divorced from practice. Effective programs of teacher preparation and professional development cannot stop at simply engaging teachers in acquiring knowledge; they must challenge teachers to develop, apply, and analyze that knowledge in the context of their own classrooms so that knowledge and practice are integrated.

In chapter 4 we identified five components or strands of mathematical proficiency. From that perspective, successful learning is characterized by comprehension of ideas; ready access to skills and procedures; an ability to formulate and solve problems; a capacity to reflect on, evaluate, and adapt one’s knowledge; the ability to reason from what is known to what is wanted; and a habitual inclination to make sense of and value what is being learned. Teaching is a complex activity and, like other complex activities, can be conceived in terms of similar components. Just as mathematical proficiency itself involves interwoven strands, teaching for mathematical proficiency requires similarly interrelated components. In the context of teaching, proficiency requires:

• conceptual understanding of the core knowledge required in the practice of teaching;

• fluency in carrying out basic instructional routines;

• strategic competence in planning effective instruction and solving problems that arise during instruction;

• adaptive reasoning in justifying and explaining one’s instructional practices and in reflecting on those practices so as to improve them; and a

• productive disposition toward mathematics, teaching, learning, and the improvement of practice.

Like the strands of mathematical proficiency, these components of mathematical teaching proficiency are interrelated. In this chapter we discuss the problems entailed in developing a proficient command of teaching. In the previous section we discussed issues relative to the knowledge base needed to develop proficiency across all components. Now we turn to specific issues that arise in the context of the components.