Role of practice. To many students, practice is as much a part of studying mathematics as of playing a sport or a musical instrument. The role of practice in mathematics, as in sports or music, is to be able to execute procedures automatically without conscious thought. That is, a procedure is practiced over and over until so-called automaticity is attained.47

The role of practice in mathematics, as in sports or music, is to be able to execute procedures automatically without conscious thought.

There are cognitive benefits to automatization. The more automatically a procedure can be executed, the less mental effort is required. Since each person has a limited amount of mental effort that he or she can expend at any one time, more complex tasks can be done well only when some of the subtasks are automatic.48 Hence, the automatization of mathematical procedures is justifiable when those procedures are regularly required to complete other tasks. For example, basic multiplication combinations such as 4×6=24 and 6×7=42 are needed for estimation, multidigit multiplication, single-digit division, multidigit division, and addition and multiplication of fractions, to name a few. Therefore, multiplication combinations need to be practiced until they can be produced quickly and effortlessly. The availability of calculators and computers raises the question of which mathematical procedures today need to be practiced to the point of automatization. Single-digit whole number addition, subtraction, multiplication, and division certainly need to be automatic, since they are used in almost all other numerical procedures. Opinions vary, however, as to which other procedures should be made automatic.

Kinds of practice. Textbook and worksheet exercises offer the most common kinds of practice used in U.S. mathematics classrooms. Such exercises are used to provide students frequent and repeated opportunities to practice what they have learned. Often the practice is directly associated with the topic of the lesson, with the teacher or other students providing assistance until the student can perform independently. Another approach distributes the practice over a longer period: On any one day, only a few of the exercises assigned might address the lesson topic, and the rest would address topics studied earlier in the year. Such distributed practice is based on the principle that mastery is achieved gradually and once achieved is maintained through regular practice. A number of studies of the U.S. curriculum have concluded that it is too repetitive.49 These criticisms are about topics being retaught year after year, not about students practicing learned concepts and procedures throughout the year to improve efficiency and retention. Ms. Lawrence’s assignment of a mixture of problems is presumably no acci-

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