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MA THEME TICS Mathematics reveals hidden patterns that help us under- stand the world around us. Now much more than arithmetic and geometry, mathematics today is a diverse discipline that deals with data, measurements, and observations from sci- ence; with inference, deduction, and proof; and with math- ematical models of natural phenomena, of human behavior, and of social systems. The cycle from data to deduction to application recurs everywhere mathematics is used, from everyday household tasks such as planning a long automobile trip to major man- agement problems such as scheduling airline traffic or man- aging investment portfolios. The process of "doing" math- ematics is far more than just calculation or deduction; it involves observation of patterns, testing of conjectures, and estimation of results. As a practical matter, mathematics is a science of pattern and order. its domain is not molecules or cells, but num- bers, chance, form, algorithms, and change. AS a science of abstract objects, mathematics relies on logic rather than on observation as its standard of truth, yet employs obser- vation, simulation, and even experimentation as means of discovering truth. M: ~ ~ ~ athematics is a science of pattern and order. The special role of mathematics in education is a con- sequence of its universal applicability. The results of mathematics theorems and theories-are both significant and useful; the best results are also elegant and deep. Through its theorems, mathematics offers science both a foundation of truth and a standard of certainty. In addition to theorems and theories, mathematics of- fers distinctive modes of thought which are both versatile and powerful, including modeling, abstraction, optimiza- tion, logical analysis, inference from data, and use of sym- bols. Experience with mathematical modes of thought builds searching for patterns Mathematical Modes of Thought Modeling Representing worldly phenomena by mental constructs, often visual or symbolic, that capture important and useful fea tures. Optimization Finding the best solution (least expensive or most efficient) by asking "what if' and exploring all possibilities. Symbolism- Extending natural language to symbolic represen- tation of abstract concepts in an economical form that makes pos- sible both communication and computation. Inference Reasoning from data, from premises, from graphs, from incomplete and inconsistent sources. Logical Analysis Seeking impli- cations of premises and searching for first principles to explain ob- served phenomena. Abstraction Singling out for spe- cial study certain properties com- mon to many different phenom- ena. 31
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Mathematics Back to School Design a dog house that can be made from a single 4 ft. by ~ ft. sheet of plywood. Make the dog house as large as possible and show how the pieces can be laid out on the plywood before cut- ting. 32 mathematical power a capacity of mind of increasing value in this technological age that enables one to read critically, to identify fallacies, to detect bias, to assess risk, and to sug- gest alternatives. Mathematics empowers us to understand better the information-laden world in which we live. Our Invisible Culture Mathematics is the invisible culture of our age. Although frequently hidden from public view, mathematical and sta- tistical ideas are embedded in the environment of technology that permeates our lives as citizens. The ideas of mathemat- ics influence the way we live and the way we work on many different levels: · Practical knowledge that can be put to immediate use in improving basic living standards. The ability to compare loans, to calculate risks, to figure unit prices, to understand scale drawings, and to appreciate the effects of various rates of inflation brings immediate real benefit. This kind of basic applied mathematics is one objective of universal elementary education. · Civic concepts that enhance understanding of public pol- icy issues. Major public debates on nuclear deterrence, tax rates, and public health frequently center on scien- tific issues expressed in numeric terms. Inferences drawn from data about crime, projections concerning population growth, and interactions among factors affecting interest rates 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 pub- lic policy. Ideally, secondary school mathematics should help create the "enlightened citizenry" that Thomas lef- ferson called the only proper foundation for democracy. · Professional skill and power necessary to use mathemat- ics as a tool. Science and industry depend increasingly on mathematics as a language of communication and as a methodology of investigation, in applications ranging from theoretical physics to business management. The principal
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...searchi1lg for patterns M athematics is a profound and powerful part of human culture. goal of most college mathematics courses is to provide stu- dents with the mathematical prerequisites for their future careers. · Leisure- disposition to enjoy mathematical and logical challenges. The popularity of games of strategy, puzzles, lotteries, and sport wagers reveals a deep vein of amateur mathematics lying just beneath the public's surface indif- ference. Although few seem eager to admit it, for a lot of people mathematics is really fun. · Cultural the role of mathematics as a major intellectual tradition, as a subject appreciated as much for its beauty as for its power. The enduring qualities of such abstract concepts as symmetry, proof, and change have been devel- oped through 3,000 years of intellectual effort. They can be understood best as part of the legacy of human culture which we must pass on to future generations. indeed, it is only when mathematics is viewed as part of the human quest that lay persons can appreciate the esoteric research of twentieth-century mathematics. Like language, religion, and music, mathematics is a universal part of human cul- ture. These layers of mathematical experience form a matrix of mathematical literacy for the economic and political fabric of society. Although this matrix is generally hidden from public view, it changes regularly in response to challenges arising in science and society. We are now in one of the periods of most active change. From Abstraction to Application During the first half of the twentieth century, mathe- matical growth was stimulated primarily by the power of l ``Ifyou want to under- stand nature, you must be conversant with the lan- guage in which nature speaks to users - Richard Feynman 33
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Mathematics Strictly Speaking MATHEMATICAL SCIENCES is a term that refers to disciplines that are inherently mathematical (for example, statistics, logic, ac- tuarial science), not to the many natural sciences (for example, physics) that employ mathemat- ics extensively. For economy of language, the word "mathemat- ics" is often used these days as a synonym for "mathematical sci- ences," as the term "science" is often used as a summary term for mathematics, science, engineer- ing, and technology. 34 abstraction and deduction, climaxing more than two cen- turies of effort to extract full benefit from the mathematical principles of physical science formulated by Isaac Newton. Now, as the century closes, the historic alliances of mathe- matics with science are expanding rapidly; the highly devel- oped legacy of classical mathematical theory is being put to broad and often stunning use in a vast mathematical land- scape. Several particular events triggered periods of explosive growth. The Second World War forced development of many new and powerful methods of applied mathematics. Postwar government investment in mathematics, fueled by Sputnik, accelerated growth in both education and research Then the development of electronic computing moved math- ematics toward an algorithmic perspective even as it pro- vided mathematicians with a powerful too! for exploring patterns and testing conjectures. At the end of the nineteenth century, the axiomatization of mathematics on a foundation of logic and sets made pos- sible grand theories of algebra, analysis, and topology whose synthesis dominated mathematics research and teaching for the first two thirds of the twentieth century. . · . ~. ~ These tradi- onal areas nave now oeen supplemented oy major develop- ments in other mathematical sciences in number theory, logic, statistics, operations research, probability, computa- tion, geometry, and combinatorics. In each of these subdiscinTines~ __ 7 applications parallel theory. Even the most esoteric and abstract parts of mathematics number theory and Tocic. for example are now used routinely in applications (for example, in com- puter science and cryptography). Fifty years ago, the leading British mathematician G. H. Hardy could boast that number theory was the most pure and least useful part of mathemat- ics. Today, Hardy's mathematics is studied as an essential prerequisite to many applications, including control of au- tomated systems, data transmission from remote satellites, protection of financial records, and efficient algorithms for computation. ., .
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...searching for patterns Mathematics is the foundation of science and technology. Without strong mathematics, there can be no strong science. in 1960, at a time when theoretical physics was the central jewel in the crown of applied mathematics, Eugene Wigner wrote about the "unreasonable effectiveness" of mathematics in the natural sciences: "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither under- stand nor deserve." Theoretical physics has continued to adopt (and occasionally invent) increasingly abstract math- ematical models as the foundation for current theories. For example, Lie groups and gauge theories exotic expressions of symmetry are fundamental tools in the nhv~icist's search for a unified theory of forces. ~, _ ~ , During this same period, however, striking applications of mathematics have emerged across the entire landscape of natural, behavioral, and social sciences. All advances in design, control, and efficiency of modern airliners depend on sophisticated mathematical models that simulate perfor- mance before prototypes are built. From medical technology (CAT scanners) to economic planning (input/output models of economic behavior), from genetics (decoding of DNA) to geology (locating of! reserves), mathematics has made an indelible imprint on every part of modern science, even as science itself has stimulated the growth of many branches of mathematics. Applications of one part of mathematics to another of geometry to analysis, of probability to number theory- provide renewed evidence of the fundamental unity of math- ematics. Despite frequent connections among problems in science and mathematics, the constant discovery of new al- liances retains a surprising degree of unpredictability and serendipity. Whether planned or unplanned, the cross- fertilization between science and mathematics in problems, "Equations are just the boring par' of mathematics. reattempt to see thi1'gs it' terms of geometry." - Stephen Hawking
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Mathematics Myth: As computers become more powerful, the need for mathematics will decline. Reality: Far from diminishing the importance of mathematics, the pervasive role of computers in science and society contributes to a greatly increased role for math- ematical ideas, both in research and in civic responsibility. Be- cause of computers, mathematical ideas play central roles in impor- tant decisions on the job, in the home, and in the voting booth. 36 theories, and concepts has rarely been greater than it is now, in this last quarter of the twentieth century. Computers Alongside the growing power of applications of mathemat- ics has been the phenomenal impact of computers. Even mathematicians who never use computers may devote their entire research careers to problems arising from use of com- puters. Across all parts of mathematics, computers have posed new problems for research, supplied new tools to solve old problems, and introduced new research strategies. Although the public often views computers as a replace- ment for mathematics, each is in reality an important too} for the other. Indeed, just as computers afford new opportu- nities for mathematics, so also it is mathematics that makes computers incredibly elective. Mathematics provides ab- stract models for natural phenomena as well as algorithms for implementing these models in computer languages. Ap- plications, computers, and mathematics form a tightly cou- pled system producing results never before possible and ideas never before imagined. Computers influence mathematics both directly- through stimulation of mathematical research and indirectly by their effect on scientific and engineering practice. Comput- ers are now an essential too} in many parts of science and engineering, from weather prediction to protein engineer- ing, from aircraft design to analysis of DNA. In every case, a mathematical mode! mediates between phenomena of sci- ence and simulation provided by the computer. Scientific computation has become so much a part of the everyday experience of scientific and engineering practice that it can be considered a third fundamental methodology of science-parallel to the more established paradigms of experimental and theoretical science. Computer models of natural, technological, or social systems employ mathemati- cally expressed principles to unfold scenarios under diverse conditions scenarios that formerly could be studied only through lengthy (and often risky) experiments or prototypes. The methodology of scientific computation embeds mathe
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...searchi1'g for patterns matical ideas in scientific models of reality as surely as do axiomatic theories or differential equations. Computer models enable scientists and engineers to reach quickly the mathematical limits permitted by their models. Robotics design, for instance, often encounters limits im- posed not by engineering details, but by incomplete under- standing of how geometry controls the degrees of freedom of robot motions. Models of weather forecasting consistently reveal uncertainties that suggest intrinsically chaotic behav- ior. These models also reveal our severely limited knowledge of the mathematical theory of turbulence. Whenever a sci- entist or engineer uses a computer mode! to explore the fron- tiers of knowledge, a new mathematical problem is likely to appear. Computer models have extended the mathematical sciences into every corner of · . ~ . · · . sclentl~c anc . engineering practice. Whereas, traditionally, scientists and engineers who were engaged primarily in experimental research could get along with a small subset of mathematical skills uniquely suited to their field, now even experimentalists need to know a wide range of mathematical methods. Small errors of approxi- mation that are intrinsic to all computer models compound, like interest, with subtle and often devastating results. Only a person who comprehends the mathematics on which com- puter models are based can use these models effectively and efficiently. Moreover, as a consequence of current limits on computer models, further advances in many areas of sci- entific and engineering knowledge now depend in essential ways on advances in mathematical research. The Mathematical Community Because of its enormous applicability, mathematics is- apart from English the most widely studied subject in 37
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Mathematics Back to School Two banks are offering car loans with monthly payments of $100. One has an interest rate of 16 percent; the other has a higher rate of 18 percent together with a premium of a free color televi- sion (worth $400~. If you need a $5,000 loan and would really like the color TV, which bank should you choose? 38 school and college. Present educational practice for mathe- matics requires approximately 1,500,000 elementary school teachers, 200,000 high school teachers, and 40,000 college and university teachers. Mathematics education takes place in each of 16,000 public school districts, in another 25,000 private schools, in 1,300 community colleges, 1,500 colleges, 400 comprehensive universities, and 200 research universi- ties. Roughly 5,000 mathematicians, principally those on the faculties of the research universities, are engaged in research. Only half of the nation's students take more than two years of high school-level mathematics; only one quarter take more than three years. That remaining quarter roughly one million enter colleges and universities with four years of mathematics. Four years later, about ~ 5,000 students emerge with majors in mathematics. One quarter of these students go on to a master's degree, but only 3 percent (about 400) complete a doctoral degree in the mathematical sciences. M athematics is the nation's second- largest academic discipline. Just to replace normal retirements and resignations of high school teachers will require about 7,000 to 8,000 new teach- ers a year, which is half of the expected pool of ~ 5,000 math- ematics graduates. Elementary school teachers, in contrast, are drawn primarily from the three quarters of the popula- tion who dropped mathematics after two or three courses in high school. For many prospective elementary school teach- ers, their high school experiences with mathematics were probably not positive. Subsequently, teachers' ambivalent feelings about mathematics are often communicated to chil- dren they teach. In sharp contrast to the eroding conditions of mathemat- ics teaching, one finds enormous vitality and diversity in the
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...searching for patterns breadth of the mathematics profession. Over 25 different or- ganizations in the United States support some facet of pro- fessional work in the mathematical sciences. Approximately 50,000 research papers 20,000 by U.S. mathematicians- are published each year in 2,000 mathematics journals around the world. At the school and college level alone, there are over 25 U.S. publications devoted to students and teachers of mathematics. Students and faculty participate in problem-solving activities sponsored by these journals as well as learn about the ways in which current research can relate to curricular change. This massive system of mathematics education has had no national standards, no global management, and no planned structure despite the facts that each step in the mathemat- ics curriculum depends in vital ways on what has been ac- complished at all earlier stages and that scores of professions depend on skills acquired by students during their study of mathematics. Both because it is so massive and because it is so unstructured, mathematics education in the United States resists change in spite of the many forces that are revolution- izing the nature and role of mathematics. Undergraduate Mathematics Undergraduate mathematics is the linchpin for revitaliza- tion of mathematics education. Not only do all the sciences depend on strong undergraduate mathematics, but also all students who prepare to teach mathematics acquire attitudes about mathematics, styles of teaching, and knowledge of content from their undergraduate experience. No reform of mathematics education is possible unless it begins with revi- talization of undergraduate mathematics in both curriculum and teaching style. During the last two decades, as undergraduate mathemat- ics enrollments have doubled, the size of the mathematics faculty has increased by less than 30 percent. Workloads are now over 50 percent higher than they were in the post- Sputnik years and are typically among the highest on many campuses. Resources generated by the vigorous demand for undergraduate mathematics are rarely used to improve un "Between now aids the year 2000, for the firs t time in history, a ma- jority of all new jobs will require postsecond~ary education." Workforce 2000 39
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Mathematics "Too many teachers over mathematics on a take it-or-leave-it basis in the universities. The result is that some of the brightest mathematical minds elect to lea ye it." Edward E. David, Jr. A Pipeline to Science The undergraduate mathematics major not only prepares students for graduate study in mathemat- ics, but also for many other sci- ences. Indeed, nearly twice as many mathematics majors go on to receive a Ph.D. in another sci- entific field rather than in the mathematical sciences them- selves. 40 dergraduate mathematics teaching. To administrators wor- ried about tight budgets, mathematics departments are often the best bargains on campus, but to students seeking stimu- lation and opportunity, mathematics departments are often the Rip Van Winkle of the academic community. R· · · . . . . . . . . . . . . form of undergraduate mathematics is the key to revitalizing mathematics education. During these same two decades, both the opportunity and the need for vital innovative mathematics instruction have increased substantially. The subject moves on, yet the cur- riculum is stagnant. Only a minority of the nation's colle- giate faculty maintains a program of significant professional activity. Even fewer are regularly engaged in mathematical research, but these few sustain a research enterprise that is the best in the world. Unfortunately, those who are most professionally active rarely teach any undergraduate course related to their scholarly work as mathematicians. Mathe- maticians seldom teach what they think about and rarely think deeply about what they teach. Departments of mathematics in colleges and universities serve several different constituencies: general education, teacher education, client departments, and future mathe- maticians. Very few departments have the intellectual and financial resources to meet well the needs of all these fre- quently conflicting groups. Worse still, most departments fait to meet the needs of any of these constituencies with energy, effectiveness, or distinction. Since almost everyone who teaches mathematics is edu- cated in our colleges and universities, many issues facing mathematics education hinge on revitalization of undergrad- uate mathematics. But critical curricular review and revital- ization take time, energy, and commitment essential in- gredients that have been stripped from the mathematics fac- ulLty by two decades of continuous deficits. Rewards of pro- motion and tenure follow research, not curricular reform;
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...searching for patterns neither institutions of higher education nor the professional community of mathematicians encourages faculty to devote time and energy to revitalization of undergraduate mathe- matics. To improve mathematics education, we must restore in- tegrity to undergraduate mathematics. This challenge pro- vides a great opportunity. With approximately 50 percent of school teachers leaving every seven years, it is feasible to make significant changes in the way school mathematics is taught simply by transforming undergraduate mathematics to reflect the new expectations for mathematics. Undergrad- uate mathematics is the bridge between research and schools and holds the power of reform in mathematics education. 41
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Representative terms from entire chapter: