teaching.33 The implications for teacher education and professional development is that teachers engage not only in learning methods of teaching but also in reflecting on them and justifying and explaining them in relation to such matters as the mathematics being taught, the goals for students, the conceptions and misconceptions that students have about the mathematics, the difficulties they have in learning it, and the representations that are most effective in communicating essential ideas.
One of the ways that the professional development programs described below foster teachers’ ability to justify and explain classroom practices is that teachers examine familiar artifacts from practice, and those artifacts help them focus their attention and develop a common language for discussion. In some cases the program leaders provide the artifacts; in others the artifacts come from the teachers’ classrooms. Teachers are often asked to pose a particular mathematical problem to their classes and to discuss the mathematical thinking that they observe.
The final component of teaching proficiency is a productive disposition about one’s own knowledge, practice, and learning. Just as students must develop a productive disposition toward mathematics such that they believe that mathematics makes sense and that they can figure it out, so too must teachers develop a similar productive disposition. Teachers should think that mathematics, their understanding of children’s thinking, and their teaching practices fit together to make sense and that they are capable of learning about mathematics, student mathematical thinking, and their own practice themselves by analyzing what goes on in their classes. Teachers whose learning becomes generative perceive themselves as in control of their own learning.34 They learn by listening to their students and by analyzing their teaching practices. Not only do they develop more elaborated conceptions of how students’ mathematical thinking develops by listening to their students, but they also learn mathematical concepts and strategies from their interactions with students. The teachers become more comfortable with mathematical ideas and ripe for a more systematic view of the subject.
Teachers whose learning becomes generative see themselves as lifelong learners who can learn from studying curriculum materials35 and from analyzing their practice and their interactions with students. Programs of teacher education and professional development that portray to the participants that they are in control of their own learning help teachers develop a productive dispo-