Japanese Lesson Study Summary
Tuesday, July 1, 2003
Each day Joyce will take notes and post them on the website. You can find these
under the section entitled "daily reports." You might also enjoy looking at the
daily reports from the previous two years [2001 and 2002] for the Japanese lesson study group.
Gail and Joyce presented a suggested schedule for our working group time. Group
members agreed that this plan seemed workable and agreeable to all. We
currently have eleven members in our group including Gail, four math supervisors
(only with us this week), and six high school math teachers.
Mon - Intro
Tue - Discussion on topic, lesson plan
Wed - Continue discussion of topic
Thu - Decide on topic and pair up
Mon - Make lesson plan
Tue - Make lesson plan
Thu - Make lesson plan
Fri - Make lesson plan
Mon - Trial run of lesson
Tue - Refine
Thu - Teach
Fri - Revise
Next, we discussed the lesson design process.
Understanding by design:
- What is your goal - choose the big essential question
- What student assessment will you use to capture understanding?
- Develop the activities to use with students.
Lesson from within the larger unit - what is the mathematical objective?
What prerequisite knowledge is required of the students?
Choose an engaging math problem - one with a hook, problem on which they will work
Student strategies - how will students do the problem?
Teacher responses - choose teacher responses for all anticipated student responses
Whole class discussion - how will this be done and what questions should be asked?
Summing up - make sure to use kids words so that the students work is validated
Evaluation - what evidence will we have that kids understood?
For our unit/lesson - what are some potential candidates? What is the important math
and why will we/students care? What kind of challenge will there be in this topic?
Some topics not interesting to everyone even though they are interesting
mathematically. Some topics are extremely rich mathematically and have good
connections to advanced math. Similarity is a good example of this.
What is our goal? What are we trying to do? Are we trying to learn something new,
experience the process, be able to use the lesson ourselves, or perhaps all of the
Some problems reintroduce over and over deepening and refining the concept.
Here are our top topics with rationale for doing them.
- Pythagorean theorem
Many different ways to prove it
Many famous people have proved it (including an ex-president and artist)
It has many uses, is taught in many places but is not always understood
At different levels "use what you know about the Pythagorean Theorem"
- Transformational geometry
This is left out, not done in many places, a great precursor for higher math
Often only touched lightly
Fun to teach, interesting applications (computer science and art)
Great as a Sketchpad lesson
- Congruence of triangles
Interesting to get at why the postulates exist, what works works and why
what doesn't work and why?
Relation between similar triangles and right triangle trig
Hard to get across where it comes from, sine is a number -
tangent is the ratio of opposite over adjacent
offers amazing connections with slope and higher math
used on unit circle, used to build a table, changed using TI92 to get sine
Where do these numbers come from when you type them into calculator?
It does not matter how big the triangle is or how small; same ratio holds up.
Tie things together - similar, congruent, triangles, properties, parallel
lines, putting it all together, need to spend time doing these things so they
don't memorize facts. Geometer's Sketchpad - what happens dynamically?
- What happens as you scale up measurements? (perimeter, area, volume?)
Reason - kids have problems with this. Not a lot of problems in text
Dimensionality? Tie into Pythagorean theorem.
Gives good questions and answers; can challenge students?
How big are some things really? A cubic yard of concrete, for example.
- Spatial geometry - can use parallelograms? Side views, etc. impossible figures
blind spots, lines of sight?
- Navigation - boats collide? Interesting and useful? Two islands in the distance -
which is farther away? Location, spherical geometry can tie to spatial geometry
similarity, trig, great circles
- Fractals - good topic for exploration, recursion to linearity, integration of
algebra and geometry, tower of Hanoi studied forever and through fractal geometry,
anyone can solve it similarity, iterations, recursions
THE BIG THREE (after our initial vote with three votes per person)
Similarity - notion of self similar
What does self similarity look like? How can we demonstrate it?
Symmetry - in lots of places in real life and in mathematics; good
projections in the future. Plays roles in group theory- some of the good
ideas involved here?
Jerry talked to Lars about our students for our lesson.
There are 10 students who are getting a credit for geometry by doing an entire
Geometry course in six weeks, 12:30 - 3:00 each day. These are the brightest and most
motivated students. They are using "Discovering Geometry, an Investigative
Approach from Key Curriculum," but will not be using computers. They have gone
through chapter 9 - Pythagorean Theorem. Test on Chapter 8 on Wednesday, the day
before our lesson. The next day they will start chapter 10 sections 1,2,3,4 (Volume,
the geometry of solids). They will finish the session with chapters 11 and 12.
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This material is based upon work supported by the National Science Foundation under Grant No. 0314808.
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