DOCUMENT RESUME ED 334 084 SE 052 182 AUTHOR Steen, Lynn Arthur, Ed. TITLE On the Shoulders of Giantst New Approaches to Numeracy. INSTITUTION National Academy of Scieices - National Research Council, Washington, DC. Mathematical Sciences Education Board. SPONS AGENCY Andrew W. Mellon Foundation, New York, N.Y.; Carnegie Corp. of Few York, N.Y. REPORT NO ISBN-0-309,-04234-8 PUB DATE 90 NOTE 239p. AVAILABLE FROM National Academy Press, 2101 Constitution Ave., N.W., Washington, DC 20418 ($17.95 U.S.; $21.75 export). PUB TYPE Books (010) -- Guides - Classroom Use- Teaching Guides (For Teacher) (052) EDRS PRICE AF01 Plus Postage. PC Not Available from EDRS. DESCRIPTORS Calculators; Computer Assisted Instructicn; Curriculum Development; *Curriculum Enrichment; *Educational Change; Educational Improvement; *Educational Innovation; Elementary School Mathematics; Elementary Secondary Education; Geometric Concepts; Mathematical Concepts; *Mathematical Enrichment; Mathematics Curriculum; Mathematics Education; Mathematics Instruction; Secondary School Mathematics IDENTIFIERS Mathematical Thinking; *Numeracy; *Patterns (Mathematics) ABSTRACT Forces created by the proliferation of computer hardware and software, by innovative methods of mattematimal modelling and applications, by broader demographicconsiderations, and bv schools themselves are protaundly changiagthe way mathematics is practiced, the way eaLhematics is taught, and theway mathematics is learned. Ir ttis volume, a vision of the richness ofmathematics is expressed and illustrated as f'..ve different possiblestrands of school mathematics through the 97:pension on the themeof mathematics as the language and science of patterns with emphases cn interconnections amd onmmunalities. Included are: (1)an introduction to the five sample strInds with examples of the continuityunderlying the fundamentals of mathematics, of the mathematicalconnections that both unite and repeat, and of the mathematicalperspective necessary to view both the variety and the regularity ofpatterns, (2) the Dimension strand, which focuses on the development ofrelationships between the different dimensions with ampnasison three-dimensional topics; (3) the Quantity strand, etich providesinsights into the three basic tasks of measuring, orderimg, and codingwith attendant symbologies and procedures; (4) the Uncertaintystrand, which develops idees about the natural themes andstrategies witLin the twin concepts of data and chance; (5)the Shape strand, which uses classification to discover similaritiesand differences among objects, analysis to discern thecomponents of form, and representation to recogaze and comprehend shapeswithin different contexts; and (6) the Change strand, which advocatesimaginative and sensitive responses to the constantly emergingnew types of i.atterns, includingpatterns ofthinkingabout mature, as well as mathematics. (31 references) (jM) iNtti led4 U OE PALMVE.4T OP EDUCATION Ofte at EdLICabortat Rodearers andtn1DfOrenIfInt EDUC rONAL RESOURCES INFORMATION CENTER IERIC) 110:4 hICIOCurbentnag been reproduced es rede..00 'tom the Pet Son Or Offpnastion ort.itteap liAno, Menges aawe been made to Improve reproduchOn cluahtir C`0'n?" 0? volt* 0t oemons staled .n !Ns docu- ment do not nacessanly seoteseni otbcfau OE RI Doildfon or gsoUcy -PERMISSION TOREPRODUCE THIS MATERIAL IN MICROFICHEONLY BY HAS BEEN GRANTEDI 4. rt. DeonKa S. tut iliavn TO THE EDUCATIONALRESOURCES INFORMATION CENTER(ERIC) ." BEST COPY AY E A Is $17.95 ON THE SHOULDERS OF GIANTS New Approaches to Numeracy For many of us and our children, mathemat- ics was (or is) a painful and difficult subject seemingly with little application to daily life. But experts agree that much of the problem lies in outdated teaching methods and tiresome,pre- dictable course work. Mathematics can be exciting for students and great fun for younger childrenif educators and parents adopt a fresh perspectiveon mathe- matical concepts and how theyare presented u our schools. Even the youngest child can experi- ence exponential phenomena by folding a sheet of paper repeatedly in half. With the comment "If I have seen farther than others, it is because I have stoodon the shoulders of giants," Isaac Newton credited his foresight in developing calculus to theaccumu- lated work of his predecessors. On the Shoulders of Giants inspires us to a :tmilar leap of vision, providing a conceptuai pathway toward a new view of mathematics for theyoung in the coming century. And now is the time to take such a giiant step forward. We must understand mathemati- cal concepts to appraise the flood ofnews and information we receive each day, from the weather report to the latest tax regulation. Mathematics affects everything from the foodwe eat, to the investments we make, to the size and shape of thP cities we build. It provides the tools for coping with the technology that increasingly penetrates our lives. On the Shoulders of Giants proposes that we help our students delve deep to find the con- cepts that underlie all mathematics and thereby better understand their significance in the world. Five essays, each exploring a different concept, present tn,lthematics as the language and science of patterns, offering countless, imaginativeways in which mathematical 'deas can be developed through formal school and college study: Change encourages us to throw away the text- books that move ploddingly from ar:thmetic to algebra to calculusand explore changeas it affects real-wolici issues suchas population dynamics. (Continued on back flav) GIANTSOn the Shoulders of New Approaches to Numeracy LYN' ARTPUR STEEN Editor Mathematical Sciences Education Board National Research Council lowse ame aw, a 11, 1.47%1M. t\I S NATIONAL ACADEMY PRESS Washington. D.C. 1990 NATIONAL ACADEMY P'kESS2101 Constitution Avenue, NW Washington, DC 20418 NOTICE: il,, peo;ect that is the subject of this report was approved by the Governing Board of the National Research Council, whose members are drawn from the ceuncils of theNation& Academy of Sciences, the National Academy of Engineering, and the Institute of Medicine, Theauthors of the report were chosen for their special competences. The National Research Council was organized by the National Academy of Sciences in1916 to associate the broad community of science and technology with the Academy's purposesof furthering knowledge and advising the federal government. Functioning in accordante with generalpolicies determined by the Academy. the Couneil has become the principal operating agency of both the National Academy of Sciences and the National Academy of Engineering in providing services to the government, the public, and the scientific and engineering communities. The Council is administered jointly by both Academies and the Institute of Medicine. Dr. Frank Press and Dr. Robert M. White are chairman and vice chairman, respectively, of the National Research Council, The Mathematical Sciences Education Board was established in 1985 to provide acontiauing national overview and assessment capability for mathematics education and is converned with mathematical sciences education for all students at all levels. The Board reports directly to the Governing Board of the National Research Council. Development, publication, and dissemination of this book were supported by a grant from the Carnegie Corporation of New York. Additional dissemination of the book was supported by a grant from the Andrew W. Mellon Foundation. The observations made herein do not necessarily reflect the views of the grantors. Library of Congress Cataloguing-in.Pubikation Data On the shoulders of giants : new approaches to numeracy / Lynn Arthur Steen, editor : Mathematical Sciences Education Board, National Research Council. p.cm. Includes bibliographical trferences and index. ISBN 0-309-042344 (hardbound) 1 MathematicsStudy and teachingUnited States,1. Steen. Lynn Arthur, 1941- II. National Research Council Mathematical Sciences Education Board. QA13.0531990 513.071073--dc20 90-41566 Copyright © PM by the National Academy of Sciences Cover; A computer-generated image of a torus tthe surface of a doughnut-shaped figure i. When projected from four-dimensional space. the torus divides all of three-dimtnsional space into two symmetrical pieces. Created by Thomas Banchoff and Nicholas Thompson at Brown University Permission for limited reproduction of portions of this book for educational purposes but not for sale may be granted on receipt of a written request to the National Academy Press, 2101 Constitution Avenue, NW. Washington, DC 20418. Copies of this report may be purchased from the National Academy Press, 2101 Constitution Avenue. NW, Washington. DC 29418. Printed in the United States of America t)f- Preface n01. PN, Today's headlines are filled with reports of illiteracy, innumeracy, and other signs of educational decay. Tomorrow's schools can be filled with evidence of renaissance if we begin now to till the soil for effective educationin mathematics, in science, and in all subjects. This vol- ume offers five visions of mathematics suitable for tomorrow's schools visions rooted in imagination, in mathematics, and in science. Ideas in this volume can pnv ide fertile soil for new approaches to tomorrow's numeracy. Forces created by computers, applications, demographics, and schools themselves are changing profoundly the way mathematics is practiced, the way it is taught. and the way it is learned. Even as we work to make incremental change in today's schools, we must think also