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by Tony Anderson, Jennifer Kumi Burkett, Gail Burrill, Joyce Frost,
Jerry Gribble, Jill Ryerson, and Celeste Williams
Background
During the summer of 2003, six high school teachers from around the United States had an experience that will forever change the way we look at lesson planning. We met 12 times for two-hour sessions at the Summer 2003 Park City Math Institute in Park City, Utah, to experience the process of Japanese Lesson Study.
According to Dr. James Hiebert, professor of education at the University of Delaware, Lesson Study originated in Japan, where Japanese admirers of John Dewey met secretly to discuss his philosophy of education. Through these meetings, they developed the process that is now known as Japanese Lesson Study. After World War II, they were free to share their ideas on education just as Japan's education system was being rebuilt. As a result, Lesson Study became an integral part of lesson planning in elementary and some middle schools. In Japan, there is a formal structure to the process of developing the lesson and to the lesson itself. The process is collaborative and iterative, and focuses on the lesson rather than the teacher.
Before the lesson is planned, an appropriate overarching goal is selected. This goal is not content specific but is rooted in a school-wide goal centered on developing the whole child. Next, a mathematical objective is chosen that is both content specific and designed to fit into the selected goal. Then a group of teachers work together and construct the lesson. Several drafts of the lesson may be prepared and refined. When the lesson is ready, it is used with a class while the members of the group and invited others observe. Immediately after the lesson is presented, teachers and observers meet to debrief and critique the lesson, after which the group revises the lesson based on the comments.
The Japanese lesson format consists of four main parts:
- hatsumon (asking a question to stimulate students' thinking)
- shu hatsuman (the key question for the day)
- neriage (polishing up, a whole class discussion)
- matome (summing up)
The hatsumon is a "hook" designed to spark students' curiosity about the day's lesson. Typically these problems relate to the students experiences outside of school. Great care is given to choosing and wording the shu hatsumon. The question is scrutinized to assure that it will lead to the desired mathematical objective. During the shu hatsumon, the teacher observes individual student work, keeping mental notes of what he or she sees. Using these notes, the teacher strategically plans not only which students will present their work but also the order in which they will present. Incorrect work is often used for class discussion if the teacher thinks it will be beneficial to the class. The teacher spends a substantial amount of time connecting student ideas during neriage. This discussion will eventually lead to the emergence of the key mathematical ideas. The matome stage, where the teacher carefully summarizes the lesson with as much mathematical rigor as possible, is considered vital to a successful lesson.
Selecting a Topic
Unlike a school building setting, the Park City Math Institute (PCMI) Japanese Lesson Study Group did not have a specific curriculum for which lessons are needed. Our first task was to decide what age level and topic we would be teaching when developing our lesson.
Two months before the start of PCMI, the members of the Japanese Lesson Study Group conducted a poll by email and decided to create a lesson for high school geometry. Gail Burrill, our group leader, and Mr. Lars Nordfelt, a high school teacher in Park City, Utah, arranged for us to teach a one hour lesson to Mr. Nordfelt's summer geometry students.
Our first assignment at PCMI was to select a geometry topic that we could use to develop a lesson. We were encouraged to consider topics that were significant in the course, paying special attention to areas in which students struggle and topics that connect to other areas of mathematics. On the second day, we generated the following top ten "big ideas":
- Pythagorean theorem
- Transformational geometry
- Congruence of triangles
- Parallelograms
- Scale factor and its impact on area and volume
- Spatial geometry
- Navigation
- Fractals
- Similar triangles and right triangle trigonometry
- Ratio problems
After some discussion about the importance of each topic and its relationship to the broader mathematics curriculum, we voted to narrow our list to the following three choices: (1) Pythagorean theorem, (2) Scaling and dimensionality, and (3) Fractals.
During our discussion on day three, several group members mentioned there were connections between these topics and perhaps two or more of them could be combined. It was at this point that Debra Ferry, one of our visiting math supervisors, demonstrated that we could cut out a 3-D stair-step fractal to illustrate that changing the dimensions of a rectangular prism affects its area and volume. The fractal also had an amazing connection to the Towers of Hanoi mathematical puzzle. We each tried to make this fractal and were completely captivated by both the beauty of the model and how it demonstrates the mathematical concept of scaling and dimensionality. After spending most of an hour investigating stair-step fractals with different scaling factors and sizes of paper, we knew that we had our math objective and a pretty exciting "hook" to intrigue our class of ninth- and tenth-grade geometry students.
Designing the Lesson
Once we had settled on the topic, we devoted one week to designing the lesson. To make the most efficient use of our time, we divided into two groups of three teachers each. Each group was asked to produce a first attempt at the lesson. Both groups thought carefully about the hatsuman (the hook). We asked ourselves how we would introduce the stair-step fractal to our students and what thought-provoking question we would pose to engage them. Once the introduction was decided on, we attempted to predict possible responses that students might have. We thought about how we might steer students toward the main ideas of our lesson if the ideas of scale factor and dimensionality did not emerge naturally. We also debated smaller details of the lesson, such as what size graph paper would be the most appropriate for building the models and what kinds of directions (written or verbal) we would use to instruct the model making. We spent time developing ideas for extensions to the lesson that would serve either as homework problems or follow up lessons.
After spending two days working separately, we began merging the documents from the two groups into a working lesson. Before we could proceed, we agreed that our lesson would be introductory. The process of merging our documents led to discussions in which we debated the exact wording the teacher would use when asking they key question, the exact moment the teacher would begin speaking, and how the mathematics within the lesson would unfold. We recognized the difficulty in predicting what students would discover, but despite this challenge, we spent time thinking about how the teacher might use the students' work to draw out our main mathematical ideas about scaling and dimensionality. In addition, we developed an extension problem that the teacher would use to determine the students' level of understanding. We also generated homework questions to serve as further assessment.
As a "trial" session, we asked 12 colleagues from other areas of our program to serve as our "students." Though they were not high school students, we felt that teaching our lesson to this group would still be beneficial in determining strengths and weaknesses in our lesson plan. Before the lesson, each member of our group was assigned a specific area to focus on as he/she observed the research lesson. One observer would be responsible for following two students' responses throughout the lesson while another observer was responsible for tracking how and when the important mathematics emerged. In addition, we had observers keep track of the time spent on each section of the lesson, how well the content matched the students' ability and how well we phrased questions and anticipated student responses when we planned the lesson.
Refining the Lesson
After the trial lesson, the members of our group debriefed. Each member of the group shared positive feedback and concerns they had based on their observations. We discussed each concern and revised the lesson plan as needed. One key issue we raised was how we would define scale factor in the lesson. Would we define scale factor as a ratio or a number? We debated this issue extensively because scale factor is such an integral part of our lesson. We watched the video of the trial lesson, pausing the tape several times to discuss interesting things we noticed as well as issues we believed should be addressed. In watching the video, we recognized weaknesses in our lesson that we had previously overlooked. Most notably, we discussed how to better facilitate the class discussion after students had explored the relationships in the fractal and presented their findings to the class. We wanted to be sure the important ideas that students presented were tied together in such a way that the mathematical objective was met. We had not discussed this section of the lesson plan in detail beforehand; it was not surprising that this section of the lesson now needed to be considered in greater depth. After watching the discussion on video, we decided how the teacher could use the students' presentations to structure the discussion so that our objective was met. Our revisions were incorporated into the revised lesson plan. The lesson was taught to a class of high school students a few days later.
Final Debriefing
The lesson did not go as well with the students as it had with our teachers in the "trial lesson." When we got together to debrief the lesson, we discussed extensively what happened. We realized that the students were not used to working in groups or exploring mathematics through discovery. Because the students rarely spoke, we suspected they were uncomfortable with the camera and the observers sitting throughout the room. Judging from the students' blank stares and lack of response to our key question, we concluded that perhaps our question was flawed. We had not anticipated how to respond if students did not understand the question. As a result, the teacher chose to use direct instruction to develop the mathematical ideas. During the debriefing, we generated a list of sub-questions and alternate methods we could employ in the future if this problem arises. These questions would give students greater direction while still allowing them to explore the model on their own.
Conclusion
This fall, several teachers in our group will teach this lesson in their own classrooms. We expect the lesson will run smoothly now that it has been further revised. In addition, we will not be teaching in an artificial setting. Our group will continue to dialogue via email and may choose to videotape and critique some of the lessons. But perhaps of greatest importance, the collaboration techniques we have learned through this intensive process will help make it possible for us to develop other lessons with our colleagues.
Reflections on Lesson Study
Written by the teacher who taught the lesson.
Teaching a lesson written by a group of teachers working with the Japanese Lesson Study model was an unusual experience. In my five years of teaching, I had only taught lessons that I designed. My everyday lessons are structured and thought out but do not involve the level of planning I experienced during the Japanese Lesson Study Process.
Our group worked hard on the phrasing of a few key questions and comments. I needed to honor that collaborative work and at the same time avoid sounding like an actress playing an artificial role. At times I found it necessary to use my own judgment and deviate from the "script" in order to maintain a natural flow in the lesson.
In my experience, teaching in the United States is a highly personal profession. I wondered what the Lesson Study process would be like, especially during the debriefing session when the lesson I taught would be critiqued. Though a few constructive or complimentary comments about my particular teaching were given, it was clear that the lesson we all worked to create, and not my teaching specifically, was the focus of the debriefing. My fears were never realized and I learned a great deal from the Lesson Study process.
Reflections on Bringing Lesson Study Home
Written by a member of the Japanese Lesson Study Group.
The collaborative lesson development process is very different from the usual experience of creating and teaching lessons in American classrooms. Since returning from PCMI, I have been much more aware of the possibilities of collaboration with teachers in my building and even across the district. Sharing lesson plans, lessons, computer activities, quizzes, tests, etc. is becoming a much more natural process for me. I'm really beginning to understand the power and necessity of ongoing collaboration in teaching.
Our group hopes that our experiences with the process of lesson study will inspire you to try it with your colleagues. We have included the lesson and its various components, linked below, to serve as an example of how the process works. Please feel free to use it with your students as well. But, keep in mind that it is the process that is important and that our lesson is still a work in progress for our group.
References
Bass, H., Z. Usiskin, and G. Burrill, eds. 2002. Studying Classroom Teaching as a Medium for Professional Development: Proceedings of a U.S. Japan Workshop. Washington, D.C.:
National Academy Press.
Lewis, C. 2002. Does Lesson Study Have a Future in the United States? Nagoya Journal of
Education and Human Development 1:1-23.
Lewis, Catherine. 2002. A Handbook of Teacher-Led Instructional Change. Philadelphia, Pa.: Research for Better Schools, Inc.
Uribe, Diego. 1994. Fractal Cuts. Norfolk, England: Tarquin Publications.
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