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CURRICULUM Mathematics is one way we make sense of things. It en- ables us to perceive patterns, to comprehend data, and to reason carefully. Truth and beauty, utility and application frame the study of mathematics like the muses of Greek theater. Together, they define mathematical power, the ob- jective of mathematics education. The transformation of mathematics from a core of ab- stract studies to a powerful family of mathematical sciences is reflected poorly, often not at all, by the traditional math- ematics curriculum. One can hardly blame students for not becoming interested in mathematics if they rarely see evi- dence of its full power and richness. M ............................... athematics curricula at all levels must introduce more of the breadth and power of the mathematical sciences. As mathematics is more than calculation, so education in mathematics must be more than mastery of arithmetic. Ge- ometry, chance, and change are as important as numbers in achieving mathematical power. Even more important is a comprehensive flexible view that embodies the intrinsic unity of mathematics: estimation supplements calculation; heuristics aid algorithms; experience balances innovation. To prepare students to use mathematics in the twenty-first century, today's curriculum must invoke the full spectrum of the mathematical sciences. Philosophy Virtually all young children like mathematics. They do mathematics naturally, discovering patterns and making con- jectures based on observation. Natural curiosity is a power- fu! teacher, especially for mathematics. ...developing mathematical power 43
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Curriculum " 'To some extent, every- body is a mathematician' . ..Schoo! mathematics must endow all students with a realization of this fundamental fruth." NCTM Standards 44 Unfortunately, as children become socialized by school and society, they begin to view mathematics as a rigid sys- tem of externally dictated rules governed by standards of accuracy, speed, and memory. Their view of mathematics shifts gradually from enthusiasm to apprehension, from con- fi(lence to fear F.ventil~liv most ct,~A-ntc learn Phi_ ~ ~^-~ ~ ^~ v ~ ~ ~ ~ ~ 1~ ~ ~ 111CI L11~111~ L . . - arcs under duress, convinced that only geniuses can learn it. Later, as parents, they pass this conviction on to their chil- dren. Some even become teachers and convey this attitude to their students. Doing mathematics is much like writing. In each, the fi- na~ product must express good ideas clearly and correctly, but the ideas must be present before the expression can take form. Good ideas poorly expressed can be revised to im- prove their form; empty ideas well expressed are not worth revising. A mathematics curriculum that emphasizes compu- tation and rules is like a writing curriculum that emphasizes grammar and spelling; both put the cart before the horse. Excessive emphasis on mechanics of mathematics not only inhibits learning, but also leads to widespread misconcep- tions among the public concerning strengths and limitations of mathematical methods. Because early school experiences suggest that all mathematics problems have a single correct answer, the public to its great risk tends to believe un- critically any expert who employs mathematical arguments. Such a misconception can lead to disastrous consequences when naive mathematical modeling is used as the basis for economic, military, or social planning. Too few of those who use mathematical models are well enough prepared to --~7 _ me, ~ ~ . ~ ~ ~ . . appreciate the limitations of the models they use. Mathematics instruction must not reinforce the common impression that the only problems amenable to mathemati- cal analysis are those that have unique correct answers. Even more, it must not leave the impression that mathematical ideas are the product of authority or wizardry. Mathematics is a natural mode of human thought, better suited to certain types of problems than to others, yet always subject to con- firmation and checking with other types of analyses. There is no place in a proper curriculum for mindless mimicry mathematics.
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...developing mathematical power ~ Self-confidence built on success is the most important objective of the mathematics curriculum. The ability of individuals to cope with mathematics wher- ever it arises in their later lives" whether as wage-earners, parents, or citizens depends on the attitudes toward math- ematics conveyed in school and college classes. Above all, mathematics curricula must avoid leaving a legacy of mis- understanding, apprehension, and fear. Standards Independence is the hallmark of U.S. education. Educa- tional policy is set not by the U.S. Department of Education, but by fifty states and thousands of independent school dis- tricts. Local control of education is embedded in the Amer- ican body politic as deeply as anything, a legacy of consti- tutional authority that reserves to the states all matters not expressly granted to the federal government. Yet this independence is largely a myth, especially for mathematics education. Effective control comes not from Washington, but from invisible local or state committees that approve textbooks and from anonymous officials who select standardized tests. Few facts stand undisputed in educa- tional research, but the dependence of teachers on textbooks and of students on tests is as firm a finding as exists in this amorphous discipline; especially in mathematics, teachers teach only what is in the textbook and students learn only what will be on the test. In practice, although not in law, we have a national cur- riculum in mathematics education. It is an "underachieving" curriculum that follows a spiral of almost constant radius, reviewing each year so much of the past that little new learn- ing takes place. Some states (for example, California, Texas, Wisconsin, and New York) have recently promulgated new standards for mathematics education, often with surprising Myth: What was good enough for me is good enough for my child. Reality: Today's world is more mathematical than yesterday's, and tomorrow's world will be more mathematical than today's. As computers increase in power, some parts of mathematics be- come less important while other parts become more important. While arithmetic proficiency may have been "good enough" for many in the middle of the cen- tury, anyone whose mathematical skills are limited to computation has little to offer today's society that is not done better by an in . ~ . expensive machine. 45
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Curriculum Myth: Arithmetic is the major goal of elementary school mathe- matics. Reality: Number sense builds on arithmetic as words build on the alphabet. Numbers arise in mea- surement, in chance, in data, and in geometry, as well as in arith- metic. Mathematics in elemen- tary school should weave all these threads together to create in chil- dren a robust sense of numbers. 46 consequences. In California, new standards led to initial rejection of ad mathematics textbook series submitted for authorized adoption. California wanted to stress as emphat- ically as possible that, in the future, mathematics textbooks must be designed to develop student capabilities to address and solve complex, subtle, and unpredictable problems. A,, erica needs to reach consensus on national standards for school mathematics. We must judge schools not by remembrances of things past, but by necessary expectations for the future. Students must learn not only arithmetic, but also estimation, measure- ment, geometry, optimization, statistics, and probability- all of the ways in which mathematics occurs in everyday life. In the process, they must gain confidence in their ability to communicate and reason about mathematics; they should become mathematical problem-solvers. Elementary Education Elementary school is where children learn the mathemat- ical skills needed for daily life. Formerly, shopkeeper arith- metic was an adequate objective since, for most people, mathematics in daily life required little more than arith- metic. This is no longer true. Calculators now do most of the arithmetic needed for dominated society requires of chance, of reasoning, of form, and of pattern. while the goal of elementary education has not changed, the mathe- matical objectives appropriate to this goal are very different now from what they were half a century ago. The major objective of elementary school mathematics should be to develop number sense. Like common sense, daily life, while a technology- that everyone have a good grasp
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...dereloping mathematical power number sense produces good and useful results with the least amount of effort. It is not mindlessly mechanical, but flexible and synthetic in attitude. It evolves from concrete experi- ence and takes shape in oral, written, and symbolic expres- sion. Links to geometry, to chance, and to calculation should reinforce formal arithmetic experience to produce multiple mental images of quantitative phenomena. Developing number sense will move children (and teach- ers) beyond narrow concern for school-certified algorithms for arithmetic. Even in the absence of calculators, neither children nor adults make much use of the specific arithmetic techniques taught in school. School children do, however, need to learn how to use mathematics for common tasks- making change, measuring quantities (food, lumber, fabric), planning schedules, estimating chances but the particular means that they use must be appropriate to the task. They need to learn not only how to estimate and calculate, but also how to decide whether to estimate or calculate. Good number sense includes common sense about how to find an answer as well as a range of choices of methods. Appropriate use of calculators enhances children's understanding and mastery of arithmetic. using calculators intelligently is an integral part of num- ber sense. Children should use calculators throughout their school work, just as adults use calculators throughout their lives. More important, children must learn when to use them and when not to do so. They must learn from experience with calculators when to estimate and when to seek an exact answer; how to estimate answers to verify the plausibility of calculator results; and how to solve modest problems men- tally when neither pencil nor calculator is convenient. Calculators create whole new opportunities for ordering the curriculum and for integrating mathematics into science. Back to School "Target Addition" is a calcula- tor game that reinforces mental arithmetic. Two children with a calculator agree on a target num- her such as 21. They take turns adding a number from 1 to 5 into the memory. The goal is to make the memory number match the target number. 47
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. Curriculum Myth: Early use of calculators will prevent children from learn- ing basic facts of arithmetic. Reality: Children learn to solve arithmetic problems by a variety of means just as adults do in ordinary activities. Sometimes, one calculates mentally (figur- ing tips); other times, one esti- mates (buying groceries). Still other times, one calculates with a pencil (doing carpentry) or with a calculator (preparing taxes). Learning basic arithmetic re- quires that one learn all these approaches-not only how to do them, but, more importantly, when to do them. Calculators are an essential part of the repertoire of arithmetic skills that any child or adult needs to use. 48 No longer need teachers be constrained by the artificial re- striction to numbers that children know how to employ in the paper-and-pencil algorithms of arithmetic. Decimals can be introduced much earlier since they arise naturally on the cal- culator. Real measurements from science experiments can be used in mathematics lessons because the calculator will be able to add or multiply the data even if the children have not yet learned how. They may learn first what addition and multiplication mean and when to use them, and only later how to perform these operations manually in all possible cases. Elementary school mathematics should reinforce a child's natural curiosity about patterns. Children must be encour- aged to perceive mathematics in the world around them. Shapes, numbers, chance the foundations of geometry, arithmetic, probability- will emerge from careful guided ob- servation. Science study will lead naturally to mathematics, following the paradigm of data, deduction, and observation. Many adults fear that early introduction of calculators will prevent children from learning basic arithmetic "properly," as their parents learned it. The experiences of many schools during the last fifteen years show that this fear is unfounded. Students who use calculators learn traditional arithmetic as well as those who do not use calculators and emerge from el- ementary school with better problem-solving skills and much better attitudes about mathematics. Although mindless cal- culation can be as destructive as mindless arithmetic, proper use of calculators can stimulate growth of a realistic and productive number sense in each child. Secondary Education Secondary education is where students begin to learn the mathematics they will need for careers as well as the mathe- matics required for elective citizenship. Whereas, tradition- ally, secondary school has been characterized by the intro- duction of algebra as an extension of arithmetic, contempo- rary society requires much greater breadth from secondary school mathematics.
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...developing mathematical power The focus of the secondary school curriculum remains- as it should on the transition from concrete to conceptual mathematics. As students' understanding moves from num- bers to variables, from description to proof, from special cases to general equations, they learn the power of mathe- matical symbols. In a very real sense, the major objective of secondary school mathematics is to develop symbol sense. All students need to leave secondary school well prepared mathematically tor leading intelligent lives as productive citizens since even many of those who go on to higher education will take little or no further mathematics. High school graduates need to know enough about chance to understand health and environmental risks; enough about change and variability to understand investments; enough about data and experiments to understand the grounds for scientific conclusions; enough about representation to inter- pret graphs; and enough about the nature of mathematics to be supportive parents to their children who will learn aspects of mathematics that their parents never studied. Students who enter the work force directly from high school will be expected to be able to read documents re- plete with technical language-computer guides, shop man- uals, financial reports. They need to be able to comprehend three-dimensional images (assembly diagrams) and logically intricate instructions (tax code); they need to be able to read symbol-laden text (computer manuals) and to plan complex systems (purchasing property). They need enough mathe- matics and enough confidence to be able to learn what they need as they need it. A I} high school students should study a common core of broadly useful mathematics. _ · , Typically, secondary school mathematics curricula are dominated by a philosophy of preparation for college calcu- lus. Few natural applications emerge from secondary school Back to School Investigate the relation between automobile mileage and age by gathering data from cars in the school parking lot. AIDS Tests How is it possible for an AIDS test that is 95 percent accurate to be misleading 90 percent of the time when used in mass testing? Issues such as this are central to public policy, yet require a basic knowledge of probability to be understood properly. 49
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Curriculum Myth: The primary goal of mathematics education is to educate mathematicians. Reality: School mathematics is part of universal education. Three of every four college ma- jors require students to study college-level mathematics. To cope with a technological age, all students should study mathemat- ics every year they are in school. 50 mathematics since the traditional topics are merely tools for the applications of calculus. Even worse, students rarely learn mathematics appropriate to enlightened citizenship or to the needs of the workplace. These deficiencies are not inherent in secondary school mathematics, but historical accidents arising from the teleo- logical influence of calculus. New mathematics with broader applications can offer much greater appeal to all students. In- stead of tracking students by curricular objective (for exam- ple, commercial, general, or precollege), school mathematics should provide for all students a core of mainstream math- ematics in which different student groups are distinguished not by curricular goals, but only by speed, depth, and ap- proach. For students planning to enter college, still more is needed: thorough grounding in mathematical methods required for calculus, statistics, and computer science. As these colle- giate subjects change in response to the impact of comput- ers, the prerequisites will shift away from formal algebra to more integrative. Droblem-solvina approaches. Versatil ~ , ~ ~ ~ ,~ . ity, confidence, experience, reasoning, and communication about mathematics are skills that will be valued as prerequi- sites to college mathematics. A ................................ 11 students should study mathematics every year they are in school. . · ~ The gradual mathematization of society has increased con- tinuously the amount of mathematics that students must learn. Universal elementary education has been replaced, for all practical purposes, by universal secondary education. As a consequence, students' study of mathematics must con- tinue throughout secondary school. · ~.
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...d~eveloping mathematical power Higher Education Most students who enter postsecondary education must study further mathematics, either as part of a general degree requirement or as a prerequisite to their particular course of study. in this regard, vocational-technical institutes, com- munity colleges, continuing education, and adult education programs are no different from four-year colleges and uni- versities. Most courses of study have some kind of mathe- matical prerequisite and most students find that they need additional study of mathematics to meet these prerequisites. Over three quarters of the degree programs at most uni- versities require courses in calculus, discrete mathematics, statistics, or other comparable mathematics. These subjects are required because they introduce students to functions, to relations among variables. The language of change and chance is conveyed by the symbolism of functions. College students need this level of mathematical literacy in order to understand with precision the mathematical ideas that form the foundations for science, business, and engineering courses. Learning mathematics entails a gradual progression from the concrete to the abstract, from the specific to the general. Numbers lead to symbols, to names for variables; relations among symbols lead to functions, to the links among objects expressed in symbols. if it does nothing else, undergraduate mathematics should help students develop function sense a familiarity with expressing relations among variables. Nine of every ten mathematics course enrollments in higher education are in elementary calculus, in elementary statistics, or in courses that are prerequisites to these sub- jects. Prerequisite courses are normal parts of the secondary school curriculum and should, if at all possible, be studied there rather than in college. Elementary statistics is not of- ten taught in secondary schools today, but should become a regular part of secondary school mathematics. Calculus, taught both in high school and in college, is the principal gateway through which most students must pass if they are to prepare for mathematics-based careers. Although discrete mathematics and statistics provide nec Undergraduate Mathematics THOUSANDS OF ENROLLMENTS FALL SEMESTER 2,800 2,400 2,000 1,600 1 ,200 800 400 Advanced Course Enrollments ~3, - 1965 1970 1975 1980 1985 Each term, nearly 3 million students enroll in postsecondary mathematics courses. About 60 percent study elemen- tary mathematics and statistics below the level of calculus, while 30 percent take calculus-level courses. The remaining 10 percent study higher (post-calculus) mathematics. 51
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Curriculum Symbolic Computer Systems As calculators have surpassed hu- man capacity for arithmetic cal- culations, so now are symbolic computer packages overtaking human ability to carry out the calculations of calculus. Until re- cently, computers could only op- erate numerically (with rounded numbersJ and graphically (with visual approximations). But now they can operate symbolically just as people do, solving equations in terms of x and y just as we teach students in school mathematics. Symbolic computer systems com- pel fundamental rethinking of what we teach and how we teach it. 52 essary foundations for computer, engineering, and social sci- ences, calculus remains the archetype of higher mathematics. It is a powerful and elegant example of the mathematical method, leading both to major applications and to major theories. The language of calculus has spread to all scientific fields; the insight it conveys about the nature of change is something that no educated person can afford to be without. Unfortunately, calculus as presently taught has little in common with the way calculus is used. Many students who enroll never complete the course; many of those who do fin- ish learn little beyond a series of memorized techniques now more commonly performed by computers. Because of the central importance of calculus for all scientific study and research, the National Science Foundation has launched an . . . . . nltlatlve to improve instruction in calculus, which is now studied by nearly one million students every year. ~ Successful calculus is essential to healthy mathematics and science. The quality of calculus instruction is a barometer of math- ematics education. Since preparation for calculus has been the organizing principle of high school mathematics, calcu- lus receives the inheritance of school practice. Changes in calculus reverberate throughout secondary school curricula, just as changes in school mathematics are magnified by the challenge of calculus. Although other courses need improv- ing as much as calculus, and although many courses are as important or as practical, the unique position of calculus as the gateway from school to college mathematics imposes on it a special burden to be attractive, compelling, and intellec- tually stimulating. For the past two decades, enrollments in mathematics ser- vice courses in colleges and universities have been rising, yet throughout most of that period the number of math- ematics majors has been declining. Part of this paradox is
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...d~eveloping mathematical power due to career competition from attractive disciplines that are new users of mathematics, notably computer science, eco- nomics, and now, increasingly, biology. But another part of the problem lies with the mathematics major itself, which in many institutions has continued a narrow focus on graduate school preparation, ignoring the many students who might (and should) major in mathematics as a versatile preparation for careers in other disciplines. Undergraduate mathematics provides a powerful platform for careers in many fields. Building and sustaining a mathematics major with broad appeal require commitment to integrity in the undergraduate program: · Freshman and sophomore courses (especially calculus and linear aIgebra) should be taught by the most able instruc- tors who can motivate students to study mathematics. . Introductory courses must be taught in a manner that re- flects the era in which we live, making full use of comput- ers as an integral too! for instruction and for mathematics. · Upper-division offerings should be designed to represent broadly the several mathematical sciences, introducing students in appropriate ways to applications, computing, modeling, and modern topics. . Diverse opportunities outside formal classroom work should be provided so that students can engage In math- ematics through projects, research, teaching, problem- solving, or independent study. Broad undergraduate mathematics programs will attract more students to extended study of mathematics, will offer these students appealing opportunities to explore mathemat- ics applicable to many fields, and will engage good students in exploring and learning mathematics on their own. Such Continuing Education Almost as many persons study mathematics outside traditional school structures as inside them. Whereas mathematics curric- ula have been conceived in the traditional structure of continu- ous grade-level education from childhood to early adulthood, to- day much mathematics is stud- ied by older adults. Some large businesses actually operate mini- school districts just for the con- tinuing education of employees; many universities and colleges- especially community colleges- attract large numbers of adults both to regular degree programs and to special short courses for professional growth or cultural enrichment. Every issue of pur- pose, quality, and electiveness concerning traditional mathe- matics education also affects con- tinuing mathematics education. The need for renewal in the non- traditional forums is just as ur- gent as it is within the walls of traditional schools. 53
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Curriculum 54 an approach, while intended as a foundation for students in many different majors, will inevitably attract good students to careers in mathematics. Demographic data and degree trends show that it is vi- tally important for undergraduate mathematics departments to offer effective, broad-based curricula. The United States needs more American graduate students in the mathematical sciences; more mathematics graduates who can teach in sec- ondary schools; and more students with better preparation in mathematics entering graduate programs in science and engineering. Redressing the serious shortage of mathemat- ically educated college graduates is a significant and urgent challenge to our mathematics faculties.
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Representative terms from entire chapter: