standing of students in the United States has resulted in increasing attention to the problems involved in teaching mathematics as a set of procedural competences.14 At the same time, students with too little knowledge of procedures do not become competent and efficient problem solvers. When instruction places too little emphasis on factual and procedural knowledge, the problem is not solved; it is only changed. Both are clearly critical.

Equally important, procedural knowledge and conceptual understandings must be closely linked. As the mathematics confronted by students becomes more complex through the school years, new knowledge and competencies require that those already mastered be brought to bear. Box 1-6 in Chapter 1, for example, describes a set of links in procedural and conceptual knowledge required to support the ability to do multidigit subtraction with regrouping—a topic encountered relatively early in elementary school. By the time a student begins algebra years later, the network of knowledge must include many new concepts and procedures (including those for rational number) that must be effectively linked and available to support new algebraic understandings. The teacher’s challenge, then, is to help students build and consolidate prerequisite competencies, understand new concepts in depth, and organize both concepts and competencies in a network of knowledge. Furthermore, teachers must provide sustained and then increasingly spaced opportunities to consolidate new understandings and procedures.

In mathematics, such networks of knowledge often are organized as learning paths from informal concrete methods to abbreviated, more general, and more abstract methods. Discussing multiple methods in the classroom—drawing attention to why different methods work and to the relative efficiency and reliability of each—can help provide a conceptual ladder that helps students move in a connected way from where they are to a more efficient and abstract approach. Students also can adopt or adapt an intermediate method with which they might feel more comfortable. Teachers can help students move at least to intermediate “good-enough” methods that can be understood and explained. Box 5-4 describes such a learning path for single-digit addition and subtraction that is seen worldwide. Teachers in some countries support students in moving through this learning path.

Developing Mathematical Proficiency

Developing mathematical proficiency requires that students master both the concepts and procedural skills needed to reason and solve problems effectively in a particular domain. Deciding which advanced methods all students should learn to attain proficiency is a policy matter involving judgments about how to use scarce instructional time. For example, the level 2 counting-on methods in Box 5-4 may be considered “good-enough” meth-



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