ing, understanding, and practicing methods; trying to learn and use concepts that look easy to adults but are challenging for children (e.g., place value); and gradually increasing their mathematical proficiency by continually seeking to make sense of number and numerical operations.
A few basic themes are critical. First, students’ progress viewed from a distance is marked by a kind of gradualness and continuity, but viewed up close it appears uneven and varied. At any given moment, students know and use a range of computation methods that may vary according to the numbers in the problem, the problem situation, and other individual and classroom variables. A student may use different methods even on very similar problems, and any new method competes for a long time with older methods and may not be used consistently. In general, however, students steadily extend methods they understand to solve a larger variety of problems, and they shape current methods into more efficient ones.
A second theme is the many ways in which the strands of proficiency can be interwoven. Initially, in classrooms focused on understanding, students’ conceptual understanding and procedural fluency are tightly connected— students use only methods they understand. Later, their learning in one strand boosts their progress in the others. As students become more fluent with multidigit algorithms, their understanding and use of the place-value notational system become more robust. As their reasoning about multidigit numbers and place-value concepts improves, they make sense of more efficient multidigit algorithms. Students also actively choose among different procedures and representations. In so doing, they strengthen their strategic knowledge and their conceptual understanding of the procedures and the representations. Not only is mathematical proficiency multidimensional, but also the path to proficient performance requires progress along each strand interactively.
A third and final theme is that there are some identifiable patterns in the development of students’ proficiency as long as the strands are allowed to develop together in mutual dependence. Students begin their study of number situations by modeling problems directly, using the context to shape their concrete and often cumbersome methods. They gradually move toward representing problems more abstractly. They apply methods that are less transparent and more embedded, abbreviated, and independent of the problem. These methods are more sophisticated mathematically, use structural properties such as commutativity, and use the place-value symbolic notation in productive ways. As students begin multidigit arithmetic, it is vital that teachers and classrooms provide support for all to build understanding of