In the reflection and revision phase of the adapted Lesson Study cycle, the teachers participated in video clubs (Sherin and Han, 2004) during study group sessions where they watched and discussed short video clips (1-4 minutes) from each other's enactments of the lesson. 

• The research team selected the video clips to be shown during the study group sessions using the criteria developed by Sherin, Linsenmeier, and van Es (2009).   This framework rates the windows into, depth of, and clarity of student thinking displayed in the video. 
• The research team created the transcript of the students' interactions and added captions to the videos.
• The research team made copies of the students' work to distribute during the video club.
• The research team viewed the video while discussing the students' thinking in relation to the mathematical task. 
• Through this discussion and sharing their own experiences implementing the lesson, the teachers were able to investigate how students' approached the task and in what ways the task engaged their prior knowledge. 
• Teacher proposed edits and adjustments to the lesson based on their observations.
• The project had teachers engage in a Lesson Study Cycle for each concept twice over two years.  After the first cycle of reflections and revisions based on what was observed during the video clubs as well as their own experiences teaching the lesson, the teachers proposed revisions to the lesson for the second lesson implementation.  In the second year, teachers participated in video clubs using videos from the implementation of the revised lesson and discussed changes to their lesson based on their observations of students' thinking.

Based on their noticing of students' thinking and prior knowledge during the video clubs, the teachers identified three ways that they wanted to modify the original student worksheet for the Fair Location Problem. 

• The number of possible "fair" locations

  • Three locations - Teachers observed that students would find the midpoint and then one additional point, and then use symmetry to copy their point to the other side of the midpoint. This solution method allowed for students to use their spatial intuition to find three possible fair locations, but it also allowed students to not heavily rely on measuring, which may make them miss that all points on the perpendicular bisector are equidistant to the two schools.  Student Worksheet Option 1 (original); Student Worksheet Option 2
     
  • Four locations - Based on their observations about how students found three points, some teachers opted to have students locate four fair locations.  This change forced students that relied on symmetry to find an additional point for a fair location, which may make them to move to using a ruler.  In the second year of lesson implementation, teachers learned that some students still relied on symmetry, but also noted that it seemed like less students used symmetry when finding four locations as opposed to three.   Student Worksheet Option 3; Student Worksheet Option 4; Student Worksheet Option 5
     

• The diagram

  • Rectangular frame around diagram - Teachers observed that students were using the border to orient their ruler, "squaring" up their ruler with the side of the diagram to find the perpendicular bisector.  Student Worksheet Option 1 (original); Student Worksheet Option 2; Student Worksheet Option 3
     
  • Irregular frame around diagram -  By using an irregular frame, students were forced students to not longer rely on the orientation of the border of the diagram.  Student Worksheet Option 4
     
  • Actual map of town - Teachers wanted students to be able to use their prior knowledge of the town, such as the streets and roads, as an entry point into the problem.   Student Worksheet Option 5

• The summary

  • Fill-in-the-blank summary-  After noticing in the video clubs that many students did not take notes of the class summary during the first years' implementation, the teachers decided to create a fill-in-the-blank statement that would prompt students to formally summarize the lesson.  While more students did complete this statement in this year, after the second implementation, the teachers wondered if the statement was too leading for the students. Student Worksheet Option 2; Student Worksheet Option 4; Student Worksheet Option 5

  • Word bank summary-  Similarly to the reasoning for the fill-in-the-blank summary, one teacher opted to use a word bank to prompt students to develop the summarizing statement on their own.  Upon reflection in the second year, while the teachers appreciated the reinforcement of vocabulary, they were concerned that it was too rigid, only allowing for the correct mathematical term instead of synonyms that would me the same thing (such as a student saying that the lines are a "right angle" instead of "perpendicular"). Student Worksheet Option 3

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