Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page 1
OPPOR TUNITY 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 pro- vides knowledge to compete in a technological economy. 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 mentally fit 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, Amer- icans need to think for a living; more than ever before, they need to think mathematically. ~ ~ uaTitv mathematics education for n11 students Is essential for a healthy economy. Yet, for lack of mathematical power, many of today's stu- dents are not prepared for tomorrow's jobs. In fact, many are not even prepared for today's jobs. Current mathe- matical achievement of U.S. students is nowhere near what is required to sustain our nation's leadership in a global technological society. AS technology has ~rIlathematlcized'' the workplace and as mathematics has permeated society, a complacent America has tolerated underachievement as the norm for mathematics education. We have inherited a mathematics curriculum conforming to the past, blind to the future, and bound by a tradition of minimum expectations. Wake up, America! Your children are at risk. Three of every four Americans stop studying mathematics before ...tapping the power of mathematics "How can students com- pete in a mathematical society whet' they leave school knowi1'g so little mathematics?" - rester Thurow* * People, reports, and dlata cited! in the margins are identified on pages 102-103. 1
OCR for page 2
Opportunity "The fastest growing jobs require much higher math, language, and rea- soning capabilities than current jobs, while slowly · e ~ ~ growlngJoos require less.', Workforce 2000 2 completing career or job prerequisites. Most students leave school without sufficient preparation in mathematics to cope either with on-thejob demands for problem-solving or with college requirements for mathematical literacy. Thus, in- dustry, universities, and the armed forces are burdened by extensive and costly demands for remedial education. Today's mathematics opens doors to tomorrow's jobs. As successive waves of immigrants have used this country's ed- ucational system to secure better lives for themselves and their children, so today's children the world over are using mathematical training as a platform on which to build up their lives. America's children deserve the same chance. Children can succeed in mathematics. Many do so in other countries and some do so in this country. The evidence from other nations shows overwhelmingly that if more is expected in mathematics education, more will be achieved. Clear expectations of success by parents, by schools, and by society can promote success by students. Children can succeed in mathematics. If more is expected, more will be achieved. In today's world, the security and wealth of nations de- pend on their human resources. So does the prosperity of individuals and businesses. As competitors get smarter, our problems get harder. Long-term investment in science and technology both for businesses and for our nation requires serious commitment to revitalizing mathematics ed- ucation. It is time to act, to ensure that ad Americans benefit from the power of mathematics. Context for Change In 1983, A Nation at Risk awoke a sleeping nation to alarming problems in our educational system. Since then,
OCR for page 3
...tapping the power oimathematics dozens of reports have analyzed virtually every aspect of this enormous problem. Some call for changes in curricu- lum, others for changes in the structure of schools; some cite deficiencies in the ways teachers are educated, while others examine signs of decay in the social and economic structures of society. All agree that the present system must change. Mathematics education takes place in the context of schools. Like other subjects, mathematics is constrained by limits of school and society, of texts and tests. Much that needs improvement must be accomplished by systemic reme- dies that affect all subjects and all schools. Much is being done by school districts and community organizations, by legislatures and universities, by corporations and teachers. Nevertheless, much remains to be done. Although mathematics is not unique in its importance to education, mathematics education tends to magnify many problems of the schools. Literature, history, science, and other subjects contribute in essential ways to a well-balanced education; no one can say that one field is intrinsically of greater worth than another. It is clear, however, that math- ematics plays a special role in school education, one that is especially sensitive to deficiencies in the effectiveness of the educational system. Since mathematics is the foundation of science and tech nology, it serves as a key to opportunity and careers. More- over, mathematics contributes to literacy certain distinctive habits of mind that are of increasing importance to an in- formed citizenry in a technological age. Because of the fundamental importance of literacy and numerary, English and mathematics are the only subjects taught continuously throughout the school years. Education in any discipline helps students learn to think, but education also must help students take responsibility for their thoughts. While this objective applies to all subjects, it is particularly apt in mathematics education because math- ematics is an area in which even young children can solve a problem and have confidence that the solution is correct- not because the teacher says it is, but because its inner logic · ~ IS SO clear. "Reading, 'Riting, and " · -- Mathematics, of course, is not the only cornerstone of opportu- nity in today's world. Reading is even more fundamental as a basis for learning and for life. What is different today is the great in- crease in the importance of math- ematics to so many areas of edu- cation, citizenship, and careers. 3
OCR for page 4
Opportunity Myth: "There is no algebra in my future." Reality: Just because students do not use algebra anywhere except in algebra class does not mean that they will not need mathemat- ics in the future. Over 75 percent of all jobs require proficiency in simple algebra and geometry, ei- ther as a prerequisite to a training program or as part of a licensure examination. 4 M athematics is a key to opportunity and careers. Educators no longer argue that mathematics trains the mind for clear thinking in other subjects. Mathematics does, however, provide one of the few disciplines in which the growing student can, by exercising only the power inherent in his or her own mind, reach conclusions with full assur- ance. More than most other school subjects, mathematics offers special opportunities for children to learn the power of thought as distinct from the power of authority. This is a very important lesson to learn, an essential step in the emergence of independent thinking. Mathematics for Tomorrow The increasing importance of mathematics to society is only one of many factors that compel special examination of mathematics education. Mathematics itself is now signif- icantly more diverse than it was several decades ago when today's leaders and educators went to school. The arith- metic, algebra, geometry, and calculus taught nowadays are mere shadow images of modern mathematics. The mathe- matical sciences of today blend deep new results from these traditional areas with methods from such applied fields as statistics, operations research, and computer science. Mod- ern mathematics provides a powerful instrument for under- standing the world in which we live. Several factors growth of technology, increased applica- tions, impact of computers, and expansion of mathematics itself-have combined in the past quarter century to extend greatly both the scope and the application of the mathemati- cal sciences. Together, these forces have created a revolution in the nature and role of mathematics a revolution that must be reflected in the schools if our students are to be well prepared for tomorrow's world. Education reflecting only
OCR for page 5
...tapping the power of mathematics the mathematics of the distant past is no longer adequate for present needs. From the accountant who explores the consequences of changes in tax law to the engineer who designs a new air- craft, the practitioner of mathematics in the computer age is more likely to solve equations by computer-generated graphs and calculations than by manual algebraic manipulations. Mathematics today involves far more than calculation; cIar- ification of the problem, deduction of consequences, formu- lation of alternatives, and development of appropriate tools are as much a part of the modern mathematician's craft as are solving equations or providing answers. Statistics, the science of data, has blossomed from roots in agriculture and genetics into a rich mathematical science that provides essential tools both for analyses of uncertainty and for forecasts of future events. From clinical research to market surveys, from enhancement of digital photographs to stock market models, statistical methods permeate policy analysis in every area of human affairs. Challenging problems in such diverse fields as computer science and social science have invigorated the discipline of discrete mathematics, a field that reflects both computer logic and human ambivalence. Moreover, new mathematical tools such as game theory and decision theory are being applied to the human sciences where one seeks to make choices, deci- sions, and coalitions on some rational and systematic basis. A pplications, computers, and new discoveries have extended greatly the landscape of mathematics. Today's mathematics is a creative counterpoint of com- putation and deduction, rooted in data while unfolding in abstraction. Mathematics today is being continually cre- ated and adapted to meet new needs. Frequent interactions
OCR for page 6
Opportu1'ity "The nafionaispotligh' is turning on mathematics as we appreciate its cen- tral role in the economic growth of this country. . . . It must become a pump instead of abetter in the pipeline." Mathematics Pipeline NUMBER OF STUDENTS 1 0.000.000 Robert M. White 1,000,000 \ 1 00,000 1 0,000 Ninth Graders \ Freshmen \ B.S. Degrees \`M.S. Degrees 1,000 ~ Ph.D. Degrees 100 1 1 1 1 1 1972 1976 1980 1984 1988 Data from enrollments in mathematics courses reveal that roughly half the stu- dents leave the pipeline each year. 6 among different specialties reveal deep connections and hid- den unity, strongly suggesting that the different mathematical sciences are merely facets of a single science of patterns. A Pump, Nof a Filter The revolution in the way that mathematics is practiced is mirrored by a similar but unfulfilled-revolution in our understanding of how students actually learn mathematics. Research on learning shows that most students cannot learn mathematics effectively by only listening and imitating; yet most teachers teach mathematics just this way. Most teach- ers teach as they were taught, not as they were taught to teach. Research in learning shows that students actually construct their own understanding based on new experiences that en- large the intellectual framework in which ideas can be cre- ated. Consequently, each individual's knowledge of math- ematics is uniquely personal. Mathematics becomes useful to a student only when it has been developed through a per- sonal intellectual engagement that creates new understand- ing. Much of the failure in school mathematics is due to a tradition of teaching that is inappropriate to the way most students learn. We require students to study mathematics for many rea- sons: to learn practical skills for daily lives, to understand quantitative aspects of public policy, to develop problem- solving skills, and to prepare for careers. None of these goals is being achieved; not only do we face a shortage of person- nel with mathematical preparation suitable to scientific and technological jobs, but also the level of mathematical literacy (or "numerary") of the general public is completely inade- quate to reach either our personal or national aspirations. Mathematics education is an immense enterprise, involv- ing over 10 percent of the nation's educational resources- about $25 billion annually. Each year, 25 million elementary school children, 10 million secondary school students, and 3 million undergraduates study mathematics; indeed, about 60 percent of school-level and 30 percent of college-level sci- ence and technology education are devoted to mathematics.
OCR for page 7
· - M. · ~ ~ ~ Mathematics must become a pump rather than a filter in the pipeline of American education. Everyone depends on the success of mathematics education; everyone is hurt when it fails. More than any other subject, mathematics filters students out of programs leading to scientific and professional careers. From high school through graduate school, the half-life of students in the mathematics pipeline is about one year; on average, we lose half the students from mathematics each year, although various requirements hold some students in class temporarily for an extra term or a year. Mathematics is the worst curricular villain in driving students to failure in school. When mathematics acts as a filter, it not only filters students out of careers, but frequently out of school itself. Low expectations and limited opportunity to learn have helped drive dropout rates among Blacks and Hispanics much higher- unacceptably high for a society committed to equality of opportunity. It is vitally important for society that ad citizens benefit equally from high-quality mathemat- ics education. Numeracy To function in today's society, mathematical literacy- what the British call "numerary"- is as essential as verbal literacy. These two kinds of literacy, although different, are not unrelated. Without the ability to read and understand, no one can become mathematically literate. Increasingly, the reverse is also true: without the ability to understand basic mathematical ideas, one cannot fully comprehend modern writing such as that which appears in the daily newspapers. Numeracy requires more than just familiarity with num- bers. To cope confidently with the demands of today's so- ciety, one must be able to grasp the implications of many ..tapping the power of mathematics "Numeracy is the abil- ify to cope confidently with the mathematical demands of adult life." - Mathematics Counts 7
OCR for page 8
Opportunity 8 mathematical concepts-for example, chance, logic, and graphs" that permeate daily news and routine decisions. Literacy is a moving target, increasing in level with the ris- ing technological demands of society. Indeed, there is some evidence that the decline in reading comprehension scores over the last several decades is due in part to the growing mathematical content of what one is required to read. It is not just computer manuals or financial reports that require an understanding of mathematical ideas; so do reports of political polls, debates about AIDS testing, and arguments over the federal deficit. Even Supreme Court decisions re- semble mathematical arguments whose subject matter is law rather than numbers; often, legal cases rest as much on prob- abilistic inferences (for example, DNA fingerprinting, fiber analysis) as on direct evidence. M ............................... athematical literacy is essential as a foundation for democracy in a technological age. Functional literacy in all of its manifestations verbal, mathematical, scientific, and cultural provides a common fabric of communication indispensable for modern civilized society. Mathematical literacy is especially crucial because mathematics is the language of science and technology. Dis- cussion of important health and environmental issues (acid rain, waste management, greenhouse effect) is impossible without using the [Language of mathematics; solutions to these problems will require a public consensus built on the social fabric of literacy. The study of mathematics can help develop critical habits of mind to distinguish evidence from anecdote, to recog- nize nonsense, to understand chance, and to value proof. Citizens in a democracy must recognize that change is a pro- cess with expected regularities; that order can beget disorder
OCR for page 9
...tapping the power oimuthematics (as in turbulence) and vice versa (as in statistical experi- ments); that similar mathematical models can represent dif- ferent phenomena (for example, patterns of growth in biol- ogy, economics, and chemistry); and that simple models can clarify complex systems (as in linear models of economic systems), even though simplistic analysis can result in mis- leading interpretations. Citizens who are bombarded daily with conflicting quantitative information need to be aware of both the power and the limitations of mathematics. The great majority of American children spend most of their school mathematics time learning only practical arith- metic. Few retain much of what they learn about geometry; fewer still learn anything about chance. Secondary education is particularly devoid of exposure to modes of mathematical thought required for intelligent citizenship. Even colleges and universities seem unable to infuse appropriate mathe- matical ideas into liberal education. There is no consen- sus whatsoever on a collegiate mathematics curriculum for students outside the preprofessional programs where math- ematics serves a well-defined yet strictly utilitarian purpose. Rarely does mathematics contribute as it should to liberal education, to the honing of values, and to effective citizen- ship. Attitudes One of the more disturbing conclusions of recent stud- ies of mathematics education is that the American public tends to assume that differences in accomplishment in school mathematics are due primarily to differences in innate ability rather than to differences in individual effort or in opportu- nity to learn. These beliefs surface in many ways in studies of parents' views, in common self-deprecating remarks ("I never could do math"), and in public infatuation with early tracking as a strategy for mathematics education. One consequence of such beliefs is that parents often accept and sometimes even expect their children's poor performance in mathematics. Another consequence is that adults who determine policy in mathematics education often measure the mathematical needs of today's students by their "We've inherited a woe- fully limited se! of expec- tations of what schools can accomplish and what children can learnt." William R. Graham 9
OCR for page 10
Opportunity Myth: Learning mathematics re- quires special ability, which most students do not have. Reality: Only in the United States do people believe that learning mathematics depends on special ability. In other countries, students, parents, and teachers all expect that most students can master mathematics if only they work hard enough. The record of accomplishment in these countries and in some inter- vention programs in the United States shows that most students can learn much more mathemat- ics than is commonly assumed in this country. 10 own meager and outdated mathematical accomplishments. From the faulty premise that most students "can't do math" and the fact that many adults who never learned mathemat- ics have succeeded without it, they rationalize that official expectations should be limited to minimal basic levels. The result is a spiral of lowered expectations in which poor per- formance in mathematics has become socially acceptable. ~u0~c acceptance of deficient standards contributes significantly to poor performance in mathematics education. Even worse, the unrestricted power of peer pressure often makes good performance in mathematics socially unaccept- able. This environment of negative expectation is strongest among minorities and women-those most at risk during high school years when students first exercise choice in cur- ricular goals. Even strong positive support of adults is often powerless to open the eyes of children who are blinded by their entertainment-dominated culture. Public attitudes about mathematics are shaped primar- ily by adults' childhood school experiences. Consequently, mathematics is seen not as something that people actually use, but as a best forgotten (and often painful) requirement of school. For most members of the public, their lasting memories of school mathematics are unpleasant since so often the last mathematics course they took convinced them to take no more. Some adults blame the "new math" for their fears, having become convinced of their own mathematical ineptitude by instruction steeped in premature abstraction. Others have been made apprehensive by a teacher's rigid view of mathe- matics as a string of procedures to be memorized, where right answers count more than right thinking. Either extreme- mindless abstraction or mindless calculation yields mind- less mathematics. These widespread fears reinforce common
OCR for page 11
...tapping the power of mathematics public perceptions that mathematicians are responsible for making mathematics hard and that only geniuses are capable of learning mathematics. Children respond to expectations of their parents and teachers. It is no wonder that many students stop study- ing mathematics as soon as it becomes optional, since soci- ety provides so few hero-images for whom perseverance in mathematics has paid off. When parents think that ability supersedes effort, most students never learn the value of ef- fort. "Hard work pays oh' should be society's message to those who study mathematics. Goals Historically, schools in the United States were designed with a dual mission: to teach all students basic skills required for a lifetime of work in an industrial and agricultural econ- omy and to educate thoroughly a small elite who would go to college en route to professional careers. As the needs of society have changed as the fraction of students preparing to work in factories or on farms has declined- the balance of these two goals has shifted. Today's schools labor un- der the legacy of a structure designed for the industrial age misapplied to educate children for the information age. Not only in mathematics but in every school subject, edu- cators are faced with rising expectations for preparing the kind of work force the country will need in the future. Tnformation-age technology will continue to grow in impor- tance; pressed by rising international competition, industry will demand increased quality and increased productivity. The world of work in the twenty-first century will be less manual but more mental; less mechanical but more elec- tronic; less routine but more verbal; and less static but more varied. The changing nature of work will make continuing educa- tion a lifelong reality for adults. Schools, therefore, will have to provide all students with a strong foundation for lifelong learning; colleges and universities will have to educate both young adults and older workers; and industry will have to focus its continuing education on areas that extend rather Back to School A newspaper interested in a tax proposal conducted a poll using a random sample of 100 voters on each of two successive Mondays. The first week they found 57 per- cent in favor; in the second week, they found 59 percent in favor. The headline for its story read: "Support Grows for Tax Hike." Write a letter to the editor about this story. 11
OCR for page 12
Opportunity "We canimprove. We can keep kids in school longer a1'dF achieve a het- fer result. We cannot fad' in education. If we fail, we fad! our kids and we fai'ourfuture." 12 I· . . - n the future, schools and colleges will need to meet goals that they now believe to be impossible. than repeat what schools provide. Education in the future must build continually from childhood to retirement on a versatile foundation provided by school education. Literacy and numeracy are the primary sources of strength and versatility in school education. As colleges, universities, and continuing education attract significant fractions of the population, schools must now prepare all students for some type of postsecondary study. The level of literacy formerly associated with the few who entered college must now be a goal for all. Quantitative literacy provides the foundation of techno- logical expertise in the workplace. The facility with math- ematics formerly required only of those preparing for sci- entific careers is now an essential ingredient in the founda- tion for lifelong work in the information age. In tomorrow's world, the best opportunities for jobs and advancement will go to those prepared to cope confidently with quantitative, scientific, and technological issues. Mathematical Dower oro- vides the key to these opportunities. Students at Risk ~. . ~ Daunting as the challenge of reform appears, its cost is insignificant when compared with the consequences of in- action. Over 25 percent of all high school students drop out before graduating; although half of these students do eventu- ally receive an equivalency diploma, their detour is costly for them and for society. Among Blacks, Hispanics, and Native Americans, the dropout rate often exceeds 50 percent. The majority of those who drop out are functionally il- literate and hardly any of them possess enough mathemati- cal skills to make productive contributions to the American economy. Dropouts and illiteracy are destroying individual Jangle Escalante
OCR for page 13
...tappi1lg the power of mathematics hope and threatening the foundation of this country's econ- omy. Disparities that divide one third of our nation from the rest compromise quality of life for all citizens. Industry spends as much on remedial mathematics educa- tion for employees as is spent on mathematics education in schools, colleges, and universities. in addition, 60 percent of college mathematics enrollments are in courses ordinar- ily taught in high school. This massive repetition is grossly inefficient, wasting resources that could be used better to im- prove rather than to repeat mathematics education. We are at risk of becoming a nation divided both economically and racially by knowledge of mathematics. Broad-brush attempts to deal with underachievement in mathematics often aggravate the very problems they are try- ing to solve. Raising standards for graduation often widens the gap between those who know mathematics and those who do not, since increased standards are rarely accompanied by program changes to provide appropriate courses for students who are not motivated to study mathematics. Requirements designed to ensure that all students reach certain minimum levels often yield results in which most students reach only the minimum level because rarely are schools given the re- sources necessary both to achieve minimum standards for all and to provide unlimited accomplishment for most. Despite massive effort, relatively little is accomplished by remediation programs. No one not educators, mathemati- cians, or researchers- knows how to reverse a consistent early pattern of low achievement and failure. Repetition rarely works; more often than not, it simply reinforces pre- vious failure. The best time to learn mathematics is when it is first taught; the best way to teach mathematics is to teach it well the first time. "America is moving backward no' forward in its efforts to achieve the fullparticipa- tion of minority citizens it' the life and prosperity of the ration." - One Third of a Nation
OCR for page 14
Opportu1'ity 14 Apart from economics, the social and political conse- quences of mathematical illiteracy provide alarming signals for the survival of democracy in America. Because mathe- matics holds the key to leadership in our information-based society, the widening gap between those who are mathemati- cally literate and those who are not coincides, to a frightening degree, with racial and economic categories. We are at risk of becoming a divided nation in which knowledge of mathe- matics supports a productive, technologically powerful elite while a dependent, semiliterate majority, disproportionately Hispanic and Black, find economic and political power be- yond reach. Unless corrected, innumeracy and illiteracy will drive America apart.
OCR for page 15
OCR for page 16
A: ~ I I' it: ~,~ A: ~ :~ ~ I_ ::: ~:~ :~: ~ If_ ~:~ :~::: _ :~ ,,~ _~ , _ ~ 1 I, A: it: ~ ~ ~ ~ ~ ~' :~ 1 _ - :~ ~ ~ } :~ ~ ~ __ ~ ~: ~ :: ~ it. ~ ': ~ ~ i~ihii~L':I ~ ~kJ~ ~ :~: ~ _ - ~ ~ ~ ~ ~: ~ :~ ~ ~: _ : ~: ':: are: ~ _ ~: :: :~ i: ~ :~ 31:: : :~ I: :: :: it: ~ I:: , hi ~ i: :~: it: ::: ::; ~ ~ I: ~ ~ I'm ~ :~ '' ~ ~ ~ ~ ~hi', ~ ~: ~21 : i~ ~ ~ ;~ ~ ~ : ~ ,: : i}~! ~ : :: ~ ~ ~ ~ ~ :i ! :: :: ~ ~ ~ ~:: : ~: .. ~:: ~ _9.
Representative terms from entire chapter: