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A Practical Philosophy Foundations for an improved mathematics curriculum must rest on analysis of the nature of mathematics and the goals of mathematics eclucation. Even the most superficial observation shows unequivocally that the nature of mathematics is chang- ing,-that the goals of mathemat- ics education are expancling, and that schools are in the micist of major transitions. Expanding Goals .... ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ .... ~ ...... . ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ I.. a....... .. . ~ ~:~:~:':~:: :':~:~:~:~:~:~:~:~:~:,:,:~:~:~:~:~:~:~:~:~:~:~:~:,:: :,:':,:,:,:':,:,:2 :'.: 4 ::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::: . .: : :::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ., ::::::::::::::::::.:.::::::.:.::.:.:.:.:.: :.: . ,:2:':': :':::: ::::: :':.:.: :.:~:.: :.:.:':':':':':':':':2:'2''''''.::: :::::::::::::::::::::::::::::::::::::::::::::: ':,:':,:': :::':::,::': :,:':,:':,:,:,:,:,:,:,:,:,:,:,:2:,:,:,:,:,:2:,: :,:::: :> . ::::::::::: :.: :.:.:.: :.:.:.:.:.:.:.:.:.:.:.:.:.:.: :::: ':': :,::: :':::,:,:,:':':': :,:,:,:,: :,:,:,:,:,:,:2:,:,:,:,:,:,: :.:,:.::::: ......................... ::::: ':':2:': :~:: :':2:':':':':':.:':':':':':':2:':':':':':':':':' .'':::::: :::::::::::::::::::::::::::::::::::::::::: ,:':: :': :':':2:':':':2:':2:2:::::': :':':':':':':':':':':.:.:::::: :2::: :::::::::: :.:.:.:.:.:::::.:.:.:.:.:.:.:.:.:.:.:.:.:.:::::::.::: '::',:,:,:,,::: a.::::::: a.::::: :< :i: :i: -: :::::::-::::::::-:::::-::-:-:::-::::::::: . '''''-',:,: :,:,: :,:':':,:,: :,:,:,:,:,:':,:2:,:':2:,:,:,:,:,:,"' :2.2222:.:2:22:.:': :2:2: :2:':.:2:,:,:: :,:j :::::::::::::::::: :~ ::::::::::::::::::: : 2'' 2'2 2 :is i:::::: :::::::::':: :'; :::::::::::::: - :::::::::::::~1 ::::::::: ~ b t ' 2'2 2: ~ ~ :':':':,:2:,: ~.,:,:,:2: :,:,.. ::::::::::::::-: :':':':':':':': :'. :':: : ::: :.:.:.:.:..::.:: : ~':,:,:2:,:,:.:,:,: :.:,: . :::::::-: .. r .~ it... ,,:,:.:,:,:,:,:,:,:,:,:.:,:,:,:,:,:,:2:,:,:,:, _ .................. ~,::::::::::::::::,::: _ . ' :,:2:,:,:,:,:2:,:2:,:2:2:2:,:2 ''::::::::::: ,:' '': :':':':':': ''::::::::: ' ,:,:,:2:,:,:,:2 it::::::: ':2:,:,:,:,:' :::::: j,:2:,:,:,:, 1::::: [2:,:,:2:2 A:::::::: go:::: a:::: go: A:::::::::: a::::: i::::: :::,:,:: ::::::. ::: : "FF:::';~: TWO. :~-"------ --- .~ ',2.''~ : ::: :2: :':,:'::::: ::~ ::::::.:::::::::::,:: ::::::::' ~ :: ::: :::::::: :. :2:::: :q ~ : . :...:..:'::: :::::::::::: :: I.:::::: ::::::::::::::::::::~ ,~,~x. : ... ::::::::::::::::::::::::::::::::::; ': ' ' . ' ', ' 'I',, ,::: ~:~:~:~: :,:,:,:::: :,:.::::: :':.. C' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ''''2 as;. ..... am'' ''''''''''''''''' '''' ''''' '''''''''' '-2,. ,,, -. 2',''','',,,'',,,,, , ~ ',' ,'',,,,,,, ,',,, ,', ,',,,,, 2, j-, ~ I' 2'~ it - .' ' ' ' ' . . .'.' '. . ' ' '.'.'. .'.'.'.'.'.'.'.'.'.'.'.'.'.'.'.'. We teach mathematics to serve several very different goals that reflect the diverse roles that mathematics plays in society: A Practical Goal: To help individuals solve problems of everyday life. A Civic Goal: To enable citizens to participate intelligently in civic affairs. A Professional Goal: To prepare stuclents for jobs, voca- tions, or professions. A Cultural Goal: To impart a major element of human culture. The mathematical knowledge needed to achieve these gocis has changed ciramaticoi~y in the twentieth century and is changing more rapicily now than ever before.

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8 Reshaping School Mathematics Perhaps most obvious is the change needed for everyday life. Whereas daily activities once required a considerable amount of paper-and-pencil calculation, virtually all routine household arithmetic is now done either mentally or with an inexpensive hancl-held calculator, It is not just that the kinds of solutions we have at our disposal have changed, but so too have the problems. Today to be mathematically literate one must be able to interpret both quantitative and spatial infor- mation in a variety of numerical, symbolic, and graphical con- texts. These changes provide an unprecedented opportunity to redirect much of current elementary school mathematics to more fruitful and important areas especially to the new world of sophisticated electronic computation, As calculators and computers diminish the role of routine computation, school mathematics can focus instead on the conceptual insights and analytic skills that have always been at the heart of math- ematics. The changes in mathematics needed for intelligent citizen- ship have been no less significant. Most obvious, perhaps, is the need to understand data presented in a variety of differ- ent formats: percentages, graphs, charts, tables, and statisti- cal analyses are commonly used to influence societal deci- sions. Largely because data are now so widely available, daily newspapers employ a considerable variety of quantitative images in ordinary reporting of news events. Citizens who cans not properly interpret quantitative data are, in this day and age, functionally illiterate. It is, however, the professional and vocational needs for mathematics that have changed most rapidly. Mathematics is essential to more disciplines than ever before. The explosive growth of technology in the twentieth century has amplified the role of mathematics. By increasing the number and variety of problems that can be solved, calculators and computers have significantly increased the need for mathematical knowl- edge and changed the kind of knowledge that is needed. Computers have moved many vocations (e.g., farming) to become more quantitative and thus more productive. The result is that people in an expanding number of vocations and professions need to know enough mathematics to be able to recognize when mathematics may be helpful to them. Because of society's preoccupation with the practical and professional roles of mathematics, schools rarely emphasize cultural or historical aspects of mathematics. Like all subjects, mathematics is dehumanized when divorced from its cultural contributions and its history. To the extent that these subjects are discussed at all, students are likely to get the impression that mathematics is static and old-fashioned. While it is com- monplace for school children to become familiar with modern concepts in the sciences such as DNA and atomic energy,

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9 A Philosophy and Framework rarely are children introduced to any mathematics (such as statistics or topology) discovered less than a century ago. Chil- dren never learn that mathematics is a dynamic, growing dis- cipline, and only rarely do they see the beauty and fascina- tion of mathematics. The mathematics curriculum can no longer ignore the twentieth century. Fundamental Questions To realize a new vision of school mathematics will require public acceptance of a realistic philosophy of mathematics that reflects both mathematical practice and pedagogical experience. One cannot properly constitute a framework for a mathematics curriculum unless one first adciresses two funda- mental questions: What is mathematics? What does it mean to know mathematics?

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10 Reshaping School Mathematics Although few mathematicians or teachers spend much time thinking about these philosophical questions, the unstat- ed answers that are embecided in public ancl professional opinion are the invisible hancis that control mathematics edu- cafion. No change in education can be effective until the hicl- den influences of these creep issues are redirected toward objectives more in tune with today's woricl. Yet even as forces for change are providing new directions for mathematics eclucation, the public instinct for restoring tra- clitional stability remains strong. Unless the 3uiclance system for mathematics education is permanently reset to new and more appropriate gocis, it will surely steer the curriculum back to its Al path once present pressure for change abates. Answers to these fundamental questions would help clarify for both eclucators and the pubic what mathematics is really about-what it stuclies, how it operates, what it is good for (Romberg, 1988~. Appropriate answers would provide a con- vincing platform on which to erect a new mofhe-matics cur- riculum of the twenty-first century in which children would be introcluecJ not only to the traditional themes of number and space, but also to many newer themes such as logic, chance, computation, and statistics. From these answers would flow a pragmatic philosophy of mathematics that could help explain the creative tension that bincis the two funclamenfal poles of mathemofical reality: Theory: That in mathematics, reasoning is the test of truth. Applications: That mathematical models are both apt and useful. One might think that the many definitions of mathematics provided by scholars in centuries past would suffice for this task. But in the past few years, as computers have begun to unfold new potentials of mathematical systems, we have been able to see mathematics in a significantly broadened context. As the Apollo missions for the. first time enabled peo- ple to see and describe the back side of the moon, so com- puters have now enabled us to grasp a much richer land- scape of the mathematical sciences. It is now time to reshape mathematics education to reflect both the significant role of computers in the practice of mathematics ant] the frans- formed role played by mathematics in modern society. Describing Mathematics We begin with a simple approximation: mathematics is a science. Observations, experiment, discovery, and conjecture

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11 A Philosophy and Framework are as much part of the practice of mathematics as of any natural science. Trial ancl error, hypothesis and investigation, and measurement and classification are part of the mathe- matician's craft and should be taught in school. Laboratory work and fieldwork are not only appropriate but necessary to a full understanding of what mathemofics is ancl how it is used. Calculators ancl computers are necessary tools in this mathe- matics lab, but so too are sources of real data (scientific experiments, demographic clata, opinion polls), objects to observe ancl measure (clice, blocks, balls), ancl tools for con- struction (rulers, string, protractors, clay, graph paper). As biology is a science of living organisms ancl physics is a science of matter ancl energy, so mathematics is a science of patterns. This description goes back at least to Descartes in a slightly different form (he called mathematics the "science of order"), and has been refined by physicist Steven Weinberg who used it to explain the uncanny ability of mathematics to anticipate nature (Steen, 1988~. A sirni~ar view of mathematics as the science of "patterns and relationships" forms the basis for the expression of mathematics in Science for All Americans (American Association for the Acivancement of Science (AAAS), 19891. By classifying, explaining, ancl describing pat- terns in all their manifestations-number, data, shape, arrange- ments, even patterns themselves mathematics ensures that any pattern encountered by scientists will be explained some- where as part of the practice of mathematics. Patterns are evident in every aspect of mathematics. Young children learn how arithmetic clepencis on the regularity of numbers; they can see order in the multiplication table and wonder about clisorder in the pattern of primes. The geometry of polyhedra exhibits a regularity that recurs throughout nature and in architecture. Even statistics, a subject which studies dis- order, depencis on exhibited patterns as a yardstick for assess- ing uncertainty. As a science of patterns, mathematics is a mode of inquiry that reveals fundamental truth about the order of our world. But mathematics is also a form of communication that com- plements natural language as a tool for describing the world in which we live. So mathematics is not only a science, but also mathematics is a language. It is, as science has revealed, the language in which nature speaks. But it is also an apt lan- guage for business ancl commerce. From its beginnings in ancient cultures, the language of mathematics has been widely used in commerce: measure- ment ancl counting-geometry ancl arithmetic-enabled trade and regularized financial transactions. In recent cen- turies, mathematics (first calculus, then statistics) provided the intellectual end inferential framework for the growth of sci- ence. The mathematical sciences (inclucling statistics) are now the founclation disciplines of natural, social, ancl behavioral sciences, Moreover, with the support of computers ancl woricl

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12 Reshaping School Mathematics wide digital communication, business and industry depend increasingly not only on traditional but also on modern mathe- matical methods of analysis. Mathematics can serve as the language of business and science precisely because mathematics is a language that describes patterns. In its symbols and syntax, its vocabulary and idioms, the language of mathematics is a universal means of communication about relationships and patterns, It is a lan- guage everybody must learn to use, Knowing Mathematics If mathematics is a science and language of patterns, then to know mathematics is to investigate and express relation- ships among patterns: to be able to discern patterns in com- plex and obscure contexts; to understand and transform rela- tions among patterns; to classify, encode, and describe patterns; to read and write in the language of patterns; and to employ knowledge of patterns for various practical purposes. To grasp The diversity of patterns-indeed, to begin to see pat- terns among patterns -it is necessary that the mathematics curriculum introduce and develop mathematical patterns of many different types. As the patterns studied by mathematics are not limited to the rules of arithmetic, so the patterns stucl- ied in school mathematics must break the bonds of this artifi- cial constraint. A person engaged in mathematics gathers, discovers, cre- ates, or expresses facts and ideas about patterns. Mathemat- ics is a creative, active process very different from passive mastery of concepts and procedures. Facts, formulas, and information have value only to the extent to which they sup- port effective mathematical activity. Although some funda- mental concepts and procedures must be known by all stu- dents, instruction should persistently emphasize that to know mathematics is to engage in a quest to understand and com- municate, not merely to calculate, By unfolding the funda- mental principles of pattern, mathematics makes the mind an effective tool for dealing with the world. From these views can flow an effective and dynamic school curriculum for the next century. Practical Effects The practical test of a philosophy is the effect it should have on practice-in this case, on the teaching of mathematics.

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13 A Philosophy and Framework ~ ABCD -EF`H=2 AIR ~ :( ~ ABED ' FHJK= ~ En. ICKY D Draw Label E Erase M Measure ~ Repeat S $~e change N New shape Q Quit I; ~ Education Dsvelopm~l Cenlor. 1989 The Geometric Supposer is a set of software learning environments deliberately designed to change school plane geometry from a closely guided museum tour (where the guide points out certain arti- facts to be "proven") to an active process of building and exploring conjectures. For example, a student who constructs the three medi- ans of a triangle and notices that they all intersect in a point might wonder if this is a fluke, or whether it might hold for other triangles. By using a repeat feature, the student can quickly execute the same construction on a series of triangles, either generated at random by the computer or produced by the student in a way designed to stress the conjecture in some particular way (e.g., on a long, thin obtuse tri- angle). In the six figures above, a student uses the Supposer to generalize a basic construction where corresponding points of adjacent sides of a square are joined and the ratio of the area of the square and the interior figure are calculated. From left to right, the construction and calculation are repeated on different quadrilaterals. In the top row, the sides of each figure are divided into two equal parts; in the bot- tom row, the sides of each figure are divided into three equal parts. What conjectures emerge? How can these conjectures be justi- fied? The Geometric Supposer

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14 Reshaping School Mathematics Many important ideas follow from the view of mathematics as a science and language of patterns. By expressing a broad view of the mathematical sci- ences, this proposed philosophy encompasses all tracli- tional topics covered by school mathematics. Arithmetic ancl geometry, algebra and calculus are richly endowed with patterns of number, shape, and measure-patterns that will supporl much of the traclitiona~ curriculum. By suggesting that mathematics encompasses all kincis of patterns wherever they arise, this perspective compels a broader vision of school mathematics that inclucles, for example, mathematical structures in probability ancl statistics, in discrete mathematics ancl optimization. By stressing that mathematics is a science, this philosophy supports a style of instruction that rewarcis exploration, encourages experiments, ancl respects conjectural approaches to solving problems. By recognizing that mathematics is an apt language of business ancl science, this view underscores the universal importance of mathematics as a subject that all students must learn to use. By invoking the metaphor of science in which experiment complements theory, the perspective of mathematics as a science of patterns helps bridge the gap between "pure" and "appliecl" mathematics. The patterns studied by mathematicians are, for all practical purposes, as real as the atomic particles studied by physicists. By emphasizing that mathematics is a process rather than a set of facts, this perspective makes clear that stu- dents need to experience genuine problems-those whose solutions have yet to be developed by the stu- dents (or even perhaps by their teachers). Problem situa- tions should be complex enough to offer challenge, but not so complex as to be insoluble, Learning should be guided by the search to answer questions-first at an intu- itive, empirical level; then by generalizing; ancl later by justifying (proving). By making clear that mathematics is the study of patterns rather than merely a craft for calculation (or an art with no evident purpose), this pragmatic view highlights the philosophical basis for using calculators in school mathematics: as microscopes are to biology ancl tele- scopes to astronomy, calculators ancl computers have become essential tools for the study of patterns. By recognizing that practical knowledge emerges from experience with problems, this view helps explain how

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15 A Philosophy and Framework experience with problems can help develop students' ability to compute. This recognition contrasts sharply with the pre- vailing expectation in schools that skill in computation should precede encounter with word problems. Present strategies for teaching need to be reversecl: students who recog- nize the need to apply partic- ular concepts have a stronger conceptual basis for recon- structing their knowledge at a later time. By stressing mathematics as a language in which students express ideas, we enable students to devel- op a framework that can be cirawn upon in the future, when rules may have been forgotten but the struc- ture of mathematical language remains embecicled in memory as a foundation for reconstruction. Learning the language of mathe- matics requires immersion in situa- tions that are sufficiently simple to be manageable, but sufficiently complex to offer diversity: incliviclu- a~, small-group, or large-group instruction; a variety of mathe- matical domains; and open and flexible methods. By affirming the importance of mathematics as a language and science of patterns we reset the gyroscopes that guide school mathematics. instead of being viewed as an immutable collection of absolute truths, mathematics will be seen as it is-as an evolving, pragmatic discipline that seeks to understand the behavior of patterns in science, in society, and in everyday life. Philosophical Perspectives Changing the public philosophy of school mathematics is an essential step in effecting reform of mathematics educa- tion. An effective practical philosophy of mathematics can be based on two considerations:

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16 Reshaping School Mathematics That mathematics is a science and language of patterns; That to know mathematics is to investigate and express relationships among patterns. Nothing in this approach implies that these are unique or necessary considerations. They are, however, sufficient to meet certain important criteria that any effective philosophy of mathematics education must satisfy: They encompass new as well as traditional topics; They provide a substantive rationale for using calculators and computers in school mathematics; They encourage experience with genuine problems; They stimulate exploration, use of real data, and appren- ticeship learning; They help bricige the gap between pure and applied mathematics; They emphasize active modes of learning; They are understandable to a broad segment of the public. The framework for mathematics education that follows from this practical philosophy provides an environment to support present efforts at curricular reform. Other philosophies can also provide similar support, and surely many others will emerge in the process of national curricular change. The counterpoint between a philosophy and a framework of mathematics edu- cation will continue as long as the process of change remains vigorous.