Mathematics teacher development and The Learning Cycle

The ITE CoP explored ways to develop teacher expertise, including the use of The Learning Cycle for development of elementary mathematics teaching expertise.

We compiled a set of Questions and Answers with one of the researchers and teacher educators responsible for The Learning Cycle, Professor Elham Kazemi, who spoke to ITE CoP participants in an April 2017 webinar. The resources Professor Kazemi refers to are housed on the tedd.org website.

Q and A with Learning Cycle Expert Professor Elham Kazemi

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“Schools should be meaningful places for students and adults, and we should be joyful in learning together. The Learning Cycle helps create a learning environment that gets you inside practice, with others, to pay careful attention to content and to students as learners and people.” – Professor Elham Kazemi

Introducing Professor Elham Kazemi

Professor Kazemi is the Geda and Phil Condit Professor in Math and Science Education at the University of Washington. Her research focuses on the challenge of designing professional learning experiences for elementary mathematics teachers and teacher educators so that teachers’ classroom practices improve in ways that are productive for student learning. Professor Kazemi began her career in education as an elementary school teacher in Arizona.

What is the Learning Cycle?

The Learning Cycle is a structure in which you both analyse and enact teaching around a core focus, usually driven by disciplinary teaching. So, we might use the Learning Cycle to come together on a particular math content and instructional activity.

The Learning Cycle has four stages:

·             Introducing and learning about the activity

·             Preparing for and rehearsing the activity

·             Enacting the activity with students

·             Analysing enactment and moving forward

Rehearsal

In a small group of candidates or teachers, we rehearse the lesson that we will teach to a small group of students. In the rehearsals we try to simulate the lesson as much as possible, and pause many times as it unfolds to ask each other questions. As a teacher educator, I will provide suggestions, highlight things, or problem solve with the group. It is a way to make sense of knowledge we have developed for teaching and to think about how to deploy that knowledge.

There are constraints on rehearsal. For example, we cannot predict how students will react, but as we continue our partnership with particular schools and students, we tailor our rehearsals to what we have learned about these students. The rehearsals become less theoretical, and more about concrete instructional decisions like which students are we going to pair together and why, what are we aiming to achieve mathematically, and what is that explanation going to sound like?

Rehearsal is supported by planning artefacts, for example a choral counting instructional activity that focuses on the structure of number.

Enactment

Once we're in the classroom with students, candidates (or teachers) will typically first see a teacher educator and the co-operating teacher lead a similar instructional activity. This modelling helps candidates see what it’s like to think aloud about one's teaching with peers. The teacher educator and co-operating teacher pause a few times (“teacher timeouts”) during these modelling sessions to ask each other questions about what they should do next and why, or to highlight an instructional decision that was made and why.

After the modelling session, the candidates enact the instructional activities with students in small groups. These sessions are video-taped.

It's powerful for me to be in the classroom as a teacher educator, because I am able to see how candidates take up ideas that we're learning in courses, and I can provide support and help them make sense of instructional decisions in the moment, and see if we’re achieving the goals we set.

Debrief

In the debrief, teacher educators and candidates think about what they learned about each individual child and review their instructional decisions. In my debriefs, we watch videos of the classroom sessions and we take notes on what we learned about each child and about the group as a whole, and what we think is important to keep in mind for next time.

It is really important that the Learning Cycle is enacted with a focus on students’ and set within efforts to advance students’ learning opportunities and equity. Teachers and teacher educators have to be willing to interrogate their own assumptions about students and strive to learn more about students’ ideas in the classroom, to come to know students as people, and the resources they bring to school. They have to think about how school policies and instructional practices empower students or not.

What is the difference between Japanese Lesson Study and the Learning Cycle?

There are similarities between Japanese Lesson Study and the Learning Cycle, but there are some key differences. The lessons that we create are nowhere near as polished as the research lessons that are part of lesson study. There’s a lot more back and forth between the educators as the lesson is unfolding because we’re trying to make sense of what we’re learning about students in this particular way. I think in Japanese Lesson Study there is a lot of learning that happens up until the enactment of the lesson and then you really let one person lead the way while the others take notes, and I think we interact together for the purposes of learning practice in a different way.

How do you use the Learning Cycle to train pre-service teachers in a mathematics methods course?

A teacher educator must think about how to design a sequence of Learning Cycles and what candidates should learn – they must make decisions to focus on particular instructional activities, mathematical content, or issues of student learning.

For example, if I'm working with primary teachers and I want them to understand how children work with early number and how they learn to understand counting, I might blend together some readings with some videos of young children counting objects. Then we might learn more about students’ thinking and how to advance their thinking through instructional activities like counting collections and choral counting. I'll think about the sequence of those and how I want to plan each course session, based on what I want the candidates to learn about instruction and content.

An example for multiplication

The content focus is how to help students understand that multiplication is about grouping, not just repeated addition, and the instructional activity is to introduce these ideas with third graders via a word problem.

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We have a protocol for how to introduce word problems and unpack them with students. Before we get into classrooms with students, we’ll talk about how to use this particular problem. Then we'll do a rehearsal that might involve posters, diagrams, and role plays about how to set up the word problem, what kind of questions to ask of students, and how they might make sense of the situation in the word problem.

During the enactment with students, we pose the word problem, and then we give the students a recess, while we analyse the student work and our observations of how students solved the problem. When the students return we discuss two examples of student work (reproduced on poster paper). We review how the examples connect back to the problem, and are connected to each other, how we know they’ve accounted for all the numbers in the problem, and how we know what the total is.

After the enactment, we debrief what we learned about each student's understanding and how our instructional decisions impacted their understanding of how to do multiplication.

How do you use the Learning Cycle to train in-service teachers in mathematics?

We use the Learning Cycle in a similar way with in-service teachers.

An example for introducing fractions

The Learning Cycle begins with the school-based coach or instructional leader selecting some student work from students in that school. At the start of the cycle, we try to make sense of what the child seems to understand and the logic behind their notation.

In the example of student work (below), the children understand the problem situation and the concept of sharing, but their notation shows that they do not completely understand how to write parts and wholes of a fraction.

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We connect the student work to a reading which challenges the teachers to introduce fractions by focusing on the size of the part relative to the whole, rather than the number of parts into which the whole is cut. So, we introduce the idea that there might need to be some changes to the language used to introduce the concept of fractions to students.

In the co-planning stage, the coaches help teachers to think through potential tasks inspired by the reading and what is in fraction curricula. We design a task and anticipate what we think the students will say about the task and how they will perform it (e.g. ask students to tell us what fractional part is shaded in some examples and how they would notate it).

We try the task out in the classroom, where we hear students say different things and try to understand their logic. Then understanding that logic, we think about how to navigate that conversation with students. We then debrief and think about what else we might try to help students’ understanding of fractions.

During the Learning Cycle, what comes quickly to teacher candidates and in which areas do they struggle?

Teacher preparation needs to prepare teachers for how learners come to understand ideas. Candidates may be able to do the questions given to elementary school students, but their professional education should prepare them for how learners come to understand those ideas.

For example, if candidates can understand how to think about fractions as quantities, it really changes the way that they make sense of them. People often have a hard time because the algorithms in fractions are so easy in comparison to borrowing across zeros, for example. The meaning of fractions is what people have usually had very little opportunity to wrestle with. You have to connect back to what division means in a whole number context, in order to make the connection to division with fractional quantities.

As a baseline, the teacher educator has to be thinking about the trajectory of children’s thinking in whatever content you’re studying, and in maths there are some really fabulous resources that lay out the trajectory. Then teacher educators have to appreciate that a lot of ideas about how students learn are going to be novel ideas for candidates because if you’ve gone through your own K-12 schooling, you didn’t need to learn how other people learned the particular topic.  

So, teacher educators have to structure learning activities for teachers that allow them to develop the specialised knowledge, and then figure out how to translate that into learning for students. It’s not enough to see what’s hard for children or what ideas they need to develop first, teachers need to know how to introduce children to an idea and where to take those ideas. It is relatively straight forward to teach a teacher how to elicit a student’s thinking, but more difficult to teach them the follow-up questions to build that student’s learning, orient a group of students to one another’s ideas, and make selections about which ideas are useful to carrying along with a group of students. All of that nuance is what takes time to learn.

Does the Learning Cycle differ for teacher candidates that have low content knowledge, or for subject areas with more difficult content knowledge?

The Learning Cycle is structured so that the first phase is some immersion in some sort of content idea.  How long you take with that depends on how much you want to bite off. You could certainly go deep for a couple of sessions before you try something out with children, or you could just take a bite, try something, take another little bite, try something. That all depends on your context and how much time you have with candidates (or teachers), and either way I think it can work.

For difficult content, I prefer to build candidates’ knowledge slowly and intersperse it with enactments if I have the time available. Trying to do too much theoretical content work in one go before getting into classrooms might actually be counter-productive. If you take the example of fractions, it’s better to take it in smaller pieces because then candidates can have the knowledge accumulate, instead of having to learn everything there is to know about fractions and then trying it out.

Sometimes, candidates need the experience of seeing students and that’s what motivates them to learn – it’s not something they’ve read about in a paper, it’s what the students actually do.

What kinds of evaluative techniques or data do you collect to inform you of where your teacher candidates are, and the effectiveness of your teacher educators?

It’s a lot of qualitative feedback and conversation in the day to day work, so you need more knowledgeable others that have the opportunity to observe and give feedback too.

Some people use classroom observation rubrics and tools to collect data and measure changes.

In my research on the impact of professional learning, we collected student learning data as a measure of how the teachers’ learning is impacting students. In math, we’ve found that you can’t only collect data on students’ accuracy because one of the things that happens as math teachers get better is that they should be affecting the range and sophistication of students’ strategies, not just whether or not they get problems right. So, we collect both accuracy and strategy data. Sometimes the accuracy data lags behind the sophistication of strategy use and you want to be able to compare those two.

I haven’t studied a large number of teacher education programs, but with people in my immediate network at the University of Washington, our evaluative techniques are a lot of observation and feedback, and asking whether we are supporting teachers in what we do. We also pair people up, more knowledgeable teacher educators with more novice teacher educators, so that there is support for planning and enactment before they do things on their own.

What sort of changes or improvements have been made in your program as a result of the work of teacher educators in mentoring, pairing or coaching?

At a pre-service level I think our teachers are better equipped to begin their careers. Teacher education is short and we cannot teach candidates everything they need to know, but our candidates see the work that we do in our courses as relevant to their teaching and they don’t experience a disconnect between what they learn in their course and what they do in the classroom, which is a perennial challenge of teacher preparation.

The connection that we must work on very explicitly, is curriculum materials. The teachers in our partner districts work with anywhere from seven to eight different curriculum materials in their schools. So, we have to very concertedly help candidates learn how to read the curriculum materials they get and how to adapt them to be in line with what they learned through the university. It’s not true that just because you’ve learned how to elicit and respond to student thinking in the field-based methods course that you’ll do that in your student teaching if you haven’t learned how to use the curriculum materials that you’re given given in your particular grade context. So that’s definitely one disconnect that can remain if you don’t attend to it.

Overall, I think we definitely see changes in people feeling like mathematics is a subject worth teaching and that they can do it. And what I hope that we effect, although I haven’t collected any systematic data, is that the Learning Cycle is a way they will pursue continued learning throughout their career.

How do teacher educators engage with the Learning Cycle?

In the same way that the Learning Cycle gets teachers to collaborate around instructional decision making, not just theoretically but both in analysing practice and in enacting it, teacher educators need to have that same kind of experience. You can’t just talk at teacher educators about what to do, they need to experience it.

On www.tedd.org we have teacher educator guides for how you would plan for a phase of the Learning Cycle. Having some insight into what it takes to prepare for each phase is important, and then we usually engage in the Learning Cycle ourselves, with our colleagues or new early-career coaches we work with. Because we use instructional activities as a major tool to carry practice and ideas about student thinking across the Learning Cycle, we often encourage the coach or the instructional facilitator to try those activities out with children to appreciate some of the instructional dilemmas.

I cannot teach someone else to do something if I haven’t tried it myself. I was a fifth-grade teacher, but I am teaching candidates and teachers from Pre-K up to middle school, so I can’t rely on my own teaching experience. So I have to expand my own learning and the Learning Cycle helps me to do this with students and other teachers.

What are the common mistakes teacher educators make when they’re trying to use this Learning Cycle approach and instructional activities?

One mistake is if the teacher educator does not take the time to pay attention to the productive group norms needed to inquire into practice. Another mistake is if the teacher educator sets him or herself up as too much of the expert in the interactions. You have to pay attention to status in the group and whether you are giving everybody equal voice.

You need to approach this work with an asset-based perspective on students. For me, it is not just about managing students, it is about increasing the intellectual work done by students in the classroom.

The other thing that’s really important is that we centre everything on some shared experiences around an instructional activity. I don’t think that you can do rehearsals of something if the teacher educator doesn’t know what’s being rehearsed. As a teacher educator, if I don’t know what you’ve planned your lesson around and what you’re trying to accomplish, I don’t really know where we are going together. The purpose of the Learning Cycle is to have a collective focus of what is being worked on.

How has the Common Core State Standards impacted what you’re doing?

In terms of math, we had to think about when and how to incorporate the domain [content] standards versus the practice standards into what we did. It’s not hard to do. As a teacher educator, you have to become knowledgeable about it yourself and then you have to think about where and how to make those connections. To do everything is impossible, so I pick usually two or three practice standards, ones that I think have a lot of power for disrupting how people think about mathematics normally. Then we try to connect the language and the meaning [of the standards] to whatever activities that we do. We don’t cover the full terrain, but we try to pick what we think are some really big ideas that will be consequential for teachers’ early careers. That’s a judgement based on knowledge of the field and our work with schools.

How did you go about designing the instructional activities you’ve got on the www.tedd.org website so that they provide a robust exercise for candidates and teachers to enact in classrooms?

You have to select instructional activities that can be adapted across grade levels for different content, and that are worth repeating because they are routines that provide a skeleton for a broader lesson or topic. What you have to do is think, ‘is this the right activity for the kind of mathematical work that I want to do?’

In elementary math, counting and operations are important ideas, and so several of the instructional activities are around understanding numbers and the structure of numbers. We also pick them because we want to make mathematics an experience that students find joyful and meaningful, that they can do, and doesn’t reify deficit-based perspectives about students. For example, labelling some students as being high and others as being low.

So, the activities and the way they are structured allow for many ways to be successful and are structured so that teachers will see their students, and are designed to make students’ thinking visible. There’s nothing ‘magical’ about the ones that we have chosen, we accept that those things matter, and it matters that they are not too complicated to learn how to do as a beginning teacher.

It does take some time to develop the sort of insight needed to select or design instructional activities. The real challenge is how to use instructional activities strategically and intentionally, in relation to the concept you’re trying to teach about mathematics and the stage of learning of your candidates.

Further reading

Core Practices and Pedagogies of Teacher Education: A Call for a Common Language and Collective Activity

Teacher Education By Design website