In the United States, and to some extent in Germany, teachers usually teach a math lesson by explaining a topic and demonstrating a procedure. Then the students practice solving problems while the teacher goes around the room helping those who are having trouble. Problems that are not done by the end of class are often assigned as homework. In Japan, students spend less time practicing routine procedures and more time analyzing and proving than their American counterparts. A typical lesson in Japan focuses on just one or two carefully selected problems, and forms a complete story with a beginning, middle, and end.
Students work on a challenging problem for part of the class period, then share their solutions.
The Japanese teacher uses lecture and demonstration to highlight important aspects of the students' solutions or to show another solution. Before the end of class, the teacher summarizes the main point of the day. This is not to suggest that Japanese teachers are less active or directive than United States or German teachers. After allowing students time to struggle with challenging problems, they often follow up with direct explanations and summaries of what students have learned. Seventy-one percent of Japanese lessons contained at least some lecturing, compared with only about 15 percent of U.S. and German lessons.