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Adding + It Up: Helping Children Learn Mathematics 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
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Adding + It Up: Helping Children Learn Mathematics 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 mid-1990s. 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 eighth-grade 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 high-achieving 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
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Adding + It Up: Helping Children Learn Mathematics 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 high-scoring 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-
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Adding + It Up: Helping Children Learn Mathematics 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 Mathematics5 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 so-called 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 low-income 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 K-4), but some states (e.g., California, Texas, North Carolina, and Virginia) have recently developed grade-by-grade 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-
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Adding + It Up: Helping Children Learn Mathematics 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 B-plus 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 pre-K-2) 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 grade-level 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 standards-based mathematics curricula for grades pre-K to 8 with challenging grade-level 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 grade-level goals:
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Adding + It Up: Helping Children Learn Mathematics 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, well-articulated, 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 goal-setting 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
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Adding + It Up: Helping Children Learn Mathematics 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 test-oriented 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 pre-K 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-
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Adding + It Up: Helping Children Learn Mathematics 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 grade-level 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 NSF-funded 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
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Adding + It Up: Helping Children Learn Mathematics In some countries, including England, France, Hong Kong, Singapore, and the Netherlands, there are permanent national centers or institutes that conduct multi-year 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 research-based 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 even-more-ambitious 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 twenty-first century. The nation must develop a greater capacity for producing high-quality 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
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Adding + It Up: Helping Children Learn Mathematics instructional decisions. Such assessments range from the informal questions a teacher might ask about a student’s work to an end-of-year examination. They arise from the teaching-learning 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 up-to-date 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 teacher-made 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 multiple-choice testing in favor of tests on which students supply written responses.39 Multiple-choice 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 multiple-choice 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 multiple-choice items on their own tests. The form of multiple-choice 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 multiple-choice items, yet it was judged substantially more challenging than the 1992 NAEP mathematics assessment given to U.S. eighth graders, which contained both multiple-choice 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.
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Adding + It Up: Helping Children Learn Mathematics 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 high-stakes 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 grade-level 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 highest-scoring 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 lowest-scoring states,…10 had extensive student testing programs in place prior to 1990, some of which were associated
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Adding + It Up: Helping Children Learn Mathematics 23. Schwille, Porter, Belli, Floden, Freeman, Knappen, Kuhs, and Schmidt, 1983; Stodolsky, 1988; Sosniak and Stodolsky, 1993. 24. Tyson-Bernstein, 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 Wisconsin-Madison. 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. Tyson-Bernstein, 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. Darling-Hammond, 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.
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Adding + It Up: Helping Children Learn Mathematics 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 Ferrini-Mundy 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. Darling-Hammond, 1997. 87. Darling-Hammond, 1999, p. 15. 88. Blank and Langeson, 1999, p. 66.
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Adding + It Up: Helping Children Learn Mathematics 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. Darling-Hammond, 1999, pp. 38–39. 95. Darling-Hammond, 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. B-3. 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. B-3. 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. References American Association for the Advancement of Science. (2000a). Algebra for all—Not with today’s textbooks, says AAAS [On-line]. Available: http://www.project2061.org/newsinfo/press/r1000426.htm. [July 10, 2001].
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Representative terms from entire chapter: