of learning, and methods of teaching need to be developed and evaluated to see whether prospective and practicing teachers from such programs can draw appropriate connections and apply the knowledge they have acquired to teach mathematics effectively.
The second basic component of teaching proficiency is the development of instructional routines. Just as students who have acquired procedural fluency can perform calculations with numbers efficiently, accurately, and flexibly with minimal effort, teachers who have acquired a repertoire of instructional routines can readily draw upon them as they interact with students in teaching mathematics. Some routines concern classroom management, such as how to get the class started each day and procedures for correcting and collecting homework. Other routines are more grounded in mathematical activity. For example, teachers need to know how to respond to a student who gives an answer the teacher does not understand or who demonstrates a serious misconception. They need to know how to deal with students who lack critical prerequisite skills for the day’s lesson. Teachers need businesslike ways of dealing with situations like these that occur on a regular basis so that they can devote more of their attention to the more serious issues facing them. When teachers have several ways of approaching teaching problems, they can try a different approach if one does not work.
Researchers have shown that expert teachers have a large repertoire of routines at their disposal.29 They can choose among a number of approaches for teaching a given topic or responding to a situation that arises in their classes. Novice teachers, in contrast, have a limited range of routines and often cannot respond appropriately to situations. Expert teachers not only have access to a range of routines, they also can apply them flexibly, know when they are appropriate, and can adapt them to fit different situations.
The third component of teaching proficiency is strategic competence. Although teachers need a range of routines, teaching is very much a problem-solving activity.30 Like other professionals, teachers are constantly faced with decisions in planning instruction, implementing those plans, and interacting with students.31 Useful guidelines are seldom available for figuring out what to teach when, how to teach it, how to adapt material so that it is appropriate for a given group of students, or how much time to allow for an activity. On