weak in applying mathematical skills to solve even simple problems (National Research Council, 2001c). These results have generally been attributed to the shallow and diffuse treatment of topics in elementary mathematics relative to other countries (National Research Council, 2001c).

The panel had the benefit of drawing on a recent synthesis of research on elementary mathematics (National Research Council, 2001c) and on the work of a RAND study group that produced a mathematics research agenda (RAND, 2002b). The National Research Council report presents a view of what elementary schoolchildren should know and be able to do in mathematics that draws on a solid research base in cognitive psychology and mathematics education. It includes mastery of procedures as a critical element of mathematics competence, but it places far more emphasis on conditional knowledge: understanding when and how to apply those procedures than is common in mathematics classrooms today. Conditional knowledge is rooted in a deeper understanding of mathematical concepts and a facility with mathematical reasoning. The NRC committee summarized its view in five intertwining strands that constitute mathematical proficiency (National Research Council, 2001c:5):

• Conceptual understanding: comprehension of mathematical concepts, operations, and relations;

• Procedural fluency: skill in carrying out procedures flexibly, accurately, efficiently, and appropriately;

• Strategic competence: ability to formulate, represent, and solve mathematical problems;

• Adaptive reasoning: capacity for logical thought, reflection, explanation, and justification;

• Productive disposition: habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

A well-articulated portrait of mathematical proficiency is an important first step; it provides a well-defined goal for mathematics instruction. But important questions remain regarding the allocation of time and attention to the separate strands, as well as the approach to instruction that best supports the proficiency goal.