ABSTRACT
This chapter investigates how the teachers approached a mathematical idea that was new to them: the relationship between the perimeter and the area of a rectangle. The perimeter and area of a figure are two different measures. The perimeter is a measure of the length of the boundary of a figure, while the area is a measure of the size of the figure. The Chinese and the United States teachers’ responses differed in many ways. The perimeter of a rectangle can increase while two of the sides of the rectangle decrease in length. Eight teachers explored the various possible relationships between perimeter and area. The teachers explored the student’s claim and reached an understanding of the mathematical issues at various conceptual levels: finding a counter-example, identifying the possible relationships between area and perimeter, clarifying the conditions under which those relationships hold, and explaining the relationships.
