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A VISION OF SCHOOL MATHEMATICS

Mathematics is the key to opportunity. No longer just the language of science, mathematics now contributes in direct and fundamental ways to business, finance, health, and defense. For students, it opens doors to careers. For citizens, it enables informed decisions. For nations, it provides knowledge to compete in a technological community. To participate fully in the world of the future, America must tap the power of mathematics.

Communication has created a world economy in which working smarter is more important than merely working harder. Jobs that contribute to this world economy require workers … who are prepared to absorb new ideas, to adapt to change, to cope with ambiguity, to perceive patterns, and to solve unconventional problems. It is these needs, not just the need for calculation (which is now done mostly by machines), that make mathematics a prerequisite to so many jobs. More than ever before, Americans need to think for a living; more than ever before, they need to think mathematically.1

So opens the first chapter of Everybody Counts: A Report to the Nation on the Future of Mathematics Education, which describes a vision of the mathematics that should guide education so that students will work smarter and think more mathematically. The vision calls for changes in the mathematics taught, in the way it is taught, and in how it is assessed. Changes in mathematics assessment, the subject of this report, should be seen as one piece of the larger picture of reform in school mathematics.

Inside the classroom, teachers are working to change the mathematics they teach and how they teach it for many reasons, some of which they can find in their own classrooms. Far too many



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Measuring What Counts: A Conceptual Guide for Mathematics Assessment 1 A VISION OF SCHOOL MATHEMATICS Mathematics is the key to opportunity. No longer just the language of science, mathematics now contributes in direct and fundamental ways to business, finance, health, and defense. For students, it opens doors to careers. For citizens, it enables informed decisions. For nations, it provides knowledge to compete in a technological community. To participate fully in the world of the future, America must tap the power of mathematics. Communication has created a world economy in which working smarter is more important than merely working harder. Jobs that contribute to this world economy require workers … who are prepared to absorb new ideas, to adapt to change, to cope with ambiguity, to perceive patterns, and to solve unconventional problems. It is these needs, not just the need for calculation (which is now done mostly by machines), that make mathematics a prerequisite to so many jobs. More than ever before, Americans need to think for a living; more than ever before, they need to think mathematically.1 So opens the first chapter of Everybody Counts: A Report to the Nation on the Future of Mathematics Education, which describes a vision of the mathematics that should guide education so that students will work smarter and think more mathematically. The vision calls for changes in the mathematics taught, in the way it is taught, and in how it is assessed. Changes in mathematics assessment, the subject of this report, should be seen as one piece of the larger picture of reform in school mathematics. Inside the classroom, teachers are working to change the mathematics they teach and how they teach it for many reasons, some of which they can find in their own classrooms. Far too many

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Measuring What Counts: A Conceptual Guide for Mathematics Assessment Teachers' assessment of student learning should be attuned not just to judging but to helping students learn. of their students—especially members of traditionally underserved groups—are turning away from mathematics, dropping out either physically or mentally.2 Few students who stay with mathematics show much enthusiasm for it. It seems too abstract, too unrelated to either their present lives or their futures. Teachers are unhappy that students remember little of the mathematics they have been taught and seem incapable of using it. Outside the classroom, politicians and school administrators, backed by the public, express dismay over low scores on mathematics achievement tests. They worry about deteriorating American competitiveness in international markets when students' mathematics skills seem to be declining.3 They want teachers to teach more mathematics to more students while maintaining or increasing test scores. At the same time, teachers are being told by their professional associations that the mathematics they teach should be more applicable to life than is now common, that their teaching should generate active learning, and that their assessment of student learning be attuned not just to judging but to helping students learn.4 On the surface, the pressures to change mathematics instruction look inconsistent, with teachers caught in the middle. Nevertheless, all the pressures reflect disappointment with the lack of interest and accomplishment so many students show. The message is the same: School mathematics is out of step with today's world and is neither well taught nor well learned. Three pivotal forces are moving mathematics teachers toward a different approach to their teaching. These forces are changing ideas about what should be taught, how it should be taught, and to whom it should be taught. Motivating the first is a more comprehensive view of mathematics and its expanding role in society. Motivating the second is a resurgence of the view that mathematics must be made meaningful to students if it is to be learned, retained, and used. Motivating the third is the growing belief that all students can and should learn more mathematics.

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Measuring What Counts: A Conceptual Guide for Mathematics Assessment CHANGES IN MATHEMATICS AND IN MATHEMATICS EDUCATION The mathematics taught in school must change in support of the way mathematics is used in our society. Since 1900, the growth of the mathematical sciences—in scope and in application—has been explosive.5 The last 40 years have been especially productive, as advances in high-speed computing have opened up new lines of research and new ways mathematics can be applied. Problems in economics, social science, and life science, as well as large-scale problems in natural science and engineering, used to be unapproachable through mathematics. Suddenly, with the aid of computers and the new tools provided by research, many of these problems have become accessible to mathematical analysis. Applications derived from data analysis and statistics, combinatorics and discrete mathematics, and information theory and computing have greatly extended the definition and reach of the mathematical sciences. An explosion in the way mathematics is used in society mirrors the explosion in mathematics itself. Today we encounter uses of mathematics in every corner of our lives. Graphs, charts, and statistical data appear on television and in newspapers. The results of opinion polls are reported along with their margins of error. Lending institutions advertise variously computed interest rates for loans. We listen to music composed and performed with the aid of computers, and we watch the fantastically detailed pictures of imaginary worlds that computers draw. Computers also do a host of ordinary tasks. They scan bar codes on purchases, keep track of inventories, make travel reservations, and fill out income tax forms. The citizen's need to perform simple calculations may have decreased, but there has been a dramatic increase in the need to interpret, evaluate, and understand quantitative information presented in a variety of contexts. Although some people do not need or use highly technical mathematics in their daily jobs, many others do. The complexity of daily life requires that we all be able to reason with numbers. Any car or home buyer ought to understand how interest rates work even though a computer may be doing the calculation. Anyone building a house or redecorating a room should be able to make and read a scale drawing. Newspaper readers and television viewers

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Measuring What Counts: A Conceptual Guide for Mathematics Assessment should be able to draw correct inferences from data on social problems such as pollution, crime, drugs, and disease. Buyers of insurance and purchasers of stock need to know something about the calculation of risk. All adults should be both able and disposed to use mathematics to make practical decisions, to understand public policy issues, to do their job better, to enhance their leisure time, and to understand their culture.6 Mathematics and the ways it is used are changing. The mathematics taught in school must change in support of the way mathematics is used in our society. WHAT MATHEMATICS SHOULD BE LEARNED Over the years, professionals concerned with mathematics education have developed a coherent view of what mathematics is important, despite some disagreements along the way. At the turn of this century, the mathematician E. H. Moore called for a refurbished school mathematics curriculum. He expressed the hope that twentieth century students would at least encounter ''in thoroughly concrete and captivating form, the wonderful new notions of the seventeenth century,"7 particularly an introduction to calculus. He wanted primary school children to make models and study intuitive geometry along with arithmetic and algebra. He argued for tight connections and a blurring of the distinctions between all parts of school mathematics but especially between its pure and applied sides. The prestigious National Committee on Mathematical Requirements of the Mathematical Association of America, reporting in 1923, formulated the aims of mathematical instruction as practical, disciplinary, and cultural. They viewed the idea of relationship or dependence, which can be expressed in the mathematical concept of function, as encompassing many of the disciplinary aims. They also deemed fundamental an appreciation of the power of mathematics. In words that have a contemporary ring, they said The primary purpose of the teaching of mathematics should be to develop those powers of understanding and of analyzing relations of quantity and of space which are necessary to an insight into and control over our environment and to an appreciation of the progress of civilization in its various aspects, and to develop those habits of thought and of action which will make these powers effective in the life of the individual.8

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Measuring What Counts: A Conceptual Guide for Mathematics Assessment Launching the movement that became known as "the new math," the College Board's Commission on Mathematics in 1959 made the case for revamping the school mathematics curriculum: The traditional curriculum fails to reflect adequately the spirit of contemporary mathematics, which seeks to study all possible patterns recognizable by the mind, and by so striving has tremendously increased the power of mathematics as a tool of modern life. Nor does the traditional curriculum give proper emphasis to the fact that the developments and applications of mathematics have always been not only important but indispensable to human progress.9 Recently, in its Curriculum and Evaluation Standards for School Mathematics, the National Council of Teachers of Mathematics (NCTM) contended that the traditional sequence of mathematics courses leading to the calculus is inadequate: Students should be exposed to numerous and varied interrelated experiences that encourage them to value the mathematical enterprise, to develop mathematical habits of mind, and to understand and appreciate the role of mathematics in human affairs; that they should be encouraged to explore, to guess, and even to make and correct errors so that they gain confidence in their ability to solve complex problems; that they should read, write, and discuss mathematics; and that they should conjecture, test, and build arguments about a conjecture's validity.10 The Standards delineate the mathematics students need to learn under various headings, some familiar (measurement, algebra, probability, problem solving), some perhaps less so (communication, spatial sense, discrete mathematics). In an echo of the 1923 report, the Standards emphasize mathematical power and outline experiences designed to help all students gain that power. Particularly important are the processes of mathematical thinking whereby students learn problem solving, communicating, reasoning, and making connections. Concurrent with the dramatic changes in our society over the last century—including the revolution in information technology and the recent increase in economic competitiveness—the profession's view of what mathematics is important is evolving consistently. As these statements from mathematics educators show, the profession has long sought to move instruction beyond a narrow focus on calculation to a deeper consideration of the meaning, process, and uses of mathematics. Criticism has been aimed at the so-called traditional curriculum, with its stress on

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Measuring What Counts: A Conceptual Guide for Mathematics Assessment Students learn important mathematics when they are using it in relevant contexts that require them to apply what they know and to extend how they think. symbol manipulation, its fragmentation, and its artificial treatment of applications. Today's consensus on mathematics learning embraces several important components. Students should be involved in finding, making, and describing patterns. They should construct mathematical models for both practical and theoretical situations—using technology when appropriate—learning to represent and reason about quantities and shapes, to devise and solve challenging problems, and to communicate what they have learned. Students also should encounter mathematics as a human endeavor, learning something of its history in various cultures, coming to appreciate its aesthetic side, and understanding its role in contemporary society and its connections to other disciplines and areas of knowledge. School mathematics from kindergarten to 12th grade should offer much more than procedural skills. It should equip students not only for the further study of mathematics and other subjects but also to use mathematics creatively and effectively in their daily lives and subsequent careers. Elementary school mathematics should go well beyond computational arithmetic, which is only one aspect of mathematics. Instead, it should also include topics such as number sense, estimation, and an introduction to geometry, probability, statistics, and algebra, all treated in ways that deemphasize the boundaries between these strands and developed through activities that use physical objects. Middle and high school mathematics should continue the development of the strands begun in elementary school mathematics and in addition should include combinatorics, discrete mathematics, logic, number theory, trigonometry, and some basic ideas from calculus. Fundamental mathematical structures (relations, functions, operational systems) should guide the selection of topics and serve as unifying themes. Students learn important mathematics when they are using mathematics in relevant contexts that require them to apply what they know and to extend how they think. The context may be fanciful or it may resemble the real world, but the content should make sense to students and involve mathematics they need to know.

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Measuring What Counts: A Conceptual Guide for Mathematics Assessment HOW MATHEMATICS SHOULD BE TAUGHT The phenomenal growth in mathematics and the way it now permeates our world require a fresh look at what it means to understand and to mathematics. The advent of powerful computing technology has made mathematics, more than ever before, an experimental science, with the same need for observation and inquiry. Reshaping School Mathematics: A Philosophy and Framework for Curriculum, from the Mathematical Sciences Education Board (MSEB), puts it as follows: Mathematics is a science. Observations, experiment, discovery, and conjecture are as much part of the practice of mathematics as of any natural science. Trial and error, hypothesis and investigation, and measurement and classification are part of the mathematician's craft and should be taught in school.11 Moreover, the availability of computers has renewed the emphasis on realistic applications, greatly simplifying the treatment of data in the classroom and permitting dynamic representations of complex processes. With the aid of computers, students can have experiences heretofore impossible in representing patterns, estimating solutions, and exploring how changes in one representation affect another. Technology such as powerful hand-held graphing calculators—in reality, hand-held computers—allow real-life problems to be explored in the classroom in all their complexity. Just as when they study the natural, physical, and social sciences, students of mathematics should be given opportunities to pose problems and advance hypotheses after they have examined a situation for the patterns and relationships it contains. They need to learn how to construct and use mathematical models of real phenomena. They should be taught to make and test their inferences, using estimates and the mathematics of uncertainty as well as the more familiar techniques of arithmetic, algebra, geometry, and calculus. Such activities will help them understand both how their model works and how that model falls short of capturing the complexity of the situation.

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Measuring What Counts: A Conceptual Guide for Mathematics Assessment Students learn best and most enduringly by reflecting on their experience and by communicating with others about it. Mathematics is a science, "the science of patterns."12 It enables scientists to analyze the regularities in their data and fourth graders to understand and make sense of the multiplication table. More than that, mathematics is also a language, the language of much of today's business and commerce as well as "the language in which nature speaks."13 To teach mathematics as both a useful science and a living language, rather than simply as a collection of arbitrary rules to be memorized, demands a different approach to the subject. The interrelation of mathematical ideas from all branches of the mathematical sciences needs to be stressed from kindergarten through high school and beyond. For more than a century, educators have bemoaned the compartmentalization of school mathematics into arithmetic, algebra, trigonometry, and so on.14 If their mathematics is to be both understandable and usable, students must learn to apply ideas from algebra and geometry together with statistics and discrete mathematics to the analysis of data. They can then use this analysis to pose problems, test hypotheses, construct mathematical models, and communicate their findings. In such activities, it is pointless to maintain a separation between mathematical topics and contradictory to the process of mathematical exploration. Good teachers have long recognized that mathematics comes alive for students when it is learned through experiences they find meaningful and valuable. Students want to make sense of their world. Mathematics becomes part of that world when it is seen both as sensible in itself and as a tool for making sense out of otherwise confusing situations. Research from cognitive science and instructional psychology supports the view that successful learners build their own understanding of subject matter. Much of this research uses mathematics as a discipline for exploring issues of learning.15 According to this research, even the youngest learners take nothing ready-made; instead, they filter what they learn through their own sensibilities and through what they already know. Students learn best and most enduringly by reflecting on their experience and by communicating with others about it. A new view of mathematical performance is developing in which the focus is on the concrete tasks students perform in a specific social context rather than on abstract abilities that students are assumed to possess.16 Learners benefit from performing a

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Measuring What Counts: A Conceptual Guide for Mathematics Assessment challenging task and developing their understanding of it in interaction with others. Learning mathematics along with others helps students develop confidence in doing mathematics and a positive disposition toward the subject.17 WHO SHOULD LEARN MATHEMATICS Evidence is rapidly accumulating that the challenging levels of mathematics needed for the future can be learned by all students. Our curriculum is still organized in ways that prevent many students from gaining access to the mathematics they need. In America, we have developed a two-tiered system in which "poor and minority students are underrepresented in college-preparatory classes such as algebra and geometry and overrepresented in dead-end classes such as consumer math and general math."18 Research has demonstrated that practices of ability grouping for instruction deny many students the opportunity to learn valuable mathematics.19 Further complicating the problem is that many classes—including those labeled college prep—do not provide the mathematics education needed by today's students. Just gaining access to such classes is not enough. All students must learn important mathematics in these classes. Americans have tended to view achievement in mathematics as a product of special talent rather than effort.20 Yet most young children like mathematics and see it as something they are capable of doing. Indeed, most children enter school with quite sophisticated theories about how numbers are used in their world. In elementary school, however, mathematics begins to look different to them. Students judge mathematics as harder to learn than other kinds of content, see themselves as less capable of learning mathematics on their own, and feel more dependent on direct instruction from teachers and others.21 These views stem in large part from experiences the students have in school. Too much of students' mathematics instruction is divorced from any context that connects with their lives and with the intuitive understanding they bring with them to class.22 These views are also shaped and supported by the attitudes of the adults around them: Many parents who would find it completely unacceptable if their children did not learn to read are content to accept and excuse low performance in mathematics.

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Measuring What Counts: A Conceptual Guide for Mathematics Assessment Fortunately, evidence is rapidly accumulating that the challenging levels of mathematics needed for the future can be learned by all students.23 From elementary school to college, programs are emerging that reduce drop-out rates and, more importantly, record significant gains by traditionally underrepresented students in challenging mathematics classes.24 A report from the Algebra Project stated that The best example is the King School [in Cambridge, MA], where the program has been in place for 10 years. Before the Algebra Project, few students took the optional advanced-placement qualifying test in ninth grade, and virtually none passed. By 1991, the school's graduates ranked second in Cambridge on the test.25 Such programs invalidate the myth that only a talented few can learn important mathematics. Results from studies around the world, particularly from countries with high achievement in mathematics, also invalidate the myth. In many Asian countries, for example, academic standards are high, everyone has high expectations for all children, and people believe that all children can learn to those standards if they are taught well and work hard.26 To produce an adult population with the knowledge it will need, mathematics education must reflect and support the view that all students can learn significantly more mathematics than is currently the case. Assessment plays a critical role in this process because assessment will measure and influence students' learning. CURRENT EFFORTS AT REFORM Efforts to reform school mathematics are proceeding along three lines: revitalizing the curriculum, redesigning the professional development of teachers, and reconceptualizing the assessment of learning. Strategies aimed at curriculum and professional development are set out in some detail in recent reports that include Everybody Counts, Curriculum and Evaluation Standards for School Mathematics, Reshaping School Mathematics, Professional Standards for Teaching Mathematics, A Call for Change, and Counting on You.27 These reports agree that goals for student performance are shifting from a narrow focus on routine skills to the provision of a variety of experiences aimed at developing students' mathematical power. They encourage the movement from teaching as the

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Measuring What Counts: A Conceptual Guide for Mathematics Assessment Reform in mathematics assessment must be based not simply on what is easy to assess but must more importantly on what needs to be assessed. transmission of knowledge to teaching as the stimulation of learning. They view teachers as the central agents for changing school mathematics and ask that teachers be given continuing support and adequate resources. Learning as sense-making and teaching as providing experiences in which sense can be made are at the crux of the vision of school mathematics emerging today in American society.28 Yet, these ideas are far from new. Thinkers as diverse as Aristotle, Dewey, and Piaget or as similar as Pestalozzi, Froebel, and Pólya have all expressed such thoughts. Mathematics teachers have for centuries found it difficult to lead students to a deep understanding of how and why mathematics works as it does. What is different now that makes successful reform in mathematics education more likely? Part of the answer can be found in the reports noted above, as they document how efforts are moving ahead together, for perhaps the first time, on three fronts—curriculum, professional development, and assessment—to ensure the necessary transformation of mathematics learning. Another part of the answer can be found in the widespread consensus that change in assessment is critical to improving education. Content and measurement experts alike have been exploring ways of creating assessments that promote and support educational reform. Until recently, however, there was little collaboration and very few points of cross-fertilization between the two fields on how this might be accomplished. This picture is changing. As mathematics experts are grappling with educational principles to guide assessment, so measurement experts are re-examining the criteria by which the technical quality of assessments is evaluated. Like new views of mathematics teaching and learning, new technical criteria and procedures for making them operational are being refined at the present time.29

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Measuring What Counts: A Conceptual Guide for Mathematics Assessment ENDNOTES 1   National Research Council, Mathematical Sciences Education Board, Board on Mathematical Sciences, and Committee on the Mathematical Sciences in the Year 2000, Everybody Counts: A Report to the Nation on the Future of Mathematics Education (Washington, D.C.: National Academy Press, 1989), 1. 2   Ibid., 17-29. 3   National Commission on Excellence in Education, A Nation at Risk: The Imperative for Educational Reform (Washington, D.C.: U.S. Government Printing Office, 1983); U.S. Department of Education, America 2000: An Education Strategy Sourcebook (Washington, D.C.: U.S. Government Printing Office, 1991); U.S. Department of Labor, Secretary's Commission on Achieving Necessary Skills, What Work Requires of Schools: A SCANS Report for America 2000 (Washington, D.C.: U.S. Department of Labor, 1991); James G. Glimm, ed., Mathematical Sciences, Technology, and Economic Competitiveness (Washington, D.C.: National Academy Press, 1991). 4   National Council of Teachers of Mathematics, Curriculum and Evaluation Standards for School Mathematics (Reston, VA: Author, 1989); National Council of Teachers of Mathematics, Professional Standards for Teaching Mathematics (Reston, VA: Author, 199 I). 5   Everybody Counts, 34; See also National Research Council, Committee on Resources for the Mathematical Sciences, Renewing U.S. Mathematics: Critical Resource for the Future (Washington, D.C.: National Academy Press, 1984). 6   Everybody Counts, 32-33. 7   Eliakim H. Moore, "On the Foundations of Mathematics," presidential address to the ninth annual meeting of the American Mathematical Society, Science, 13 March 1903. 8   Mathematical Association of America, National Committee on Mathematical Requirements, The Reorganization of Mathematics in Secondary Education cited in National Council of Teachers of Mathematics, The Teaching of Secondary School Mathematics, NCTM Yearbook (Washington, D.C.: Author, 1970). 9   College Entrance Examination Board, Commission on Mathematics, Program for College Preparatory Mathematics (New York, NY: Author, 1959). 10   Curriculum and Evaluation Standards for School Mathematics, 5. 11   National Research Council, Mathematical Sciences Education Board, Reshaping School Mathematics: A Philosophy and Framework for Curriculum (Washington, D.C.: National Academy Press, 1990), 10-11. 12   Lynn Arthur Steen, "The Science of Patterns," Science, 29 April 1988, 611-616. 13   Reshaping School Mathematics, 11. 14   Thomas Hill, The True Order of Studies (New York, NY: G. P. Putman's Sons, 1876); See also Solberg E. Sigurdson, "The Development of the Idea of Unified Mathematics in the Secondary School Curriculum 1890-1930" (Ph.D. diss., University of Wisconsin, 1962). 15   Two classroom-based research projects in mathematics have provided

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Measuring What Counts: A Conceptual Guide for Mathematics Assessment     some strong evidence supporting the constructivist approach: The first is the "Second Grade Classroom Teaching Project" that has been reported extensively by Paul Cobb, Terry Wood, Erna Yackel and their colleagues [see, for example, Cobb, Wood, and Yackel "Classrooms as Learning Environments for Teachers and Researchers," in Robert Davis, Carolyn Maher and Nel Noddings, eds., Constructivist Views on the Teaching and Learning of Mathematics, monograph, no. 4 (Reston, VA: National Curriculum of Teachers of Mathematics, 1990), 125-146]. The second is the Cognitively Guided Instruction project, which has been described by Elizabeth Fennema, Thomas Carpenter, and Penelope Peterson [see, for example, Fennema, Carpenter, and Peterson, "Learning Mathematics with Understanding: Cognitively Guided Instruction," in J. Brophy, ed., Advances in Research in Teaching (Greenwich, CT: JAI Press, 1989), 195-221]. For a more general discussion, see Lauren B. Resnick, Education and Learning to Think, National Research Council, Committee on Mathematics, Science, and Technology Education, Commission on Behavioral and Social Sciences and Education (Washington, D.C.: National Academy Press, 1987); and Everybody Counts, 58-59. 16   Yvette Solomon, The Practice of Mathematics (London, England: Routledge, 1989), 179-187. 17   Thomas L. Good, Catherine Mulryan, and Mary McCaslin, "Grouping for Instruction in Mathematics: A Call for Programmatic Research on Small Group Processes," in Douglas A. Grouws, ed., Handbook of Research on Mathematics Teaching and Learning (New York, NY: Macmillan, 1992), 165-196. Neil Davidson and Toni Worshem, "Enhancing Thinking Through Cooperative Learning" (New York, NY: Teachers College, 1992). 18   Vinetta Jones, "Responses to Three Questions" (Paper prepared for The State of American Public Education: Views on the State of Public Schools, Washington, D.C., 4-5 February 1993). 19   Jeannie Oakes, Keeping Track: How Schools Structure Inequality (New Haven, CT: Yale University Press, 1985); Jeannie Oakes, Multiplying Inequalities: The Effect of Race, Social Class, and Tracking on Opportunities to Learn Mathematics (Santa Monica, CA: Rand Corporation, 1990); Leigh Burstein, ed., Student Growth and Classroom Processes, vol. 3, IEA Study of Mathematics (Oxford, England: Pergamon Press, 1992). 20   Harold W., Stevenson and James W. Stigler, The Learning Gap: Why Our Schools Are Failing and What We Can Learn from Japanese and Chinese Education (New York, NY: Summit Books, 1992), 113-129. 21   S. Stodolsky, S. Salk, and B. Glaessner, "Student Views About Learning Math and Social Studies," American Educational Research Journal 28:1 (1991), 89-116. 22   Heibert Ginsburg, Children's Arithmetic: The Learning Process (New York, NY: D. Van Nostrand Co., 1977); Lauren B. Resnick and Wendy W. Ford, The Psychology of Mathematics for Instruction (Hillsdale, NJ: Lawrence Erlbaum Associates, 1981). 23   Michael Cole and Peg Griffin, eds., Contextual Factors in Education: Improving

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Measuring What Counts: A Conceptual Guide for Mathematics Assessment     Science and Mathematics Education for Minorities and Women (Madison, WI: Wisconsin Center for Education Research, 1987); Edward A. Silver, Margaret S. Smith, and Barbara S. Nelson, "The QUASAR Project: Equity Concerns Meet Mathematics Education Reform in the Middle School," to appear in Walter G. Secada, Elizabeth Fennema, and Lisa Byrd, eds., New Directions in Equity for Mathematics Education, (Draft version, April 1993); Deborah A. Carey et al., "Cognitively Guided Instruction: Towards Equitable Classrooms" to appear in New Directions in Equity for Mathematics Education. 24   Cynthia M. Silva and Robert P. Moses, "The Algebra Project: Making Middle School Mathematics Count," Journal of Negro Education 59:3 (1990), 388; P. Uri Treisman, "Studying Students Studying Calculus: A Look at the Lives of Minority Mathematics Students in College," College Mathematics Journal 23:5 (1992), 362-372. 25   Alexis Jetter, "Mississippi Learning," The New York Times Magazine, 21 Feb 1993, 35. 26   This case is made most extensively by Harold W. Stevenson and James W. Stigler in The Learning Gap: Why Our Schools are Failing and What We Can Learn from Japanese and Chinese Education. 27   The Mathematical Association of America, A Call for Change: Recommendations for the Mathematical Preparation of Teachers of Mathematics, (Washington, D.C.: Author, 1991); The National Research Council, Mathematical Sciences Education Board, Counting on You (Washington, D.C.: National Academy Press, 1991). See endnotes 1, 4, and 11 for additional citations. 28   Curriculum and Evaluation Standards for School Mathematics, 15-19, 38-40; Everybody Counts, 46-48, 61. 29   See Lee J. Cronbach, "Five Perspectives on Validity Argument," in Howard Warner and Henry I. Baum, Test Validity (Hillsdale, NJ: Lawrence Erlbaum Associates, 1988), 3-17; Robert L Linn, Eva L Baker, and Stephen B. Dunbar, "Complex, Performance-Based Assessment: Expectations and Validation Criteria," Educational Researcher 20:8 (1991), 15-21; John R. Frederiksen and Allan Collins, "A Systems Approach to Educational Testing," Educational Researcher 18:9 (1989), 27-32; Samuel Messick, "Validity," in R. L Linn, ed., Educational Measurement (New York, NY: American Council on Education/ Macmillan, 1989), 13; Pamela Moss, "Shifting Conceptions of Validity in Educational Measurement" (Paper presented at the annual meeting of AERA, San Francisco, April 1992.); Lorrie Shepard, "Psychometricians' Beliefs about Learning," Educational Researcher 20:6 (1991), 33-42; Eva L. Baker and Robert L. Linn, "The Technical Merit of Performance Assessments," The CRESST Line, Newsletter of the UCLA Center for Research on Evaluation, Standards, and Student Testing, Spring 1993, 1); Stephen B. Dunbar, Daniel Koretz, and H. D. Hoover, "Quality Control in Development and Use of Performance Assess­ments,'' Applied Measurement in Education 4:4 (1991), 289-304; Eva L. Baker, Harold F. O'Neil, and Robert L. Linn, What Works in Alternative Assessment? (Draft version, September 1992); Stephen B. Dunbar and Elizabeth A. Witt, "Design Innovations in Measuring Mathematics Achievement" (Paper commissioned by the Mathematical Sciences Education Board, September 1993, appended to this report).