The difficulty of “length × width”: Is a square the unit of measurement?

In individual interviews, 220 students in grades 4, 6, 8, and 9 were given one task, and 72 eighth graders were given three tasks to answer two questions: (a) Is a square the unit of measurement for an area for students in grades 4–8? and (b) Does a square have a space-covering characteristic for students in grade 8? The answers to both questions were No (except for eighth (and ninth) graders in advanced sections of mathematics). The difficulty of “length × width” is explained in light of Piaget's theory, and educational implications are discussed.