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. 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. 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.

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. 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. These connections are more obvious in some other languages. 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 base number system.

That insight is important to learning and. 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 base 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.

The number systems of pre-K-8 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.

### International Journal of Research in Education and Science

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 pre-K-8 can use the numbers flu ently and flexibly to solve challenging but accessible problems.

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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. For adults the simplicity of calculating with single-digit 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 well-defined sequence of solution methods in learning to perform operations with single-digit 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.

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 single-digit numbers. Instruction that takes such research into account is needed if students are to become proficient:. Children should learn single-digit 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.

We believe that algorithms and their properties are important mathematical ideas that all students need to understand. An algorithm is a reliable step-by-step 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 place-value 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 pre-K 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. 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 base structure of numerals allows certain sums, differences, products, and quotients to be computed mentally.

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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. 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:.

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. 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.

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.

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. Mastery of that system does not come easily, however.

## Visual Representation in Mathematics

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 base quantities represented by number words and number notation.

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 place-value system, to using its features in calculating and problem solving, and to explaining calculation and problem- solving methods with decimal fractions. 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.

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 pre-K-8 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. 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. 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. 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 long-term 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.

## Resource Center

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 pre-K 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 pre-K-8 that would help students move from arithmetic to algebraic ways of thinking.

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. In the United States, however, some efforts to promote algebra for all have involved simply offering a standard first-year 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:.

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. 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:. 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. 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.

Problem-solving 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-. We believe problem solving is vital because it calls on all strands of proficiency, thus increasing the chances of students integrating them.

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 one-sided character to learning and inhibit the formation of connections among the strands. 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. Analyses of the U. How teachers might understand and use instructional materials to help students develop mathematical proficiency is not well understood.

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## Making Mathematics: Mathematics Research Teacher Handbook

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