Lesson Study works best with rich subject content. This course offers step-by-step support to conduct a Lesson Study cycle on fractions, by providing fractions resources and research to examine.

## Prepare for Unit and Lesson Planning

In this module, your team will agree on the flow of the fractions unit and plan one lesson within the unit in depth, to serve as a research lesson. The research lesson will be taught by one team member to their students while other team members collect data on student responses, to be examined and discussed during the post-lesson discussion.

Closely examining lessons developed by others is an excellent way to dive into the Lesson Study process and to build the teaching profession. If you are intrigued by the potential of linear measurement context to introduce fractions or to build understanding of equivalence, you may want to use the video and materials from Study Part 3 and Part 4 as the starting point for your research lesson. Or perhaps you would like to use insights about fractions gained from your work thus far to reshape your own curriculum’s lessons. In either case, this is a good time to:

Review Standards and complete the Relationship to the Standards section (#4) in the Teaching-Learning Plan*. *We recommend that you revisit the Common Core State Standards or your state standards as you begin planning for your research lesson. The Standards should help you identify aspects of student understanding you want to work on. Your state may also have a Progressions document related to your standards.

Revisit your insights from your team’s work so far. What have you learned so far about challenges in understanding fractions? Are there instructional tasks or approaches you are eager to try out with your students? Has your investigation so far illuminated elements of your instruction that you would like to work on?

The table below in the resource tile below is an expanded version of the table first encountered in Study Part 1. It includes, in the right-hand column, some new ideas about how the linear measurement context might support students in overcoming key challenges in understanding fractions. Reviewing this table may help you look back on your work so far and finalize the design of your unit and research lesson within it.

Consider where your students are mathematically. If the students who will take part in the research lesson have not yet had an in-depth introduction to fractions, your team might want to introduce fractions using a linear measurement context, drawing on the grade 3 fractions lessons from the Japanese curriculum as replacement lessons.

If your students are in grade 4 or 5, the grade 3 Japanese lessons may still provide a good starting point from which you can go on to build understanding of equivalent fractions and the other grade 4 and 5 standards. Or you can use your work so far to consider the key understandings you want to build using your own curriculum.

Write the Lesson Rationale (#3) section of the *Teaching-Learning Plan* by briefly summarizing your current thoughts on what you want your research lesson to accomplish. Don’t worry about writing a perfect lesson rationale. The point is to consolidate your thoughts before you begin detailed planning of the lesson. You can revise your lesson rationale later.

Review the chart "Lesson Flow: Teaching Through Problem-Solving" below.When you watched the fractions lessons on video, you probably noticed that students built new mathematical understandings by solving and discussing a challenging problem, and that the new mathematical ideas were drawn from discussion of the students’ own work. The chart *Lesson Flow:* *Teaching through Problem-Solving* summarizes the flow of a Japanese teaching through problem-solving lesson, which is very similar to the lesson flow recommended in *5 Practices for Orchestrating Productive Mathematical Discussions*, by Margaret S. Smith and Mary Kay Stein. Lesson study is an ideal opportunity to work with colleagues on a lesson structure that challenges students to develop as mathematical thinkers and problem-solvers.

#### Lesson Flow: Teaching Through Problem-Solving (PDF)

Lesson Phase |
Activity and Purpose |

Introduction and Posing the Task (brief) |
Teacher poses the problem. Students grasp it, become interested in solving it, and recall related ideas. |

(7-20 min) |
Students bring their own prior knowledge to bear, trying to solve the problem. There may be input from classmates after students work for a few minutes on their own, but students are individually exerting effort to come up with a solution approach. Students are not simply following the teacher’s solution. Teacher circulates, noting student solution methods on a seating chart for teacher’s reference during next phase of the lesson. Teacher may question some students (e.g., “What is the problem asking?” to struggling students; “Can you write an equation to go with your diagram?” to students who think they are finished). |

(15-30 min) |
Teacher designates several students to present their work on the blackboard and explain it. Choice and sequence of the student work is planned by teacher in order to support development of the important mathematical understandings. (Incorrect approaches are sometimes included in the presentations.) Class members actively study the solutions, supported by teacher questions such as “How many solved it this way?” and “Do you agree with this method?” Students contrast solutions, supported by teacher questions, such as “What is the same and different about Sam’s and Marika’s solutions?” and “What are the good points and difficulties of each solution method?” Discussion focuses on the thinking and reasoning used in problem solving and the central mathematical ideas. |

(brief) |
Teacher draws on student thinking to summarize what has been learned (usually on blackboard). Students use the blackboard record and math journals to organize, reflect on, and consolidate their thinking. Class often ends with a journal writing prompt such as “What I learned today.” |

**Develop the Lesson Flow**

Using the left and middle columns of the Research Lesson Plan (Section #7 in the Fractions Course Teaching-Learning Plan), capture the learning flow for your research lesson.

This should include the tasks or activities of the lesson, anticipated student ideas, the puzzles or tensions that will arise through discussion and comparison of student ideas and how students will refine or expand their understanding as they confront these tensions. Allocate an amount of time for each lesson element, from introduction through summary. Specifying time helps makes your team’s thinking about lesson design visible. The following prompts may be helpful in outlining the learning flow:

- What is the “drama” of the lesson? What is the sequence of experiences that will propel students from their initial understanding to the desired understanding?
- How might the student responses you anticipated, including misconceptions, be highlighted or compared to spark student learning?
- What might students notice that moves their thinking forward? What insights or actions would we expect from students who have a breakthrough in their understanding?

While the left column captures the lesson flow, the middle column of the Teaching-Learning Plan captures Teacher Support. For example, you might note key questions the teacher will ask the class, questions that will be posed to students who do not get started, etc.

As you plan the research lesson, group members may be tempted to micromanage each move and comment the lesson instructor will make. Instead, train your focus on the content and how students will interact with it to build the new understandings of the lesson. If a lesson element is likely to affect students’ learning in important ways, then it is legitimate territory for group discussion. Problem wording and content, choice of manipulatives, key teacher questions and design of graphic organizers are all examples of lesson elements that may affect student learning.

Other decisions, such as whether to have students at desks or gathered on the rug, may be best left to the instructor, unless you think they will shape student learning–for example, it may be important for students to be at desks so they can update their thinking in their journals as they listen to classmates present.

Dr. Takahashi’s lesson plans for the series of California lessons you watched on video may provide a useful reference as you write your plan. They appear in the resource tile below.

Complete the Data to Collect section (#8) of the Teaching-Learning Plan.

One particularly valuable form of data is narrative observations of several focal students of different achievement levels across the entire lesson, so you understand the lesson through their eyes. We recommend that you anticipate the thinking of these students and note it in the *Teaching-Learning Plan,* so you can later compare your ideas with their actual responses.

Use your work in section #8 to complete the right column in the Lesson Design table (section #7).

The “Points to Notice" column notes what observers should look for in students at each stage of the lesson–for example, how students interpreted the problem posed at the outset, what solution strategies they initially tried, and how and why their thinking changed during the lesson.