11
CONCLUSIONS AND RECOMMENDATIONS
To many people, school mathematics is virtually a phenomenon of nature. It seems timeless, set in stone—ard to change and perhaps not needing to change. But the school mathematics education of yesterday, which had a practical basis, is no longer viable. Rote learning of arithmetic procedures no longer has the clear value it once had. The widespread availability of technological tools for computation means that people are less dependent on their own powers of computation. At the same time, people are much more exposed to numbers and quantitative ideas and so need to deal with mathematics on a higher level than they did just 20 years ago. Too few U.S. students, however, leave elementary and middle school with adequate mathematical knowledge, skill, and confidence for anyone to be satisfied that all is well in school mathematics. Moreover, certain segments of the U.S. population are not well represented among those who succeed in learning mathematics. Widespread failure to learn mathematics limits individual possibilities and hampers national growth. Our experiences, discussions, and review of the literature have convinced us that school mathematics demands substantial change. We recognize that such change needs to be undertaken carefully and deliberately, so that every child has both the opportunity and support necessary to become proficient in mathematics.
Our experiences, discussions, and review of the literature have convinced us that school mathematics demands substantial change.
In this chapter, we present conclusions and recommendations to help move the nation toward the change needed in school mathematics. In the preceding chapters, we have offered citations of research studies and of theoretical analyses, but we recognize that clear, unambiguous evidence is not available to address many of the important issues we have raised. It should
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 preK 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
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 developed in grades preK 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 preK 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

ematical proficiency. Schools should support, as a central part of teachers’ work, engagement in sustained efforts to improve their mathematics instruc tion. This support requires the provision of time and resources.

The coordination of curriculum, instructional materials, assessment, instruction, professional development, and school organization around the development of mathematical proficiency should drive school improvement efforts.

Efforts to improve students’ mathematics learning should be informed by scientific evidence, and their effectiveness should be evaluated system atically. Such efforts should be coordinated, continual, and cumulative.

Additional research should be undertaken on the nature, develop ment, and assessment of mathematical proficiency.
These recommendations are augmented in the discussion below. In that discussion we propose additional recommendations that detail some of the policies and practices needed if all children are to be mathematically proficient.
Curriculum
The balanced and integrated development of all five strands of mathematical proficiency requires that various elements of the school curriculum— goals, core content, learning activities, and assessment efforts—be coordinated toward the same end. Achieving that coordination puts heavy demands on instructional programs, on the materials used in instruction, and on the way in which instructional time is managed. The curriculum has to be organized within and across grades so that time for learning is used effectively. Instead of cursory and repeated treatments of a topic, the curriculum should be focused on important ideas, allowing them to be developed thoroughly and treated in depth. The unproductive recycling of mathematical content is to be avoided, but students need ample opportunities to review and consolidate their knowledge.
Instead of cursory and repeated treatments of a topic, the curriculum should be focused on important ideas, allowing them to be developed thoroughly and treated in depth.
Building an Informal Knowledge
Most children in the United States enter school with an extensive stock of informal knowledge about numbers from the counting they have done, from hearing number words and seeing number symbols used in everyday
life, and from various experiences in judging and comparing quantities. Many are also familiar with various patterns and some geometric shapes. This knowledge serves as a basis for developing mathematical proficiency in the early grades. The level of children’s knowledge, however, varies greatly across socioeconomic and ethnic groups. Some children have not had the experiences necessary to build the informal knowledge they need before they enter school.
A number of interventions have demonstrated that any immaturity of mathematical development can be overcome with targeted instructional activities. Parents and other caregivers, through games, puzzles, and other activities in the home, can also help children develop their informal knowledge and can augment the school’s efforts. Just as adults in the home can help children avoid reading difficulties through activities that promote language and literacy growth, so too can they help children avoid difficulties in mathematics by helping them develop their informal knowledge of number, pattern, shape, and space. Support from home and school can have a catalytic effect on children’s mathematical development, and the sooner that support is provided, the better:

School and preschool programs should provide rich activities with numbers and operations from the very beginning, especially for children who enter without these experiences.

Efforts should be made to educate parents and other caregivers as to why they should, and how they can, help their children develop a sense of number and shape.
Learning Number Names
Research has shown that the English number names can inhibit children’s understanding of base10 properties of the decimal system and learning to use numerals meaningfully. Names such as “twelve” and “fifteen” do not make clear to children that 12=10+2 and 15=10+5. These connections are more obvious in some other languages.
U.S. children, therefore, often need extra help in understanding the baseten organization underlying number names and in seeing quantities organized into hundreds, tens, and ones. Conceptual supports (objects or diagrams) that show the magnitude of the quantities and connect them to the number names and written numerals have been found to help children acquire insight into the base10 number system. That insight is important to learning and
understanding numerals and also to developing strategies for solving problems in arithmetic. So that number names will be understood and used correctly, we recommend the following:

Mathematics programs in the early grades should make extensive use of appropriate objects, diagrams, and other aids to ensure that all children understand and are able to use number words and the base10 properties of numerals, that all children can use the language of quantity (hundreds, tens, and ones) in solving problems, and that all children can explain their reason ing in obtaining solutions.
Learning About Numbers
The number systems of preK8 mathematics—the whole numbers, integers, and rational numbers—form a coherent structure. For each of these systems, there are various ways to represent the numbers themselves and the operations on them. For example, a rational number might be represented by a decimal or in fractional form. It might be represented by a word, a symbol, a letter, a point or length on a line, or a portion of a figure. Proficiency with numbers in the elementary and middle grades implies that students can not only appreciate these different notations for a number but also can translate freely from one to another. It also means that they see connections among numbers and operations in the different number systems. As a consequence of many instructional programs, students have had severe difficulty representing, connecting, and using numbers other than whole numbers. Innovations that link various representations of numbers and situations in which numbers are used have been shown to produce learning with understanding. Creating this kind of learning will require changes in all parts of school mathematics to ensure that the following recommendations are implemented:

An integrated approach should be taken to the development of all five strands of proficiency with whole numbers, integers, and rational numbers to ensure that students in grades preK8 can use the numbers flu ently and flexibly to solve challenging but accessible problems. In particular, procedures for calculation should frequently be linked to various represen tations and to situations in which they are used so that all strands are brought into play.

The conceptual bases for operations with numbers and how those operations relate to real situations should be a major focus of the curricu lum. Addition, subtraction, multiplication, and division should be presented initially with real situations. Students should encounter a wide range of situations in which those operations are used.

Different ways of representing numbers, when to use a specific rep resentation, and how to translate from one representation to another should be included in the curriculum. Students should be given opportunities to use these different representations to carry out operations and to understand and explain these operations. Instructional materials should include visual and linguistic supports to help students develop this representational ability.
Operating with SingleDigit Numbers
Learning to operate with singledigit numbers has long been characterized in the United States as “learning basic facts,” and the emphasis has been on rote memorization of those facts, also known as basic number combinations. For adults the simplicity of calculating with singledigit numbers often masks the complexity of learning those combinations and the many different methods children can use in carrying out such calculations. Research has shown that children move through a fairly welldefined sequence of solution methods in learning to perform operations with singledigit numbers, particularly for addition and subtraction, where rapid general procedures exist. Children progress from using physical objects for representing problem situations to using more sophisticated counting and reasoning strategies, such as deriving one number combination from another (e.g., finding 7+8 by knowing that it is 1 more than 7+7 or, similarly, finding 7×6 as 7 more than 7×5). They know that addition and multiplication are commutative and that there is a relation between addition and subtraction and between multiplication and division. They use patterns in the multiplication table as the basis for learning the products of singledigit numbers. Instruction that takes such research into account is needed if students are to become proficient:

Children should learn singledigit number combinations with un derstanding.

Instructional materials and classroom teaching should help students learn increasingly abbreviated procedures for producing number combinations rapidly and accurately without always having to refer to tables or other aids.
Learning Numerical Algorithms
We believe that algorithms and their properties are important mathematical ideas that all students need to understand. An algorithm is a reliable stepbystep procedure for solving problems. To perform arithmetic calculations, children must learn how numerical algorithms work. Some algorithms have been well established through centuries of use; others may be invented by children on their own. The widespread availability of calculators for performing calculations has greatly reduced the level of skill people need to acquire in performing multidigit calculations with paper and pencil. Anyone who needs to perform such calculations routinely today will have a calculator, or even a computer, at hand. But the technology has not made obsolete the need to understand and be able to perform basic written algorithms for addition, subtraction, multiplication, and division of numbers, whether expressed as whole numbers, fractions, or decimals. Beyond providing tools for computation, algorithms can be analyzed and compared, which can help students understand the nature and properties of operations and of placevalue notation for numbers. In our view, algorithms, when well understood, can serve as a valuable basis for reasoning about mathematics.
Students acquire proficiency with multidigit numerical algorithms through a progression of experiences that begin with the students modeling various problem situations. They then can learn algorithms that are easily understood because of obvious connections to the quantities involved. Eventually, students can learn and use methods that are more efficient and general, though perhaps less transparent. Proficiency with numerical algorithms is built on understanding and reasoning, as well as frequent opportunity for use.
Two recommendations reflect our view of the role of numerical algorithms in grades preK8:

For addition, subtraction, multiplication, and division, all students should understand and be able to carry out an algorithm that is general and reasonably efficient.

Students should be able to use adaptive reasoning to analyze and compare algorithms, to grasp their underlying principles, and to choose with discrimination algorithms for use in different contexts.
Using Estimation and Mental Arithmetic
The accurate and efficient use of an algorithm rests on having a sense of the magnitude of the result. Estimation techniques enable students not only to check whether they are performing an operation correctly but also to decide whether that operation makes sense for the problem they are solving.
The base10 structure of numerals allows certain sums, differences, products, and quotients to be computed mentally. Activities using mental arithmetic develop number sense and increase flexibility in using numbers. Mental arithmetic also simplifies other computations and estimations. For example, dividing by 0.25 is the same as multiplying by 4, which can be found by doubling twice. Whether or not students are performing a written algorithm, they can use mental arithmetic to simplify certain operations with numbers. Techniques of estimation and of mental arithmetic are particularly important when students are checking results obtained from a calculator or computer. If children are not encouraged to use the mental computational procedures they have when entering school, those procedures will erode. But when instruction emphasizes estimation and mental arithmetic, conceptual understanding and fluency with mental procedures can be enhanced. Our recommendation about estimation and computation, whether mental or written, is as follows:
Whether or not students are performing a written algorithm, they can use mental arithmetic to simplify certain operations with numbers.

The curriculum should provide opportunities for students to develop and use techniques for mental arithmetic and estimation as a means of pro moting a deeper number sense.
Representing and Operating with Rational Numbers
Rational numbers provide the first number system in which all the operations of arithmetic, including division, are possible. These numbers pose a major challenge to young learners, in part because each rational number can represent so many different situations and because there are several different notational schemes for representing the same rational number, each with its own method of calculation.
An important part of learning about rational numbers is developing a clear sense of what they are. Children need to learn that rational numbers are numbers in the same way that whole numbers are numbers. For children to use rational numbers to solve problems, they need to learn that the same rational number may be represented in different ways, as a fraction, a decimal, or a percent. Fraction concepts and representations need to be related
to those of division, measurement, and ratio. Decimal and fractional representations need to be connected and understood. Building these connections takes extensive experience with rational numbers over a substantial period of time. Researchers have documented that difficulties in working with rational numbers can often be traced to weak conceptual understanding. For example, the idea that a fraction gets smaller when its denominator becomes larger is difficult for children to accept when they do not understand what the fraction represents. Children may try to apply ideas they have about whole numbers to rational numbers and run into trouble. Instructional sequences in which more time is spent at the outset on developing meaning for the various representations of rational numbers and the concept of unit have been shown to promote mathematical proficiency.
Research reveals that the kinds of errors students make when beginning to operate with rational numbers often come because they have not yet developed meaning for these numbers and are applying poorly understood rules for whole numbers. Operations with rational numbers challenge students’ naïve understanding of multiplication and division that multiplication “makes bigger” and division “makes smaller.” Although there is limited research on instructional programs for developing proficiency with computations involving rational numbers, approaches that build on students’ intuitive understanding and that use objects or contexts that help students make sense of the operations offer more promise than rulebased approaches.
We make the following recommendation concerning the rational numbers:

The curriculum should provide opportunities for students to develop a thorough understanding of rational numbers, their various representa tions including common fractions, decimal fractions, and percents, and operations on rational numbers. These opportunities should involve con necting symbolic representations and operations with physical or pictorial representations, as well as translating between various symbolic represen tations.
Extending the PlaceValue System
The system of HinduArabic numerals—in which there is a decimal point and each place to the right and the left is associated with a different power of 10—is one of humanity’s greatest inventions for thinking about and operating with numbers. Mastery of that system does not come easily, however. Students need assistance not only in using the decimal system but also in understanding its structure and how it works.
Conceptual understanding and procedural fluency with multidigit numbers and decimal fractions require that students understand and use the base10 quantities represented by number words and number notation. Research indicates that much of students’ difficulty with decimal fractions stems from their failure to understand the base10 representations. Decimal representations need to be connected to multidigit whole numbers as groups getting 10 times larger (to the left) and one tenth as large (to the right). Referents (diagrams or objects) showing the size of the quantities in different decimal places can be helpful in understanding decimal fractions and calculations with them. The following recommendation expresses our concern that the decimal system be given a central place in the curriculum:

The curriculum should devote substantial attention to developing an understanding of the decimal placevalue system, to using its features in calculating and problem solving, and to explaining calculation and problem solving methods with decimal fractions.
Developing Proportional Reasoning
The concept of ratio is much more difficult than many people realize. Proportional reasoning is the term given to reasoning that involves the equality and manipulation of ratios. Children often have difficulty comparing ratios and using them to solve problems. Many school mathematics programs fail to develop children’s understanding of ratio comparisons and move directly to formal procedures for solving missingvalue proportion problems. Research tracing the development of proportional reasoning shows that proficiency grows as students develop and connect different aspects of proportional reasoning. Further, the development of proportional reasoning can be supported by having students explore proportional situations in a variety of problem contexts using concrete materials or through data collection activities. We see ratio and proportion as underdeveloped components of grades preK8 mathematics:

The curriculum should provide extensive opportunities over time for students to explore proportional situations concretely, and these situa tions should be linked to formal procedures for solving proportion problems whenever such procedures are introduced.
Using the Number Line
Students often view the study of whole numbers, decimal fractions, common fractions, and integers as disconnected topics. One tool that we believe may be useful in developing numerical understanding and in making connections across number systems is the number line, a geometric representation of numbers that gives each number a unique point on the line and an oriented distance from the origin, depicting its magnitude and direction. Although it may be difficult to learn, the number line gives a unified geometric representation of integers and rational numbers within the real number system, later to be encountered in geometry, algebra, and calculus. The geometric models of operations afforded by the number line apply uniformly to all real numbers, thus presenting one unified number system. The number line may become particularly useful as students are learning about integers and rational numbers, for it may help students develop a sense of the magnitudes and relationships of those numbers in a way that is less clear in other representations:

Because it can serve as a tool for simultaneously representing whole numbers, integers, and rational numbers, teachers and researchers should explore effective uses of the number line representation when students learn about operations with numbers, relations among number systems, and more formal symbolic representations of numbers.
Expanding the Number Domain
Students currently encounter the expansion of the number domain by starting with whole numbers, gradually incorporating fractions, and only much later expanding the domain to include negative integers and irrational numbers. That sequence has a long history, but there are arguments for an alternative. For example, expanding the whole numbers to take in the negative integers in the early grades would allow students to do more with addition and subtraction before venturing into the rational number system, which requires multiplication and division. Systematic study of this alternative is needed:

Teachers, curriculum developers, and researchers should explore the possibility of introducing integers before rational numbers. Ways to engage younger children in meaningful uses of negative integers should be devel oped and tested.
Developing Algebraic Thinking
The formal study of algebra is both the gateway into advanced mathematics and a stumbling block for many students. The transition from arithmetic to algebra is often not an easy one. The difficulties associated with the transition from the activities typically associated with school arithmetic to those typically associated with school algebra (representational activities, transformational activities, and generalizing and justifying activities) have been extensively studied. Research has documented that the visual and numerical supports provided for symbolic expressions by computers and graphing calculators help students create meaning for expressions and equations. The research, however, has shed less light on the longterm acquisition and retention of transformational fluency. Although through generalizing and justifying, students can learn to use and appreciate algebraic expressions as general statements, more research is need on how students develop such awareness.
The formal study of algebra is both the gateway into advanced mathematics and a stumbling block for many students.
The study of algebra, however, does not have to begin with a formal course in the subject. New lines of research and development are focusing on ways that the elementary and middle school curriculum can be used to support the development of algebraic reasoning. These efforts attempt to avoid the difficulties many students now experience and to lay a better foundation for secondary school mathematics. We believe that from the earliest grades of elementary school, students can be acquiring the rudiments of algebra, particularly its representational aspects and the notion of variable and function. By emphasizing both the relationships among quantities and ways of representing these relationships, instruction can introduce students to the basic ideas of algebra as a generalization of arithmetic. They can come to value the roles of definitions and see how the laws of arithmetic can be expressed algebraically and be used to support their reasoning. We recommend that algebra be explicitly connected to number in grades preK8:

The basic ideas of algebra as generalized arithmetic should be anticipated by activities in the early elementary grades and learned by the end of middle school.

Teachers and researchers should investigate the effectiveness of instructional strategies in grades preK8 that would help students move from arithmetic to algebraic ways of thinking.
Promoting Algebra for All
In some countries by the end of eighth grade, all students have been studying algebra for several years, although not ordinarily in a separate course. “Algebra for all” is a worthwhile and attainable goal for middle school students. In the United States, however, some efforts to promote algebra for all have involved simply offering a standard firstyear algebra course (algebra through quadratics) to everyone. We believe such efforts are virtually guaranteed to result in many students failing to develop proficiency in algebra, in part because the transition to algebra is so abrupt. Instead, a different curriculum is needed for algebra in middle school:
“Algebra for all” is a worthwhile and attainable goal for middle school students.

Teachers, researchers, and curriculum developers should explore ways to offer a middle school curriculum in which algebraic ideas are devel oped in a robust way and connected to the rest of mathematics.
A different curriculum is needed for algebra in middle school.
Using Technology to Learn Algebra
Research has shown that instruction that makes productive use of computer and calculator technology has beneficial effects on understanding and learning algebraic representation. It is not clear, however, what role the newer symbol manipulation technologies might play in developing proficiency with the transformational aspects of algebra. We recommend the following:

Research should be conducted on the effects on students’ learning of using the symbolmanipulating capacities of calculators and computers to study algebraic concepts and to transform algebraic expressions and equa tions.
Solving Problems as a Context for Learning
An important part of our conception of mathematical proficiency involves the ability to formulate and solve problems coming from daily life or other domains, including mathematics itself. That ability is not being developed well in U.S. preK to grade 8 classrooms. Studies in almost every domain of mathematics have demonstrated that problem solving provides an important context in which students can learn about number and other mathematical topics.
Problemsolving ability is enhanced when students have opportunities to solve problems themselves and to see problems being solved. Further, problem solving can provide the site for learning new concepts and for prac
ticing learned skills. We believe problem solving is vital because it calls on all strands of proficiency, thus increasing the chances of students integrating them. Problem solving also provides opportunities for teachers to assess students’ performance on all of the strands. Other activities, such as listening to an explanation or practicing solution methods, can help develop specific strands of proficiency, but too much emphasis on them, to the exclusion of solving problems, may give a onesided character to learning and inhibit the formation of connections among the strands. We see problem solving as central to school mathematics:
We see problem solving as central to school mathematics.

Problem solving should be the site in which all of the strands of math ematics proficiency converge. It should provide opportunities for students to weave together the strands of proficiency and for teachers to assess students’ performance on all of the strands.
Improving Materials for Instruction
Analyses of the U.S. curriculum reveal much repetition from grade to grade and many topics, few of which are treated in much depth. Further, instructional materials in preK to grade 8 mathematics seldom provide the guidance and assistance that teachers in other countries find helpful, such as discussions of children’s typical misconceptions or alternative solution methods. How teachers might understand and use instructional materials to help students develop mathematical proficiency is not well understood. On the basis of our reasoned judgment, we offer the following recommendations for improving instructional materials in school mathematics:

Textbooks and other instructional materials should develop the core content of school mathematics in a focused way, in depth, and with continu ity in and across grades, supporting all strands of mathematical proficiency.

Textbooks and other instructional materials should support teacher understanding of mathematical concepts, of student thinking and student errors, and of effective pedagogical supports and techniques.

Activities and strategies should be developed and incorporated into instructional materials to assist teachers in helping all students become proficient in mathematics, including students low in socioeconomic status, English language learners, special education students, and students with a special interest or talent in mathematics.

Efforts to develop textbooks and other instructional materials should include research into how teachers can understand and use those materials effectively.

A government agency or research foundation should fund an inde pendent group to analyze textbooks and other instructional materials for the extent to which they promote mathematical proficiency. The group should recommend how these materials might be modified to promote greater math ematical proficiency.
Giving Time to Instruction
Research indicates that a key requirement for developing proficiency is the opportunity to learn. In many U.S. elementary and middle school classrooms, students are not engaged in sustained study of mathematics. On some days in some classes they are spending little or no time at all on the subject. Mathematical proficiency as we have defined it cannot be developed unless regular time (say, one hour each school day) is allocated to and used for mathematics instruction in every grade of elementary and middle school. Further, we believe the strands of proficiency will not develop in a coordinated fashion unless continual attention is given to every strand. The following recommendation expresses our concern that mathematics be given its rightful place in the curriculum:
Mathematical proficiency as we have defined it cannot be developed unless regular time is allocated to and used for mathematics instruction in every grade of elementary and middle school.

Substantial time should be devoted to mathematics instruction each school day, with enough time devoted to each unit and topic to enable stu dents to develop understanding of the concepts and procedures involved. Time should be apportioned so that all strands of mathematical proficiency together receive adequate attention.
Giving Students Time to Practice
Practice is important in the development of mathematical proficiency. When students have multiple opportunities to use the computational procedures, reasoning processes, and problemsolving strategies they are learning, the methods they are using become smoother, more reliable, and better understood. Practice alone does not suffice; it needs to be built on understanding and accompanied by feedback. In fact, premature practice has been shown to be harmful. The following recommendation reflects our view of the role of practice:

Practice should be used with feedback to support all strands of math ematical proficiency and not just procedural fluency. In particular, practice on computational procedures should be designed to build on and extend under standing.
Using Assessment Effectively
At present, substantial time every year is taken away from mathematics instruction in U.S. classrooms to prepare for and take externally mandated assessments, usually in the form of tests. Often, those tests are not well articulated with the mathematics curriculum, testing content that has not been taught during the year or that is not central to the development of mathematical proficiency. Preparation for such tests, moreover, does not ordinarily focus on the development of proficiency. Instead, much time is given to practicing calculation procedures and reviewing a multitude of topics. Teachers and students often waste valuable learning time because they are not informed about the content to be tested or the form that test items will take.
We believe that assessment, whether externally mandated or developed by the teacher, should support the development of students’ mathematical proficiency. It needs to provide opportunities for students to learn rather than taking time away from their learning. Assessments in which students are learning as well as showing what they have already learned can provide valuable information to teachers, schools, districts, and states, as well as the students themselves. Such assessments help teachers modify their instruction to support better learning at each grade level.
Time and money spent on assessment need to be used more effectively so that students have the opportunity to show what they know and can do. Teachers need to receive timely and detailed information about students’ performance on each external assessment. In that way, students and teachers alike can learn from assessments instead of having assessments used only to rank students, teachers, or schools. The following recommendations will help make assessment more effective in developing mathematical proficiency:
Students and teachers alike can learn from assessments instead of having assessments used only to rank students, teachers, or schools.

Assessment, whether internal or external, should be focused on the development and achievement of mathematical proficiency. In particular, assessments used to determine qualification for state and federal funding should reflect the definition of mathematics proficiency presented in this report.

Information about the content and form of each external assessment should be provided so that teachers and students can prepare appropriately and efficiently.

The results of each external assessment should be reported so as to provide feedback useful for teachers and learners rather than simply a set of rankings.

A government agency or research foundation should fund an inde pendent group to analyze external assessment programs for the extent to which they promote mathematical proficiency. The group should recommend how programs might be modified to promote greater mathematical proficiency.
Instruction
Effective teaching—teaching that fosters the development of mathematical proficiency over time—can take a variety of forms. Consequently, we endorse no single approach. All forms of instruction configure relations among teachers, students, and content. The quality of instruction is a function of teachers’ knowledge and use of mathematical content, teachers’ attention to and handling of students, and students’ engagement in and use of mathematical tasks. The development of mathematical proficiency requires thoughtful planning, careful execution, and continual improvement of instruction. It depends critically on teachers who understand mathematics, how students learn, and the classroom practices that support that learning. They also need to know their students: who they are, what their backgrounds are, and what they know.
The development of mathematical proficiency requires thoughtful planning, careful execution, and continual improvement of instruction.
Planning for Instruction
Planning, whether for one lesson or a year, is often viewed as routine and straightforward. However, plans seldom elaborate the content that the students are to learn or develop good maps of paths to take to reach learning goals. We believe that planning needs to reflect a deep and thorough consideration of the mathematical content of a lesson and of students’ thinking and learning. Instructional materials need to support teachers in their planning, and teachers need to have time to plan. Instruction needs to be planned with the development of mathematical proficiency in mind:

Content, representations, tasks, and materials should be chosen so as to develop all five strands of proficiency toward the big ideas of math ematics and the goals for instruction.

Planning for instruction should take into account what students know, and instruction should provide ways of ascertaining what students know and think as well as their interests and needs.

Rather than simply listing problems and exercises, teachers should plan for instruction by focusing on the learning goals for their students, keep ing in mind how the goals for each lesson fit with those of past and future lessons. Their planning should anticipate the events in the lesson, the ways in which the students will respond, and how those responses can be used to further the lesson goals.
Managing Classroom Discourse
Mathematics classrooms are more likely to be places in which mathematical proficiency develops when they are communities of learners and not collections of isolated individuals. Research on creating classrooms that function as communities of learners has identified several important features of these classrooms: ideas and methods are valued, students have autonomy in choosing and sharing solution methods, mistakes are valued as sites of learning for everyone, and the authority for correctness lies in logic and the structure of the subject, not in the teacher. In such classrooms the teacher plays a key role as the orchestrator of the discourse students engage in about mathematical ideas. Teachers are responsible for moving the mathematics along while affording students opportunities to offer solutions, make claims, answer questions, and provide explanations to their peers. Teachers need to help bring a mathematical discussion to a close, making sure that gaps have been filled and errors addressed. To develop mathematical proficiency, we believe that students require more than just the demonstration of procedures. They need experience in investigating mathematical properties, justifying solution methods, and analyzing problem situations. We recommend the following:

A significant amount of class time should be spent in developing math ematical ideas and methods rather than only practicing skills.

Questioning and discussion should elicit students’ thinking and solu tion strategies and should build on them, leading to greater clarity and precision.

Discourse should not be confined to answers only but should include discussion of connections to other problems, alternative representations and solution methods, the nature of justification and argumentation, and the like.
Linking Experience to Abstraction
Students acquire higher levels of mathematical proficiency when they have opportunities to use mathematics to solve significant problems as well as to learn the key concepts and procedures of that mathematics. Although mathematics gains power and generality through abstraction, it finds both its sources and applications in concrete settings, where it is made meaningful to the learner. There is an inevitable dialectic between concrete and abstract in which each helps shape the other. Exhortations to “begin with the concrete” need to consider carefully what is meant by concrete. Research reveals that various kinds of physical materials commonly used to help children learn mathematics are often no more concrete to them than symbols on paper might be. Concrete is not the same as physical. Learning begins with the concrete when meaningful items in the child’s immediate experience are used as scaffolding with which to erect abstract ideas. To ensure that progress is made toward mathematical abstraction, we recommend the following:

Links among written and oral mathematical expressions, concrete problem settings, and students’ solution methods should be continually and explicitly made during school mathematics instruction.
Assigning Independent Work
Part of becoming proficient in mathematics is becoming an independent learner. For that purpose, many teachers give homework. The limited research on homework in mathematics has been confined to investigations of the relation between the quantity of homework assigned and students’ achievement test scores. Neither the quality nor the function of homework has been studied. Homework can have different purposes. For example, it might be used to practice skills or to prepare the student for the next lesson. We believe that independent work serves several useful purposes. Regarding independence and homework, we make the following recommendations:

Students should be provided opportunities to work independently of the teacher both individually and in pairs or groups.

When homework is assigned for the purpose of developing skill, stu dents should be sufficiently familiar with the skill and the tasks so that they are not practicing incorrect procedures.
Using Calculators and Computers
In the discussion above, we mention the special role that calculators and computers can play in learning algebra. But they have many other roles to play throughout instruction in grades preK8. Using calculators and computers does not replace the need for fluency with other methods. Confronted with a complex arithmetic problem, students can use calculators and computers to see beyond tedious calculations to the strategies needed to solve the problem. Technology can relieve the computational burden and free working memory for higherlevel thinking so that there can be a sharper focus on an important idea. Further, skillfully planned calculator investigations may reveal subtle or interesting mathematical ideas, such as the rules for order of operations.
A large number of empirical studies of calculator use, including longterm studies, have generally shown that the use of calculators does not threaten the development of basic skills and that it can enhance conceptual understanding, strategic competence, and disposition toward mathematics. For example, students who use calculators tend to show improved conceptual understanding, greater ability to choose the correct operation, and greater skill in estimation and mental arithmetic without a loss of basic computational skills. They are also familiar with a wider range of numbers than students who do not use calculators and are better able to tackle realistic mathematics problems.
Just like any instructional tool, calculators and computers can be used more or less effectively. Our concern is that, when computing technology is used, it needs to contribute positively:
When computing technology is used, it needs to contribute positively.

In all grades of elementary and middle school, any use of calculators and computers should be done in ways that help develop all strands of stu dents’ mathematical proficiency.
Teacher Preparation and Professional Development
One critical component of any plan to improve mathematics learning is the preparation and professional development of teachers. If the goal of mathematical proficiency as portrayed in this report is to be reached by all students in grades preK to 8, their teachers will need to understand and practice techniques of teaching for that proficiency. Our view of mathematics proficiency requires teachers to act in new ways and to have understanding that they once were not expected to have. In particular, it is not a teacher’s fault that he or she does not know enough to teach in the way we are asking. It is a far from trivial task to acquire such understanding—something that cannot reasonably be expected to happen in one’s spare time and something that will require major policy changes to support and promote. Teacher preparation and professional development programs will need to develop proficiency in mathematics teaching, which has many parallels to proficiency in mathematics.
Developing Specialized Knowledge
The knowledge required to teach mathematics well is specialized knowledge. It includes an integrated knowledge of mathematics, knowledge of the development of students’ mathematical understanding, and a repertoire of pedagogical practices that take into account the mathematics being taught and the students learning it. The evidence indicates that these forms of knowledge are not acquired in conventional undergraduate mathematics courses, whether they are general survey courses or specialized courses for mathematics majors. The implications for teacher preparation and professional development are that teachers need to learn these forms of knowledge in ways that help them forge connections.
Very few teachers currently have the specialized knowledge needed to teach mathematics in the way envisioned in this report.
Mathematical knowledge is a critical resource for teaching. Therefore, teacher preparation and professional development must provide significant and continuing opportunities for teachers to develop profound and useful mathematical knowledge. Teachers need to know the mathematics of the curriculum and where the curriculum is headed. They need to understand the connections among mathematical ideas and how they develop. Teachers also need to be able to unpack mathematical content and make visible to students the ideas behind the concepts and procedures. Finally, teachers need not only mathematical proficiency but also the ability to use it in guiding discussions, modifying problems, and making decisions about what matters to pursue in class and what to let drop. Very few teachers currently have
the specialized knowledge needed to teach mathematics in the way envisioned in this report. Although it is not reasonable in the short term to expect all teachers to acquire such knowledge, every school needs access to expertise in mathematics teaching.
Teachers’ opportunities to learn can help them develop their own knowledge about mathematics, about children’s thinking about mathematics, and about mathematics teaching. Such opportunities can also help teachers learn how to solve the sorts of problems that are central to the practice of teaching. The following recommendations reflect our judgment concerning the specialized knowledge that teachers need:

Teachers of grades preK8 should have a deep understanding of the mathematics of the school curriculum and the principles behind it.

Programs and courses that emphasize “classroom mathematical knowledge” should be established specifically to prepare teachers to teach mathematics to students in such grades as preK2, 3–5, and 6–8.

Teachers should learn how children’s mathematical knowledge develops and what their students are likely to bring with them to school.

To provide a basis for continued learning by teachers, their prepa ration to teach, their professional development activities, and the instruc tional materials they use should engage them, individually and collectively, in developing a greater understanding of mathematics and of student thinking and in finding ways to put that understanding into practice. All teachers, whether preservice or inservice, should engage in inquiry as part of their teaching practice (e.g., by interacting with students and analyzing their work).

Through their preparation and professional development, teachers should develop a repertoire of pedagogical techniques and the ability to use those techniques to accomplish lesson goals.

Mathematics specialists—teachers who have special training and interest in mathematics—should be available in every elementary school.
Working Together
Elementary and middle school teachers in the United States report spending relatively little time, compared with their counterparts in other countries, discussing the mathematics they are teaching or the methods they are using. They seldom plan lessons together, observe one another teach, or analyze students’ work collectively. Studies of programs that require teachers to teach mathematically demanding curricula suggest that success is greater when teachers help one another not only learn the mathematics and learn about student thinking but also practice new teaching strategies. Our recommendation concerning time is not just about how much is available but how it is used:

Teachers should be provided with more time for planning and con ferring with each other on mathematics instruction with appropriate sup port and guidance.
Capitalizing on Professional Meetings
Teachers need more mathematically focused opportunities to learn mathematics, and they need to be prepared to manage changes in the field. Mathematics teachers already come together at meetings of professional societies such as the National Council of Teachers of Mathematics (NCTM), its affiliated groups, or other organizations. These occasions can provide opportunities for professional development of the sort discussed above. For example, portions of national or regional meetings of the NCTM could be organized into minicourses or institutes, without competing sessions being held at the same time. Professional development needs to grow out of current activities:

Professional meetings and other occasions when teachers come together to work on their practice should be used as opportunities for more serious and substantive professional development than has commonly been available.
Sustaining Professional Development
Preparing to teach is a careerlong activity. Teachers need to continue to learn. But rather than being focused on isolated facts and skills, teacher learning needs to be generative. That is, what teachers learn needs to serve as a basis for them to continue to learn from their practice. They need to see that practice as demanding continual review, analysis, and improvement. Studies of teacher change indicate that shortterm, fragmented professional development is ineffective for developing teaching proficiency.
More resources of all types—money, time, leadership, attention—need to be invested in professional development for teachers of mathematics, and those resources already available could be used more wisely and productively. Each year a substantial amount of money is invested in professional development programs for teachers. Individual schools and districts fund some programs locally. Others are sponsored and funded by state agencies, federal agencies, or professional organizations. Much of the time and money invested in such programs, however, is not used effectively. Sponsors generally fund shortterm, even oneshot, activities such as daylong workshops or twoday institutes that collectively do not form a cohesive and cumulative program of professional development. Furthermore, these activities are often conducted by an array of professional developers with minimal qualifications in mathematics and mathematics teaching. Professional development in mathematics needs to be sustained over time that is measured in years, not weeks or months, and it needs to involve a substantial amount of time each year. Our recommendations to raise the level of professional development are as follows:
Professional development in mathematics needs to be sustained over time that is measured in years, not weeks or months.

Local education authorities should give teachers support, including stipends and released time, for sustained professional development.

Providers of professional development should know mathematics and should know about students’ mathematical thinking, how mathematics is taught, and teachers’ thinking about mathematics and their own practice.

Organizations and agencies that fund professional development in mathematics should focus resources on multiyear, coherent programs. Resources of agencies at every level should be marshaled to support substan tial and sustained professional development.
Monitoring Progress Toward Mathematical Proficiency
In this report we have set forth a variety of observations, conclusions, and recommendations that are designed to bring greater coherence and balance to the learning and teaching of mathematics. In particular, we have described five strands of mathematical proficiency that should frame all efforts to improve school mathematics.
Over the past decades, various visions have been put forward for improving curriculum, instruction, and assessment in mathematics, and many of those ideas have been tried in schools. Unfortunately, new programs are tried but
then abandoned before their effectiveness has been well tested, and lessons learned from program evaluations are often lost. Although aspects of mathematics proficiency have been studied, other aspects such as productive disposition have received less attention; and no one, including the National Assessment of Educational Progress (NAEP), has studied the integrated portrait of mathematics proficiency set forth in this report. In order that efforts to improve U.S. school mathematics might be more cumulative and coordinated, we make the following recommendation:

An independent group of recognized standing should be constituted to assess the progress made in meeting the goal of mathematical proficiency for all U.S. schoolchildren.
Supporting the Development of Mathematical Proficiency
The mathematics students need to learn today is not the same mathematics that their parents and grandparents needed to learn. Moreover, mathematics is a domain no longer limited to a select few. All students need to be mathematically proficient to the levels discussed in this report. The mathematics of grades preK8 today involves much more than speed in pencilandpaper arithmetic. Students need to understand mathematics, use it to solve problems, reason logically, compute fluently, and use it to make sense of their world. For that to happen, each student will need to develop the strands of proficiency in an integrated fashion.
No country—not even those performing highest on international surveys of mathematics achievement—has attained the goal of mathematical proficiency for all its students. It is an extremely ambitious goal, and the United States will never reach it by continuing to tinker with the controls of educational policy, pushing one button at a time. Adopting mathematics textbooks from other countries, testing teachers, holding students back a grade, putting schools under state sanctions—none of these alone will advance school mathematics very far toward mathematical proficiency for all. Instead, coordinated, systematic, and sustained modifications will need to be made in how school mathematics instruction has commonly proceeded, and support of new and different kinds will be required. Leadership and attention to the teaching of mathematics are needed in the formulation and implementation of policies at all levels of the educational system.