helpful suggestions. LONG-TERM DEVELOPMENT OF MATHEMATICAL THINKING AND LESSON STUDY

In recent years, international comparisons have revealed wide differences in mathematical performance around the world, generating a general concern for improving mathematics teaching and learning. Yet, while teachers strive to improve performances on tests, there is a growing realization that in many cases this leads to the teaching of procedures to increase test scores rather than encouraging a flexible understanding of mathematics for wider use. The development of comparisons that focus on a broader range of competencies, such as PISA, reveal continuing international differences. In Japan, where students have consistently performed well on international assessments, teachers have collaboratively developed ways to foster students ’ flexible understanding of mathematics. Lesson study has been used in Japan for over a century to improve the preparation of a lesson or sequence of lessons, to predict how the children will react, to have a substantial number of observers watch the lesson, to review and improve the lesson plan in cycles to share with others. The lessons are based on a problem-solving approach in which teachers educate children to think fo