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2 Science and Mathematics Education American public schools are populated by 40 million students and 2 million teachers. The magnitude and diversity of American schooling make it impossible to deliver a complete and exact picture of how well the schools are doing: simplification and abstraction are necessary and inevitable. In attempting to abstract from the complexities of American schooling and to propose new indicators of the quality of science and mathematics education, the committee has taken on a task that requires a clear sense of definitions, methods, and goals. This chapter first defines what we take to be the purposes of science and mathematics education and then presents a conception of schooling that has shaped our choice of indicators. SCIENTIFIC AND MATHEMATICAL LITERACY The recognition that societies are changing rapidly is wide- spread witness such terms as information age, postindustrial so- ciety, and gioloal economy that have come into common usage. These changes are spurred by and give rise to the development of ever more powerful technologies. Several writers who have pondered the impli- cations of these changes have made the point that the accelerating change that characterizes U.S. and other Western societies may well require a higher degree of scientific and mathematical literacy than 15
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16 INDICATORS OF SCIENCE AND M24TTIEM:A TICS EDUCATION ever before (Toffler, 1980; Naisbitt, 1982; Zarinnia and Romberg, 1986~. In general, the committee agrees with this view: we believe that all students should have the tools of knowledge and judgment that science and mathematics provide. These tools will enable them to prepare for careers and cope with rapid changes in tomorrow's labor markets, make informed choices in their private and family lives, and understand issues of broader scope. And all students, as they become adults, should be free to enjoy the enlargement of the world that comes with scientific and mathematical literacy. Clearly, then, not only should students leave school literate in science and mathematics, but they should also have acquired the mental tools with which they can renew that literacy throughout their lives. Any useful attempt to determine to what extent schools are meeting this broad goal must reflect an informed view of what con- stitutes literacy in science and mathematics. The committee suggests that there are several dimensions of scientific and mathematical lit- eracy, each of which needs to be addressed seriously. The following descriptions of these dimensions are intended to be normative: they are the ideals against which science and mathematics curricula and instruction should be compared. Because the sciences and math- ematics, despite their many interconnections, are quite differently constituted, literacy in each of these domains is discussed separately. Literacy in Science' The dimensions of scientific literacy that should be integral to any educational program include the nature of the scientific world view, the nature of the scientific enterprise, scientific habits of mind, and the role of science in human affairs. The first three of these dimensions deal with knowledge of science and intellectual skills; the fourth deals with the relation between science and society. Each is characterized in somewhat greater detail below. The Nature of the Scientific World View Over the last three centuries or so, scientific activity in many fields has resulted in a set of interconnected and testable notions about the nature of the world *We thank F. James Rutherford, American Association for the Advance- ment of Science, for suggesting the four dimensions of scientific literacy discussed in this section.
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SCIENCE AND MATHEMATICS EDUCATION 17 and its parts. While the details continue to change with time, these notions amount to a rather robust and useful construct of ideas that deserves the name scientific world view. The ideas that constitute its elements have various levels of complexity or abstraction. . At the highest level of abstraction, there are grand concep- tual schemes that bring together and bring order to large numbers of observations, concepts, and theories, each of which is of lesser generality and applies only to some narrower field of science. Some of these grand schemes can be expressed in words or numbers; oth- ers can only be understood mathematically. Examples include the Newtonian universe, organic evolution, and plate tectonics. ~ At a more operational level, the scientific world view finds expression in theories and mathematical models that organize facts and laws in ways that help one understand a particular aspect of the world. Examples include gravitation, solid-state science, statistics, weather systems, and economic determinism. ~ Particular concepts, mathematical tools, or techniques can reappear in various scientific specialties, thereby not only helping to suggest new advances but also providing syntheses among different parts of the total scientific world view. Examples of such recurring concepts or tools include scale, cycles, waves, estimation, energy, antibodies, and probability. The scientific world view also consists of general beliefs that have shown their worth over time. These include the following notions: . The world of phenomena is rationally understandable and not capricious; causal relations may be found. . Good scientific theories permit deductions that can be checked against experience. This testing frequently takes the form of comparing measurements of the real world to numerical predictions. . Individual data and observations are subject to some uncer- tainty, but phenomena are consistent. ~ Throughout the history of scientific growth, and despite ma- jor advances along the way, scientists have often found a few fun- damental thematic notions useful and motivating: for example, the search for unity or unification among diverse phenomena; the use of mechanical or mathematical models; and such notions as simplicity, parsimony, symmetry, evolution, causality, order or hierarchy, and continuity or discontinuity.
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18 INDICATORS OF SCIENCE AND MATHEMATICS EDUCATION The Nature of the Scientific Enterprise The scientific enterprise subscribes to a set of value commitments in principle an ethos of science that can be explicitly formulated. For individuals to understand the conclusions that scientific research yields concerning the natural world, they must understand these principles and some of the characteristics of the way science works: ~ Science is both theoretical and empirical, and these two aspects reinforce each other. Mathematical models without data are sterile; observations without measurement of some kind remain largely impressionistic. . Science is not only a personal, individual calling but also a social activity carried out by individuals who collaborate over time and space. Some of this accessibility is a consequence of the univer- sality of mathematics. This characteristic makes scientific activity, next to mathematics, one of the most international and shareable experiences and opens scientific research and teaching to all talents everywhere. . The substance of science at any given time is found in the consensus among scientists, as reflected largely in current writings, data bases, and mathematical formulations. As scientific knowledge becomes more developed and inclusive, findings reached at any par- ticular time are seen as tentative. As mathematical models attain greater generality, they provide novel tests of the current formula- tions. Thus, science is an enterprise concerned with discovery of the new and with testing (and correction when needed) of the old. . Necessary conditions for understanding the processes of sci- ence include familiarity with a wide range of natural phenomena; asking questions and forming hypotheses; understanding the need for tests or controlled comparisons; embracing theories of measure- ment, evidence, and data; and accepting a method of notation or formalism, most often mathematical, that allows an unambiguous and replicable depiction of a set of phenomena. ~ Pure scientific research often points the way to practical ap- plications through engineering development, and the latter in turn can help make basic experimental science much more effective. Scientific Habits of Mind Individuals, scientists or not, need to be capable of analytical thinking in the context of the various sciences and mathematics. This capability can be taken to include:
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SCIENCE AND MA THEMA TICS ED UCA TION . 19 Identifying questions or formulating hypotheses relative to a problem, recognizing when such questions ought to be quantitative, and being able to express them mathematically if that is appropriate. . Identifying and seeking out information (numerical or other- wise) relevant to a problem or issue. ~ Using that information to test the hypothesis or answer the question, while appreciating the limits placed by samples or intrinsic uncertainty. . Playing with information, in the sense of solving puzzles and raising hypotheses. . Offering arguments and counterarguments that can be tested by reference to data or accepted principles. . Communicating, collaborating, and building consensus in or- der to develop a common language and common models. The historic study of most scientific advances shows, however, that other, more individual, and even aesthetic, elements enter to assist the purely logical-critical faculty during the creative phase of scientific work. There is, in short, no single "scientific method." Moreover, accepting the scientific world view does not disqualify an individual from sensitivity to, or the appreciation of, artistic and humanistic achievements. In fact, one of the values of science is cultural. Participation in science can be satisfying in much the same way as participation in music and art. Science and Human Affairs Science and mathematics are im- portant to society because of their deep connections with and ef- fects on human events and ideas. For example, mathematical con- cepts such as estimation, statistics, sampling, and risk underlie most public-policy concerns. The application of science and mathematics to such matters as health, industrial processes, agriculture, and the environment engage a large number of policy makers and individual citizens. From this it follows that, for a society to be scientifically literate, people need to understand some of the relationships between science, policy, and society. . Tensions exist between science and society, because science must sometimes assert the presence of uncertainty and ambiguity when facts are needed by organizations that must make policy deci- sions or individuals who must make personal ones. . Scientists may behave differently when they are involved in public policy decisions than when they are acting as researchers. The
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20 INDICATORS OF SCIENCE AND MATHEMATICS EDUCATION ethics of science in public policy calls for objectivity, but this cannot always be expected in cases in which the self-interest of science itself is at stake. ~ Science and technology have helped to better the human con- dition. But progress in science and technology can also have unan- ticipated negative effects on society. These effects can be monitored and in some instances modified by the action of alert citizens, pro- vided they have the necessary educational background to obtain and interpret information on the ways in which science and technology influence personal, local, and national affairs. A historical sense of the way science grows can be useful. One cannot always predict what science will be of practical value in the future, because several seemingly unrelated lines of basic research may come together over time (sometimes decades) and contribute to breakthroughs. For example, cell culture was initiated to permit studies of development and other cellular behaviors; it subsequently made possible such medical advances as the polio vaccine. Much mathematics originally developed for its own beauty or interest has later provided tools for scientific or technological applications that were inconceivable earlier. Literacy in Mathematics ~ The types of mathematical literacy practical arithmetic, civic application, professional use of mathematics, and cultural apprecia- tion-correspond roughly to the central objectives of the four hier- archical tiers in the educational system: primary, secondary, under- graduate, and graduate. Although it is useful to postulate levels of mathematical literacy corresponding to levels of education, some key elements are integral to all levels: . Understanding the fundamental ideas of mathematics. For example, the Pythagorean theorem has theoretical and practical importance in all levels of mathematics learning, as does the notion of symmetry. These (and other) intrinsic mathematical ideas could provide benchmarks of literacy that transcend educational levels. . Understanding the role of mathematics as the language of science and its role in describing the nature of complex systems. *We thank Lynn Arthur Steen, St. Olaf College, for formulating the four levels of mathematical literacy discussed in this section.
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SCIENCE AND MATNEMA TICS EDUCATION 21 Understanding that order can beget disorder (as in turbulence) and vice versa (as in statistical experiments); that mathematical mod- els for growth can represent phenomena in biology, economics, and chemistry; and that mathematics is still being created to meet new needs are examples of perceptions about the nature of mathematics that should be part of mathematical literacy at every level. . Recognizing that mathematics is a dynamic and changing field, not, as it is generally taught, a static and bounded disci- pline reflecting recorded knowledge (Confrey, 1985~. Three current trends have deep implications for what it means to be literate in mathematics (Hilton, 1986~: the increasing variety of applications in many other fields, which need to be recognized and understood at some level by nonmathematicians; a new unification of mathematics, which calls for breaking down artificial barriers between topics in a student's education; and the changes that the computer is bringing about in mathematics (the relative importance of topics, how some mathematics is done, and the creation of new topics), which need to infuse mathematical knowledge and understanding at all levels. Practical Literacy in Mathematics Practical literacy is knowI- edge that can be put to immediate use in improving basic living standards. The ability to compare loans, to figure unit prices, to manipulate household measurements, and to estimate the erects of various rates of inflation brings immediate real benefit. This kind of applied arithmetic is one objective of universal primary education. Civic Literacy in Mathematics Civic literacy involves more so- phisticated concepts, which enhance public understanding of leg- isTative issues. Major public debates on nuclear deterrence and nu- clearpower, economic policy, public health, and the use of resources frequently center on scientific issues. Inferences drawn from data, projections concerning future behavior, and interactions among vari- ables in complex systems involve issues with essentially mathematical content. A public afraid or unable to reason with figures is unable to discriminate between rational and reckless claims in the technological arena. Ideally, secondary education should provide all students with the mathematical knowledge and understanding needed by today's "enlightened citizenry" that Thomas Peterson called the only proper foundation for democracy.
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22 INDICATORS OF SCIENCE AND MATHEMATICS EDUCATION Using Mathematics as a Too! Literacy that involves using math- ematics as a too! encompasses the mathematics necessary to study and work in science, engineeering, and other fields that employ mathematical language, ideas, and models. It refers to all uses of mathematics whether in theoretical physics or business man- agement. As science and industry come to depend increasingly on mathematical tools, professionals in ever more diverse fields will need to learn this universal language. The basis for the mathematics that constitutes use-related literacy must be laid at the secondary school level, even though these tools are greatly extended and enhanced in college mathematics courses. Cultural Literacy in Mathematics Cultural literacy in mathe- matics, the most sophisticated of these levels, pertains to the role of mathematics as a major intellectual achievement. Because cul- tural literacy lacks an immediate, practical purpose, its appeal may be limited. Yet the simpler and historically earlier parts of mathe- matical invention, like the invention of zero or of negative numbers, are accessible to many people, including quite young students. At this level of difficulty, an appreciation of mathematics as an intel- lectual activity engaged in by one's fellows should be part of any concept of mathematical literacy. As one progresses through the more complex developments in mathematics, however, the size of the interested audience may decrease, to an audience perhaps something like the readership of Scientific American. Pursuing cultural literacy in mathematics to the more advanced stages enables one to appreci- ate the seemingly arcane research of twentieth-century mathematics not only for its potential and unknown practical application but also, and more important, as an invaluable and profound contribution to the heritage of human culture. For the most part, individuals at- tain this sort of literacy through intensive study in some advanced subject, not necessarily mathematics itself. A CONCEPTION OF SCHOOLING How do schools produce the learning that is entailed in scientific and mathematical literacy? A central principle that guides this re- port is that teachers and students are the most important resources in the educational process and that their behaviors determine schooling outcomes. A second, related principle is that incentives and con- straints influence the behavior of students and teachers. A third
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SCIENCE AND MATNEMA TICS EDUCATION 23 principle concerns the question "Excellence for whom?" The com- mittee is concerned not only with the achievement levels of the most able students, but also with the distribution of knowledge and skills among students from different backgrounds. We expand our concep- tion of these three principles in the sections that follow. Schooling as the Behavior of Students and Teachers The committee's formulation of indicators is based on the view that what students and teachers do determines how much learning takes place. This principle may seem obvious and not worth empha- sizing. To appreciate its significance, it is useful to review how it evolved from earlier work on the determinants of children's academic achievement. Such work comes from several different disciplinary approaches psychology, sociology, and economics. In educational psychology, there is a long history of research on how students learn and how teachers teach. Research on learn- ing dates back to the behaviorist theories of Thorndike (1932) and Skinner (1953, 1968), was followed by theories that emphasized the interaction of the student with the structure of the subject matter (Brownell, 1947; Piaget, 1954; Bruner, 1960, 1966; Gagne, 1965; Ausubel, 1968; Dienes and Golding, 1971), and is currently devel- oping into theories of how children actively construct knowledge for themselves through their interaction with the environment, including the formal and informal teaching to which they are exposed (Resnick, 19873. Each of these theories has implications for the behavior of teachers as they shape their instruction. Sociology and anthropology also have contributed insights on the effects of teachers' (and administrators') behavior as they set the con- text for learning by the way classroom lessons are presented, children are grouped within the classroom for instruction, and classrooms and schools are organized. (For a review, see Committee on Research in Mathematics, Science, and Technology Education, 1985:26-34.) The 1960s saw the application of economics to the study of education, sometimes referred to as the estimation of educational production function models. The goals of this line of research, as exemplified by the widely known report by Coleman et al. (1966) on equality of educational opportunity, is to find schooling inputs that are systematically related to student learning. Initially, this research treated in parallel fashion such inputs as physical facilities, teaching materials, and the attributes of teachers and students.
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24 INDICATORS OF SCIENCE AND MATHEMATICS EDUCATION These different streams of research in education have provided a great deal of knowledge about the kinds of variables that are impor- tant in explaining student achievement, including the finding that the most important resources in the educational process are human beings, whose behavior influences what is learned in school. The aspects of human behavior that influence students' achievement are wide-ranging: they include the decisions of talented college graduates about whether to become teachers and how long to stay in teach- ing (SchIechty and Vance, 1983), the decisions of elementary school teachers about how much time to allocate to mathematics and sci- ence tWeiss, 1978), and the decisions of students about whether to take science and mathematics courses (Welch et al., 1982; Bryk et al., 1984) and how much homework to do or how much television to watch (Walberg et al., 1986~. Although the results of educational research studies regarding the critical importance of the behavior of students and teachers have been informative, it has been difficult to make linkages between these results and policies to improve schooling. One reason is that, as these very studies indicate, the resources most important in explaining chil- dren's achievement are the human beings whose behavior influences what is learned in school. And human behavior is not subject to easy adjustment by managers and policy makers who wish to improve learning. Policy makers can change the behavior of teachers and students only to a limited degree. Incentives and Constrairz~s In emphasizing that the behavior of students and teachers is difficult to alter, we do not mean to imply that it cannot be influ- enced. In fact, a second principle underlying the recommendations in this report is that the behavior of teachers and students is indeed influenced by the incentives and constraints they face. Examples of such incentives include teachers' salaries relative to those offered in other professions, which may attract or dicourage talented individu- als, and the quality of the mathematics and science courses available in a school, which may increase or decrease student enrollment. These two principles the importance of the behavior of teachers and students and the responsiveness of the behavior of teachers and students to the incentives and constraints they face have influenced both the design of this report and our recommendations. They have led us to recommend the collection of information on many aspects
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SCIENCE AND MATHEMATICS EDUCATION 25 of the behavior of teachers and students that influence the quality of mathematics and science instruction and that ultimately influence the level of science and mathematics literacy in the population. And they have lect us to recommend the collection of information on many incentives and constraints that influence the behavior of teachers and students. From this perspective, what is the importance of physical re- sources devoted to mathematics and science instruction, such as laboratories, teaching materials, and, most important, curriculum? Don't they matter? Indeed they do. However, we believe that they matter primarily through their influences on the behavior of teachers and students. For example, the lack of adequate laboratory facilities may make it difficult for a school to attract teachers who really want to teach science and may force teachers who do teach science in that school to base instruction on memorizing facts rather than on de- veloping an understanding of scientific principles through hands-on experiments. By the same token, the lack of facilities and the conse- quent dullness of the instruction may lead students to avoid taking science courses. Our emphasis on looking at physical facilities and curriculum from the perspective of examining how they influence the behavior of teachers and students is not intended to downplay the importance of these resources. The opposite is in fact the case. Some of the early production function research concluded that physical facilities do not matter, because the research was based on a design that im- plicitly held constant which teachers worked in a school and which courses students took. This research design eliminated some of the most important mechanisms through which facilities do matter: by influencing the quality of teachers who are attracted to the school and the number of students who take science courses. (For a discus- sion of research on the effects of instructional resources, see Carey, 1986.) Thus, our emphasis is intended to highlight the potential importance of facilities in influencing the behavior of the key actors in the educational process. Similarly, understanding the effects of curriculum on student learning is often clouded by the lack of distinction between the cur- riculum laid out in state and school district manuals, what has been called the mandated or intended curriculum, and the curriculum that children actually experience, the de facto or actual curriculum. The difference between the intended curriculum and the actual curricu- lum stems from the decisions teachers make about what aspects of the
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26 INDICATORS OF SCIENCE AND MATHEA!A TICS EDUCATION intended curriculum to emphasize and how to adapt the curriculum (including the textbook) to accommodate their own skills and inter- ests and their perceptions of their students' skills and interests. As a result of these decisions by individual teachers about how to use the intended curriculum, children in different classrooms and in different schools experience different actual curricula and consequently learn different things, even when they all attend schools using the same in- tended curriculum. For this reason, the recommendations presented in Chapter 7 on indicators of curriculum quality are sensitive to the distinction between intended curricula and actual curricula. Another implication of our perspective is that it is important to pay attention not only to the quality of the physical resources and curricula in schools, but also to the role teachers play in shaping curricula and in deciding what supplies and materials are purchased. For example, teachers are much more likely to use new curricula and new teaching materials if they have had a hand in the planning and decision processes (Berman and McLaughlin, 1974-1975~. Therefore, some of our recommendations include ideas for learning more about what influences teachers' responses to changes in resources and the intended curriculum. The Distribution of Excellence A third principle underlying the recommendations in this report is that, in addition to describing the extent to which schools are mak- ing progress in promoting excellent mathematics and science educa- tion, indicators should address the question: Excellence for whom? This is central to promoting scientific and mathematical literacy for all students and to ensuring that talent will be nurtured wherever it is found. An example of the comm~ttee's concern regards teacher qualifications: one needs to know not only about changes in the qual- ifications of the nation's science teachers as a whole, but also about the qualifications of science teachers who teach identifiable groups of children, such as minority group children, urban children, rural chil- dren, and children not in advanced-placement science courses. This principle underlies many of our specific recommendations for how data should be collected and reported, especially data on teacher qualifications and on student behavior.
Representative terms from entire chapter: