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LOOKING AT MATHEMATICS AND LEARNING

Children today are growing up in a world permeated by mathematics. The technologies used in homes, schools, and the workplace are all built on mathematical knowledge. Many educational opportunities and good jobs require high levels of mathematical expertise. Mathematical topics arise in newspaper and magazine articles, popular entertainment, and everyday conversation.

Mathematics is a universal, utilitarian subject—so much a part of modern life that anyone who wishes to be a fully participating member of society must know basic mathematics. Mathematics also has a more specialized, esoteric, and esthetic side. It epitomizes the beauty and power of deductive reasoning. Mathematics embodies the efforts made over thousands of years by every civilization to comprehend nature and bring order to human affairs.

These dual aspects of mathematics, the practical and the theoretical, have earned the subject a place at the center of education throughout history. Even simple systems for counting have to be passed on to the next generation. Every literate society has needed people who knew how to read the heavens and measure the earth. Farmers have wanted to calculate crop production, and merchants to record their transactions.

As mathematics became more formal and abstract in the hands of the ancient Greeks, it also became enshrined among the liberal arts. The mastery of its forms of reasoning became a hallmark of the educated person. Its study was seen as bringing the discipline of logical thinking to the apprentice scholar.

Despite the value of mathematics as a model of deductive reasoning, the teaching of mathematics has often taken quite a different form. For centuries,



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Adding + It Up: Helping Children Learn Mathematics 1 LOOKING AT MATHEMATICS AND LEARNING Children today are growing up in a world permeated by mathematics. The technologies used in homes, schools, and the workplace are all built on mathematical knowledge. Many educational opportunities and good jobs require high levels of mathematical expertise. Mathematical topics arise in newspaper and magazine articles, popular entertainment, and everyday conversation. Mathematics is a universal, utilitarian subject—so much a part of modern life that anyone who wishes to be a fully participating member of society must know basic mathematics. Mathematics also has a more specialized, esoteric, and esthetic side. It epitomizes the beauty and power of deductive reasoning. Mathematics embodies the efforts made over thousands of years by every civilization to comprehend nature and bring order to human affairs. These dual aspects of mathematics, the practical and the theoretical, have earned the subject a place at the center of education throughout history. Even simple systems for counting have to be passed on to the next generation. Every literate society has needed people who knew how to read the heavens and measure the earth. Farmers have wanted to calculate crop production, and merchants to record their transactions. As mathematics became more formal and abstract in the hands of the ancient Greeks, it also became enshrined among the liberal arts. The mastery of its forms of reasoning became a hallmark of the educated person. Its study was seen as bringing the discipline of logical thinking to the apprentice scholar. Despite the value of mathematics as a model of deductive reasoning, the teaching of mathematics has often taken quite a different form. For centuries,

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Adding + It Up: Helping Children Learn Mathematics 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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Adding + It Up: Helping Children Learn Mathematics how children learn mathematics and the kinds of teaching that supports that learning. Research continues to expand our understanding of the teaching-learning process. All of this taken together makes us believe that our goal is in large measure achievable. Mathematics and Reading A comparison of mathematics with reading leads to several important observations. First, competence in both domains is important in determining children’s later educational and occupational prospects. Children who fail to develop a high level of skill in either one are precluded from the most interesting and rewarding careers. As a recent report on reading from the National Research Council put it, “To be employable in the modern economy, high school graduates need to be more than merely literate. They must be able to read challenging material, to perform sophisticated computations, and to solve problems independently.”2 Second, there are important similarities as well as differences in the problems children face in developing competence in reading and mathematics. Understanding the common features of reading development and mathematical development is as important as understanding the special characteristics of learning in each domain. Finally, international comparisons suggest that U.S. schools have been relatively successful in developing skilled reading, with improvements in both instruction and achievement occurring in a large number of schools.3 Unfortunately, the same cannot be said of mathematics. International comparisons discussed in the next chapter suggest that by eighth grade the mathematics performance of U.S. children is well below that of other industrialized countries. Furthermore, this performance has been relatively low in a variety of comparisons conducted at intervals over several decades. The organizational and instructional factors that U.S. schools have used in developing skilled reading performance may be equally important in improving the learning of mathematics. Learning to read and developing mathematical proficiency both rest on a foundation of concepts and skills that are acquired by many children before they leave kindergarten. In the case of reading, children are expected to enter school with a basic understanding of the sound structure of their native language, a conscious awareness of the units (phonemes) that are represented by an alphabetic writing system, and skill in handling basic language concepts. Likewise in mathematics, students should possess a toolkit of basic mathematical concepts and skills when they enter first grade. (These

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Adding + It Up: Helping Children Learn Mathematics are reviewed in chapter 5.) In both reading and mathematics, some children enter school without the knowledge and experience that school instruction presumes they possess. In both domains, there is evidence that early intervention can prevent full-blown problems in school.4 For both reading and mathematics, children’s performance at the end of elementary school is an important predictor of their ultimate educational success. If they have not mastered certain basic skills, they can expect problems throughout their schooling and later. Research on reading indicates that all but a very small number of children can learn to read proficiently, though they may learn at different rates and may require different amounts and types of instructional support. Furthermore, experiences in pre-kindergarten and the early elementary grades serve as a crucial foundation for students’ emerging proficiency. Similar observations can be made for mathematics. For example, nearly all second graders might be expected to make a useful drawing of the situation portrayed in an arithmetic word problem as a step toward solving it. Representing numbers by means of a drawing is a task that few children find difficult. Other tasks, however, depend much more heavily on children’s knowledge and experience. For example, in Roman numerals, the value of V is five regardless of where it is located in the numeral, whether IV, VI, or VII. The Hindu-Arabic numerals used in everyday life are different; a digit’s value depends on the place it occupies. For example, the 5 in 115 denotes five, whereas in 151 it denotes fifty, and in 511, five hundred. Also, a special symbol, 0, is used to hold a place that would otherwise be unoccupied. Although adults may view this place-value system as simple and straightforward, it is actually quite sophisticated and challenging to learn (see chapters 5 and 6). To make progress in school mathematics, children must understand Hindu-Arabic numerals and be able to use them fluently. But the children in, say, a second-grade class can be expected to differ considerably in the rate at which they grasp place value. It is a complex system of representation that functions almost like a foreign language that a child is learning to use and simultaneously using to learn other things. Much of school mathematics has this mutually dependent quality. Abstractions at one level are used to develop abstractions at a higher level, and abstractions at a higher level are used to gain insights into abstractions at a lower level. To ensure that students having reading difficulties get prompt and effective assistance outside the regular school program, the reading community has developed a variety of intervention programs designed to address the problems students are having and to bring them back rapidly into the regular

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Adding + It Up: Helping Children Learn Mathematics program.5 Although there is much “remediation” done as part of school mathematics instruction in grades K to 8 and beyond, there are not nearly so many supplementary interventions in mathematics as there are in reading. There is very little in the way of “mathematics recovery” that provides early targeted enrichment in mathematics to help students overcome special difficulties. One difference between reading and mathematics is that, after a certain point, reading requires little explicit instruction: Once children have acquired basic principles and skills for reading, they use those skills in the service of other activities, to learn about history, literature, or mathematics, for example. Their skills can always be polished and instruction given on interpreting a text, but they need no further explanations and demonstrations of reading by others. Furthermore, they practice and develop their reading throughout their lives, both inside and outside of school. As is the case for reading, students develop some basic concepts and practices in mathematics outside of school, but a new and unfamiliar topic in mathematics—say, the division of fractions— usually cannot be fully grasped without some assistance from a text or a teacher. Reading uses a core set of representations. In U.S. schools, the English alphabetic writing system, once learned, enables the student to read and decode any English sentence, although of course not necessarily to understand its meaning. Graphs, pictures, and signs also need to be read, but the core symbols are the alphabet. Mathematics, in contrast, has many types and levels of representation. In fact, mathematics can be said to be about levels of representation, which build on one another as the mathematical ideas become more abstract. For example, the increasing focus on algebra during the school years builds facility with more abstract levels of representation. Another characteristic of learning to read is the vast variation among children in their exposure to literature outside of school, as well as in the amount of time they spend reading. Studies on the development of reading6 have shown that variations in children’s reading skill are associated with large differences in reading experience. Children at the 80th percentile in reading level were estimated to average more than 20 times as much reading per day as children at the 20th percentile.7 Similar data are not available for mathematics, but differences in the amount of time spent doing mathematics are likely to be less than for reading. This suggests that direct school-based instruction may play a larger part in most children’s mathematical experience than it does in their reading experience. If so, the consequences of good or poor mathematics instruction

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Adding + It Up: Helping Children Learn Mathematics may have an even greater effect on children’s proficiency than is the case with reading. An important recent change in American education is the increased emphasis on ensuring that all children achieve a basic level of competence in reading during the course of elementary school. Success in school also depends on establishing good mathematical competence in the early elementary grades, yet mathematics instruction has not received the same sustained emphasis. Schools generally lack a mathematics specialist corresponding to the reading specialists who provide instruction and assist children having difficulties with the subject. Many school districts have revised their schedules and their curriculum programs to ensure that adequate reading instruction is given in the elementary grades; mathematics instruction has yet to receive similar attention. The recommendations we give at the end of this report attempt to take into account the progress made in homes and at school in achieving reading proficiency. Looking at Mathematics The mathematics to which U.S. schoolchildren are exposed from preschool through eighth grade has many aspects. However, at the heart of preschool, elementary school, and middle school mathematics is the set of concepts associated with the term number.8 Children learn to count, and they learn to keep track of their counting by writing numerals for the natural numbers. They learn to add, subtract, multiply, and divide whole numbers, and later in elementary school they learn to perform these same operations with common fractions and decimal fractions. They use numbers in measuring a variety of quantities, including the lengths, areas, and volumes of geometric figures. From various sources, children collect data that they learn to represent and analyze using numerical methods. The study of algebra begins as they observe how numbers form systems and as they generalize number patterns. We have focused much of this report on the domain of number. Most of the controversy over how and what mathematics should be taught in elementary and middle school revolves around number. Should children learn computational methods before they understand the concepts involved? Should they be introduced to standard algorithms for arithmetic computation, or should they be encouraged to develop their own algorithms first? How much time should be spent learning long division or how to add common fractions? Should decimals be introduced before or after fractions? How proficient do children need to be at paper-and-pencil arithmetic before they are taught

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Adding + It Up: Helping Children Learn Mathematics algebra or geometry? Such questions are controversial partly because they touch on the third R—arithmetic—that parents want their children to master, and also because they deal with topics on which reformers have taken some of their strongest stands in opposition to current practice. Furthermore, much more research has been conducted in the domain of number than in most other areas of the mathematics curriculum. For most controversial questions involving number, at least some related research is available, and many of these questions have been studied extensively. Our attention to number and operations is certainly not meant to imply that the elementary and middle school curriculum is or should be limited to number. Mathematics is a broad discipline, and children need to learn about its many aspects. Although the amount of research that is available is less, we have also reviewed what is known from research about how students develop proficiency with some of the central concepts of measurement, geometry, descriptive statistics, and probability. Further, we have reviewed the research on beginning algebra learning. Nevertheless, our review of the research on mathematics learning paints an incomplete picture of the nature of mathematics, even elementary and middle school mathematics. Many facets of the discipline are not covered or not covered adequately by the research or our review. Further, our review does not capture the many connections both between various topics in mathematics and between mathematics and its uses in the world around us. Hence, in describing what is known about how children learn mathematics, we are not indirectly prescribing what mathematics children should learn. In describing what is known about how children learn mathematics, we are not indirectly prescribing what mathematics children should learn. Nature of the Evidence For every generation of students, the mathematics curriculum and the methods used to deliver that curriculum are products of many choices. Some of these choices reflect the fact that the volume of knowledge in any subject greatly exceeds the time available for teaching it. Decisions always must be made as to what topics to teach and how much time to spend on them. Choices about the teaching and learning of mathematics also depend on what society wants educated adults to know. Questions of what needs to be taught are essentially questions of what knowledge is most preferred. Research can inform these decisions—for example, studies of modern workplaces can reveal what mathematics employees most need to know.9 However, ideas about what children today need to know also depend on value judgments based on previous experience and convictions, and these judgments often fall outside the domain of research.

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Adding + It Up: Helping Children Learn Mathematics Once choices have been made regarding the mathematics that students should know, the goals for instruction can be framed. The available evidence from research can be used to analyze the feasibility of the goals as well as to contribute to decisions about how to help children achieve them. The task then becomes, first, to identify the research that can be used to inform these analyses and decisions and, second, to figure out how best to use that research. The experience that people know and understand best is their own. To establish policies for school mathematics, however, it is essential to look beyond one’s own experience to the evidence obtained through a systematic examination of what others have seen and reported. Some of this evidence is analytical or conceptual, such as analyses of mathematical representations and strategies. This research might describe and categorize mathematical situations, analyze attributes of mathematical representations, or design conceptual supports to increase student learning. The value of this research depends on the strength of its analytical framework and its accessibility to others. Other evidence is more empirical. The essence of empirical research is that evidence has been gathered and analyzed in a systematic, focused way so as to address a clearly formulated question. Researchers make public the assumptions they have made and the methods they have used to gather and analyze their data. They explain how their conclusions follow from a careful analysis of those data. They report their methods and findings in a way that makes informed critique possible. In many cases—though not all—adherence to these methods allows others to repeat their work. Some empirical studies are largely descriptive. They can illuminate how learning occurs under various conditions, suggest what the learner brings to the teaching situation, or describe how the learner understands what is being taught. Some studies portray relationships. They can suggest how differences in conditions under which learning occurs might be related to differences in what is learned. Other studies are experimental. Through the manipulation of learning conditions, they can suggest how changes in those conditions might cause changes in learning. Whether a study is a tightly controlled experiment or an observation of a single child’s performance, it can be of high or low quality. Box 1–1 describes several determinants of quality in research. In turn, the quality of the evidence determines the level of confidence with which a conclusion, observation, or recommendation is made. In addition, no single study can provide conclusive evidence on broad educational issues. It is therefore necessary to look at as many studies as

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Adding + It Up: Helping Children Learn Mathematics Box 1–1 The Quality of Research Studies Several indicators of quality must be evaluated in assessing studies of mathematics education. This report is based on research that meets standards of relevance, soundness, and generalizability. Relevance A research study is relevant if it addresses or produces data that speak to any of a number of components of mathematics learning. The teaching and learning of mathematics involve both desired goals and various mental processes. These goals and processes include the content to be learned, materials for teaching, activities undertaken by teachers and students to promote learning, and assessment of what has been learned. Teaching and learning also take place in a social context ranging from the classroom to the nation as a whole. Teaching and learning depend not only on teachers and students but also on support from a variety of enablers: policy makers, teacher educators, publishers, researchers, administrators, and others. A relevant study of mathematics learning might, for example, lead to a sharper understanding of desired learning processes and outcomes. It might reveal features of good practice or evaluate tradeoffs among various educational alternatives. Soundness The soundness of a research study concerns the extent to which the study supplied the data needed to address the research question. A study’s soundness therefore depends on the suitability of the methods used to achieve the results obtained. Were the groups of participants adequate in size and composition, or were they biased or limited in some fashion? Did the methods generate credible, reliable, and valid data? Were the methods specified so that they could be repeated? Was the data analysis appropriate to the methods, carefully conducted, replicable, and penetrating? Was the data presentation clear and complete? Were the conclusions warranted by the results and appropriately qualified? Generalizability The generalizability of a study concerns the extent to which its findings can be applied to circumstances beyond those of the study itself. Was the class typical in size and composition? Were the time allocated to mathematics and the materials and equipment used in the study characteristic of today’s mathematics instruction? Did the conditions of the study depart from those of an ordinary classroom? Were the teachers or students somehow anomalous?

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Adding + It Up: Helping Children Learn Mathematics possible that are relevant to a particular question. The confidence with which an observation, conclusion, or recommendation is made is increased when all the relevant evidence supports the same point. This feature of convergence is reinforced when the evidence has been collected in different places, under different circumstances, and by different researchers working independently. In particular, findings should stand up across different groups of students and teachers, and ideally they should have been obtained using different methods for gathering data. Findings also should fit well within a larger network of evidence that makes good common and theoretical sense. Determining the degree of convergence in existing evidence demands discrimination and judgment. It cannot be ascertained simply by tallying studies. One problem in weighing the evidence on a given issue in education is that a fully convergent database that speaks directly to the issue and yields unequivocal findings is seldom, if ever, available. The findings from experimental studies of mathematics learning often conflict. Data from non-experimental studies of relationships generally are ambiguous with respect to causality. Descriptive data can help frame an issue but usually do not address the question of which processes might lead to which learning outcomes. Ostensibly comparable studies can differ in key features, making it difficult to decide whether the data are really comparable. Much of the evidence is still in the form of demonstrations that selected children can learn certain topics in certain ways, and large-scale studies have not yet been done. All these factors require that the research evidence be interpreted. Arguments and recommendations have to be constructed by drawing on professional judgment. Inductive reasoning must be used to make connections among studies, note patterns, fill in gaps, and attempt to explain why contradictory findings should be ignored or downplayed. We have sought to identify in this report conclusions that depend on such interpretations of the available evidence. The Role of Research in Improving School Mathematics A premise of this report is that sound research can help guide the design of effective mathematics instruction. Yet research cannot be the only basis for making instructional decisions in mathematics. First, as we stated earlier, research, by itself, cannot tell educators which of their learning goals are most important or how they should set priorities. Only after such goals have been established can research generate information to help educators decide

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Adding + It Up: Helping Children Learn Mathematics whether goals are feasible and, if so, how to accomplish them. In short, instructional decisions, as well as the research supporting them, must be guided by values. Second, decisions about how to help students reach learning goals can never be made with absolute certainty. As the famous American psychologist William James noted at the end of the nineteenth century, psychology’s description of “the elements of the mental machine…and their workings”10 does not translate directly into a prescription for educational practice. James warned: “You make a great, a very great mistake, if you think that psychology, being the science of the mind’s laws, is something from which you can deduce definite programmes and schemes and methods of instruction for immediate schoolroom use.”11 Education is an applied field: no matter what the state of theoretical knowledge from psychology or elsewhere, the conditions of practice make the success of any procedure contingent. Just as a doctor cannot be 100 percent sure that this operation will cure that patient, or an engineer that this design cannot fail, so a teacher cannot know exactly what approach will work with a particular student or class. Decisions about procedures can be made with greater confidence when high-quality empirical evidence is available, but decisions about educational practice always require judgment, experience, and reasoned argument, as well as evidence. Third, the research base for mathematics learning is diverse in the methods used and contains diverse kinds of results. For example, observational methods—including clinical interviews with students—are faithful to actual conditions and environments. But they may have trouble controlling irrelevant variables that might have been responsible for the results. It can be challenging to draw scientifically sound conclusions from a selected set of observations. In contrast, experimental methods—including studies comparing an experimental and control group—establish stronger bases for drawing conclusions, although even these conclusions have important limitations and qualifications. Experimental control is a challenge because the classroom teaching of mathematics constitutes a system of mutually dependent elements that cannot easily be disentangled so that each element can be controlled. Experimental rigor often requires narrowing one’s focus to a single feature of an instructional method or to a limited amount of mathematical content. Furthermore, evidence that an instructional method produced a certain result in a controlled situation does not guarantee that it would produce the same result in a situation when, for example, different mathematical content were being taught or the students had different backgrounds and experience. There are pros and cons for each methodological approach, and we believe that the great-

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Adding + It Up: Helping Children Learn Mathematics est progress is made when together they offer converging evidence, that is, a coherent picture of how mathematics learning occurs. The interpretation and use of research always require a search for commonalities in evidence from diverse sources. Finally, most published studies in education confirm the predictions made by the investigators. Information obtained from research therefore is particularly useful when it goes beyond the sought-after effects. The interpretation and use of such information require an examination of the conditions under which the effects were obtained and other possible effects. For example, the students in the groups under investigation may have met other learning goals than those targeted by the instructional methods. High-quality research should play a central role in any effort to improve mathematics learning. In summary, high-quality research should play a central role in any effort to improve mathematics learning. That research can never provide prescriptions, but it can be used to help guide skilled teachers in crafting methods that will work in their particular circumstances. For many important issues in mathematics education, the body of evidence is simply too thin at present to warrant a comprehensive synthesis. Where convergent evidence is not available, we have attempted in this report to suggest the sorts of evidence that would be needed for good inferences to be drawn. About This Report The Committee on Mathematics Learning was created at the request of the Division of Elementary, Secondary, and Informal Education in the National Science Foundation’s Directorate for Education and Human Resources and the U.S. Department of Education’s Office of Educational Research and Improvement. The sponsors were concerned about the shortage of reliable information on the learning of mathematics by schoolchildren that could be used to guide best practice in the early years of schooling. The charge to the committee lists three goals: To synthesize the rich and diverse research on pre-kindergarten through eighth-grade mathematics learning. To provide research-based recommendations for teaching, teacher education, and curriculum for improving student learning and to identify areas where research is needed. To give advice and guidance to educators, researchers, publishers, policy makers, and parents.

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Adding + It Up: Helping Children Learn Mathematics Additionally, the committee was charged with describing the context of the study with respect to what is meant by successful mathematics learning, what areas of mathematics are important as foundations in grades pre-K-8 for building continued learning, and the nature of evidence and the role of research in influencing and informing education practice, programs, and policies. The goals for the study cover a broad grade span and a number of different facets of mathematics education—learning, teaching, teacher education, and curriculum. Further, the report is to provide guidance to a diverse audience. The complexity of the task and the time constraints imposed led the committee to make some judicious choices and decisions. First, as indicated earlier, we chose to focus primarily on the domain of number in order to make our task manageable and to present findings on the area of mathematics of most interest to our audience. Second, because we could not assume a common background, necessary background had to be included in the report. Finally, we decided to limit the detail reported on individual studies in order to make the report more accessible. To meets its charge, the committee conducted an extensive examination of the research literature relevant to the learning of mathematics in the pre-kindergarten through eighth-grade years. We did not review other bodies of literature that have an impact on learning such as textbooks, curriculum projects, assessments, and standards documents. In reviewing the research, we asked ourselves what promising changes in practice the evidence suggests and what else needs to be known to improve practice. We then concluded how teaching, curricula, and teacher education should change to improve mathematics learning in these critical years. In chapter 2, we describe the current status of mathematics curricula, teaching practices, assessments, and student achievement. In response to the charge to describe what areas of mathematics are important, chapter 3 outlines the domain of number and discusses what it means to learn about number in the pre-kindergarten to eighth-grade years. Chapter 4 details the strands of what we refer to as “mathematical proficiency,” which we have established as what is meant by successful mathematics learning in the elementary school and middle school years. Chapters 5, 6, 7, and 8 then present a portrait of mathematics learning that spans the grade levels considered in this report. Chapter 5 considers what students learn outside school and bring with them to the formal study of mathematics. Chapter 6 describes the process by which students acquire mathematical proficiency with whole numbers, and chapter 7 addresses proficiency with other number systems. Chapter 8 describes the process by which

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Adding + It Up: Helping Children Learn Mathematics students achieve proficiency in domains other than number, including beginning algebra, measurement and geometry, and statistics and probability. Chapters 9 and 10 focus on the teaching of mathematics. Chapter 9 describes what we know from research about teaching for mathematical proficiency. Chapter 10 discusses what it means to be a proficient teacher of mathematics and describes the kinds of experiences teachers need to develop this proficiency. Finally, chapter 11 presents the committee’s recommendations for teaching practices, curricula, and teacher education, offering some suggestions for parents, educators, and others. Chapter 11 also recommends the various types of research needed if both practice and policy are to be improved. Notes 1.   Butts, 1955, p. 454; Cubberley, 1920, pp. 17, 235; Kouba and Wearne, 2000; Thorndike, 1922. 2.   Snow, Burns, and Griffin, 1998, p. 20. The case for critical reading skill and literacy by adolescence is addressed by Moore, Bean, Birdyshaw, and Rycik, 1999. 3.   Binkley and Williams, 1996; Elley, 1992. 4.   Fuson, Smith, and Lo Cicero, 1997; Griffin, Case, and Siegler, 1994; Snow, Burns, and Griffin, 1998. 5.   One well-known program is called Reading Recovery (see Snow, Burns, and Griffin, 1998, pp. 255–258), which is designed for the lowest fifth of a first-grade class. In that program, the teacher, who has received extensive instruction in the reading process and its implications for teaching, notes an individual child’s literacy strategies and knowledge and then engages the child in a structured series of activities. Each child is tutored individually for a half hour a day for up to 20 weeks. 6.   Wagner and Stanovich, 1996. 7.   Anderson, Wilson, and Fielding, 1988. 8.   See chapter 2 for data on the level of instructional emphasis fourth- and eighth-grade teachers reported giving to number and operations. 9.   See, for example, the SCANS study (U.S. Department of Labor, Secretary’s Commission on Achieving Necessary Skills, 1991). 10.   James, 1899/1958, p. 26. 11.   James, 1899/1958, p. 23. References Anderson, R.C., Wilson, P.T., & Fielding, L.G. (1988). Growth in reading and how children spend their time outside of school. Reading Research Quarterly, 23, 285–303. Binkley, M., & Williams, T. (1996). Reading literacy in the United States: Findings from the IEA Reading Literacy Study (NCES-96–258). Washington, DC: National Center for Education Statistics. Available: http://nces.ed.gov/spider/webspider/96258.shtml. [July 10, 2001].

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Adding + It Up: Helping Children Learn Mathematics Butts, R.F. (1955). A cultural history of Western education. New York: McGraw-Hill. Cubberley, E.P. (1920). The history of education. Boston: Houghton Mifflin. Elley, R. (1992). How in the world do students read? The Hague, The Netherlands: International Association for the Evaluation of Educational Achievement. Fuson, K.C., Smith, S.T., & Lo Cicero, A.M. (1997). Supporting Latino first graders’ ten-structured thinking in urban classrooms. Journal for Research in Mathematics Education, 28, 738–766. Griffin, S., Case, R., & Siegler, R.S. (1994). Rightstart: Providing the central conceptual prerequisites for first formal learning of arithmetic to students at risk for school failure. In K.McGilly (Ed.), Classroom lessons: Integrating cognitive theory and classroom practice. Cambridge, MA: MIT Press/Bradford Books. James, W. (1958). Talks to teachers. New York: Norton. (Original work published 1899) Kouba, V.L., & Wearne, D. (2000). Whole number properties and operations. In E.A. Silver & P.A.Kenney (Eds.), Results from the seventh mathematics assessment of the National Assessment of Educational Progress (pp. 141–161). Reston, VA: National Council of Teachers of Mathematics. Moore, D.W., Bean, T.W., Birdyshaw, D., & Rycik, J.A. (1999). Adolescent literacy: A position statement for the Commission on Adolescent Literacy of the International Reading Association. Newark, DE: International Reading Association. Summary available: http://www.reading.org/pdf/1036.pdf. [July 19, 2001]. Snow, C.E., Burns, M.S., & Griffin, P. (Eds.). (1998). Preventing reading difficulties in young children. Washington, DC: National Academy Press. Available: http://books.nap.edu/catalog/6023.html. Thorndike, E.L. (1922). The psychology of arithmetic. New York: Macmillan. Wagner, R.K., & Stanovich, K.E. (1996). Expertise in reading. In K.A.Ericsson (Ed.), The road to excellence: The acquisition of expert performance in the arts and sciences, sports, and games (pp. 189–225). Mahwah, NJ: Erlbaum. U.S. Department of Labor, Secretary’s Commission on Achieving Necessary Skills. (1991). What work requires of schools: A SCANS report for America 2000. Washington, DC: Author. (ERIC Document Reproduction Service No. ED 332 054).

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