Lesson Study Summary
Monday - Friday, June 27 - July 1, 2005
"It is hard for people to change their conviction...unless they have a conflict." - Gail Burrill
DAY 1 (6/27/05)
Participants introduced themselves by relaying where they are from, where and what they teach. Aki Takahashi, a lesson study expert and math education professor at DePaul University in Chicago, described the concept of lesson study to everyone.
Joyce gave the group two options for student groups for the actual lesson in the third week. One set of students, in Provo, could be used for any desired topic, any day of the week. The other set was a Geometry summer school class ion Park City split into two groups. One group of 25 was in their third week and was meeting from 9 to 12. The other was beginning their program that week and meeting from 1 to 4. The group decided to use the 1-4 group so they could be caught in their third instead of last week of their program.
The group decided the lesson will be taught on Tuesday, July 12th from 1:00 to 2:00. The practice lesson, for a group of volunteer teachers, will be on Friday, July 8th and thus the lesson must be ready by the Thursday before that.
Group decided to brainstorm things that Geometry students struggle with and do well. Teachers shared ideas and Gail took notes on the front chalkboard. Ideas that were emphasized by more than one participant were checked.
"If we tell them how to do it, they're not going to think." -Aki
Teachers discussed misconceptions and difficulties around proof/logical argument and spatial reasoning, but didn't come to a topic. The group decided to research the five checked topics (above) for big ideas, misconceptions, interesting problems and lessons, etc. for Tuesday.
DAY 2 (6/28/05)
Joyce began Day 2 by describing the students who will be the participants in the research lesson. There are 15 students in the class and all have finished either the 8th, 9th or 10th grade. They are enrolled in Geometry for summer school to be able to start the next year in an Advanced Algebra class. Joyce brought the textbook the students are currently using and the lesson plans for the days the group will take the class for the research lesson. The students will be examining trigonometric ideas with their teacher at that time.
Each participant researched Geometric concepts, lessons, ideas and methods around the six big ideas that had been brainstormed in Day 1. Gail recorded big ideas on the white board in the front of the room.
Jeremy described the open-ended approach to writing lessons from the book "How Students Learn" by the National Research Council. He compared the method of teaching of the teacher imparting knowledge on students versus discovery-based, student-centered learning. He felt that the lesson plans provided by the teacher looked very traditional and thought the group's approach to the research lesson should be based more on conceptual understanding than procedural understanding.
Ellie asked whether it's better to move on in a class once 75% of the class has "gotten it" or not. Allen responded by saying that using true open-ended activities, all students should have access to "getting it" at some point.
Gail relayed that the CORE+ program is very good at helping students with conceptual understanding, but its critics argue students who go through this program don't have strong algorithmic understanding.
Joyce and Mary had both found research articles that supported Jeremy's findings. Claudia found an article that supported using technology in the geometry classroom that cited research showing student engagement and achievement.
Allen found research saying that students develop their ideas about shapes early on. For example, the research suggested that asking students to look at shapes in first and second grade instead of sixth grade might be much more appropriate in terms of long-term understanding. He also shared other research describing misconceptions students have about angles and area. Amy gave an example that students often relate the size of an angle with the lengths of the two segments or rays that form it.
Ellie had examined the Geometry standards for the state of Utah. She found that the standards ask students to identify trigonometric relationships in fractional and decimal form, find missing parts of triangles using the laws of sines and cosines, solve problems using spatial and logical reasoning, model and solve equations using geometric situations and the basic tenets of proof.
After the group shared their findings and Gail had listed them on the front whiteboard, the group decided to narrow in on a specific topic for the research lesson. Gail brought up the idea of splitting the group in two to write two consecutive lessons, as two one-hour pieces of a large block. Aki shared that with two small groups writing two different lessons, you work more productively; two lessons mean you could just observe and debrief one lesson and PLAN, observe and debrief another. Gail asked everyone to give an opinion so the group could make a decision.
Mary - liked idea of two lessons if and only if they are tied together
Allen suggested a goal could be to learn the most from each other as possible in this experience. Group seemed to agree that this was a big focus. Jeremy then added that perhaps the group should decide on a topic, then decide on whether one lesson or two would be more appropriate. Gail asked teachers to break into groups of two and three and decided on a focus. Groups met and discussed ideas for about ten minutes before coming together to share.
Mary, Amy and Megan: Focus on logical argument, with possible access points tied to basic trigonometric concepts.
Claudia, Alan and Ellie: Focusing on trigonometry is limiting - want to engage students in the learning process using indirect measurement of height
Jeremy, Rey and Joyce: not focus on trigonometry, could be more important foci
Gail summarized groups and heard "focus on a big idea," "traditional idea" and "underpinnings are important to consider." She reminded the group that the standards written on the board did ask for students to know significant trig concepts and skills. Reviewing the lesson plans, the group noted that Monday's plan focused on right triangle trigonometry and Tuesday's plan (the day they will teach) focused on special triangle trigonometry.
The group discussed what the law of sines is good for and why students should care. The group talked about the law of sines, right triangle trig and indirect measurement as possible foci. Jeremy asked Aki for clarification on our lesson study's goals and the group re-focused on the big picture. Mary, Rey and Gail shared definitions and experiences and the group settled on the idea of logical reasoning via indirect measurement as the main focus of the research lesson.
The group decided that each member will look at interesting scenarios involving using indirect measurement and that Thursday they will begin writing goals for the lesson.
DAY 3 (6/30/05)
The lesson study group began the meeting looking at resources about lesson study provided by Gail. Resources included an article from Research for Better Schools about lesson study, featuring Akihiko Takahashi and an article written by Catherine Lewis from the Nagoya Journal of Education and Human Development. Gail also brought two books about lesson study, one of which described a group of Japanese and American educators sharing forms of professional development and discussing best practice. Gail wanted to make clear what the lesson should "look" like. "The lesson is not made out of "here's three sample problems...you give some kind of problem or task that you want to engage them in." She emphasized that the lesson should be built around an "interesting challenge."
Gail showed the group a common framework for lesson plans, which consisted of a table. Along the left-hand side were the steps of a lesson: posing a problem, student's problem solving on their own, whole-class discussion, summing up and possibly an exercise or extension. Along the tops of the columns were sections for the main learning activities, anticipated students' responses and remarks on teaching. Gail emphasized that anticipating student responses is one of the most important pieces of lesson study.
The group focused on making a concept map about "Reasoning." Allen recorded as a group members brainstormed what students should be able to do with regard to reasoning and problem-solving skills (see below). The group then moved to the mathematical understanding that should come out of the lesson, assuming these reasoning skills will be woven into the actual lesson plan.
Aki suggested that the group needed to find a way to connect student's prior knowledge of similar triangles to a basic understanding of trigonometry. Gail added that since the students will have seen basic trig functions prior to the lesson, that possibly the connection could be between similar triangles and the law of sines. Mary suggested the big question of the lesson could be What is the connection between similar triangles and trigonometry? Jeremy offered an answer to the question by saying "similar triangles have proportional sides. Those sides will then make equal ratios. Sin, cos and tan are those ratios."
Gail then asked the group how students are going to understand what trigonometry has to do with non-right triangles, as the law of sines will not have been introduced yet. Allen continued to suggest that perhaps the lesson could ask "why doesn't right triangle trigonometry work for all triangles?
Ellie talked about how some students might make side to side ratios of one triangle, while others might make side to side ratios of two similar triangles, and that these are two very different ways to look at the basic proportions. This led to a group discussion about the meaning of these ratios, specifically misconceptions students might have in moving from right triangle trig to non-right triangle trig. The lesson plan for the day before the group's lesson was re-examined and the lesson seemed "traditional" in that right triangle trigonometry would be covered using the mnemonic "SOHCAHTOA" and some practice problems.
The group decided that the big question of how similar triangles and trigonometry are related will be the focus for the lesson study. The group decided that as homework, each member would look into problems that get at this question and arouse the student's curiosity at the same time.
DAY 4 (7/1/05)
Kathy Kanim joined the group for the meeting and described her experience brining lesson study to her district in New Mexico last year. Gail summarized the previous meeting by describing where the group had gotten "stuck" the day before. For example, she reminded the group that Ellie had brought up how students write ratios very differently, and that this will affect the transition from these ratios to proportions to trigonometry.
Claudia relayed how difficult choosing a problem will be and Joyce added that this is a very important big idea. Allen suggested asking the students Why does a non-right triangle not work with the right triangle trig functions? Or How can we apply the right triangle trig function to a non-right triangle? Jeremy said that this would be too hard to answer, especially with students the group doesn't know. Amy continued to suggest the question could be CAN you use right triangle trig functions for any kind of triangle? Jeremy emphasized his previous point, explaining that this is a big idea that might be too hard for any student.
Teachers began sharing specific questions they had found the previous night:
The group all focused on doing the math of Rey's problem. Teachers worked individually and in pairs and put up answers on the board. Once all members understood the math behind the problem, the group began discussing the math that the problem evokes.
Though everyone agreed the problem was a good one in and of itself, they questioned the usefulness of it in addressing the big question of how similarity and trigonometry are related. Gail asked if there are an infinite number of triangles that would work with perimeters other than 1200. Kathy thought it would be just as interesting to look at the triangles with perimeters of 1200 that don't work. Joyce, in looking at the textbook the students would be using, relayed that the chapter on similarity comes at the end of the book and that the students might not have that background going into the lesson. Amy and Aki wondered how this problem would get us to non-right triangles. Mary responded by saying that the same process would be used for non-right triangles to find missing sides or angles (i.e. the law of sines).
Aki suggested posing a problem where at least two solutions involve using similarity and using trig, respectively. He relayed that he thought this might be a good way to connect the two. He continued to say "if you change the area ratio so that the area of one triangle is one-third of the other it becomes a 30-60-90 triangle," which he suggested might provide more access into the problem. Allen drew a diagram on the board to illustrate the new idea. Gail finally said that to get to the trigonometry, however, an angle needs to be given off the bat in some way.
The group re-focused on Ellie's problem, which asked students the following: You are building a house on an island. There is a utility tower on the other side of the river [creating the island]. The installation cost for electric service is based on the distance from the utility pole. The county surveyors have provided the location where the angle to the utility pole is 90 degrees and another location 30 meters away (along the same line), where the angle to the tower is 48 degrees. Where should I build the house to minimize the distance to the tower? If the installation cost is $1.50 per meter, how much would the installation charge be?
Ellie drew a diagram on the board similar to the following but the group decided the problem would hit trigonometry via indirect measurement, but not similar triangles. A few other teachers gave suggestions, but none of the problems hit both targets. The group decided to re-examine Rey's problem and focus on Aki's editing. Specifically, the group decided more research needed to be done around finding a way to make Rey's problem solvable using similarity or trigonometry.
A new problem was put on the board showing half a treasure map. The problem asked you to locate a treasure on the part of the map that is missing, specifically how far the treasure is from a point on the piece of the map given. Teachers decided that students could be asked which angles or distance measurements could be taken away for the problem to still be solvable. Aki reminded the group that students should not only think about the fact that there are two different solving methods, but also about which method is more appropriate and why.
Rey suggested a new look at the problem and the group split into three small groups. Each small group worked on fleshing out either the treasure map problem or a version of it.
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.