Aki started off today's class by noting: "Some of you have had
difficulty with the assignment from yesterday. This shouldn't be too
surprising. These difficulties are similar to what your students are
seeing."

"With parallelograms the opposite sides stay the same length as it is
shifted. The angles change, but the relationships between them stay
the same."

"What type of questions do you want to raise with your students? Why
do some of the properties stay the same? Why do some properties
change? Are the students synthesizing -- seeing the big picture? What
is the big picture? If you look at their solutions individually, you
might not be able to see the generalizations."

"The rectangle is a special case of a parallelogram. That's why
properties stay the same. To find the area of a rectangle -- just
length times width, but with parallelogram it is base times height.
Maybe we can use base times height to find the area of a rectangle?
The rectangle is a special case of parallelogram. In the rectangle
case, the height and width are the same."

"The teacher's job is to help students see the relationships between
these ideas. We need to be very careful when we design a lesson to
predict what students might come up with. Traditionally teachers
talk, talk, talk. 'Teacher, teacher, teacher, teacher, student.' It
should be 'student, student, student, student, teacher.' If you
really want the students to speak up, you need to stop talking. I
just want to let you struggle with the problem. If I talk, you just
listen. I'm very happy some of you are uncomfortable."

Aki showed us a page from a traditional American geometry textbook and compared it with a page on the same topic, area of a parallelogram, from a Japanese 5th grade textbook:

The Japanese book was considerably smaller than the American one. The material was treated quite differently.

He handed out a diagram of a parallelogram drawn on a grid (where the
"height" was not inside the parallelogram) and asked us to figure out
different ways to find the area. Some used scissors to cut it up and
rearrange it. One wrote "main idea: subtract the area of the smaller
rectangle from the area of the larger rectangle to find the area of
the parallelogram."

Aki showed us pictures of students showing their solutions. These
students hadn't learned how to find the area of a triangle.

After Aki discussed a number of the methods, he said -- "Now: let's
think about a triangle." A number of different student driven methods
were described. One student wrote: "I think we can change the shape
into another shape that we already know how to find the area of"

Aki: "What happens if your students read the textbook before you teach
them? Actually, it never happens!"

Tomorrow we are going to divide into three groups -- to find the area
of a trapezoid.