Lesson Study in Japan is a model of teacher-led research in which a group of teachers collaborate to target a particular area for development in their students’ learning. Based on their prior teaching, the group of teachers work together to research, plan, teach and observe a series of lessons, using ongoing discussion, reflection and expert input to monitor and improve their teaching.

Here in the U.K., as elsewhere in the world, there is a lot of interest in Lesson Study as a mode of professional learning, which quite often focuses on mathematics. However, there are many variations on a theme emerging, and there is a danger that the true essence of what showed initial promise might become corrupted.

The first step to ensure this is mitigated against is to really understand the fundamentals of what constitutes high quality in the innovation.

My research with colleagues, both in our Centre for Research in Mathematics Education in Nottingham and at the Tokyo Gakugei University in Japan, suggests that the Lesson Study should be informed by expert knowledge of the curriculum and carefully structured practitioner research.

Importantly, at the very centre of the process is a well posed question that asks how we might better teach and support students learning of a particular *mathematical* concept or behaviour. For example, we may wish to focus students’ attention on how they might develop better or more convincing mathematical arguments. In another lesson we might consider how to best develop conceptual understanding of averages as measures of location.

Whatever the focus, the teacher orchestrates whole-class discussion to construct a careful argument that seeks to ensure that students’ different thoughts are exposed, confronted, considered and resolved in ways that ensure as many students as possible leave the lesson having developed their understanding successfully. Central to this is a well-designed task that ensures that whole-class discussion can take place.

To illustrate, consider how a particular problem-solving lesson might play out. The diagram below shows a solid with constant cross-section. Students are asked to calculate its volume using one or more methods. This is a task that my colleague Professor Keiichi Nishimura used with his Teacher Research Group in Tokyo. To this point students will have learnt that the volume of a prism, such as a cuboid, is found by calculating *base area x height*. Importantly, all diagrams of prisms the students will have met to date will have been drawn carefully with the constant cross section of the prism positioned as base and often shaded.

As in all Japanese Problem Solving lessons, at the start of the lesson the task is introduced, and students are given 10-15 minutes to work on it, either on their own or possibly in pairs or groups.

An important part of Lesson Study in Japan is that the teachers think carefully about all of the possible ways in which students might go about solving the problem. This allows the teacher to identify the solutions they want to share with the whole class and the order in which they would like to do this. As the students work on the problem the teacher identifies which students’ work they will draw upon according to their plan, selecting those examples that will provide mathematical insight and concept development. At this point you may like to think about what you might expect students to do in this first phase of the lesson when working with this particular task: think about all of the different ways in which students might respond.

This next diagram shows the students’ methods that the teacher identified to share with the class in the class discussion phase of the lesson, where shared understanding is developed. At this stage, given that students are used to calculating the volume of a prism as *(base area) x height,* you may like to consider the different ways in which the students are ‘seeing’ the problem: what are they taking as the base? How have they split the problem up so as to find the volumes of different cuboids?

These questions were raised with students in a lesson focusing on how different students were thinking about the problem and how these different approaches all eventually lead to the same calculated value. An important issue the teacher drew attention to, after gathering these different expressions, was how each of these can be expressed as 44 x 8. This allowed the teacher to emphasise that in this case the volume can be found by multiplying the area of the constant cross-section, BFGLKC, considered as the ‘base’ by the height of the solid. This shows students a method that will work for more complex composite solids in the future.

Overall this lesson focused on what is effectively developing understanding of how we can calculate the volume of composite solids. It also provides insight into how the ‘maths sentences’ we write can provide a window on our thinking processes. This latter issue is something that students do not always pay attention to and is something that deserves more time and focus.

The common practice of sharing students’ work in this lesson format can be used to encourage students to think carefully about how they communicate their mathematical thinking clearly in their writing and diagrams in ways that reveal their understanding of mathematical structure. In Japan the whole practice is informed by carefully considered curriculum expertise in relation to concept development and the learning of maths.

As a final thought you might like to think about (1) how the dimensions of the L-shaped prism have been carefully chosen so that by judicious splitting into two cuboids and realignment of these a single cuboid can be achieved, and (2) how the teacher might work with their class to find the volume of the solid by finding the volume of a surrounding cuboid and subtracting the missing volume CDJK x KL.

The benefits achieved through teachers working together in groups are also seen in Singapore, where they work collaboratively in groups called PLCs: Professional Learning Communities. This is discussed in Professor Berinderjeet Kaur‘s blogpost __here__.

You can learn more about the use of lesson study to inform task design from our colleague in Japan, Professor Toshiakira Fujii in this article.

*Geoff Wake leads the work of the Centre for Research in Mathematics Education at the University of Nottingham. Following a teaching career in schools and colleges he has carried out substantial work in curriculum and assessment research and development working with partners across Europe, the United States, South Africa and Japan. Geoff is currently collaborating with Japanese colleagues exploring curriculum design for coherence.*