composite, comprehensive view of successful mathematics learning. This view, admittedly, represents no more than a single committee’s consensus. Yet our various backgrounds have led us to formulate, in a way that we hope others can and will accept, the goals toward which mathematics learning should be aimed. In this chapter, we describe the kinds of cognitive changes that we want to promote in children so that they can be successful in learning mathematics.

Recognizing that no term captures completely all aspects of expertise, competence, knowledge, and facility in mathematics, we have chosen mathematical proficiency to capture what we believe is necessary for anyone to learn mathematics successfully. Mathematical proficiency, as we see it, has five components, or strands:

  • 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.

These strands are not independent; they represent different aspects of a complex whole. Each is discussed in more detail below. The most important observation we make here, one stressed throughout this report, is that the five strands are interwoven and interdependent in the development of profi ciency in mathematics (see Box 4–1). Mathematical proficiency is not a one-dimensional trait, and it cannot be achieved by focusing on just one or two of these strands. In later chapters, we argue that helping children acquire mathematical proficiency calls for instructional programs that address all its strands. As they go from pre-kindergarten to eighth grade, all students should become increasingly proficient in mathematics. That proficiency should enable them to cope with the mathematical challenges of daily life and enable them to continue their study of mathematics in high school and beyond.

The five strands are interwoven and interdependent in the development of proficiency in mathematics.

The five strands provide a framework for discussing the knowledge, skills, abilities, and beliefs that constitute mathematical proficiency. This frame-



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