## Lesson Study Summary

### Monday - Friday, July 11 - 15, 2005

DAY 10 (7/11/05)

The group looked at the edited lesson and brainstormed possible ways to pose the big question.

Possible Questions:

1. What information do you need to find the distance?
2. What information do we need to find the treasure?
3. Do you think the Math you've been studying can help us?
4. Do you think Geometry can help us?
5. Do you think Math can help us?
6. I tried walking down Main St. but realized I didn't know where to stop.

The group talked about the posing of the problem and the discussion the entire meeting.

The group split up preparations for the next day's lesson:

Megan - clean up lesson plan, small maps
Jeremy and Mary - fleshing out the big questions, the discussion at the end of the lesson
Amy - create BIG poster to show at the beginning of class with clues
Alan - create 8 smaller posters for each pair
Megan - photocopy small maps
Claudia - name cards, markers, tape

"A good lesson is one where the teacher tries to understand what the students are thinking; a bad lesson is one where the students try to understand what the teacher is thinking."
- Aki, in describing a philosophy of his mentor.

ACTUAL LESSON (7/12/05)

The actual lesson took place at Park City High School Learning Center, in Park City. Jeremy taught the first hour of a two-hour class with 16 students. All students were seated randomly in pairs. The lesson lasted approximately 70 minutes.

After the lesson, the group spent thirty minutes reflecting and discussing it in the style of a formal debrief. Aki facilitated the discussion by first explaining how it would be structured and allowing Gail to introduce two guests. Jeremy began a general reflection of the lesson. His thoughts included the following opinions:

• 10 minutes passed until any kid had something written on his/her paper
• students asked for a huge variety of information
• students were obviously not used to working in groups
• asking "Why do you need that?" took a long time for students to answer and no pair could explain why the two triangles were similar; many groups were waiting, too
• decided to give students the 75 paces length and would give it at the beginning of the lesson if it were taught again
• all groups started with trigonometry, possible because it is what they had done the day before; after the 75 paces length was given, many pairs saw the scale factor of 5
• would stress more in beginning that only a few pieces of information were known
• would have liked to get around to other groups more
• students were frustrated about why triangles were similar
• at end, it was difficult to make the connection between similarity and trig and thus lost the attention of most of the kids

Joyce briefly restated the goal of the lesson, for the most part for the two guests at the debrief. Aki added that with more time, this should be more of an explanation of the rationale for choosing the goals and topics of the lesson.

Aki focused on Jeremy's thoughts, restating big points that could be discussed by the entire group. Specifically, he mentioned the issue of when or whether to give the 75 paces length to the students and how the final connection at the end of the lesson could be made. Members of the group shared opinions on these two topics but no decisions were made.

Aki invited the two guests to share their observations or ask questions of the group. Annette, a teacher from a middle school in Provo, Utah, said that she liked how the class was asked what they had been doing in the class previously at the beginning of the lesson, but that this prior knowledge was never accessed. Blake, a professor of Math Education at BYU, also in Provo, made may points about the lesson, including that the culture of the classroom was very positive overall in terms of engagement and willingness to jump in to the Math. He ended by asking "do you have to have closure in the end for it to have been a good lesson?"

DAY 12 (7/14/05)

The group met for the last time to finish the debrief, reflect on the lesson study process and product and prepare for the presentation to the rest of the teacher program the next day.

Gail began by pointing out some of the really positive outcomes from the three-week process. She relayed that these things are easy to overlook at the end of a long, tiring research lesson like this but are so important to examine. She first talked about the specificity and appropriateness of the objectives that were written in the first week. "We articulated the objectives well enough that we could all say where the kids were at the end of the lesson [in terms of understanding]." She said that the lesson had been planned so that the task was always framed to meet the objectives. She emphasized that the discussions about the lesson always were always under the umbrella of how the activities would help students make the connection desired. Finally, she said that although the group had left the debrief the day before thinking about what could be done at the end of the lesson to pull the kids together and make sure they understood the connection between similarity and trig, it was obvious from the final lesson that learning had occurred.

Aki added his reflection on the process by focusing on what the teachers could think about when designing lessons on their own. He said when designing any lesson, a great deal of time should be spent focusing on the sequence of ideas, questions introduced. "This is not common in US curriculum, it's not common to think deeply about the sequential order of topics. A good lesson plan is not drawing one line from beginning to end. A good lesson plan is a map. If a student goes outside of some paths, we are comfortable seeing how they may get to the end."

He continued to say that "real mathematical discussion starts after students find the answer." He emphasized that seeing the mathematics is not simply seeing the right answer.

As the group began discussing the final editing for the lesson, Aki said that ideally, all the teachers would go back to their own classrooms and teach the lesson with their own kids, in their familiar environment, in the context of a greater curriculum. Learning from this could be shared with everyone and the lesson could be adjusted again and again. The group discussed the lesson for over an hour, focusing on the discussion and the summary sections. Specifically, a great deal of time was spent talking about how to get kids to the big connection.

When the group got stuck on whether or not to give the right angle to the students, Aki said that this is something that can be researched in the teacher's own classrooms. "This is the purpose of a research lesson, not the lesson plan," Aki said. Mary said that she would be interested in doing it one way with one class and another way with a different one to compare and decide on the best approach.

The last topic the group discussed in terms of the lesson was how to avoid students waiting for the teacher to come around and give clues. Suggestions included a systematic movement of the teacher around the room, less discussion with each pair about why they needed the info or having three pieces of information given at the beginning. The group agreed that this change would be up to each teacher's discretion when using the lesson. Aki suggested putting these kind of discussion topics in a separate section after the lesson.

As the group finished editing the lesson, Gail asked what the presentation would look like for the rest of the teachers. Ideas included sharing struggles, defining lesson study and giving the teachers the problem used in the lesson. The group split up the long-term project of writing about pieces of the lesson study (rationale, reflection, etc.) and decided that all pieces would be due to Joyce by September 1. The group worked to plan the short presentation the following day.

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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.