Park City Mathematics Institute
Secondary School Teacher Program

Reflection on Practice Class: Day 3
Akihiko Takahashi

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.

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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.