be obvious that much additional research will be needed to fill out the picture, and we have recommended some directions for that research to take. The remaining recommendations reflect our consensus that the relevant data and theory are sufficiently persuasive to warrant movement in the direction indicated, with the proviso that more evidence will need to be collected along the way.

Information is now becoming available as to the effects on students’ learning in new curriculum programs in mathematics that are different from those programs common today. Over the coming years, the volume of that information is certain to increase. The community of people concerned with mathematics education will need to pay continued attention to studies of the effectiveness of new programs and will need to examine the available data carefully. In writing this report we were able to use few such studies because they were just beginning to be published. We expect them collectively to provide valuable information that will warrant careful review at a later date by a committee like ours.

Our report has concentrated on learning about numbers, their properties, and operations on them. Although number is the centerpiece of pre-K to grade 8 mathematics, it is not the whole story, as we have noted more than once. Our reading of the scholarly literature on number, together with our experience as teachers, creators, and users of mathematics, has yielded observations that might be applied to other components of school mathematics such as measurement, geometry, algebra, probability, and data analysis. Number is used in learning concepts and processes from all these domains.

Below we present some comprehensive recommendations concerning mathematical proficiency that cut across all domains of policy, practice, and research. Then we propose changes needed in the curriculum if students are to develop mathematical proficiency, and we offer some recommendations for instruction. Finally, we discuss teacher preparation and professional development related to mathematics teaching, setting out recommendations designed to help teachers be more proficient in their work.

Mathematical Proficiency

As a goal of instruction, mathematical proficiency provides a better way to think about mathematics learning than narrower views that leave out key features of what it means to know and be able to do mathematics. Mathematical proficiency, as defined in chapter 4, implies expertise in handling mathematical ideas. Students with mathematical proficiency understand basic

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