers’ knowledge, but as we all know, content knowledge by itself is not sufficient for quality teaching and teaching expertise. Rather, we need knowledge of pedagogy and knowledge of students that enables the knowledge that teachers have, the content knowledge that teachers have to be grappled with by students in a way that is intellectually honest. And so all of that happens, as you’ll see on the screen, in the bottom domain of practice when the teachers facilitate learning rather than tell students what to do; when they scaffold by design, but also in the moment, that’s related to reflection in the moment, reflection in action; and as they engage in practices of formative assessment in which they are adjusting in order to bring students’ performances, bring students’ work closer to where they want to be. And so the collaborative lesson inquiry cycles that Monique will describe for you in much greater detail are theoretically driven by this idea that reflection on practice and in practice is what drives the development of knowledge over time.

And that knowledge is not just content knowledge, although content knowledge is important, it is knowledge of teaching and learning and how to make these ideas accessible and meaningful for students. So that in a nutshell is what are the relevant domains of teacher expertise. I want to turn then to, if we want to expand teacher expertise in these dimensions, what is necessary for their collaboration? Well, first of all, teachers need to view the enterprise, the task of educating English learners not as the job of one specialist who happens to have the title of English as a second language, but rather as something that all educators contribute to and add to in this joint enterprise of educating English learners in mathematics, in secondary mathematics. And so for this to happen, ideally, mathematics educators and all of the educators who work in the classroom with English learners are mutually engaged. They’re working in the same place and at the same time. Space-time is funny that way. You have to be in the same place and at the same time, working on and with students at the point of learning. That’s why we think of this as a form of coaching because it’s happening at the interface of practice.

And so what will happen over time is teachers, groups of teachers, teams of teachers will develop what is known as a shared repertoire. They will develop common approaches and common tools for planning lessons, for assessing what students are talking about and thinking and what they’re learning, and ultimately to adjust and refine instructions, sort of collectively. So it’s these conditions that we need to put in place at a systems level as a group of leaders of educators, this is what we need to place put in place at a systems level to then be able to foster the deep kinds of collaboration that Monique will now describe in the particular instance of a collaborative lesson inquiry cycle. So I turn it over to Monique, who will take it away.

Monique Evans:

Thank you, Haiwen, for grounding us in the idea that joint and mutual enterprise and the theoretical background. Today, we’ll be talking about a powerful model for teacher collaboration, which we call collaborative lesson inquiry cycles. We also sometimes use the term CLICs to refer to our collaborative lesson inquiry cycles. I’ll share how these cycles foster a teacher’s capacity to challenge and support our English learners in secondary mathematics. I’ll be taking you through the design and the process of the collaborative lesson inquiry cycle, and we’ll be talking about the impact on teachers, specifically on teacher practice, and also how the CLIC impacts students within the classroom. At the end, we’ll take a look at an actual lesson that was designed and implemented by a group of teachers in California that I was working with. The design of the CLIC, or the collaborative lesson inquiry cycle, is based around the Japanese lesson study. We take a team of four to six teachers along with a facilitator coach, and we work together to collaborate during the CLIC over the course of two days.

There are four major parts that a team of teachers collaborate in. I’ll go more in depth about the process of those two days on the next slide. But the major idea is that first, teachers collaboratively plan a lesson together that will be implemented with students. Then teachers get the opportunity both to implement the lesson and also to observe the lesson. So we teach the lesson more than once, or we implement the lesson more than once. After they’ve implemented the lesson, we reflect on how the lesson went. And then we refine that lesson based on what we saw within the reflection. So let’s take a more in-depth look at the actual cycle. On day one, we consider that our planning day. On that day one, the team of teachers gets together, and we develop a focus for observation and planning, which I’ll discuss more in depth on the next slide. Once we have a shared focus, we begin to identify goals for the lesson.

So teachers typically choose a topic to bring to our planning day, and as a team, we decide on the conceptual understandings of the lesson. So moving away from just rote procedures, focusing more on those cross-cutting concepts that are going to drive the learning. And then we focus on those math practices that students will engage in as well as the language practices we’ll have students engage in. So what will students do in interaction with others, and how will they talk about mathematics? Then once the team has honed in and focused on those three pieces, the team works together to design the flow of the lesson. So we’re really thinking about, what are the tasks? How will students be grouped within each of those tasks? Will they be in pairs? Will they work into groups of four? Then once the team has a clear shared vision of the lesson, we divide and conquer to develop the actual student-facing pieces: the handouts, the slide deck, and the language that we will provide to students.

Once we feel we have a draft ready for the entire lesson, teachers then rehearse the lesson. Typically, we have one or two teachers teach it to the rest of the team of teachers, and the rest of the teachers act as students to see what might need to be revised. Do we need more language support? Are the instructions clear before we put it in front of students? So that by the time we get to the end of day one, our teacher team has a lesson that’s ready to implement in class. As we’re thinking about that focus for observation and planning, I’m gonna give you a sample of what that might look like. So for example, one sample focus that we might have is what is necessary for quality talk. When we’re talking quality talk, we’re specifically referring to quality interactions within the classroom. In the left-hand column, you can see what we might consider typical interactions in the classroom, and the right hand column is what we would consider quality interactions in the classroom.

So in a typical classroom, students might report understandings that they have already achieved, and what we wanna do is build lessons that move students to think through speech, to think aloud, and to use talk as a vehicle for new ideas. Instead of giving factual responses to closed questions like, what is the square root of 16, we wanna build lessons so that students are exploring ideas in depth and at breadth in interaction with each other so that student talk is sustained. In a typical classroom where the teacher dominates talk and tells students whether the answers they were providing are right or wrong, instead, we are focusing on engaging in reciprocal interactions so that students are building upon one another’s ideas, they’re challenging each other, or they’re revising their ideas through speech.

The way we engage teachers to calibrate what we mean by quality talk and quality interactions, first, we read about what quality talk is. Then we watch a video of students interacting in the classroom. We have teachers script and take notes about what students in the video are saying. This allows us to have a rich discussion and a common definition and look for what quality talk is in the classroom. We keep this in mind as we plan our lessons to provide opportunities for those quality interactions, as well as how to observe for quality talk in the classroom. We’re keeping this in mind. So day one is our planning day. We’ve completed this lesson flow. We have all of our tasks ready. So day two takes us into our implementation day. Typically, we implement the lesson twice, and the way this works is that team of four to six teachers get split into smaller teams of team A and team B. In the morning, what typically happens is that team A implements the lesson first, and team B is going to observe. And they’re observing students, looking for that quality talk. We’re not observing teachers. We’re observing what students are doing for that quality talk and to see where we might refine the lesson to provide more opportunities for that quality talk.

After team A has implemented the lesson, we come back together, and the observing team shares first where they saw evidence of student talk, and then the implementing teachers share their overall impressions. Then we begin to look forward. What needs to be revised and refined for more opportunities of that quality talk? Do we need to change the language that we’re providing to students? Do we need to restructure one of the tasks? Once we’ve debriefed and made some decisions, we go back and refine the lesson to make sure that the lesson still flows, and we provide more opportunities for that quality student talk. From there, the teacher team switch roles. So now team B implements the lesson where team A observes for that quality talk. Then once again, we come back together, we debrief the lesson and choose any refinements that might need to be made. By the end of the day, we have a polished lesson that teachers will be able to utilize in the future. Then to close out the cycle, each teacher names something they learned from the process and a takeaway or a next step for their own practice. We say all of these things out loud to discuss our collective learnings.

So I wanna go back to this idea of observation. In order to set ourselves up for success, we wanna make sure that we are setting up specific guidelines for what those observation procedures actually look like. When each team is taking turns observing students, we give them guidelines so they can have a common experience. We ask each observing teacher to sit with a pair or a small group of students. We have them follow that individual or pair throughout the whole lesson. We have the teacher watch how the students’ language and understanding of the content evolve over the course of the lesson. This also helps us to think through the flow of the lesson. And also, the idea is that if student groupings change at all the lesson, we have our observer following around one specific student to really see how their language changes over the course of the lesson.

We set up the expectation that teachers are true observers. They are not to interfere with either the student experience or the teacher’s implementation. So we can see if the instructions from teachers are clear and that we can pick up any misunderstandings from students. While teachers are observing, we ask them to record all evidence of student conversation and any hand gestures that students might use to interact with one another. Sometimes we’ll even ask them to take pictures of student work. Teachers typically script what students are saying and doing in interaction with their partner or small group so they can share that out during the debrief. The idea is they are strictly recording what happens during the lesson. There’s no inferences and no judgments. Lastly, we also set up the expectation that observers are not participating in side conversation with other observers. Through their participation in the collaborative lesson inquiry cycle, we are expanding teacher knowledge. They’re experiencing planning in a community of practice, and they’re bringing that practice back to their own classroom.

First, we are able to define what we mean by quality interactions as opposed to those typical interactions, and teachers have the opportunity to collaborate with one another to plan for those quality interactions, offering those opportunities to English learners to participate and use language in the classroom. In addition, the collaborative lesson inquiry cycle provides teachers opportunities to sustain a language focus in mathematics, which may mean language to deepen mathematical ideas, to complete a task, or to engage in mathematical practices. So I’ve taken you through the design and the process. Now I’d like to discuss and share with you a sample lesson in action. This was a lesson that happened in April of this year in the Bay area of California. It’s a team that I had been working with all year. This was their third CLIC with a team of six teachers. It took place in a math AI class. In this case, AI stands for applications and interpretation. It’s an IB course that students were participating in. It was mostly 11th and 12th graders.

The team chose the topic of rational functions, and on day one, we planned a lesson focusing on the concept of discontinuities, thinking about vertical asymptotes and removable discontinuities within those rational functions. In our collaborative planning, we created a lesson flow that included the tasks that you see here that are done in interaction with a partner or with a group of four. On the left-hand side, you can see the conceptual “what” of the lesson, and on the right-hand side is the pedagogical “how”. So first, we have students describe features of rational functions using a sort-and-label, and I’ll show you an example of what that looks like on the next slide. Then students find domain and discontinuities of rational functions using an all-four-corners. Then students read tables to find vertical asymptotes or removable discontinuities using a read-and-do. Then they described graphs and connect the graphs with the equations using a describe-and-connect.

And last, students work together to sketch a graph of a rational function using specific constraints, using a collaborative problem. All of this is done in interaction with other students, and again, the team of teachers is split into two teams where one team is implementing while the other is looking for quality talk. Then we came back together and switched. So now I’d like to show you an example of our sort-and-label task. On the screen here, the picture that you see is a picture of graphs. This is our card sort. There’s a bunch of different rational functions, and students have grouped and labeled them. So for the first task of the lesson, which is our sort-and-label, teachers wanted to surface ideas about how students might describe graphs of rational functions. In a sort-and-label, students work in a group of four. Each student takes a graph, describes what they see without showing their graph to others. Then they place the graph down for the rest of the group to see. Then students group the graphs based on characteristics that matter to them. It’s very open-ended.

Typically, students use everyday language to describe the graphs. Here you can see that students grouped the rational functions into four groups: the U group, L group, table group, and boomerang group. The idea was that this task would access student prior knowledge and focus attention on some of the key ideas that would be introduced in the lesson. At the beginning of the lesson, students tend to use more everyday language to describe the graphs. From here, the implementing teacher helped to define a vertical asymptote and a removable discontinuity for students. On this slide, we see a picture of a student paper with her work from her group, and on the right-hand side, we see graphs that students then need to connect the functions in the table.

So as teachers planned the rest of the lesson, they wanted to make sure students had the opportunity to coordinate representations while talking about them within their group, so matching the graph, the equation, and a table, and considering the types of discontinuities, specifically those vertical asymptotes and removable discontinuities. From the sort-and-label, that same group of four students were given four rational functions, and they found the domain of each function. They identified the undefined values of the function. They examined a table to label the undefined values of the function as either vertical asymptotes or removable discontinuities. In the picture shown here, students are using information they found from their table to decide which graph matches with which equation. As students are sharing their reasonings, their language grew more technical and more precise. They moved from using everyday terminology, like in the sort-and-label with U group and table group, to using words like vertical asymptote and hole in the graph to talk about their graphs and sharing their reasoning of why they think the graph matches the equation.

And I mentioned that at the end of our cycle, the teachers were able to share their collective learnings. So after we’ve implemented and observed and refined, the teachers were asked to share their collective learnings and their takeaways from the end of our two days of collaborating together. And when I asked teachers to do that, most teachers described what it is that they had previously done and what they wanted to do now based around what they had seen students capable of doing within their classroom. So we found that teachers wanted to move away from telling or modeling just rote procedure to supporting students to make sense of the ideas. So putting more emphasis on the learning experiences rather than on specifically just teaching. One teacher noted that they wanted to move away from assigning individual worksheets to students, and instead focus on structuring tasks that would invite students to collaborate and co-construct understanding together.

And last but not least, teachers wanted to move away from prioritizing completion and compliance, particularly thinking about emphasizing that procedural fluency and just completing worksheets. Instead, the teachers really wanted to start to prioritize a conceptual focus and a deep understanding of concepts. These shifts in teacher practice came about through our work together in this collaborative lesson inquiry cycle over the course of these two days and throughout the course of the year in the three collaborative lesson inquiry cycles they got to participate in. Teachers were able to work together, plan lessons, implement those lessons within their classes with students, reflect on and revise those lessons, all to challenge and support English learners in their classroom. It’s a powerful model of teacher collaboration that has a big impact on teacher practice. And I’m gonna go ahead and turn it back over to Haiwen to talk to us about our resources and next steps.

Haiwen Chu:

Thank you for that, Monique. Yes, indeed. We hope you’ve enjoyed a quick snapshot of one lesson, which is one instance of how this kind of generative model for collaboration has the potential to transform learning for English learners alongside all of their classmates. I wanna emphasize that the particular instance that we sort of chose was from upper high school, and so the students there are what is known in California as quote unquote “long-term” English learners for which there’s a long definition we don’t have time for. And so the idea is that this group of students did not happen to be the very newcomer students that I myself used to work with back when I was in the classroom. So in terms of resources and next steps, we wanted to offer you, upon request, if anyone wants to take a look at another example of a lesson, we have, as you’ll see on the next slide…

[NARRATOR]:

On the next slide, a screenshot of another sample lesson titled From Talk to Tables, Multilingual Learners Discussing Independence is shown.

Haiwen Chu:

…an example of a lesson that we created with some teachers in the Jurupa Unified School District around notions of independence. Again, an upper high school lesson.

We have other examples for more newcomer examples in other places, but this is one example of the kind of artifact that is possible from a collaborative lesson inquiry cycle. And more broadly, we would like to direct you to our website, which is Quality Teaching for English Learners, where we offer multiple ways to partner with districts and schools and educators around how this may actually look like in practice in terms of the development of teacher expertise, of which, once again in the collaborative lesson inquiry cycle, is one model that should be part of a balanced and rich diet of opportunities for educators. And so with that, we’re gonna flash for you on the screen our contact info. We know this was a super short webinar, but we would love to hear from you on email. And Danny’s gonna now take us out in terms of some of the closing logistics. Monique and I once again thank you for making time in your busy day to join us to hear a little bit about collaborative lesson inquiry cycles, and we hope to hear from you directly soon as well. Thanks again.

Danny Torres:

Thank you, Monique and Haiwen, for a great session today. And thank you to all our participants for joining us. We really, really appreciate you being here. For those of you interested in learning more about WestEd’s Quality Teaching for English Learner Services and professional learning opportunities, visit us online at qtel.wested.org. That’s Q-T-E-L dot WestEd dot O-R-G. And feel free to reach out to Haiwen and Monique via email if you have questions about the work we discussed today. You can reach Haiwen at [email protected], and you can reach Monique at [email protected].

And there’s still time to register for our upcoming Leading Together webinars. We’re covering a range of topics including science, literacy, assessment, and more. During these webinars, we’re sharing insights and evidence-based practices to improve teaching, leading, and learning. For more information about our Leading Together webinar series, visit us online at wested.org/leading-together-2025. And finally, you can also sign up for WestEd’s email newsletter to receive updates. Subscribe online at wested.org/subscribe, or you can scan the QR code displayed on the screen here. You can also follow us on LinkedIn and Bluesky. With that, thank you all very, very much. We’ll see you next time.