ematics and children’s thinking are set in a context that relates to their practice. The mathematical ideas and how children think about them are seen in classroom interactions. The problems discussed in the workshop are problems that the teachers can and do use in their classes; the interactions about mathematics that occur in the teachers’ classes provide a setting for workshop discussion of mathematical ideas and children’s thinking. The activities taking place in the workshop and in the teachers’ classrooms have the same goals. In both places the teachers engage in inquiry to gain a deeper understanding of mathematics, students’ thinking about that mathematics, and how to plan their instruction so as to foster the development of students’ mathematical thinking.

Before beginning a professional development program similar to the one described above, teachers participating in the program found that fewer than 10% of their students at any grade demonstrated a relational concept of equality. After one year of the program, the percentage of students in their classes who demonstrated a relational concept of equality ranged from 66% in first and second grades to 84% in sixth grade.43

Although these programs place a heavy emphasis on children’s thinking, understanding children’s mathematical thinking depends upon understanding the mathematics with which that thinking is engaged. The programs do not deal with general theories of learning. They concentrate instead on understanding children’s thinking in specific domains of mathematical content. Understanding the mathematics of the domain being studied is a prerequisite to understanding children’s thinking in that domain. For example, to understand the different strategies that children use to solve different problems, teachers must understand the semantic differences between problems represented by the same operation, as illustrated by the sharing and measurement examples of dividing cookies described above in Box 10–1. In programs focusing on children’s mathematical thinking, teachers learn to recognize and appreciate the mathematical significance of children’s informal methods for solving problems, how these methods evolve into more abstract and more powerful methods, and how the informal methods could serve as a basis for students to learn formal concepts and procedures with understanding.

Professional development programs focusing on helping teachers understand both the mathematics of specific content domains and students’ mathematical thinking in that domain have consistently been found to contribute to major changes in teachers’ instructional practices that have resulted in significant gains in students’ achievement.44 For example, in an experimental study of CGI with first-grade teachers, teachers who had taken a month-long

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