assigned, and students complete exercises like those they have been shown. The teacher often ends the lesson by checking some of the seatwork problems and assigning similar problems for homework.

Typical lessons in Germany and Japan contain many of the same components, but the components are arranged differently and aim at different goals. For example, most lessons in all three countries include an early segment in which the teacher presents one or more problems for the day. But that activity has a different purpose in each country. In Germany, presenting the problem initiates a relatively lengthy development of advanced solution techniques. The teacher guides, through questioning, the process of solving the problem, which is often quite challenging. In Japan, presenting the carefully chosen problem sets the stage for the students to work, individually and in groups, on developing solution procedures that they then report to the class. About half the time, the procedures are expected to be original constructions. As described above, presenting problems in the United States leads to students practicing procedures that have been demonstrated by the teacher.

The different patterns of teaching generated a set of findings that illustrated the dramatic differences in classroom practice across the three countries. For example, 78% of the mathematical topics in the U.S. lessons contain concepts that were stated by the teacher rather than developed through examples or explanations. In contrast, that practice occurred for 23% of the concepts in Germany and only 17% in Japan; at least some of the concepts from the remaining topics in these countries were developed and elaborated in some way.75 Moreover, the quality of the mathematical content of the U.S. lessons was independently rated as being much lower than that of the German and Japanese lessons.76

The descriptions from the TIMSS Video Study match other reports of classroom practice in mathematics. For example, a 1998 report to the California State Board of Education summarizes the conventional method of mathematics teaching in the United States, often used as the control treatment in experimental studies of new teaching approaches.77 The summary divides the conventional method into two phases. In the first phase, the teacher demonstrates, often working one to four problems, and the students observe passively; in the second phase, the students work independently, with the teacher possibly monitoring their work and giving feedback.

That description might easily have been written to describe U.S. mathematics lessons in 1900. Mathematics teaching in the United States clearly has not changed a great deal in a century. It continues to emphasize the



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