change to improve mathematics learning during these critical years. The committee identifies five interdependent components of mathematical proficiency and describes how students develop this proficiency. With examples and illustrations, the book presents a portrait of mathematics learning: Research findings on what children know about numbers by the time they arrive in pre-K and the implications for mathematics instruction. Details on the processes by which students acquire mathematical proficiency with whole numbers, rational numbers, and integers, as well as beginning algebra, geometry, measurement, and probability and statistics. The committee discusses what is known from research about teaching for mathematics proficiency, focusing on the interactions between teachers and students around educational materials and how teachers develop proficiency in teaching mathematics. Table of ContentsFront MatterEXECUTIVE SUMMARY1 LOOKING AT MATHEMATICS AND LEARNING2 THE STATE OF SCHOOL MATHEMATICS IN THE UNITED STATES3 NUMBER: WHAT IS THERE TO KNOW?4 THE STRANDS OF MATHEMATICAL PROFICIENCY5 THE MATHEMATICAL KNOWLEDGE CHILDREN BRING TO SCHOOL6 DEVELOPING PROFICIENCY WITH WHOLE NUMBERS7 DEVELOPING PROFICIENCY WITH OTHER NUMBERS8 DEVELOPING MATHEMATICAL PROFICIENCY BEYOND NUMBER9 TEACHING FOR MATHEMATICAL PROFICIENCY10 DEVELOPING PROFICIENCY IN TEACHING MATHEMATICS11 CONCLUSIONS AND RECOMMENDATIONSBIOGRAPHICAL SKETCHES

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