concepts, are fluent in performing basic operations, exercise a repertoire of strategic knowledge, reason clearly and flexibly, and maintain a positive outlook toward mathematics. Moreover, they possess and use these strands of mathematical proficiency in an integrated manner, so that each reinforces the others. It takes time for proficiency to develop fully, but in every grade in school students can demonstrate mathematical proficiency in some form. In this report we have concentrated on those ideas about number that are devel-oped in grades pre-K through 8. We must stress, however, that proficiency spans all parts of school mathematics and that it can and should be developed every year that students are in school.

In every grade in school, students can demonstrate mathematical proficiency in some form.

All young Americans must learn to think mathematically, and they must think mathematically to learn. We have elaborated on what such learning and thinking entail by proposing five strands of mathematical proficiency to be developed in school. The overriding premise of our work is that throughout the grades from pre-K through 8 all students can and should be mathematically proficient. That means they understand mathematical ideas, compute fluently, solve problems, and engage in logical reasoning. They believe they can make sense out of mathematics and can use it to make sense out of things in their world. For them mathematics is personal and is important to their future.

School mathematics in the United States does not now enable most students to develop the strands of mathematical proficiency in a sound fashion. Proficiency for all demands that fundamental changes be made concurrently in curriculum, instructional materials, classroom practice, teacher preparation, and professional development. These changes will require continuing, coordinated action on the part of policy makers, teacher educators, teachers, and parents. Although some readers may feel that substantial advances are already being made in reforming mathematics teaching and learning, we find real progress toward mathematical proficiency to be woefully inadequate. These observations led us to five general recommendations regarding mathematical proficiency that reflect our vision for school mathematics.

School mathematics in the United States does not now enable most students to develop the strands of mathematical proficiency in a sound fashion.

  • The integrated and balanced development of all five strands of math ematical proficiency should guide the teaching and learning of school math ematics. Instruction should not be based on extreme positions that students learn, on the one hand, solely by internalizing what a teacher or book says or, on the other hand, solely by inventing mathematics on their own.

  • Teachers’ professional development should be high quality, sustained, and systematically designed and deployed to help all students develop math-



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