2
THE STATE OF SCHOOL MATHEMATICS IN THE UNITED STATES
The U.S. system for teaching children mathematics is large, is complex, and has numerous components. Children’s mathematical achievement, however, is ultimately determined and constrained by the opportunities they have had to learn. Those opportunities are determined by several major components of school mathematics. The curriculum contains learning goals spelling out the mathematics to be studied. It also includes instructional programs and materials that organize the mathematical content, together with assessments for determining what has been learned. In addition, and of primary importance, it is through teaching that students encounter the mathematical content afforded by the curriculum.
In every country, the complex system of school mathematics is situated in a cultural matrix. Mathematics teaching is not the same in the United States as in, say, Japan or Germany,^{1} and the curricula are different as well.^{2} Countries differ in such global characteristics as the centralization of educational policies, the organization and types of schools, and the success of efforts to provide universal access to education. The status of teachers in the society, the composition and mobility of the student population, and the extent to which external examinations determine one’s life chances all constrain the ways in which mathematics is taught and learned. Countries also differ in more specific ways: parents, teachers, and students have different beliefs about the value of hard work and the importance of mathematics for one’s education; whether and how students are grouped for mathematics instruction varies; mathematics textbooks are written, distributed, and used in diverse ways; and there is variation in the prevalence of tutors or special schools to coach
students for mathematics tests. Each country provides a unique setting for school mathematics, one that very much determines how students are taught, what they learn, how successful they are, and how satisfied society is with the products of the system.
Education in the United States is marked by a diverse, mobile population of students and teachers, a variety of organizational structures, and minimal centralized control over policies and practices. The U.S. system of school mathematics has evolved over several centuries in accordance with these characteristics. Not only do the components of the U.S. system differ from those of other countries, but they are organized and operate differently. To understand the possibilities for improving children’s learning of mathematics, one needs a sense of how the elements of U.S. school mathematics currently function.
To understand the possibilities for improving children’s learning of mathematics, one needs a sense of how the elements of U.S. school mathematics currently function.
In the past half century, a number of research studies have examined differences in the mathematics learned by students in various educational systems. Some of these studies have also looked at various features of the systems that might help researchers understand and interpret the pattern of results. To date, the most comprehensive study to be analyzed in detail has been the Third International Mathematics and Science Study (TIMSS), which was conducted in the mid1990s. Over 40 countries participated in TIMSS. Tests in science and mathematics, as well as questionnaires about their studies and their beliefs, were given to students midway through elementary school (grade 4 in the United States), midway through lower secondary school (U.S. grade 8), and at the end of upper secondary school (U.S. grade 12). Questionnaires about beliefs, practices, and policies were also given to these students’ teachers and school administrators. Unique features of TIMSS included an extensive examination of textbooks and curriculum guides from many of the participating countries, a video study of eighthgrade mathematics classes in three countries, and case studies of educational policies in those three countries.
The results from TIMSS have been widely reported in the media, catching the attention of politicians, policy makers, and the general public. Many people have compared various practices, programs, and policies in the United States with those of highachieving countries. Such comparisons are interesting but at best can only be suggestive of the sources of achievement differences. TIMSS provides no evidence that a single practice—say, the amount of homework assigned, the particular textbook used, or how periods of mathematics instruction are arranged during the school day—is responsible for higher mathematics test scores in one country than in another. The countries
participating in TIMSS vary in many respects—educationally, socially, economically, historically, culturally—and in each of those respects, they vary along many different dimensions. In the absence of more evidence than TIMSS can provide, one cannot select one practice and claim that if it were changed to be more like that of highscoring countries, scores in the United States would rise.^{3} Studies like TIMSS can at best generate conjectures that need to be tested in the complex system of school mathematics that exists in any county. In this report, we use data from TIMSS and other international studies to help describe practice and performance in the United States— sometimes in contrast to that of other countries but never assuming a simple causal relation between a specific practice and performance.
This chapter is intended primarily to give an overall picture of U.S. mathematics education, describing the experiences and achievement of most students. But it should be emphasized that U.S. education is quite diverse, partly because of an unequal distribution of needs and resources, and partly because of the principle of local control. Thus, this chapter also attempts to describe that diversity, particularly with respect to student achievement.
In this chapter, we first take up in turn four central elements of school mathematics—learning goals, instructional programs and materials, assessment, and teaching—discussing the current status of each in the United States. We then examine the preparation and professional development of U.S. teachers of mathematics. Finally, we look at a major indicator of the health of the whole system, student achievement results, both across time and internationally.
Learning Goals
The U.S. Constitution leaves to the separate states the responsibility for public education. State and local boards of education have the authority to determine the mathematics that students learn as well as the conditions under which they learn it. Many state boards of education have created curriculum standards and frameworks, and some have specified criteria that educational materials (principally textbooks) must meet if they are to be approved. Thus, each state can, in principle, specify quite different goals for learning mathematics at each grade level, and each local district can make adjustments as long as they fall within the state guidelines.
A major effort to set comprehensive learning goals for school mathematics at the national level was undertaken in 1989 by the National Council of Teachers of Mathematics (NCTM) with the release of Curriculum and Evalu
ation Standards for School Mathematics.^{4} The document outlined and illustrated goals in the form of standards to be met by school mathematics programs. It called for a broadened view of mathematics and its teaching and learning, emphasizing the development of students’ “mathematical power” alongside more traditional skill and content goals. The NCTM later produced Professional Standards for Teaching Mathematics^{5} and Assessment Standards for School Mathematics.^{6} Beginning in 1995, it embarked on a process to revise all three documents, resulting in Principles and Standards for School Mathematics,^{7} which was released in April 2000.
Although none of the NCTM documents established national standards for school mathematics in an official sense, much of the activity in U.S. mathematics education since 1989 has been based on or informed by the ideas in those documents. Many school mathematics textbooks claim to be aligned with the NCTM standards, and 13 curriculum projects were funded by the National Science Foundation to produce materials for elementary, middle, or high school that embodied the ideas expressed in the standards documents.^{8} The NCTM standards of 1989 launched the socalled standards movement, with standards for other school subjects appearing over the following decade.^{9} In 1994 the reauthorization of Title I of the Elementary and Secondary Education Act furthered boosted the movement. Title I provides supplemental financial assistance to local educational agencies to improve teaching and learning in schools with high concentrations of children from lowincome families. The reauthorization “requires states to develop challenging standards for performance and assessments that measure student performance against the standards.”^{10} It should also be noted that A Nation at Risk, America 2000, and Goals 2000 (under Presidents Reagan, Bush, and Clinton, respectively) all called for higher, measurable standards in education.^{11}
As of 1999, 49 states reported having content standards in mathematics and several states were in the process of revising their standards.^{12} These standards (sometimes called curriculum frameworks) describe what students should know and be able to do in mathematics. Most of the state standards reflect the 1989 NCTM standards and either repeated verbatim or were adapted from the document. Early versions of these state standards were organized into grade clusters (e.g., grades K4), but some states (e.g., California, Texas, North Carolina, and Virginia) have recently developed gradebygrade standards.^{13}
Current state standards and curriculum frameworks vary considerably in their specificity, difficulty, and character, as illustrated by the widely divergent ratings they received in three reviews conducted by the American Federa
tion of Teachers, the Fordham Foundation, and the Council for Basic Education.^{14} The conflicting reports have created confusion among parents, teachers, and policy makers alike. According to one analysis of the reviews:
While…multiple analyses of state standards are better than no analyses, the grade differentials among the three reports are confounding—enough so to make state leaders either throw up their hands in utter bewilderment or embrace a high mark and ignore the others. Both responses threaten to defeat the very purpose of the reports. For example, Florida received a D from one appraiser and the equivalent of an A from another in mathematics. In both English and mathematics, Michigan received an F from one appraiser and a Bplus from another.^{15}
Often missing from the public discussion of such reports are the processes and criteria that gave rise to the ratings, which has only added to the confusion.
Some caveats about standards deserve mention. First, most groups charged with developing standards for a school subject have strong expectations for learning in that subject. They may spend more time devising the standards than checking the feasibility of achieving them in the time available for learning. One analysis of standards for 14 subjects found that it would take nine additional years of schooling to achieve them all.^{16} Thus, it is important that states and districts avoid long lists that are not feasible and that would contribute to an unfocused and shallow mathematics curriculum.
Second, when grade bands (e.g., grades preK2) are used in specifying standards, it is important to clarify that each goal does not have to be addressed at every grade in a band. Such redundancy again contributes to the dissipation of learning efforts and interferes with the acquisition of proficiency.
Third, states and districts need to decide what they will do when students do not meet gradelevel goals. Children enter school with quite different levels of mathematical experience and knowledge. Some need additional learning time and support for learning if they are to meet the goals. As schools shift to standardsbased mathematics curricula for grades preK to 8 with challenging gradelevel goals, thorny questions arise as to whether and how special accommodations will be made for some students and what criteria will be imposed for promotion to the next grade.
A recent comparative analysis of mathematics assessments given to U.S. and Japanese eighth graders revealed some striking differences in the expectations held for each group, with much lower expectations in the United States. The author concluded by pointing to the need for gradelevel goals:
To achieve the coherence and focus observed in the Japanese materials, the Curriculum and Evaluation Standards for School Mathematics need to be further extended to provide grade level guidance about focus and primary activities for given years. This step to achievement and delivery standards for school mathematics is curricularly achievable within the framework outlined by the NCTM content standards. Whether it is politically acceptable or systematically implementable are larger and more volatile questions.^{17}
On balance, we see the efforts made since 1989 to develop standards for teaching and learning mathematics as worthwhile. Many schools have been led to rethink their mathematics programs, and many teachers to reflect on their practice. Nonetheless, the fragmentation of these standards, their multiple sources, and the limited conceptual frameworks on which they rest have not resulted in a coherent, wellarticulated, widely accepted set of learning goals for U.S. school mathematics that would detail what students at each grade should know and be able to do. Part of our purpose in this report is to present a conceptual framework for school mathematics that could be used to move the goalsetting process forward.
Instructional Programs and Materials
Learning goals are inert until they are translated into specific programs and materials for instruction. What is actually taught in classrooms is strongly influenced by the available textbooks because most teachers use textbooks as their primary instructional materials.^{18} As of 1998, 12 states—including the very large markets of California and Texas—had policies in which the state either chose the materials that students would use or drew up a list of textbooks and materials from which districts had to choose, though sometimes only if they wanted to use state funds for the purchase. Another seven states recommended materials for use.^{19}
Surveys of U.S. teachers have consistently shown that nearly all their instructional time is structured around textbooks or other commercially produced materials, even though teachers vary substantially in the extent to which they follow a book’s organization and suggested activities.^{20} In 1980 one researcher maintained that the chalkboard and printed textbooks were the predominant instructional media in mathematics classes,^{21} a verdict substantiated by recent data from the National Assessment of Educational Progress (NAEP). Responding to a questionnaire in 1996, teachers of three fifths of the fourth graders and of almost three fourths of the eighth graders in the
NAEP sample said that they used the mathematics textbook almost every day.^{22} Observational studies of elementary school classrooms, however, reveal that at least some teachers pick and choose from the mathematics textbook even as they follow its core content.^{23}
The American textbook system is notable for being heavily market driven. In that market, publishers must contend with multiple and sometimes contradictory specifications:
If we lived in a country with one national curriculum, then textbook publishers could compete with each other in the effort to produce a book that would best mirror that one curriculum. But we are not such a country. Instead, we have dozens of powerful ministries of education issuing undisciplined lists of particulars that publishers must include in the textbooks. Since publishers must sell in as many jurisdictions as possible in order to turn a profit, their books must incorporate this melange of testoriented trivia, pedagogical faddism, and inconsistent social messages.^{24}
To be sold nationwide, a textbook needs to include all the topics from the standards and curriculum frameworks of at least those influential states that officially adopt lists of approved materials. Consequently, the major U.S. school mathematics textbooks, which collectively constitute a de facto national curriculum, are bulky, address many different topics, and explore few topics in depth.
In comparison with the curricula of countries achieving well on international comparisons, the U.S. elementary and middle school mathematics curriculum has been characterized as superficial, “underachieving,” and diffuse in content coverage.^{25} Successful countries tend to select a few critical topics for each grade and then devote enough time to developing each topic for students to master it. Rather than returning to the same topics the following year, they select new, more advanced topics and develop those in depth. In the United States, not a single topic in the grade preK to 8 mathematics curriculum is seen as the province of one grade, to be learned there once and for all. Instead, topics such as multidigit computations are distributed over several years, with one digit added to the numbers each year. Students invariably spend considerable time on topics they encountered in the previous grade.^{26} At the beginning of each year and of each new topic, numerous lessons are devoted to teaching what was not learned or was learned inadequately the year before. Because the curriculum is consequently so crowded, depth is seldom achieved, and mastery is deferred. Not surprisingly, inter
national curriculum analyses have found that U.S. mathematics textbooks cover more topics, but more superficially, than do their counterparts in other countries.^{27}
The massive amount of review created by the inadvertent de facto curriculum set by textbooks wastes learning time and may bore those students who have already mastered the content. Such constant review is also counterproductive. It is much easier to help students build correct mathematical methods at the start than to correct errors that have been learned and practiced for a year or more. As the following chapters show, the lack of concentrated attention to core topics militates against powerful learning.
Further attributes of this de facto curriculum also are problematic. For example, even with their supplementary materials, many textbooks fail to discuss student strategies or progressions in student thinking. They also frequently omit explanations of mathematical processes. Further, decorative artwork with little connection to textbook content sometimes confuses or distracts students.^{28} Research indicates that students can learn more mathematics than is usually offered them in the early grades, so the U.S. elementary school mathematics curriculum could be made more challenging. If the curriculum of the early grades were more ambitious, and if instruction were focused on mastery of topics rather than unwarranted review, teachers of the middle and upper grades could concentrate on teaching core gradelevel topics more thoroughly.
The short timelines between the formulation of state learning goals and the selection of textbooks create a textbook production schedule that seldom permits both consultation of research about student learning and field testing followed by revision based on actual use in schools.^{29} Most students today are using materials that were produced under heavy (perceived or actual) market constraints. In contrast, some recent school curriculum development projects that were supported by the National Science Foundation built research and pilot testing into their design.
An expert panel convened by the Department of Education recently evaluated materials from these NSFfunded projects as well as from other programs. The panel labeled some curriculum programs as “exemplary” and others as “promising” based on a review process that examined evidence of the programs’ effectiveness.^{30} Almost immediately, the panel’s conclusions were called into question.^{31} Just as with ratings of standards, evaluations of curriculum materials have led to divergent ratings depending on the group doing the evaluating.^{32}
In some countries, including England, France, Hong Kong, Singapore, and the Netherlands, there are permanent national centers or institutes that conduct multiyear research and curriculum development efforts in school mathematics. In the United States, the government has funded both a research center for mathematics learning at a single institution and projects to develop materials for teaching and learning mathematics at a number of other institutions.^{33} Typically, the curriculum development programs have required, as part of the project, both pilot testing of the materials while they are under development and the collection of evidence on the effectiveness of the materials, once developed. In some cases, the evaluation studies have been only perfunctory and the evidence gathered of poor quality. In others the support has resulted in sustained researchbased curriculum development that systematically uses evidence as to what U.S. students can learn.^{34} Such a development program can be interactive, with improved learning materials yielding improved student learning that, in turn, yields improved and evenmoreambitious learning materials.
Developing teachers’ capacity to acquire and use good instructional materials is also a problem. Textbook selection processes can be overwhelming. Committee members usually do not have time to examine carefully the continuity of treatment of topics or the depth and clarity of the conceptual development facilitated by the materials. Instead, their focus is often on superficial features such as the appearance of the materials and whether all goals on a checklist are addressed. The problems created by checklists are especially keen in states and local districts with large numbers of specified special criteria. Failure to meet even a few of these criteria can eliminate an otherwise strong program.^{35}
The methods used in the United States in the twentieth century for producing school mathematics textbooks and for choosing which textbooks and other materials to use are not sufficient for the goals of the twentyfirst century. The nation must develop a greater capacity for producing highquality materials and for using effectively those that are produced. In subsequent chapters, we cite research on children’s learning that can guide that production and use.
Assessments
In general, assessments of children’s mathematics learning fall into two categories: internal and external. Internal assessments are those used by teachers in monitoring and evaluating their students’ progress and in making
instructional decisions. Such assessments range from the informal questions a teacher might ask about a student’s work to an endofyear examination. They arise from the teachinglearning process in the classroom. External assessments, in contrast, come from outside, from projects gathering comparative research data or mandated by state or local districts as part of their evaluation programs.
Relative to the vast literature on external assessments and their results, little uptodate information is available on how U.S. teachers conduct internal assessments in mathematics, particularly those activities such as classroom questioning, quizzes, projects, and informal observations. Even less attention appears to have been paid to how teachers’ assessments might help improve mathematics learning. According to one analysis, “Aside from teachermade classroom tests, the integration of assessment and learning as an interacting system has been too little explored.”^{36}
As part of the 1996 NAEP mathematics assessment, teachers responded to several questions about their testing practices.^{37} Fourth graders were usually tested in mathematics once or twice a month, with about a third being tested once or twice a week. More frequent testing was associated with lower achievement.^{38} Eighth graders were somewhat more likely to be tested weekly. At both grades, teachers appeared to be responding to calls arising from the standards movement for less multiplechoice testing in favor of tests on which students supply written responses.^{39} Multiplechoice testing is still prevalent, however, stimulated perhaps by the increased number of such tests provided by publishers to accompany their textbooks. Two thirds of fourth and eighth graders had teachers who reported that they used multiplechoice tests to assess students’ progress at least once or twice a year, most as often as once or twice a month.^{40} In part, teachers are attempting to prepare students for external assessments by using multiplechoice items on their own tests.
The form of multiplechoice test items appears not to be as big a problem as the nature of the items and the conditions under which they are typically administered in the United States. An examination given to a national sample of eighth graders in Japan as part of a Special Study on Essential Skills in Mathematics was composed entirely of multiplechoice items, yet it was judged substantially more challenging than the 1992 NAEP mathematics assessment given to U.S. eighth graders, which contained both multiplechoice items and items on which students had to write either a brief or lengthy response.^{41} The difference was that the Japanese exam contained about half as many items as the U.S. exam; the items were longer, demanded more reading and analysis, and were more focused on strategies for problem solving.
Exhortations to change assessments, whether internal or external, clearly need to focus on more than just item format. In the remainder of this section, we examine current external assessment practices and results.
In recent years, largely because of language in the reauthorization of Title I, many states have designed and implemented their own assessments, usually aligned with newly developed state standards or curriculum frameworks. Many of these assessments are intended to have high stakes. They may have financial or other consequences for districts, schools, teachers, or individual students. In some cases, promotion or even a high school diploma may depend on a student achieving a passing score. As of 1998, 48 states and the District of Columbia had instituted testing programs, typically at grades 4, 8, and 11, and usually in mathematics, language arts, science, and technology.^{42}
Many states report the results of their highstakes assessments by school or by district to identify places that are most in need of improvement. The states’ responses to those results vary. Some states have the authority to close, take over, or “reconstitute” a failing school. To date, only a few states have ever used such sanctions.^{43} Florida awards additional funds to schools that perform near the bottom and also to schools that perform near the top.^{44} When schools or districts with poor results do not show sufficiently rapid improvement, some states revoke accreditation, close down the school, seize control of the school, or grant vouchers so that students may choose to enroll elsewhere.
Currently, 19 states require that in order to graduate from high school, students must pass a mandated assessment, and several other states are phasing in such a requirement.^{45} In TIMSS, countries with rigorous assessments at the end of secondary education outperformed other countries at a comparable level of economic development; such assessments, however, were probably not the most important determinant of achievement levels.^{46} In response to calls for an end to social promotion, some states and districts have begun requiring gradelevel mastery tests for promotion, typically in grades 4 and 8. Interestingly, there is some evidence to suggest that there is an almost inverse relationship between statewide testing policies and students’ mathematics achievement:
Among the 12 highestscoring states in 8th grade mathematics in 1996, …none had mandatory statewide testing programs in place during the 1980s or early 1990s. Only two of the top 12 states in the 4th grade mathematics had statewide programs prior to 1995. By contrast, among the 12 lowestscoring states,…10 had extensive student testing programs in place prior to 1990, some of which were associated
with highly specified state curricula and an extensive menu of rewards and sanctions.^{47}
Of course, this relationship does not imply that simply easing statewide test policies would improve achievement.
To give teachers, students, parents, and other caregivers sufficient time to prepare for highstakes assessments, states typically administer them for several years before the consequences take effect. During these trial runs, the failure rates are sometimes alarmingly high. In Arizona, for example, only 1 in 10 sophomores passed the mathematics test first given in the spring of 1999. That same spring, only 7% of Virginia schools were able to achieve a 70% passing rate, which was to become the condition for accreditation in 2007. In response to these results, some states have begun to relax their expectations, reconsider the test, or withdraw it altogether. Wisconsin, for example, yielded to pressure from parents and withdrew its high school graduation test. Massachusetts and New York set lower passing scores for their exams.^{48}
Most states report the level of student results on their assessments by setting socalled cut scores to define categories with such labels as advanced, proficient, needs improvement, and failing,^{49} terms similar to those used in NAEP: advanced, proficient, and basic. When results on state assessments are compared with the state results in NAEP, the proportions of students reaching the proficient level are often higher.^{50} Some researchers, politicians, and policy makers have concluded from this discrepancy that most state tests do not reflect sufficiently high expectations.^{51} Others argue instead that minimum competence and high expectations are different goals that cannot be measured by the same assessment and certainly not with the same cut scores. Thus, the results appear discrepant because the same categories are used to describe performance on assessments with very different goals.
Many states and school districts use standardized tests^{52} (which may or may not coincide with the state assessments discussed above) to assess how their students are achieving. Commercially published standardized mathematics achievement tests are quite variable in the topics they cover and in the proportion of these topics emphasized at each grade level.^{53} The tests frequently are not aligned with the teaching materials used in a district or even with the goals of the district. This misalignment further dilutes teaching efforts, as teachers must add to their long list of goals coverage of the major topics emphasized on a specific standardized test.
Standardized tests can have other negative consequences. The word standardized is likely to carry certain connotations: that such a test is more objective than other instruments, that it contains mostly gradelevel items, that it
was developed or sanctioned by experts in the domain, that it reflects important learning goals in a balanced way, and that it represents and assesses what students know about the content that the state or district has prescribed for that grade level. In fact, many standardized tests have few or none of these characteristics.^{54}
Most standardized tests might be called “comparison” tests because their function is to rank order students, schools, and districts or to compare them with another group that was selected as typical. Items are chosen to range widely in difficulty in part to disperse students’ scores. That range allows for half the students to be classified as “below average” and the other half as “above average.” The tests do not include many items that only a few students get right or that only a few get wrong, because such items do not help distinguish among students.^{55} The omission of these items may mean that some important aspects of mathematics that students have or have not learned are not tested. For tests designed for making comparisons, however, the omission is necessary.
If the purpose of a test is to assess whether students have met specific goals, test designers can choose items to span the important mathematics to be learned.
In contrast, if the purpose of a test is to assess whether students have met specific goals, test designers can choose items to span the important mathematics to be learned. When the goal is to determine students’ proficiency with gradelevel topics, the cut scores are then set to indicate various levels of proficiency. Students and teachers know where to aim their efforts, and students can study for the test with the goals in mind. If the students have learned well, large proportions of them can achieve high proficiency, and there is no need to label half of them as below average (or even to rank them at all). Standardized tests have traditionally been kept secret so that questions can be reused. In recent years, this practice has come under fire. If students are to reach publicly accepted standards, the argument goes, they need to know what type of performance will be expected of them.^{56} They should have an opportunity to learn the mathematical content and processes on which they will be examined. At the same time, they need to become familiar with the instructions, the organization of the assessment, and the format of the items, so that such nonmathematical considerations do not prevent them from showing what they know. Legally and ethically, when the stakes are high, students should be provided with sample assessments or at least sample items that are representative of the actual assessments.^{57}
The movement over the past four decades to hold schools accountable for students’ performance has resulted in increased highstakes testing of “minimum competency” in mathematics and other subjects. Many states give competency tests at several grade levels, including high school exit exams. Performance on the mathematics portions of such tests has often been con
siderably below what was anticipated or desired. Many districts meanwhile have continued to use standardized comparison tests that were not necessarily aligned with their textbooks, their state goals, or their state competency tests. The combination of standardized comparison tests and state competency tests can overwhelm teachers, who have to prepare students for two kinds of highstakes tests about which they often know very little.
State competency tests in mathematics are often given first at a grade level at which many students are already far behind and likely to have difficulty catching up. If such tests are to be used, they need to be accompanied in earlier grades—and throughout all grades—by other assessments that would enable teachers to make their instruction more effective. In particular, such assessments could identify students who are not achieving and need special help so that they do not fall further behind. This linking of assessment to instructional efforts is consistent with the recent NRC report Testing, Teaching, and Learning,^{58} which focuses on recommendations for Title I students. Two of the central recommendations of that report concerning assessment and instruction are as follows:

Teachers should administer assessments frequently and regularly in classrooms for the purpose of monitoring individual students’ performance and adapting instruction to improve their performance, (p. 47)

Teachers should monitor the progress of individual children in grades preK3 to improve the quality and appropriateness of instruction. Such assessments should be conducted at multiple points in time, in children’s natural settings, and should use direct assessments, portfolios, checklists, and other work sampling devices. The assessments should measure all domains of children’s development, particularly social development, reading, and mathematics, (p. 53)
The current national focus on standardsbased testing is a definite improvement on the past focus on comparison testing. But standardsbased assessment needs to be accompanied by a clear set of gradelevel goals so that teachers, parents, and the whole community can work together to help all children in a school achieve those goals. (And the goals need to aim at more than skills, as we argue in chapter 4.) Continuing informal assessments throughout the year can help teachers adjust their teaching and identify students who need additional help. More such help might be available if money formerly spent on comparison testing were reallocated to help children learn.
Teaching
Even with high standards, exemplary textbooks, and powerful assessments, what really matters for mathematics learning are the interactions that take place in classrooms. The literature on mathematics education, perhaps surprisingly, contains little reliable data about those interactions. Most of the available research evidence consists of reports by teachers of their practice, but an increasing amount comes from systematic observations of lessons. The discussion in this section addresses both types of evidence.
Reported Practices
The emphasis in U.S. elementary and middle school mathematics teaching seems to be predominantly on number and operations. Teachers of 93% of the fourth graders and 88% of the eighth graders in the 1996 NAEP mathematics assessment reported that they gave the topic “a lot” of instructional emphasis.^{59} At grade 8, algebra also received a lot of emphasis (for 57% of the students), but that was the only other curriculum strand to receive much attention. Fourthgrade teachers reported giving considerable emphasis to facts, concepts, skills, and procedures (over 90% of the students got “a lot”), with less emphasis on reasoning processes (52%) and even less attention to communication (38%). Eighthgrade teachers’ responses followed a similar pattern, with somewhat less attention to facts, concepts, skills, and procedures (79%). In a recent study comparing schools participating in state initiatives in mathematics and science with schools not involved in such initiatives, elementary school teachers in the initiatives schools spent significantly more time than their counterparts on reasoning and problemsolving activities.^{60}
For decades, mathematics educators have been exhorting teachers to allow children to use manipulatives—counting blocks, geometric shapes, and other objects—to support their thinking. The use of manipulatives, however, is not a common classroom practice. In 1996, teachers of 27% of the fourth graders in NAEP reported that their students used counting blocks and geometric shapes at least once a week; 74% used them at least once a month, leaving 26% who seldom if ever used them. Teachers of 8% of the eighth graders said that their students used such manipulatives at least once a week, and teachers of more than half the students reported essentially no use. Data were not available on how this use was connected to mathematical ideas, words, and notations.
Materials such as rulers and calculators are apparently used much more frequently than manipulatives in mathematics teaching. Teachers of almost
half the fourth graders in the 1996 NAEP sample reported that their students used rulers or related tools at least once a week, and teachers of 95% of the fourth graders reported frequencies of at least once a month. Teachers of a quarter of the eighth graders reported that their students used objects such as rulers at least once a week, and teachers of almost 80% said their students used them at least once a month.
Eighthgrade teachers reported considerably greater use of calculators in their teaching than fourthgrade teachers did. Teachers of over half of the eighth graders in the 1996 NAEP sample reported that their students used calculators almost every day, and teachers of less than a tenth claimed never or hardly ever to use calculators. Teachers of less than a third of the fourth graders, in contrast, said their students used a calculator in class at least once a week, teachers of only 5% said almost every day, and teachers of more than a quarter said never or hardly ever. Eighth graders enrolled in algebra were reported to use calculators more frequently than those in prealgebra or eighthgrade mathematics, and at both grades 4 and 8 the reported frequency of calculator use increased from 1992 to 1996.
The teachers of about a quarter of the 1996 NAEP sample at both grades 4 and 8 reported that their students worked in small groups or with a partner almost every day, and teachers of more than 90% of the students had them working that way at least once a month. Teachers of about a third of each sample said that at least once a week their students wrote a few sentences about how to solve a mathematics problem, but teachers of another third said their students never or hardly ever wrote up their solutions. Few students at either grade wrote reports or worked on projects more than once a week, and teachers of about two thirds said their students hardly ever did project work. For nearly half of the eighth graders and more than a third of the fourth graders, their teachers reported that almost every day they had students discuss solutions with one another, and teachers of almost all students held such discussions at least once a month. According to these survey data, standardsbased efforts to increase attention to realistic mathematics problems may be having some effect:
In 1996, substantial proportions of students from grades 4 and 8 were working and discussing mathematics that reflected reallife situations at least “once or twice a week.” Teachers of 29 percent of fourthgrade students reported that their students did this “almost every day,” while teachers of 45 percent reported that their students did this “once or twice a week.”
The percentages were similar for eighthgrade students: teachers of 27 percent reported that students worked and discussed mathematics problems that reflected reallife situations “almost every day,” and teachers of 47 percent reported working and discussing these types of problems “once or twice a week.”^{61}
As part of the 1996 NAEP, teachers were asked about their knowledge of the 1989 NCTM standards. The teachers of 46% of the fourth graders professed little or no knowledge of the standards, and only 5% of the fourth graders had teachers who indicated that they were very knowledgeable. In contrast, only 19% of the eighth graders had teachers who claimed to have little or no knowledge of the standards, and 16% had teachers claiming to be very knowledgeable.^{62}
The accuracy of teachers’ selfreports of their practice can of course be questioned. Teachers have their own meanings for what they do. For example, in a recent survey of 85 elementary school teachers in two districts, 93% said that they were using cooperative learning, a practice in which students are grouped for instruction, are assigned roles in the group, work together on a task, are each assessed on their performance, are each held accountable for contributing to the work, and, in some versions, are taught skills for working together, promote each other’s contributions, and work collectively to improve their effectiveness.^{63} Interviews with 21 of the teachers who had indicated they were using cooperative learning (17 of whom said they used it for mathematics) revealed that all but one had their own version of the practice, which they distinguished from the “more formal” version. Primarily, they almost never attempted to make sure that individual students were held accountable for contributing to the work. From their own descriptions, the majority of the teachers were using a form of cooperative learning that differed substantially from the forms described in the literature by the researchers who had developed the practice. Similar discrepancies have been documented between teachers’ reports of implementation of other reform practices and the observation of those practices in their video lessons.^{64}
Overall, teachers’ reports give at best a mixed picture of mathematics teaching in U.S. elementary and middle schools: heavy attention to traditional content accompanied by modest and possibly idiosyncratic use of practices endorsed by advocates of standardsbased instruction. Regardless of how teachers are interpreting these practices, most do appear to be at least somewhat aware of recent proposals for change. Selfreport data address isolated practices only, however; observational data are needed if one is to get a sense of how lessons are organized and conducted.
Observed Lessons
For more than a century, observers have been looking into classrooms and emerging with descriptions of how U.S. teachers teach.^{65} What is most striking in these observers’ reports is that the core of teaching—the way in which the teacher and students interact about the subject being taught—has changed very little over that time. The commonest form of teaching in U.S. schools has been called recitation.^{66} Recitation means that the teacher leads the class of students through the lesson material by asking questions that can be answered with brief responses, often one word. The teacher acknowledges and evaluates each response, usually as right or wrong, and asks the next question. The cycle of question, response, and acknowledgment continues, often at a quick pace, until the material for the day has been reviewed. New material is presented by the teacher through telling or demonstrating. After the recitation part of the lesson, the students often are asked to work independently on the day’s assignment, practicing skills that were demonstrated or reviewed earlier. U.S. readers will recognize this pattern from their own school experience because it has been popular in all parts of the country, for teaching all school subjects.
Although there are some differences in the way different subjects are taught,^{67} the description of recitation teaching is consistent with more recent descriptions of mathematics lessons. In the mid1970s, the National Science Foundation funded a set of studies on classroom practice, including a national survey of teaching practices^{68} and a series of case studies.^{69} After observing a number of mathematics classrooms, one researcher said:
In all math classes I visited, the sequence of activities was the same. First, answers were given for the previous day’s assignment. The more difficult problems were worked by the teacher or a student at the chalkboard. A brief explanation, sometimes none at all, was given of the new material, and problems were assigned for the next day. The remainder of the class was devoted to working on the homework while the teacher moved about the room answering questions. The most noticeable thing about math classes was the repetition of this routine.^{70}
The findings for the full set of case studies are not easily summarized because there were some substantial differences between teachers, but a commissioned synthesis noted that the most common pattern in mathematics classrooms was “extensive teacherdirected explanation and questioning followed by student seatwork on paperandpencil assignments.”^{71}
At about the same time, the National Advisory Committee on Mathematical Education (NACOME) commissioned a study of elementary school mathematics instruction. Their report was entirely consistent with that of the National Science Foundation studies. In fact, NACOME expressed some concern that teaching had changed so little over the previous 10 to 15 years, a time of concentrated curriculum development in mathematics. The NACOME report’s concluding remarks reviewed the committee’s findings:
The median [elementary school] classroom is selfcontained. The mathematics period is about 43 minutes long, and about half of this time is written work. A single text is used in wholeclass instruction. The text is followed fairly closely…. Teachers are essentially teaching the same way they were taught in school.^{72}
The most extensive look into mathematics classrooms around the United States was conducted in 1995: the video study component of TIMSS.^{73} The TIMSS Video Study marked the first time that a nationally representative sample of classrooms was selected for study and that a sample of lessons was videotaped. The videotapes revealed classroom instruction that resembled the instruction described in earlier reports. Apparently, U.S. teachers are continuing to teach mathematics in the same way their predecessors taught.
The TIMSS videotapes allowed researchers to take a much more detailed look at common classroom practice than any earlier study had provided, and the availability of tapes from Germany and Japan permitted some contrasting descriptions. The full sample included 81 eighthgrade mathematics lessons in the United States, 100 such lessons in Germany, and 50 lessons in Japan.
Reports from parents and in the popular press as to how U.S. children are being taught today suggest that some teachers have their students investigating mathematical ideas almost entirely on their own, whereas others are carefully explaining those ideas and providing lots of practice. It is tempting to conclude, therefore, that methods of teaching mathematics are highly variable within the United States. In fact, the TIMSS Video Study clearly shows that such differences are quite small compared with the substantial differences that exist between countries. Each country appears to have its own dominant style of mathematics teaching.^{74}
In the videotaped lessons from the United States, a typical lesson begins by checking homework or engaging in a warmup activity. The teacher then presents a few sample problems and demonstrates how to solve them. This part of the lesson is often conducted in recitation fashion, with the teacher asking fillintheblank questions as the procedures are shown. Seatwork is
assigned, and students complete exercises like those they have been shown. The teacher often ends the lesson by checking some of the seatwork problems and assigning similar problems for homework.
Typical lessons in Germany and Japan contain many of the same components, but the components are arranged differently and aim at different goals. For example, most lessons in all three countries include an early segment in which the teacher presents one or more problems for the day. But that activity has a different purpose in each country. In Germany, presenting the problem initiates a relatively lengthy development of advanced solution techniques. The teacher guides, through questioning, the process of solving the problem, which is often quite challenging. In Japan, presenting the carefully chosen problem sets the stage for the students to work, individually and in groups, on developing solution procedures that they then report to the class. About half the time, the procedures are expected to be original constructions. As described above, presenting problems in the United States leads to students practicing procedures that have been demonstrated by the teacher.
The different patterns of teaching generated a set of findings that illustrated the dramatic differences in classroom practice across the three countries. For example, 78% of the mathematical topics in the U.S. lessons contain concepts that were stated by the teacher rather than developed through examples or explanations. In contrast, that practice occurred for 23% of the concepts in Germany and only 17% in Japan; at least some of the concepts from the remaining topics in these countries were developed and elaborated in some way.^{75} Moreover, the quality of the mathematical content of the U.S. lessons was independently rated as being much lower than that of the German and Japanese lessons.^{76}
The descriptions from the TIMSS Video Study match other reports of classroom practice in mathematics. For example, a 1998 report to the California State Board of Education summarizes the conventional method of mathematics teaching in the United States, often used as the control treatment in experimental studies of new teaching approaches.^{77} The summary divides the conventional method into two phases. In the first phase, the teacher demonstrates, often working one to four problems, and the students observe passively; in the second phase, the students work independently, with the teacher possibly monitoring their work and giving feedback.
That description might easily have been written to describe U.S. mathematics lessons in 1900. Mathematics teaching in the United States clearly has not changed a great deal in a century. It continues to emphasize the
execution of paperandpencil skills through demonstrations of procedures and repeated practice.
Teacher Preparation, Certification, and Professional Development
A bachelor’s degree and a teaching certificate are required to teach in most public schools in the United States. Teaching certificates are granted by states, usually based on the completion of specific undergraduate coursework and field experience in schools. Some states also require that candidates pass an examination. A teaching certificate from one state is occasionally honored across state lines; states without reciprocity of certification commonly offer a provisional certificate to outofstate teachers until they have met all the requirements. Some states also offer alternative routes to certification for prospective teachers with a bachelor’s degree but lacking some of the requisite coursework or field experience.
Programs of teacher education have traditionally separated knowledge of mathematics from knowledge of pedagogy by offering separate courses in each.^{78} A common practice in universitybased programs has been for prospective teachers to take courses in mathematics from the mathematics department and courses in pedagogy from the college or department of education, which is where they also get field experience and do supervised teaching practice. The standards for both types of courses have, in recent years, been influenced by reports such as A Call for Change,^{79} which listed expectations for the mathematics courses required in teacher preparation, and the Professional Standards for Teaching Mathematics,^{80} which concentrated more on issues of pedagogy.
Nationally, twoyear colleges have been urged to play a larger role in recruiting future elementary and middle school teachers and providing collegelevel mathematics courses for them.^{81} At the same time, universities are exploring different ways of connecting courses on mathematics content and pedagogy and on giving students earlier and more intensive experience in school mathematics classes. Some recent programs have attempted to bring content and pedagogy together in both teacher preparation and professional development by considering the actual mathematical work of teaching.^{82}
Although states have long set such requirements for teachers seeking certification, some have recently begun to impose higher standards for the knowledge teachers should have to teach children at a given age or grade level, requiring teachers to take specified courses and to pass assessments of their subject matter knowledge.^{83} There is considerable variation across states
as to how rigorous these requirements are. As of 1998, 31 states reported having standards for teacher certification, although in several the standards were not yet in effect. In 12 of the 31, there were specific standards for mathematics. Six other states were still developing standards.^{84}
To be certified to teach elementary school, only 12 states require a minimum number of credits in mathematics (from 6 to 12 semester hours). The other states either specify a total number of credits drawn from five to eight fields (often with a major in one of the fields), impose their own standards rather than specifying courses, require a minimum number of credits in one unspecified field, or require the completion of an approved teacher education program. Thirtyseven states grant middle school certification, and the requirements fall into categories similar to those for elementary school. Eight of those states require a minimum number of credits in mathematics to teach in middle school (from 6 to 21 semester hours).
A highly influential report on the reform of teacher education was issued in 1986 by the Holmes Group, later the Holmes Partnership, a consortium of major research universities.^{85} The report recommended that prospective teachers get a solid grounding in academic subjects as undergraduates, learning pedagogy as postgraduates. The report also encouraged the development of socalled professional development schools and other forms of cooperative partnerships between schools and universities. In part because of the Holmes report, some 300 schools of education created programs that went beyond the traditional fouryear degree programs, included more study of subject matter, and gave more clinical training in schools.^{86} Also, during the 1990s, more states began to require new teachers to have an undergraduate or graduate major in an academic subject they would be teaching rather than a major in education. As of 1998, 21 states required a major in the teaching field, and another 10 required either a major or a minor. In most states the requirement applies to teachers applying for middle or secondary certification, which usually cover grades 7 to 12. In four states an academic major is required for teachers at all grades K to 12.
In line with the trend toward more mandated assessments of students, as of 1998, 38 states required that prospective teachers pass an assessment, sometimes to be admitted to a program and other times after completing the program but before certification. Almost all of these states assess new teachers’ “basic skills,” and most of the others also assess “professional knowledge of teaching,” “subject matter knowledge” (e.g., mathematics), or both. Eight states use portfolio assessment, with some requiring the portfolio at the end of preservice education and others requiring it during the first or second year
of teaching. Thirteen states require classroom observation as part of the assessment for certification.
Despite the establishment of these increased standards, there is wide variation in the extent to which they are enforced:
Whereas some states do not allow districts to hire unqualified teachers, others routinely allow the hiring of candidates who have not met their standards, even when qualified teachers are available. In Wisconsin and eleven other states, for example, no new elementary or secondary teachers were hired without a license in their field in 1994. By contrast, in Louisiana, 31% of new entrants were unlicensed and another 15% were hired on substandard licenses. At least six other states allowed 20% or more of new public school teachers to be hired without a license in their field.^{87}
Of the 26 states reporting data in 1998 on the certification of their teachers at grades 7 and 8, only 6 states reported that 90% or more of these teachers were certified in mathematics, and only 10 states reported that more than 80% were certified. In response to urgent needs for teachers, states often issue socalled emergency credentials that bypass their own requirements. These credentials typically require only a bachelor’s degree and enrollment in an approved program leading to some form of alternative certification. Many districts respond to the need for mathematics and science teachers by assigning teachers to teach outside their field.^{88}
The evidence is mixed as to whether relatively fewer teachers are teaching outside their field today than a decade ago; data from different sources yield different numbers and contrasting evidence of change. In the 1996 NAEP mathematics assessment, teachers of 81% of the eighth graders in the sample reported that they were certified in mathematics, and the corresponding figure for fourth graders was 32%. Those numbers were not significantly different from what teachers had reported in 1992.^{89} In contrast, the Council of Chief State School Officers reported in 1998 that 72% of all mathematics teachers at grades 7 and 8 in the 26 states providing data were reported as certified, 22% as not certified, and the remainder as having elementary school certification. In a corresponding survey in 1994, the percentage of certified teachers at those grades had been only 54, a significantly smaller number.^{90} In other words, to judge by teachers’ own reports, the situation has not changed, but to judge by reports from the states, it has improved at grades 7 and 8.
In the 1996 NAEP mathematics assessment, teachers were asked how many hours of professional development they had received in the previous 12 months. Nationally, 28% of the fourth graders in the sample had teachers who had received 16 or more hours of professional development in mathematics; for eighth graders, the percentage was 48. In 16 states, over half the eighth graders were taught by mathematics teachers who had received that much professional development.^{91}
The number of states requiring that teachers participate in professional development activities for renewal of certification has been on the increase over the past decade. Currently, only Hawaii, Illinois, New Jersey, New Mexico, and New York do not have a policy on professional development for renewing certification. In half the states the policy is 6 semester credits every five years. Several states have higher requirements. North Carolina requires 15 credits every five years, and in Oregon, teachers must earn 24 quarter hours in their first three years of teaching.^{92}
In an effort to encourage teachers to extend their professional development efforts, 30 states have adopted incentives for teachers certified by the National Board for Professional Teaching Standards, such as portability of certification, certification renewal, fee supports, and pay supplements.^{93} Standards for National Board certification are available in mathematics for teachers of students ages 11 to 15. Certification at the elementary school level is general. Teachers seeking a certificate must submit a portfolio documenting their classroom practice and must go to an assessment center for a oneday series of exercises in which they demonstrate their knowledge of mathematical content and analyze student work.
There is a growing body of evidence suggesting that states and local districts “interested in improving student achievement may be welladvised to attend, at least in part, to the preparation and qualifications of the teachers they hire and retain in the profession.”^{94} A qualitative and quantitative analysis of data from a 50state survey of policies, state case study analyses, the 1993–94 Schools and Staffing Surveys, and NAEP identified the percentage of teachers with full certification and a major in the field they teach as a strong and consistent predictor of student achievement in mathematics, considerably stronger than such factors as class sizes, pupilteacher ratios, state perpupil spending, or teachers’ salaries.^{95} This link between teacher qualification and student achievement raises the question of how good that achievement is.
Achievement
Since the early 1970s, a series of national and international assessments have provided a reasonably consistent picture of U.S. students’ achievement in mathematics. As one analysis of these assessments puts it, the results “evoke both a sense of despair and of hope.”^{96} The despair comes from the generally low level of performance, the hope from signs that performance in some areas of mathematics and by some groups of students has been improving over the last decade.
The many mathematics assessments conducted since 1973 by NAEP demonstrate that student performance at each of the grade levels assessed is considerably below what mathematics teachers and the public would prefer. Since 1990, NAEP has included two separate components for mathematics: main NAEP and longterm trend NAEP. The longterm trend assessments use the same sets of questions first used in 1973, allowing comparison across time. The main assessments reflect more contemporary educational objectives and are used to collect both national and state data, including contextual data such as teaching practices, some of which are reported earlier in this chapter.^{97} Except when we refer explicitly to the longterm trend assessments, the data reported here are from the main assessments.
In the 1996 mathematics assessment—the most recent main assessment to be thoroughly analyzed—across grades 4, 8, and 12, roughly 35% of the students were below the basic level of achievement and another 45% or so were at that level, which is defined as denoting “partial mastery of knowledge and skills that are fundamental for proficient work.” In the same assessment, 21% of fourth graders and 24% of eighth graders were at or above the “proficient” level, where proficiency is defined as students having “demonstrated competency over challenging subject matter” and being “well prepared for the next level of schooling.” Only 2% and 4% of fourthgrade and eighthgrade students, respectively, were doing advanced work significantly “beyond proficient gradelevel mastery.”^{98}
Although overall levels of achievement are low, the main NAEP assessments in the 1990s revealed significant gains.^{99} The gains between 1990 and 1996 have been estimated to be about one grade level.^{100} According to the NAEP longterm trend, mathematics achievement improved between 1973 and 1996 at both the fourthgrade and eighthgrade levels.^{101} Performance improved even more sharply from 1973 to 1996 among black and Hispanic students.^{102} Although the gap between black students and white students had narrowed through the 1980s, it widened between 1990 and 1999, especially among students of the besteducated parents.^{103} This disparity repre
sents a serious challenge to U.S. education. In 1994, NAEP began collecting information on participation in Title I programs, programs designed to help disadvantaged students, and in 1996 on eligibility for free or reducedpriced lunches. At both grades 4 and 8, students who participated in Title I programs and students who were eligible for free or reducedpriced lunches scored lower than their nonparticipating or noneligible classmates.^{104} The low mathematics achievement of poor children is embedded in the larger social issues of poverty and poses another serious challenge to U.S. education.
International comparisons of mathematics achievement demonstrate many of the same findings as the NAEP results. On several international mathematics assessments conducted since the 1970s, the overall performance of U.S. students has lagged behind the performance of students in other countries. In TIMSS, U.S. fourth graders performed above the international average of the 26 participating countries at fourth grade but still significantly below the levels of the topperforming countries. U.S. eighth graders performed slightly below the international average in mathematics among the 41 participating countries.
As this volume went to press, the results of TIMSSR (Third International Mathematics and Science StudyRepeat), the 1999 version of TIMSS, had just been released. Between 1995 and 1999, there was no significant change in the mathematics achievement of U.S. eighth graders. Furthermore, the eighth graders in 1999, who compared quite well internationally in 1995 as fourth graders, were very much like the 1995 eighth graders, performing near the international average.^{105}
One way to quantify U.S. students’ performance is in terms of the average number of points they scored on the 1995 TIMSS assessment. Each student answered a subset of the TIMSS questions, and an average score was calculated for each question, with some questions worth more than one point. The U.S. fourth graders scored, on average, 71 out of the 113 points available on the TIMSS achievement test, which contained 102 questions.^{106} That was about 4 points above the performance across all 26 countries, but it was 11 to 15 points below the performance of students in the top four countries (Singapore, Korea, Japan, and Hong Kong) and was in a band of performance comparable with that found in the Czech Republic, Ireland, and Canada. In the assessment of eighth graders, U.S. students scored, on average, 86 points out of the 162 available on the 151 TIMSS items, which was 3 points below the 41country average. Students in the four topscoring countries—Singapore, Japan, Korea, and Hong Kong—scored, on average, between 113 and 128 points.^{107}
The performance of U.S. students in TIMSS differed markedly across core domains of mathematics. U.S. performance was above the international average on data representation, analysis, and probability and not significantly different from the international average on fractions, number sense, and algebra. Performance was below the international average on geometry, measurement, and proportionality.^{108} For example, U.S. eighth graders had much weaker abilities, overall, than their counterparts in other countries to conceptualize measurement relationships, perform geometric transformations, and engage in other complex mathematical tasks. These kinds of abilities are among the learning goals called for by national documents setting forth standards and benchmarks for school mathematics and by many sets of state standards, indicating that many U.S. students are not now achieving the objectives of those standards.^{109}
Interestingly, the variance of U.S. scores in the TIMSS results was not markedly greater than in other countries. There was, however, considerable variability in scores between states. A study linking state NAEP scores at grade 8 with TIMSS scores showed that the topscoring states on NAEP performed quite well internationally, with only 6 of 41 countries scoring significantly higher. In contrast, lowscoring states scored significantly higher than as few as 3 of 41 countries.^{110} These results suggest that national averages may miss important aspects of U.S. mathematics education.
Even state averages do not tell the whole story, however. A consortium of districts in suburban Chicago participated in TIMSS so that they might be treated as a country in the analysis. Their performance was exceptional on the mathematics assessments at both grades 4 and 8, with only Singapore scoring significantly higher. Although some of their success is clearly attributable to being relatively wealthy districts, socioeconomic factors explained only 25% of the differences in scores at fourth grade and 50% of the differences in scores at eighth grade.^{111}
More generally, variance in student scores was strongly linked to the specific classes a student took (for example, regular mathematics versus algebra in middle school or junior high) and to differences among schools. In particular, 64% of the variance in U.S. student mathematics achievement at eighth grade can be explained by differences between schools or classes. In Japan, in contrast, only 7% of the variance in student mathematics achievement was between schools or classes.^{112} These findings suggest that many U.S. students are not being given the educational opportunities they need to achieve at high levels.^{113}
Coordinating Improvement Efforts
In the late 1850s, the city of Chicago started a massive project to replace its dirt (and often mud) streets with a more permanent road and sidewalk system. The city had to raise the roadbed substantially and lift the existing buildings so that they were level with the new sidewalks. The zenith of this undertaking was the lifting of the Tremont Hotel in 1858, organized by George Pullman. While hotel patrons ate breakfast, Pullman’s crew of 1,200 men carefully turned some 5,000 jackscrews to raise the building evenly.
Improving the U.S. system of school mathematics demands not simply effort but coordination.
It requires a thorough, methodical overhaul.
As with raising the Tremont Hotel, improving the U.S. system of school mathematics demands not simply effort but coordination. Although many individuals have worked diligently over the past several decades to change the ways in which mathematics is taught and learned, the evidence clearly indicates that considerable improvement is still necessary. Across the country, schools and teachers face the substantial challenge of providing all children with the opportunity to become mathematically proficient. Much of the difficulty in meeting that challenge arises because the effort to date has not been concerted. The U.S. system of school mathematics cannot be made to operate better by fixing one tiny piece at a time; it requires a thorough, methodical overhaul.^{114}
Authority in the U.S. system is widely dispersed, with states, districts, the federal government, textbook and test publishers, professional and political organizations, teachers, and parents and other caregivers each trying to exercise control of the part of the system within their purview. We urge, therefore, all who are attempting to improve mathematics learning in grades preK to 8 to reflect on the observations made in this report and to consider how they might connect and coordinate their efforts with those of others.
In subsequent chapters we set forth important research, theory, and organizing principles intended to ground future efforts in fact and principled argument, to make assumptions more explicit, and to bring greater coherence to the system. We would like to see an independent group of recognized standing conduct continuing, ongoing assessment of the progress made over the coming years in meeting the goal of mathematical proficiency for all U.S. schoolchildren. Such an assessment would help enormously in the coordination of efforts to make school mathematics a better functioning system for everyone.
Before considering the issues of learning and teaching that contribute to the development of mathematical proficiency, we devote the next chapter to considering the mathematical landscape upon which our later analyses are built. To understand how it is that students become proficient and the chal
lenges they face in doing so, it is important to understand the mathematics with which they are engaged. Because we have chosen to focus on proficiency with number, chapter 3 lays out the mathematics of number.
Notes
1. 
Robitaille, 1997; Stigler and Hiebert, 1999; U.S. Department of Education, 1998b, 1999a, 1999b. 
2. 
Howson, 1995; Schmidt, McKnight, and Raizen, 1997. 
3. 
An analysis of data from the Second International Mathematics Study (SIMS) examined features such as time for mathematics instruction, class size, and teacher preparation, and other instructional variables and concluded that none of them alone could explain differences in achievement across countries (McKnight, Crosswhite, Dossey, Kifer, Swafford, Travers, and Cooney, 1987). 
4. 
National Council of Teachers of Mathematics, 1989. 
5. 
National Council of Teachers of Mathematics, 1991. 
6. 
National Council of Teachers of Mathematics, 1995. 
7. 
National Council of Teachers of Mathematics, 2000. 
8. 
See http://www.edc.org/mcc/currcula.htm for information on the 13 NSF projects. 
9. 
See Jennings, 1998. In making the case for national standards and describing the background behind the movement, Ravitch, 1995, emphasizes that when the president and the governors established national education goals in 1990, mathematics was the only subject matter for which “educators were ready to say what children should learn and teachers should teach” (p. 121). 
10. 
Elmore and Rothman, 1999, p. 1. 
11. 
A Nation at Risk: National Commission on Excellence in Education, 1983; America 2000: U.S. Department of Education, 1991; Goals 2000: U.S. Department of Education, 1998a. 
12. 
Blank, Manise, and Brathwaite, 2000, pp. viii–xi. See also Orlofsky and Olson, 2001. 
13. 
See the individual state reports in Raimi and Braden, 1998. 
14. 
Fordham Foundation, 1997–98; Gandal, 1997; Joftus and Berman, 1998; Raimi and Braden, 1998; for an analysis of the divergence across the three sets of ratings, see Camilli and Firestone, 1999. 
15. 
Pimentel and Arsht, 1998. 
16. 
Marzano, Kendall, and Gaddy, 1999. 
17. 
Dossey, 1997, p. 40. 
18. 
McKnight, Crosswhite, Dossey, Kifer, Swafford, Travers, and Cooney, 1987, p. 74; Suydam, 1985; Tyson and Woodward, 1989; Woodward and Elliott, 1990. 
19. 
Council of Chief State School Officers, 1998. 
20. 
Woodward and Elliot, 1990; Tyson and Woodward, 1989. The observations in this paragraph are based on a review by Grouws and Cebulla, 2000. 
21. 
Fey, 1980. 
22. 
Grouws and Smith, 2000. 
23. 
Schwille, Porter, Belli, Floden, Freeman, Knappen, Kuhs, and Schmidt, 1983; Stodolsky, 1988; Sosniak and Stodolsky, 1993. 
24. 
TysonBernstein, 1988, p. 7. 
25. 
Fuson, Stigler, and Bartsch, 1988; McKnight, Crosswhite, Dossey, Kifer, Swafford, Travers, and Cooney, 1987; McKnight and Schmidt, 1998; Peak, 1996. 
26. 
Flanders, 1987; Fuson, Stigler, and Bartsch, 1988; Schmidt, McKnight, and Raizen, 1997. 
27. 
Fuson, Stigler, and Bartsch, 1988; Schmidt, McKnight, Cogan, Jakwerth, and Houang, 1999; Schmidt, McKnight, and Raizen, 1997. 
28. 
Levin, 1989; Levin and Mayer, 1993; Mayer, 1993. 
29. 
Reys, 2000. 
30. 
U.S. Department of Education, Mathematics and Science Expert Panel, 1999. 
31. 
Mathematically Correct, 2000. 
32. 
American Association for the Advancement of Science, 2000a, 2000b; Clopton, McKeown, McKeown, and Clopton, 2000a, 2000b. 
33. 
The current center is the National Center for Improving Student Learning and Achievement in Mathematics and Science at the University of WisconsinMadison. For information on currently funded projects, see http://forum.swarthmore.edu/mathed/curriculum.dev.html. [July 20, 2001]. 
34. 
For example, the University of Chicago School Mathematics project and the Mathematics in Context project at the University of Wisconsin. 
35. 
TysonBernstein, 1988, pp. 17–36. 
36. 
Glaser and Silver, 1994, p. 403. 
37. 
Mitchell, Hawkins, Jakwerth, Stancavage, and Dossey, 1999, pp. 260–264. 
38. 
Mitchell, Hawkins, Jakwerth, Stancavage, and Dossey, 1999, p. 261. Moderate testing is associated with higher achievement even when controlling for socioeconomic factors. See Mullis, Jenkins, and Johnson, 1994, p. 61. 
39. 
For a discussion of these calls, see Elmore and Rothman, 1999. 
40. 
Mitchell, Hawkins, Jakwerth, Stancavage, and Dossey, 1999, p. 262. 
41. 
Dossey, 1997, p. 37. 
42. 
Council of Chief State School Officers, 1998. 
43. 
Jerald, Curran, and Boser, 1999, p. 81. See Education Commission of the States, 2000, for a thorough description of state policies and actions. 
44. 
Sandham, 1999. 
45. 
Gehring, 2000. 
46. 
Bishop, 1997. 
47. 
DarlingHammond, 1999, p. 33. 
48. 
Steinberg, 1999. 
49. 
This terminology was part of the Title I law; Elmore and Rothman, 1999. 
50. 
Archer, 1997. 
51. 
Musick, 1997. 
52. 
Standardized tests are tests that are “administered and scored under conditions uniform to all students” (U.S. Congress, Office of Technology Assessment, 1992, p. 5). 
53. 
Romberg and Wilson, 1992. 
54. 
Rothman, 1995; U.S. Congress, Office of Technology Assessment, 1992, chap. 6. 
55. 
Anastasi, 1988; Crocker, and Algina, 1986. 
56. 
Rothman, 1995, p. 5. 
57. 
Heubert and Hauser, 1998; Pullin, 1993. 
58. 
Elmore and Rothman, 1999. 
59. 
Except for the data on teachers’ knowledge of the 1989 NCTM standards, the remaining data in this section are taken from Mitchell, Hawkins, Jakwerth, Stancavage, and Dossey, 1999. 
60. 
Council of Chief State School Officers, 2000, p. 10. 
61. 
Mitchell, Hawkins, Jakwerth, Stancavage, and Dossey, 1999, pp. 251–252. 
62. 
Hawkins, Stancavage, and Dossey, 1998, p. 41. 
63. 
Antil, Jenkins, Wayne, and Vadasy, 1998. 
64. 
Stigler and Hiebert, 1999, pp. 104–106. 
65. 
Cuban, 1993; Hoetker and Ahlbrand, 1969. 
66. 
Hoetker and Ahlbrand, 1969; Tharp and Gallimore, 1988. 
67. 
Stodolsky, 1988. 
68. 
Weiss, 1978. 
69. 
Stake and Easley, 1978. 
70. 
Welch, 1978, p. 6. 
71. 
Fey, 1979, p. 494. 
72. 
National Advisory Committee on Mathematical Education, 1975, p. 77. 
73. 
Stigler, Gonzales, Kawanaka, Knoll, and Serrano, 1999. 
74. 
Stigler and Hiebert, 1999. 
75. 
Stigler and Hiebert, 1999, p. 61. 
76. 
Stigler and Hiebert, 1999, p. 57. 
77. 
Dixon, Carnine, Kameenui, Simmons, Lee, Wallin, and Chard, 1998a, 1998b. 
78. 
Swafford, 1995. 
79. 
Leitzel, 1991. 
80. 
National Council of Teachers of Mathematics, 1991. 
81. 
Raychowdhury, 1998. 
82. 
See, for example, National Research Council, 2001; Conference Board of the Mathematical Sciences, 2000. See FerriniMundy and Findell, 2001, for a discussion of the principles behind these and other approaches to improving the connection between the mathematical education of teachers and the mathematics used in classrooms. 
83. 
See http://www.ccsso.org/intasc.html [July 20, 2001] for information on model standards and assessments of beginning teachers promoted by the Interstate New Teacher Assessment and Standards Consortium. 
84. 
Council of Chief State School Officers, 1998. Unless otherwise indicated, the data on certification come from this document. 
85. 
Holmes Group, 1986. 
86. 
DarlingHammond, 1997. 
87. 
DarlingHammond, 1999, p. 15. 
88. 
Blank and Langeson, 1999, p. 66. 
89. 
Hawkins, Stancavage, and Dossey, 1998, p. 19. 
90. 
Blank and Langeson, 1999, p. 64. 
91. 
Blank and Langeson, 1999, p. 73. 
92. 
Council of Chief State School Officers, 1998, p. 26. 
93. 
Jerald, Curran, and Boser, 1999, p. 116. For information on the National Board for Professional Teaching Standards, see http://www.nbpts.org [July 20, 2001] or Kelly, 1995. 
94. 
DarlingHammond, 1999, pp. 38–39. 
95. 
DarlingHammond, 1999, p. 29. 
96. 
Dossey and Mullis, 1997, p. 20. 
97. 
Campbell, Voelkl, and Donahue, 2000. 
98. 
Reese, Miller, Mazzeo, and Dossey, 1997, p. 53. 
99. 
Reese, Miller, Mazzeo, and Dossey, 1997. 
100. 
Dossey, 2000, p. 31. 
101. 
Campbell, Voelkl, and Donahue, 2000. 
102. 
Campbell, Voelkl, and Donahue, 2000, p. 62–64. See also Secada, 1992; Silver, Strutchens, and Zawojewski, 1997; Strutchens and Silver, 2000. 
103. 
Zernike, 2000. 
104. 
Reese, Miller, Mazzeo, and Dossey, 1997, pp. 38–39. 
105. 
U.S. Department of Education, 2000b. 
106. 
The values in the text are computed from Mullis, Martin, Beaton, Gonzalez, Kelly, & Smith, 1997, p. B3. For similar discussions, see National Research Council, 1999a, p. 21; National Council of Teachers of Mathematics, 1997. 
107. 
The values in the text are computed from Beaton, Mullis, Martin, Gonzalez, Kelly, & Smith, 1996, p. B3. For similar discussions, see National Research Council, 1999a, p. 21; National Council of Teachers of Mathematics, 1996. 
108. 
U.S. Department of Education, 2000a. 
109. 
National Research Council, 1999a, p. 27; Wilson and Blank, 1999, pp. 2–3. 
110. 
National Education Goals Panel, 1998. 
111. 
Kimmelman, Kroeze, Schmidt, van der Ploeg, McNeely, and Tan, 1999. 
112. 
Martin, Mullis, Gregory, Hoyle, and Shen, in press. The Second International Mathematics Study produced similar results (McKnight, Crosswhite, Dossey, Kifer, Swafford, Travers, and Cooney, 1987, pp. 108–109). 
113. 
National Research Council, 1999a, p. 20. 
114. 
The National Research Council, 1999b, put forward a Strategic Education Research Program that aims to coordinate improvement efforts through networks of committed education researchers, practitioners, and policy makers. 
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