the spot, teachers need to find out what a student knows, choose how to respond to a student’s question or statement, and decide whether to follow a student’s idea. These are problems that every teacher faces every day, and most do not have readymade solutions.

Conceptual understanding of the knowledge required to teach for proficiency can help equip teachers to deal intelligently with these problems. It is misleading to claim that teachers actually solve such problems in the sense of solving a mathematical problem. There is never an ideal solution to the more difficult problems of teaching, but teachers can learn to contend with these problems in reasonable ways that take into account the mathematics that students are to learn; what their students understand and how they may best learn it; and representations, activities, and teaching practices that have proven most effective in teaching the mathematics in question or that have been effective in teaching related topics.

Teacher education and professional development programs that take into account the strategic decision making in teaching can help prepare teachers to be more effective in solving instructional problems. Rather than being designed to resolve teachers’ problems, programs of teacher education and professional development can engage prospective and practicing teachers in the analysis of instructional problems and potential ways of dealing with them. Teachers can learn to recognize that teaching involves solving problems and that they can address these problems in reasonable and intelligent ways.

Adaptive Reasoning

The fourth component of teaching proficiency is adaptive reasoning. Teachers can learn from their teaching by analyzing it: the difficulties their students have encountered in learning a particular topic; what the students have learned; how the students responded to particular representations, questions, and activities; and the like.32 Teachers can become reflective practitioners, and reflection is essential in improving their practice. The focus of teachers’ reflection and the tools they use shape the nature of that reflection and affect whether, what, and how they learn from it. Many successful programs of teacher education and professional development engage teachers in reflection, but the reflection, or perhaps more appropriately the analysis, is grounded in specific examples. In those programs, teachers engage in analyses in which they are asked to provide evidence to justify claims and assertions. As with other complex activities, teacher learning can be enhanced by making more visible the goals, assumptions, and decisions involved in the practice of

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