## Implementing Lesson Study Summary## Monday - Friday, June 26 - June 30, 2006Participants - Gail Burrill (Supervisor)
- Akihiko Takahashi (Supervisor)
- Anita Aysola
- Jo Guido
- Irma Kirpes
- Susana Salamanca-Riba
- James Stallworth
- Eileen Teel
- Donald Williams
The group began by introducing themselves and where they were from. We then started a discussion on what Lesson Study is and what it look like. The goal of Lesson Study is to improve the effectiveness and quality of the learning experiences that teachers provide. The lesson should consist of the following essential components: - A well planned lesson plan with a clear hypothesis
- Live lesson observation with various participants
- Focused post-lesson discussion based on the participants' observations
- Reteaching of the lesson with modifications or changes from the discussion
It is important to take into consideration the most important question: What did the students learn? The follow-up to this question is: What evidence do you have that students learned it? We watched a PowerPoint presentation that illustrated different forms of lesson study. These forms ranged from team-based lesson study, school-based lesson study, and district-based lesson study. Many of us had heard of the team- and school-based lesson study, but we were surprised that the district-based lesson study existed. This form of lesson study had the class and teacher on a stage in an auditorium. The auditorium was filled to capacity with teachers observing the lesson. We asked several questions about how the students performed under this level of scrutiny, compared to other forms of lesson study. Aki informed us that this type of lesson study really focuses on the teacher and the lesson. We then watched a 15-minute video clip demonstrating lesson study in an inner-city Chicago school. The lesson was entitled Patterns in Hexagon Tables. It appears that the lesson was held in a gymnasium and there were at least 10 observers along with the instructor. The lesson was for 6th grade students and met specific Chicago Academic Standards on extending geometric and number patterns. It also allowed the students to describe trends and patterns using a variety of methods. After watching the video clip, we had a discussion about some of the benefits of the observation. We were taken by particular student who typically was one of the lowest achieving students in the class. But during this lesson, the student was engaged and leading his group towards a solution of the problem. Even though the solution that he provided ended up being incorrect, it was important to see how involved this student was in the entire process.
On this day we discovered that we would be working with Lori Halls from Park City Public Schools for our lesson. She is currently teaching a six week Geometry class for rising 9th grade students or 10th graders already enrolled in Geometry. We looked at her teaching schedule and discovered that we did not have much choice in the lesson. Since she is nearing the end of the course, she has certain topics that she has to cover. Lori is currently using the Discovering Geometry by Key Curriculum Press. On the day that we will be teaching the lesson, her plan was to cover Chapter 8 on area. This means that she will be instructing the students on how to find the area of rectangles, parallelograms, triangles, trapezoids and kites. So we felt that it was important for the students to have an understanding of where the area formulas come from. This way instead of just memorizing the formulas for the different shapes, they would learn how to derive the formulas for themselves and understand the connections between the different shapes. We address some things in our discussion that we deemed "important concepts." Some of the important concepts were - square units
- the notion of cover
- the description of the base and height
- what happens when we change the dimensions
Through discussion these points led us to consider how to extend the area formulas to different shapes. It became important for us to pay attention to the differing measures. We talked about the classification of quadrilaterals and how triangles are related to quadrilaterals. These discussions then led us to concentrate on the perpendicular measures of the figures. If we look at the all of the polygons that we have previously mentioned, the consistent values in the formulas are the base and the height. So we decided that it would be important for the students to have a rich understanding of the relationship between base and height. This understanding allows the students to understand and to derive the formulas, and not aimlessly plug numbers into a formula.
On this day we had to address some housekeeping issues. If we are going to teach the lesson to the students on Monday July 10th, then we will have to do a practice run through on Friday July 7th with volunteers from SSTP. We felt that it was then important to get into the process of planning the lesson since we would be presenting in just over a week. Gail and Aki talked to us about the six parts of a lesson created using a lesson study approach. Those components are as follows: - Launch - a grabbing entry to engage the students
- Working on the problem - the "what" and "how" of the lesson
- Student Strategies - list all of the possible student responses and teacher actions
- Massaging the student work - taking the ideas created by the students and placing it into a whole
- Evaluation or Assessment - a clear understanding of what the evidence of learning will look like
- Summary - a statement of the mathematics that the students learned
After some discussion and clarification about these ideas, we began to decide what would be the overall goals for our lesson. We brainstormed some ideas about the possible goals for the lesson. We came up with the following list: - Recognize how to make rectangles and use that to find area of any shape
- Be able to fine perpendicular lines for every figure to derive altitudes
- Identify altitude in a figure
- Use basic understanding to allow them to derive formulas for shapes
- Apply knowledge to find area
This is the way that the goals were displayed on the board. Once we finished brainstorming, we went into further discussion over the goals that were listed. There was a lot of conversation about the language that was used, whether we would be able to do all of these goals in a one hour period, and which of the listed goals our main focus was. We were given the task to look at the goals overnight and come in the next day ready to begin the writing of the lesson.
We looked at the goals from the previous day. At this point we did not change them, but we used that as a guide for creating an initial draft of the lesson. Aki broke us up into two groups of three participants. Each group was given the task of creating an initial lesson on the topic we had chosen. Both of the groups started the lesson from the students' understanding of the area of a rectangle. One group began the lesson by having the students construct a rectangle with side lengths of n and m, where n and m were integers to be determined later. Then there would be a discussion about the number of different rectangles that could be constructed and the area of these various rectangles. Then the students would be asked to construct parallelograms with side lengths n and m. The students would be asked to share their responses again. This time the teacher would be looking for a variety of responses both about the look of the shape and the area of the shape that the students constructed. The second group came up with a variety of methods to approach the goals. One group member wanted to look at 3 parallelograms with different side measures and orientations and compute the areas. The side measures and orientations would be selected so that the areas would be the same. Then the students would be presented with an image of triangles with a base on one line and a vertex on a line parallel to the initial line. The vertex could slide along the parallel line, but as long as the base remains fixed the area will remain the same. This would lead the students to a discussion about the values that remained the same in all of the figures. Another member felt that it was important for the students to derive the formula for the parallelogram by reconstructing the figure into rectangles. This would allow the students to have a variety of approaches and discover the relationship between the perpendicular measures and the area of the figure. The groups came back together and shared ideas. We decided as a group that we wanted to start with the rectangle because this is the previous unit and it allows the students to build on prior knowledge. We also like having the students to create rectangles of certain lengths and check the areas. This should demonstrate to the students that there were a fixed number of rectangles that they could create with those side lengths. Also the students would see that there was only one possible area for the rectangle that they created. Then the students would be instructed to construct a parallelogram with the same fixed side lengths. This time there would be a variety of shapes that the student could construct. The student should also see that the area of the parallelogram did not depend on the side lengths. While all of the constructed figures had the same side lengths, they did not have the same area. This would lead them to find another way of determining the area of the shape, using the base and height. One group member suggested an approach called a "placemat" as a brainstorming device for the students. The placemat would be divided into five regions, four trapezoidal regions around the outside and a square region in the center. Four students would be assigned to one placemat. Each of the students would be assigned to one of the trapezoid regions and proceed to do their work individually. Then once the teacher called time, the students would share their work. The students would then place whatever it was that they wanted to present to the class in the center region. Many members of the working group felt that this would be a good method to help manage class discussions. PCMI@MathForum Home || IAS/PCMI Home
With program support provided by Math for America This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. |