Japanese Lesson Study Summary

Friday, July 11, 2003

We had a busy and productive day. We first talked about release forms so that our lesson can be videotaped. We had found forms on the New Mexico site that Lisa had suggested, but that form didn't really seem to fit our situation. Jennifer had also found a form on the National Board Certification site. Gail looked over both of them and suggested modifying one that had been used this week for Deborah Ball's class of 5th graders. Here is Gail's revision:

Dear Parent:
This letter will provide you with information about the special mathematics class in which your child will be participating at the Park City Mathematics Institute on July 17. We are writing to ask your permission to use your child’s written work and the class videos in which your child may appear, to discuss teaching with teachers and other professionals. Please read this letter carefully.

We have designed the mathematics lesson to provide your child with an opportunity to learn some mathematics as well as to provide a professional learning opportunity for mathematics educators who will be attending the lesson. The class will be observed and documented - video and audio records, as well as copies of the students’ written work, will be made - to allow the teacher and the participating educators to assess the lesson.

This practice of collecting and studying records of classes is something many teachers routinely do. Such records allow for professional analysis that is important for improving teaching and learning. However, the work that is done in the class session can be a potentially invaluable source of information for other educators. With your permission, we would like to use the materials we collect in other professional education settings - to help teachers develop the quality of their own teaching practices.

The inclusion of your child's written work and her/his recorded images and sounds in these professional settings is entirely voluntary. Please note that your child will not be identified by full name to anyone who views or listens to these recordings; however, your child might be recognizable to someone who knows your child. Please note that your child's full name will not appear on any work that will be seen by people outside those actually observing the lesson.

In order to protect confidentially, pseudonyms will be used in place of real names in any published report or discussion about the lesson. Additionally, you are free to withdraw your consent at any time.

If you desire further information about these records and their use, you may call Gail Burrill at (435) 649-7100 or write email to burrill@msu.edu.

Permission for video, audio, and written records to be used for educational purposes:

____ I have read the above statement and agree for my child to be included in materials development projects by giving my permission for project staff to use the video, audio, and written information collected during the Park City Mathematics Institute for educational purposes. I understand that my child's full name will never be used in any materials that are developed and her/his identity will be treated with confidentiality to anyone outside the project staff. I also understand that my child may be recognizable to people who know my child. I understand that my child's inclusion in such projects is strictly voluntary. Finally. I understand that I may withdraw my permission for my child to be included such projects at any time without penalty, although the existing materials will stand.

____ I do not give permission for my child to be included in any materials development projects.

___________________________________________
Signature

____________________
Date

____________________________________________
Printed name

Next, we decided on what our roles as observers should be for both the practice lesson and for the model lesson on Thursday. We decided upon the following:

Teacher: Celeste
Moderator for debriefing: Gail 
·	How can we improve the questions and better anticipate student responses - 
Jennifer
·	Pick 3 students record everything they do - Tony
·	Time each section of the lesson  - Jerry
·	How did kids use the resources
·	When / how did the “math appear”?  When were the key statements made?- Joyce
·	Did we match the level of difficulty correctly? / How did students apply 
their previous knowledge - Jill

We decided to have Celeste teach the lesson (both she and Tony had volunteered)
 because Celeste really thought it would help to go observe the class on 
Wednesday afternoon and she was willing to do that (it's our afternoon off).  

We worked on the lesson some more and made some great progress.  

Japanese Lesson Study Lesson
Topic: Scaling, Measurement and Dimensionality – Stair-Step Fractal
I.	Overarching Goal: 
II.	Mathematical Objective- When considering the measurements length, area, and 
volume, students will be able to clearly articulate the effect of scaling one 
measurement on the two remaining measurements.
III.	How does this objective fit into a unit? Students will understand the 
relationship between dimensionality and scale factor
IV.	What is the pre-requisite knowledge?
How to find the area of a rectangle
How to find the volume of a rectangular solid
V.	Math problem – hook, problem on which they will work
A.	When students enter the class, they will be given printed directions 
(including a sketch) on how to construct the 1st iteration of the stair step 
fractal.  After students finish construction, then the teacher will address the 
class and say, “ You can make some amazing things with paper.  Mathematicians often 
make models to describe or understand mathematical concepts.  Is this amazing?   
Well, why not? So, what happens if we repeat the process?”  --Walk the students 
through the 2nd and 3rd iterations — Pop it up again?  Amazing?  What interesting 
things do you see and what do you wonder when you look at this figure?”
1.	Brainstorm individually before taking student answers.
    Potential responses:
ii.	It looks like stairs – Response: are all the stairs the same? 
iii.	I see squares / boxes
iv.	The stairs are getting bigger
v.	  The figure is symmetrical – Response” What type of symmetry? or how? 
vi.	It looks like how you make snowflakes
vii.	It’s a cool design. – What makes it cool?
viii.	It looks like a building – What type of building
ix.	It looks like stairs – what might these stairs be used to model?
x.	The little steps are half the size of the medium steps, medium steps are 
half of larger steps
xi.	There are small, medium, and large squares
xii.	All the shapes are similar and the same kind of shape
xiii.	There are different sized steps.
xiv.	It looks like a fractal.
xv.	The middle step is centered on the large step in the same way that the small 
step is centered on the middle step.
		
B.	Key Question: What relationships exist between the stairs in this model?  
(Individual work)
Find a partner and compile your conclusions in an organized way (In pairs).  Record 
your answers on butcher paper.  Be prepared to explain your answers and how you got 
them.
VI.	 Student strategies – How might students solve the problem?
(In teacher responses be sure to be clear about which dimensions students are 
describing)
A.	Students might use inspection as their strategy.  
i.	The teacher can then push them to explain, “ What do you mean by small, 
medium, or large?  Can you quantify that? Are they all rectangles?  Are there any 
squares?
ii.	Students may say 1 large, 2 medium, 4 small.  In this case the teacher 
should  affirm it but not lead students to believe that it is the key idea. 
B.	Students might measure side length and/or area and/or volume.
i.	If students do not mention all three, after discussing their observations, 
ask what measurement is missing.
C.	Units – the small one is how many of the medium etc.
VII.	Teacher responses – anticipate how you will respond to student questions.  
See above.
VIII.	Evaluation – what evidence will we have that kids understood?
A.	Present students with a model that is scaled by a factor of 3.  Have the 
students work independently to figure out how area and volume change in this case.  
Q: “The scale factor in this figure is different.  How do you think the scale factor 
in this model effects the area and volume?” With the students conclusions, create a 
table and ask students to fill in the row for scaling  by 3.  Deal with 
misconceptions ( the funny nature of 2).  In a whole class discussion, fill in the 
table for scale factors n > 3 and the general case.
**Note introduce the vocabulary of scale after students see the need – 5thng, 6thng 
etc.
**You can keep scaling up to the n case and then go to cases where n < 1
IX.	Summing up – use the kids words not a pre-written script.  (Otherwise they 
have no reason to do the activity.)
**Mention that this fractal can be used to solve the famous Tower of Hanoi problem.

Finally, we spent some time refining our homework questions with some great suggestions from Gail. Here they are.

Name ___________________________________
Homework
Geometry

  1. Describe examples of scaling you find in your world. You may want to consider examples from architecture, literature, toys, film, science, etc.
  2. There is a rule which states that your height at age 2 is ˝ your adult height.
    1. How would you expect your weight to change over the same period?
    2. If you want your weight to double from age 2 to adult, how would you expect your height to change? Does this seem reasonable?
  3. Suppose the figure you created in class is a set of stairs. How many people can fit on the stairs if 15 people fit on the top stair?
  4. Suppose that each dimension of a figure is scaled with a different scale factor. Without actually doing extensive computation, can you predict the relationship between the volume of the original figure and the scaled figure?

We plan on using the room near the main theater past the women's bathroom. We will be using butcher paper sheets to record student responses and will have a bag or box for Celeste to have models ready but out of sight. Joyce will prepare some cubes in case the students need help visualizing volume increasing by eight when the side length is doubled and increasing by 27 when the side length is tripled. We will be using 4 squares per inch graph paper which either Jill or Joyce will supply. Joyce will also round up enough scissors. Art has agreed to videotape both days and Jim says he has a video camera that we may use. We still need to find name tags for the students.

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This material is based upon work supported by the National Science Foundation under Grant No. 0314808.
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.