Implementing Lesson Study Summary

Monday - Friday, July 2 - 6, 2007



Gail Burrill
Joyce Frost
Rudy Penczer
Anastasia Rodriguez
Jim Town
Oscar Saenz
Judy Lauper
Gloria Brooks
Connie Sherk
Rosemary Bertucci

We began with a presentation from Gail on the essence of Lesson Study. In summary, there are four main components to the lesson study. An introduction includes a "hook", ties in experiences of students to activate their learning through a question. The second part of the process is identifying the key question followed by student solutions. Finally, a lesson study involves summing up the lesson with connections and highlights of the lesson.

In this short period of three weeks, we will be completing a lesson study from start to at least the first revision of the lesson. We will work on the pre-lesson preparation during the first two weeks. In the second week, a group of "test subjects" will be "taught" our lesson as a trial run for revision before traveling to Park City High School, where we will be teaching our lesson to high school geometry students.

Important considerations are the characteristics of the lesson. How will it be delivered? Will it include the use of the blackboard? What sort of language and word choice will be used? Consider taking care with symbols used.

Our first main goal will be to flush out the pre-lesson process, which includes defining the goal, mathematical objective, relation to curriculum, and place in the lesson. To begin, we brainstormed topics relating to geometry and others more general. Following are the topics we discussed. Before we left, we narrowed the topics to a few that related to area, logic, proof, communication of ideas, real world application, and explanation.

Brainstorming Session for Topic of Lesson Study

Logic (Converse)
Similarity ~ Congruence
Special Triangles (30,60,90)
Area Model - Fractions
Angles - Alternate Interior
Application of Geometry Outside World
Area of Irregular Figures
Drawings not to scale (diagramming)
3D Geometry
Deductive Reasoning vs. Inductive Reasoning
Systems of Equations
Numbers into Words (Communicating math ideas)
Students Explaining themselves
Words, Numbers, and Pictures
Calculator to find roots
Write down - not working out
Thinking Skills
Specific Ideas
Area of Irregular Figures
PCMI on the hill, breaking up the area, purchasing materials to cover it
Carpet a staircase (and 3D geometry)
3D geometry
Cubic figure created by students, they are given a base and build upon it
    Proof - logic statement about how they created their figures     Surface area vs. volume


Participants: Gail, Joyce, Gloria, Oscar, Jim, Judy, Rudy, Rosemary, Anastasia, Connie

In continuation from yesterday's brainstorming session, we began with further exploration of the geometry topic and problem that we will explore for our lesson study. The purpose today is to come up with a defined objective for our lesson. What will the students be able to do? What will they learn? Several problem ideas were discussed, with an emphasis on context and mathematical objectives.

Proposed Objective
How to use areas of shapes you know to figure out shapes you don't know.

We want a lesson that contributes in a substantive way to advancing their knowledge. Think of this problem as the opener or closing problem. We broke up into smaller groups to develop our ideas of what the lesson might look like. Each group shared about the 20 minute break out session.

Group ideas:
Introduce very open-ended shapes that can be broken down differently. Generalize the discussion to keep students away from numbers and more focused on the idea of "how to use shapes you know to find area of shapes you don't know."

Hook - Comparing shadow shapes broken down into tangrams. Can include the school mascot and the rival school mascot. Which shape has the biggest area?
Question for discussion: What geometric ideas of formulas would be helpful in finding the areas of these shapes?
Students share solutions with the group.
Introduce 3 circle problem.... Have the same discussion.

Transitional activity, areas of several shapes (triangle, rectangle, trapezoid, circle)
Approach would be to create a figure on the shape. Build it from the elements, adding shapes on to whatever is drawn on the board. (What about using paper cutouts of shapes and adding them on to whatever was posted on the board already.)

To think about:
- Dropping the straws in the cup. Very fun, interesting, and engaging way to get involved in the problem.
- Narrow in on a direction on Thursday.



The Hook. What are we looking to do in our hook? Is the hook more appropriate to our objective as a geometric shape that will be broken apart by groups, or is the geometric probability problem using cups and straws a better hook? If we use the probability hook, should we revisit our objective. Would the new objective involve geometric probability, with area being a subgoal and a stepping stone to find the probability?

Our geometry students:
Course starts on July 9th. The geometry course is exactly 13 days long with 3-hour class each day. We present on July 17th, 10:00-11:00AM. Class size is approximately 14 students.
Class will have covered the Pythagorean theorem. It is a very compact course, with little extra time for proofs, group work, and constructions.

Our direction so far is sort of a synthesis of several geometric concepts and objectives.


  1. How to use area of shapes you know to find area of shapes you don't know
  2. Teaching geometric probability.
  3. Which arrangement of the four cans/cups gives the least open closed space?
  4. Which allows you to "pack" more?

Concern: Will students know enough about circles for the purpose of the lesson? What are the tools we will have in the classroom?

Possible Anticipated Student Responses:

  • Create a triangle in the space between the circles, without even using the circles.
  • Find the area of the circles first.
  • With centers, connect the dots.
  • Draw a triangle on the outside of all three circles.
  • Draw a circle around all three circles.

Discussion points:

  • What part of the cup is being measured? Distinguish between top and bottom. The straws are dropping in through the top. Visualize that plane and entry point. Imagine cylinders or cans.
  • Grouping - pairs. Students are unaccustomed to cooperative grouping in this course. They are with a new teacher for one hour.

New directions:
What about presenting materials of different sizes?
Give them a perspective, like the game guy at the county fair that wants you to lose. You are considering a redesign for your game.

Closing thoughts:
As a probability question:
How likely is it that a straw will fall into the space between the cups?
As an area question:
Which arrangement has the least space? The most space?
As a contextual problem:
How does the arrangement affect the probability? How would you arrange this game as a player? How would you arrange the cups as the host of the game?

Maximize: Contestant of games with cups
Minimize: Packaging materials and space between

TSW be able to apply areas of common shapes to find areas of weird shapes. (weird and not irregular... composite shapes)

If we continue with probability, will we want to incorporate the experimental probability?

Idea for researching this problem: Use the problem of the four cups as a five minute short. Observe reactions of teachers as part of our anticipating student response.


Participants: Gail, Joyce, Jim, Anastasia, Judy, Oscar, Rudy, Gloria, Connie, Rosemary

This morning, our problem was presented as a morning short to the PCMI SSTP group of teachers. The problem was presented with modeling of four cups and then participants were asked to figure the probability using trials or other methods with four cups. Cups stacked at tables were used by some groups. Our working group members saw many groups start with different arrangements, some with a square configuration and some with two spaces and center cups touching.

How will we lead them to four cups? Do we want to show them the three cups first, or will this sway their responses?

Assumption has to be made about the problem, such as with probability. What is included as part of the board? Perhaps establish a point system, with a certain point for the cups, and a certain point for the inner space, leaving the outer space valued at zero.

Goal in the next three days:
Develop the lesson itself.

  • Launch - Key Question. How will we do this? Cups? How many? What do they get? What question is posed? Straws?
  • Student Responses and Solutions. What strategies might they use? What might they misunderstand? What would kids say/do?
  • Summing Up - Closure. How do we want to bring it together? What kind of closure? Could this be an activity?

We discussed the questions from the previous day, analyzing their wording and the order we may consider incorporating them into the lesson. Two main questions we debated were:

  • What is the question we are asking?
  • What is the mathematics involved?

To approach the task of developing our lesson, we worked on large chart paper to develop solutions to the problem we are going to present. Exploration includes solutions using three, four, or six cups.

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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.