Understanding Lesson Study for Mathematics

A Practical Guide for Improving Teaching and Learning

Rosa Archer, Siân Morgan, David Swanson

• Publisher: Routledge
• Year: 2020
• eBook ISBN: 9781351048262
• Edition: 1
• Topic: Education
• Subtopic: Education General

Using the latest research, this book provides an insight into how learning in mathematics can be improved through a lesson study approach. This highly practical resource explores the research and theory that underpins lesson study, and shows the significant impact it can have on teacher development.

Divided into ten accessible main chapters that focus in depth on an individual mathematics lesson, each chapter provides research and background to the lesson, an outline of key features, a detailed description and analysis of the lesson in practice, post-lesson discussions and reflections which generalise from the experience, as well as links to helpful resources. Some of the key topics explored include:

• Fractions
• Proportional relationships
• Probability and statistics
• Geometry
• Modelling
• Algebra
• Dialogic reasoning.

Understanding Lesson Study for Mathematics is the perfect resource for all mathematics teachers, trainee teachers, and professional developers who are looking to develop the use of lesson study in their own practice or for those simply seeking new inspiring ideas for the mathematics classroom.

1

An introduction to lesson study

We begin this chapter with a background to lesson study (LS), its origins and current popularity. Also included is a description of the variety of LS ‘research foci’ that we discuss within the book. This is followed by a discussion of benefits of and risks to LS, the phases of the LS cycle and how to overcome some of the possible obstacles. We provide an overview of a typical Japanese lesson and finally reflect on our experiences of working with LS in a variety of contexts.

Background to lesson study

LS is a long-established research-based approach to the continuing development of teachers’ professional practice. It is widely accepted that LS assisted in transforming teaching and learning in Japan from teacher-centred to student-centred learning aimed at developing mathematical thinking and problem solving (Takahashi & McDougal, 2016). It has been used in some form for over 140 years in Japanese schools. Stigler and Hiebert (1999) popularised Japanese mathematics teaching and LS in the US through their book The Teaching Gap: Best Ideas from the World’s Teachers for Improving Education in the Classroom. Since then, LS has been developed elsewhere including in the West, initially being used in the US and Australia and, more recently, in the UK (Hart, Alston & Murata, 2011).

LS is usually described as some or all of the following: teachers working to research and develop a particular practice (the focus); researching and planning a lesson together; collaboratively teaching and observing the planned lesson; and collaboratively analysing and critically reflecting on that lesson. It is not primarily about perfecting lessons but mainly about researching and developing pedagogical understanding and practice. Ideally, this implies that findings from the lesson will be generalised beyond the immediate context and experience, and understandings will be shared with others through reports and potentially change pedagogy and even the curriculum.

In some cases the research lesson is refined and retaught before the stage of communicating and generalising. We will analyse later in this chapter the advantages and disadvantages of re-teaching the lesson in particular contexts. For example, in Initial Teacher Education (ITE) we will see how, in our experience, re-teaching is an essential part of the cycle for pre-service teachers. Fernandez (2010) also advocated the relevance of ‘repeated cycles’ in LS with ITE students. However in Japan, re-teaching the lesson is not common practice (Fujii, 2014) and some Japanese researchers believe the re-teaching of the lesson to be unethical, since it positions teaching as a science experiment, and children as experimental objects (Seleznyov, 2018).

Teachers begin the LS process by deciding on a LS focus and a problem or content area that they wish to develop. Previous research (Fernandez & Yoshida, 2004; Lewis & Tsuchida, 1998) has indicated the need for an identified focus when conducting LS, the Japanese calling this focus the research goal and the lesson plan the research proposal. Some examples of goals from lessons include ‘take initiative as learners’, ‘be active problem-solvers’, ‘be active problem-seekers’, ‘develop scientific ways of thinking’ and ‘develop their individuality’ (Lewis & Tsuchida, 1998, p. 14). Equally goals may be focused round topics that are difficult to teach, such as ‘How does one divide fractions?’ The research foci for the chapters in this book can be found in the preface.

Hart, Alston and Murata (2011, p. v) highlight that LS is a ‘complex professional learning approach’ with a variety of interpretations, and thus there is a ‘need to identify what is essential for an experience to be LS’. We would agree with this. In the authors’ experiences, LS is not just seen differently in different cultures/nations, but also within the same cultures/nations, with participants adapting some of the key elements to suit local conditions. Through this book and the following ten LS cycles, we will highlight various elements within LS, and in Chapter 12, we will analyse our experiences and summarise what may be useful in other contexts.

LS is aimed at developing the pedagogic practice of the profession as a whole, though, of course, in doing so the individuals involved would be expected to develop their own professional practice too. A great deal of time is spent reflecting on how learners think and learn, with general implications for better teaching practice, rather than a focus on the errors of the teacher and how a particular teacher might improve (as in coaching or other more judgemental forms of lesson observation). As Stepanek, Appel, Leong, Mangan and Mitchell (2006, p. 2) say: ‘Developing a new approach requires deep thought, inquiry and collaboration with a collective focus on teaching rather than teachers.’

Benefits of lesson study

LS has been seen to develop and support teachers in a non-threatening collaborative way, which is important in the current context in schools in the UK (and elsewhere) where ‘performance management and performativity are so dominant in schools and in professional learning’ (Williams, Ryan & Morgan, 2014, p. 151). LS is essentially a collaborative form of professional development. Cajkler, Wood, Norton, Pedder and Xu (2015) analysed 200 studies into LS, and they claim/state that there were four principal benefits for teachers:

• Greater teacher collaboration
• Sharper focus among teachers on students’ learning
• Development of teacher knowledge, practice and professionalism
• Improved quality of classroom teaching and pupil learning outcomes.

(pp. 193–194)

We have also observed, by working on LS for several years, that it can provide a powerful way to develop teachers’ mathematical content and pedagogical knowledge. Teachers question their own content knowledge and pedagogical knowledge by discussing the lesson plan with colleagues and by reflecting on learners’ responses. Alongside this, LS can help develop a culture of talking about mathematics between colleagues who, with time, consequently gain the confidence to discuss content and pedagogy. As we observed above, the LS process is not aimed at achieving the perfect lesson. Participants engage in LS in Japan to learn something new and extend their professional knowledge (Takahashi & McDougal, 2016).

Risks

There are some risks associated with conducting LS cycles and it is important to note them here in order to be prepared. Norwich and Jones (2014) discuss two central risks: (1) that those involved in the LS may not fully appreciate the complexities of the process, particularly if they are new to the practice; and (2) that ‘the LS strategy could be used selectively to make it fit current practices, perhaps because of external pressures’ (p. 152). Such risks might hinder reflections and observations so they may become more superficial/surface level and not achieve the deep analysis and reflection needed. We have observed this particularly when not enough time is dedicated to the preparatory work. Also, in relation to the second point, in our experience lesson study can be misused as a performativity/accountability tool negating the associated benefits of LS.

We, however, believe that the LS process is beneficial when it can help develop a climate of enquiry within schools, even if initially, for the inexperienced team, the gain might be more limited. LS is not a quick fix; it takes time and dedication and the authors have observed that it works better when school management/leadership are committed to devoting, or at least allowing, time and energy to it.

Different phases of lesson study

In this section we discuss the different elements of the lesson study process, all of which support the development of pedagogical understanding and practice.

In order to develop expertise, learning by reading, listening and seeing is not sufficient. It also requires learning through planning, doing and reflecting.

(Takahashi, 2015, p. 52)

As we outline the different phases we will include the Japanese names for them since some of these words are commonly used in some LS practice, including in schools in our own context. These phases describe the Japanese LS model that is theorised in the literature. This ideal form is not necessarily followed exactly in other contexts or even in all Japanese practice, but the elements described below will always be present in one way or another. Later in this chapter we will describe some variations within the contexts we operate in, and how we have followed these elements within the experiences described within this book.

Select a research focus

As stated above, in order for the LS to be meaningful there must be a chosen research focus for the lesson. The focus allows observers to collect data during the lesson so the discussion can be centred round the research focus instead of being a more general ‘what went well’ type of reflection. The data collection and analysis of learners’ responses allows us to distinguish between a research lesson and a lesson observation (for evaluation or for demonstration). We have included the research focus for all chapters within the preface and within the introduction or lesson outline of each chapter.

Researching and planning the lesson: Kyozai Kenkyu and the development of the research lesson proposal

Kyozai Kenkyu literately means ‘research of teaching materials’ in Japanese:

• Kyozai – teaching materials
• Kenkyu – research

During this phase teachers get together and use the literature to study the themes related to the chosen topic/focus/problem. This is an opportunity for teachers to deepen their own understanding of the chosen topic and related pedagogy (Stepanek et al., 2006). Kyozai kenkyu also involves a study of the curriculum, and requires a reflection on how the selected lesson fits within it, what the children need to know and what they will learn in the lesson and its aftermath. Including provision for this has been identified as a weakness in the teaching of mathematics in English schools, while a lot of thought is given to the sequencing of topics within the Japanese curriculum (Robutti et al., 2016). This phase also involves a study of the literature in order to identify learning issues such as typical misunderstandings and misconceptions around the topic. Existing literature is also used to consider possible tools, manipulatives, or materials that may be used and possible tasks that may be presented to students. This process is what we intend to model, by presenting a background section within each chapter that reviews ideas from the literature related to the lesson. According to Takahashi and McDougal (2016) a LS cycle cannot be successful if this phase of the cycle is overlooked or done too quickly.

In our experience with the ten lesson studies in this book, the main part of this research and planning of the lesson was completed by the authors prior to meeting with the teachers, (as university educators we are immersed in mathematics educational research and this day-to-day practice, and referring to associated literature, informs what we did at this stage). We then met as a planning team to further plan the lesson and consider the context; this was generally audio recorded, in order to improve our reflections of the process.

The teachers then created a detailed plan for the lesson, basing their decisions on what they learned (Takahashi & McDougal, 2016). The lesson plan is a research proposal that must include:

• The research focus
• The needs of all students
• The place of the lesson in the sequence
• The stages within the lesson and their purpose
• The key questions that will be posed
• Anticipated student responses
• Planned teacher responses to anticipated students’ responses.

In our book, each chapter’s ‘lesson outline’ section presents these elements in summary form.

Live research lesson (Kenkyu Jugyou)

Usually in Japan one teacher, from the planning team, teaches the lesson while the rest of the planning team observe alongside other teachers interested in the focus of the lesson. The number of observers varies considerably according to the context of the LS. In some district lessons there could be up to 100 observers watching a lesson happening on a stage while in smaller-scale events the lessons happen in the classroom with a smaller number of observers. Often, each observer focuses on a small group of learners, perhaps so as to cover the whole class, or a representative sample of the class between them, with the intent to understand students’ thinking and capture students’ reactions to the lesson. This happens also in larger-scale LS where, for practical reasons, the audience is divided into inner and outer circle. The inner circle is given priority in accessing children’s working during the lesson, but in all cases participants do this only by observing rather than interacting with the learners. The planning team prepares a number of questions, linked to the research focus, to help guide the observations and these are given to all observers. In our experience of lesson studies for this book, in most cases the lesson and discussions are recorded, observers make copious notes; photos are taken of board work and student work is collected. In each chapter, the ‘lesson reflection and analysis’ section attempts to capture the experience of the lesson but it also begins to analyse the lesson in a way that would normally occur post-lesson.

Post-lesson discussion (collecting the evidence) Kenkyu Kyougikai

Often, the board work is not deleted and is used to initiate the post-LS discussions and reflections; students’ work is also brought to the table for the purpose of understanding learners’ thinking (Robutti et al., 2016). As stated above the purpose of the discussion is not to refine the lesson but to understand students’ responses, by analysing the data (video recording, audio recording of students’ discussions, board work, student work) from the lesson. We have provided some examples of pictures of students’ work and ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Acknowledgements
• 1. An introduction to lesson study
• 2. Perimeter and mastery
• 3. Developing understanding of fractions through context
• 4. Proportional relationships and the double number line
• 5. Communicating mathematically about shape
• 6. Constructing deep geometrical ­understanding
• 7. Modelling through open-ended tasks
• 8. Early algebra using real-life scenarios
• 9. Sustaining argument about randomness
• 10. Experimental design: connecting statistics to student experience
• 11. Teaching logarithms through ­problem solving
• 12. Conclusion(s): mathematics, pedagogy, professional development and research
• Index