by Akihiko Takahashi
This paper takes the reader, step by step, through the lesson-planning process of lesson study. Takahashi offers clear and specific suggestions, such as parameters for the lesson study group meetings and a lesson plan template to support the lesson-writing process. Lesson study goals are explored and explained in the context of a sample lesson. The paper discusses lesson procedure, including presenting the problem, discussing solution methods, and summarizing the major point, as well as aligning the lesson with state and national standards.
Most of my knowledge about teaching mathematics over the past 20 years is from lesson study. Many teachers in Japan learn this way. One way in which lesson study helps teachers grow professionally is through the lesson-planning and lesson-writing components. With lesson study, a team is responsible for planning and writing a research lesson that will later be observed by other lesson study participants.
Being part of a writing team provides an opportunity to take a deeper look at the content and concepts that teachers intend to teach and to work toward understanding the ways in which students learn. Using a collaborative approach to planning and writing a research lesson takes advantage of the strengths of the individual members of the team. It also pools the teaching resources that individual teachers use and helps to combat teacher isolation.
Groups usually meet three to five times over the course of several months to develop a lesson plan. The first meeting is spent agreeing on a lesson study goal. The lesson study goal should be aligned with school-wide research theme, with a specific focus on the team’s grade level or subject area. (The conference paper Reflections on Implementing Lesson Study in the United States: “Incidental” versus “Purposeful” Learning discusses the notion of a research theme in more detail.)
The second meeting is used to select the topic of the research lesson and examine textbooks, teaching materials, standards, and other resources that will be used. By the end of the second meeting, one or two teachers in the group agree to develop the first draft of the lesson plan. This draft is used at the next meeting. Other members take responsibility for logistics of the observation and debriefing session, such as inviting other staff to observe the research lesson and coordinating the schedule with the principal.
During the third and fourth meetings, the team works together to revise the plan. The shared experiences of the entire group can help to improve the research lesson during this stage. The group can identify anticipated student responses, common student misconceptions, and ways to improve the research lesson so that it capitalizes on student responses, considers the ways in which students learn, and furthers the group’s lesson study goal. Becoming better at anticipating student response is one outcome of lesson study. By spending a good deal of planning time discussing anticipated student reaction, teachers develop a lesson centered on the ways in which students learn, and the teachers can “see through the students’ eyes.”
At the last meeting, the group finalizes the lesson procedure that the teacher will follow, including learning activities, blackboard organization, teaching materials to use, use of manipulatives, teacher questions to ask, assessment questions, and development of a student seating chart.
To help keep the process organized, a lesson plan template can be used. This template can be tailored as needed by writing teams. The template is designed especially to help teachers who have little or no experience with lesson study, and it represents a plan for just one lesson that might fall within a larger unit of study. As teachers become more comfortable and experienced, this type of lesson writing can be done for an entire unit. Most of the research lessons taught at this conference follow this template, but all of them modify it slightly. There is no one standard lesson plan template in lesson study.
I will use the lesson To Open a Cube that I developed, which follows the lesson template I have provided, as my example of the lesson-writing process for this paper. The full text of the plan, as well as its blackboard plans, can be found at the Lesson Study Group at Mills College Web site.
Goal of the Research Lesson
The goals of the research lesson are the learning objectives. They should be aligned with the research theme of the school and the lesson study goal of the group. (For more information on choosing a research theme, read the conference paper Whole-School Lesson Study as the Basis for Whole-School Research.) One way for lesson study teams to choose topics might be to agree upon concepts that are difficult for students to grasp. Choosing a difficult concept encourages teachers to stretch their own understanding of the content as well as to dig deep into ways to help students understand and learn the concept.
To Open a Cube’s objective is to help strengthen the concept of three-dimensional objects by challenging students with an open-ended question that tests their prior knowledge and deepens their problem-solving abilities. It has the following goals:
- to deepen students’ understanding of three-dimensional geometric objects through problem-solving activities
- to help students become good problem-solvers by providing a challenging open-ended problem and encouraging them to:
- use their existing knowledge to solve a challenging problem
- find common properties and relationships among various patterns by comparing peers’ solutions in order to find a solution to the problem
- consider their solutions from a different perspective, so that they can make reasoned conjectures
- to provide students with opportunities to discover the importance of working with peers to deepen their understanding of mathematics.
Aligning the Lesson with the Standards
Using the National Council of Teachers of Mathematics’ (NCTM) Principles and Standards for School Mathematics (PSSM), state standards, teacher reference books, and journal articles, the team can ensure that the research lesson is aligned with standards and research. Lesson study helps make a connection among standards, research and daily practice. In Japan, this connection seems to be clearer than it is in the United States, where standards can become a political issue rather than one of pedagogy. With the collaborative processes involved in planning a research lesson, teachers have the opportunity to make the standards “come alive.”
The lesson plan for To Open a Cube is aligned with the following California fifth-grade standards:
- Construct a cube and rectangular box from two-dimensional patterns, and use those patterns to compute the surface area for those objects.
- Visualize and draw two-dimensional views of three-dimensional objects made from rectangular solids.
About This Lesson
In the “about this lesson” section of the template, the writers explain their rationale for selecting the lesson topic and how the research lesson relates to the lesson study goal. It should be written from the point of view of what students will learn, what skills they will develop, and how this lesson’s activities will help them learn. It should also discuss the instructional strategies that the teacher will use. The section should answer the questions:
- Why did you choose this topic for lesson study?
- Why is it important to have this unit/lesson at this particular time in the students’ learning?
- Why/how did you choose the main activities?
- What are the key instructional strategies that are needed for this lesson?
One reason I chose the problem in To Open a Cube is to provide students with an opportunity to extend their problem-solving strategies. Another reason is to provide students with an opportunity to see what they have learned through the end of grade four. In this way the lesson can serve as a bridge between fourth- and fifth-grade learning.
Writing down the rationale and instructional strategies is important for several reasons. First, putting the rationale and strategies into writing helps to clarify the research lesson for the lesson-writing team. In addition, research lessons are often published or shared, and if the instructional strategies and rationale are not fully spelled out, some of the lesson’s intentions may be missed by others who practice the lesson.
Relationship of This Lesson to Its Unit and Other Units
Because each lesson is a very small part of a larger scope and sequence of learning, it is important to establish where a particular research lesson fits in the sequence of units. This way observers and teachers understand the students’ previous knowledge and future learning. This is also why it helps to have teachers from other grade levels on the planning team.
The other lesson plans from the conference show examples of ways to describe the relationship between the research lesson and other units. To Open a Cube explains the relationship by describing the California standards from the previous and future grades.
This lesson is related to the following California fourth-grade standards:
- Identify, describe, and classify common three-dimensional geometric objects (including cube, rectangular solid, sphere, prism, pyramid, cone, cylinder).
- Identify common solid objects that are the components needed to make a more complex solid object.
This lesson is related to the following California sixth-grade standards:
- Visualize, describe, and make models of geometric solids in terms of number and shape of faces, edges, and vertices; interpret two-dimensional representations of three-dimensional objects; and draw patterns (of faces) for a solid that, when cut and folded, will make a model of the solid.
Lesson Procedure
The lesson procedure is the “road map” for the research lesson, and it includes the instructional sequence as well as the specific learning activities. Each learning activity is accompanied by questions that the teacher should ask of students, expected student reactions, teacher support for students, and the points of evaluation for each activity.
In writing the lesson procedure, it is very important to specify who is responsible for an action. Be sure to say “teacher questions” for questions that the teacher should ask and “student responses” for responses that you think students may have. When reading a research lesson written by someone else, it is very helpful to know who is doing what.
The full lesson procedure of To Open a Cube is available at the Lesson Study Group at Mills College Web site. Rather than reproduce it here, this section will discuss the sequence of mathematics instructional strategies used in To Open a Cube. It is a sequence that is common in Japan. It includes four activities:
- presenting the problem for the lesson
- students working individually or in a group
- discussing solution methods
- highlighting and summarizing the major points of the lesson.
The Third International Mathematics and Science Study-Repeat (1999) found that approximately 54 percent of the mathematics lessons presented in the Japanese classrooms it studied used a sequence similar to this one (Hiebert, et al., 2003). The movement to gear lessons toward a problem-solving approach arose from research that was conducted in the 1980s in the U.S. but never widely implemented here.
Presenting the Problem
The practice of presenting an open-ended problem (Becker, 1997) at the beginning of the lesson differs from many American lessons where formulas and algorithms are presented as the keys for problem-solving. Presenting the problems first asks the students to solve the problems using prior knowledge, and in doing so, discover the formulas themselves. Devising effective problem-solving activities can be one of the outcomes of lesson study.
In To Open a Cube, the problem is: “How many edges of a cube do you need to cut in order to open a cube completely and make a net? Find the least number of edges that need to be cut.” After presenting this problem to the students, the teacher begins by cutting a cube as the students watch. After each cut, the teacher asks if the students see any difference. The teacher keeps cutting along the edges, first one, then the next until the cube falls open.
Next the teacher gives the problem to the students. The students have no manipulatives at this time. The teacher asks, “What kinds of things would you like to use to solve this problem?” Generally the students respond that they need a cube and a plastic knife (the same tools the teacher had been using).
The Students Work Individually or in Groups
Giving students time to problem-solve on their own or in groups tends to be more common in Japanese lessons than in American lessons (Hiebert, et al., 2003). During this time, the teacher should facilitate as needed, rather than continuing to speak or lecture. Teachers should aim to guide students as they make conjectures, discover mathematical relationships, and draw conclusions. Again, designing effective group activities can be an outcome of the lesson study process.
In To Open a Cube, as the students gather their materials, the teacher explains that the students can work with their partners to mark each edge with a marker before they cut it, record their number of cuts, and then sketch their flattened cube on paper. Students should have more than one cube in order to refer back to the shape.
After each student team has had a chance to collect data, the students share their findings, which in this case are the net patterns that the opened cubes make. If there are enough students in the class, most of the 11 possibilities for the shapes of the net will be identified. If not, the teacher may need to add the missing nets. This can be a very challenging problem, because no routine path is apparent. By noting that each student response took seven cuts, the teacher can lead the students to the hypothesis that seven cuts are needed in all of the different nets, and therefore seven cuts might be the answer to the problem. During these group activities and the sharing time, the teacher can assess whether the students are able to discover this common properties of all 11 nets.
Discussing Solution Methods
Now is the time for the students to test the hypothesis that seven cuts are necessary. With all their responses on the blackboard, the students can begin to prove the hypothesis. The teacher should lead student discussions and help students discover a relationship between the number of edges that a cube has (12) and the number of edges that six faces are connected at in each net (5). From this, a relationship can be derived that proves the hypothesis and solves the problem: 12 ( # of edges of a cube) - 5 (# of edges remaining uncut in each net) = 7 (# of edges to be cut to open a cube)
Highlighting and Summarizing the Major Point of the Lesson
Summarizing the lesson to help determine what students have learned is an important part of the process. To help summarize, Japanese teachers pay special attention to the presentation of materials on the blackboard, so that by the end of the lesson, the blackboard serves as a visual representation of the concepts learned. Students can take notes in their journals in a similarly organized fashion. Attention to blackboard organization and student note-taking are two outcomes of years of lesson study in Japan. The conference paper Developing Effective Use of the Blackboard Through Lesson Study examines blackboard use in further detail.
In To Open a Cube, the following items are all posted on the blackboard by the end of the lesson: the problem, student nets (including any student errors), hypothesis on the number of cuts, and the solution (12-5 = 7). Students can see the main points of the lesson at a glance and can take notes in their journals accordingly.
Assessment
Assessment questions should be included with each learning activity, using specific questions to help the teacher assess each activity. The questions should be driven by the group’s lesson study goal. Examples of assessment questions that are included in the To Open a Cube lesson plan include the following:
- Were the students able to find several ways to open a cube and find out how many edges needed to be cut?
- Were the students able to compare the 11 patterns of nets and find general properties and relationships among the nets to establish a conjecture?
- Were students able to review what they learned through the lesson and write about it in their journals?
Observation and Debriefing
After the lesson is planned, one member of the team agrees to teach the research lesson while other teachers observe and collect data. It is very important that teachers outside of the planning group are invited to observe so that the research lesson can be observed with fresh eyes.
The observation is followed by a debriefing session where participants share data and conclusions. With information from the debriefing session, the planning group can make changes to improve the research lesson.
To Open a Cube evolved as a result of lesson study debriefing sessions, during which some of the following considerations surfaced:
- Should you give the cube to individual students, pairs of students, or groups of students? Individual students should solve the problem, but students might be more comfortable with a partner; with three or four members of a group, one or two might sit back and not participate.
- Should you consider shortening the introduction to allow more time for student seatwork? This would be possible, and I revised the lesson to simply say, "How many cuts?"
To read more about the observation and debriefing processes, refer to the article Guidelines for Lesson Observation and Debriefing.
Conclusion
Being part of a lesson planning and writing team is a unique experience that allows teachers to think more deeply about lesson planning than regular day-to-day lesson planning typically allows. The templates and experiences provided in this article are only examples of what can evolve from the process of lesson study. Part of the lesson study process is developing a culture of sharing, and it is hoped that through conferences like this one, the evolution of lesson study in the U.S. can continue.
References
Becker, J.P. & Shimada, S., eds. (1997). The Open-Ended Approach: A New Proposal for Teaching Mathematics. Reston, VA: National Council of Teachers of Mathematics.
Hiebert, J., Gallimore, R., Garnier, H., Bogard Givvin, K., Hollingsworth, H., Jacobs, J., Chui, A.M.Y., Wearne, D., Smith, M., Kersting, N., Manaster, A., Tseng, E., Etterbeek, W., Manaster, C., Gonzales, P., & Stigler, J. (2003). Teaching Mathematics in Seven Countries: Results from the TIMSS 1999 Video Study. Washington, DC: U.S. Department of Education, National Center for Education Statistics. Retrieved June 26, 2003, from http://nces.ed.gov/pubs2003/2003013.pdf.
This paper was originally published online in August 2003.