ods; they are general, rapid, and sufficiently accurate that valuable school time might better be spent on topics other than mastery of the whole network of knowledge required for carrying out the level 3 methods. Decisions about which methods to teach must also take into account that some methods are clearer conceptually and procedurally than the multidigit methods usually taught in the United States (see Box 5-5). The National Research Council’s *Adding It Up* reviews these and other accessible algorithms in other domains.

This view of mathematics as involving different methods does not imply that a teacher or curriculum must teach multiple methods for every domain. However, alternative methods will frequently arise in a classroom, either because students bring them from home (e.g., Maria’s add-equal-quantities subtraction method, widely taught in other countries) or because students think differently about many mathematical problems. Frequently there are viable alternative methods for solving a problem, and discussing the advantages and disadvantages of each can facilitate flexibility and deep understanding of the mathematics involved. In some countries, teachers emphasize multiple solution methods and purposely give students problems that are conducive to such solutions, and students solve a problem in more than one way.

However, the less-advanced students in a classroom also need to be considered. It can be helpful for either a curriculum or teacher or such less-advanced students to select an accessible method that can be understood and is efficient enough for the future, and for these students to concentrate on learning that method and being able to explain it. Teachers in some countries do this while also facilitating problem solving with alternative methods.

Overall, knowing about student learning paths and knowledge networks helps teachers direct math talk along productive lines toward valued knowledge networks. Research in mathematics learning has uncovered important information on a number of typical learning paths and knowledge networks involved in acquiring knowledge about a variety of concepts in mathematics (see the next three chapters for examples).

To teach in a way that supports both conceptual understanding and procedural fluency requires that the primary concepts underlying an area of mathematics be clear to the teacher or become clear during the process of teaching for mathematical proficiency. Because mathematics has traditionally been taught with an emphasis on procedure, adults who were taught this way may initially have difficulty identifying or using the core conceptual understandings in a mathematics domain.