many students have learned mathematical knowledge—whether the rudiments of arithmetic computation or the complexities of geometric theorems— without much understanding.1 Of course, many students tried to make whatever sense they could of procedures such as adding common fractions or multiplying decimals. No doubt many students noticed underlying regularities in the computations they were asked to perform. Teachers who themselves were skilled in mathematics might have tried to explain those regularities. But mathematics learning has often been more a matter of memorizing than of understanding.

Today it is vital that young people understand the mathematics they are learning. Whether using computer graphics on the job or spreadsheets at home, people need to move fluently back and forth between graphs, tables of data, and formulas. To make good choices in the marketplace, they must know how to spot flaws in deductive and probabilistic reasoning as well as how to estimate the results of computations. In a society saturated with advanced technology, people will be called on more and more to evaluate the relevance and validity of calculations done by calculators and more sophisticated machines. Public policy issues of critical importance hinge on mathematical analyses.

The overriding premise of our work is that throughout the grades from pre-K through 8 all students should learn to think mathematically.

Citizens who cannot reason mathematically are cut off from whole realms of human endeavor. Innumeracy deprives them not only of opportunity but also of competence in everyday tasks. All young Americans must learn to think mathematically, and they must think mathematically to learn. The overriding premise of our work is that throughout the grades from pre-K through 8 all students should learn to think mathematically.

Helping all students learn to think mathematically is a new and ambitious goal, but the circumstances of modern life demand that society embrace it. Equal opportunity in education and in the workplace requires that mathematics be accessible to all learners. The growing technological sophistication of everyday life calls for universal facility with mathematics. For the United States to continue its technological leadership as a nation requires that more students pursue educational paths that enable them to become scientists, mathematicians, and engineers.

The research over the past two decades, much of which is synthesized in this report, convinces us that all students can learn to think mathematically. There are instances of schools scattered throughout the country in which a high percentage of students have high levels of achievement in mathematics. Further, there have also been special interventions in disadvantaged schools whereby students have made substantial progress. More is now known about



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