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Perspectives on School Mathematics L: Reshaping School Mathematics A Philosophy and Framework for Curriculum Mathematical Sciences Eclucation Boarcl National Research Council NATIONAL ACADEMY PRESS Washington, D.C. 1990

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NATIONAL ACADEMY PRESS 2101 Constitution Avenue, NW Washington, DC 20418 NOTICE: The project that is the subject of this report was approved by the Governing Board of the National Research Council, whose members are drawn from the councils of the National Academy of Sciences, the Notional Academy of Engineering, and the Institute of Medicine, The members of the committee responsible for the report were chosen for their special competences and with regard for appropriate balance. This report has been reviewed by a group other than the authors according to procedures approved by a Report Review Committee consisting of members of the National Academy of Sciences, the National Academy of Engineering, and the Institute of Medicine, The National Research Council was organized by the National Academy of Sciences in 1916 to associate the broad community of science and technology with the Academy's purposes of furthering knowledge and advising the federal government. Functioning in accordance with general policies determined by the Academy, the Council 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 a continuing national overview and assessment capability for mathematics education and is concerned with excellence in 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 grants from The Educational Foundation of America, Exxon Education Foundation, National Research Council, National Science Foundation (Directorates for Biological and Behavioral and Social Sciences, Computer and Information Science and Engineering, Engineering; Geosciences, Mathematical and Physical Sciences; and Science and Engineering Education), and The Teagle Foundation. The observations made herein do not necessarily reflect the views of the grantors' Library of Congress Catalog Card No. 89-64176 International Standard Book Number 0-309-04187-2 Copyright ~ 1990 by the National Academy of Sciences Permission for limited reproduction of portions of this book for educational purposes but not for sale may be granted upon 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 20418, Printed in the United States of America S085 First Printing, February 1990 Second Printing, May 1990 Third Printing, January 1991 Fourth Printing, July 1992

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Ill Preface As momentum for change in U.S. education continues to grow, questions about the nature and goals of mathematics education take on increasing urgency. Several recent reports have opened a national dialogue on these issues. The purpose of this new report is to focus that discussion on fundamental issues that transcend particular details of current curricula or assessment results. Two documents set the stage. Everybody Counts: A Report to the Nation on the Future of Mathematics Education (National Research Council, 1989) establishes the need for change in the way we teach mathematics and outlines a plan of action. Curriculum anal Evaluation Stanc/ards for School Mathematics (National Council of Teachers of Mcsthematics, 1989) provides objectives for mathematics curricula and assessment that build effectively on present practice and knowledge. Although implementation of these stanclarcis will necessarily be slow for practical reasons, the expectations expressed in the Standarcis are fully attainable toclay. Indeecl, many schools with innovative programs already implement the spirit of mathematics education expressed in the Standards. In effect, the Stanclarcis represent goals for the first stage of a national effort to reshape mathematics education, an effort expectecl to last well into the next century. This Philosophy and Framework provides a supporting rationale for the first stage and for transition to the next stage. By suggesting a new practical philosophy of mathematics, supported by fundamental but forwarcl-looking principles and goals, it opens a door to a whole new vision of mathematics eclucation, The American public expects improvement in school performance. As countless changes are tried in thousands of districts of every size and type, progress will depend on consistency of purpose. The Mathematical Sciences Education Board (MSEB) intends to keep the spotlight of reform on long- term objectives to guide the journey of all who work for beneficial change. This Philosophy and Framework illuminates important issues that must be clarified wherever curriculum reform is to take place. It should help advance the cause of reform by insuring a common foundation for our efforts. On behalf of the entire Boarcl, I want to express my appreciation to the Curriculum Framework Task Force, and especially to its chairman, Anthony Ralston, for energetic and sustained work on the early drafts of this report. The key insights into fundamental long-term goals were framed and

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iv developed by the Task Force during the early years of the MSEB. The process was guided with pragmatism and wisdom by the MSEB Curriculum Committee, chaired by Henry 0. Pollak. Special thanks are also clue to Lynn A. Steen who shaped the final document by combining Board and Task Force thinking, and to Lincla P. Rosen who served as staff director for the entire project. Shirle,~'Hill, Chairman An' ~~~ ~ Mathemati361 Sciences Eclucation Board December 1989

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v MSEB CURRICULUM FRAMEWORK TASK FORCE Anthony Ralston (Chairman), Professor of Computer Science and Mathematics, State University of New York, Buffalo Kim Alberg, Elementary School Teacher, Franklin Smith Elementary School, Blue Springs, Missouri Gail Burrill, Mathematics Teacher, Whitnall High School, Greenfielcl, Wisconsin Robert Dilworth, Professor Emeritus of Mathematics, California Institute of Technology Lames T. Fey, Professor of Curriculum & Instruction and Mathematics, University of Maryland Shirley M. Frye, Director of Curriculum and Instruction, Scottsdale School District, Arizona Steven ]. Leinwanc], Mathematics Consultant, Connecticut State Department of Eclucation Jack Lochhead, Director of the Scientific Reasoning Institute, University of Massachusetts Henry 0. Pollak, Assistant Vice Presiclent, Mathematical, Communications and Computer Sciences Research Laboratory, Bell Communications Research, New Jersey (retired) Alan H. Schoentelcl, Professor of Eclucation and Mathematics, University of California, Berkeley

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vl MSEB CURRICULUM COMMITTEE {7988- 7989) Henry 0. Poliak (Chairman), Assistant Vice President, Mathematical, Communications and Computer Science Research Laboratory, Bell Communications Research, New Jersey (retired) Wade Ellis, Jr., Mathematics Instructor, West Valley College, California Andrew M. Gleason, Hollis Professor of Mathematicks and Natural Philosophy, Harvard University Martin D. Kruskal, Professor of Mathematics, Rutgers University Leslie Hiles Paoletti, Chairman, Department of Mathematics And Computer Science, Choate Rosemary Hall, Connecticut Anthony Ralston, Professor of Computer Science And Mathematics, State University of New York, Buffalo Isacdore M. Singer, Institute Professor, Department of Mathematics, Massachusetts Institute of Technology Zalman Usiskin, Professor of Education, The University of Chicago

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vii MATHEMATICAL SCIENCES EDUCATION BOARD Shirley A. Hill (Chairman), Curators' Professor of Mathematics and Education, University of Missouri-Kansas City Alvin W. Trivelpiece (Chairman Designate), Director, Oak Riclge National Laboratory Iris M. Carl (Vice Chairman), Elementary Mathematics Instructional Supervisor, Houston Inclepenclent School District, Texas Lillian C. Barna, Superintendent of Schools, Tacoma Public Schools, Washington Licia K. Barrett, Dean, College of Arts and Sciences, Mississippi State University C. Diane Bishop, Superintendent of Public Instruction, State of Arizona Constance Clayton, Superintendent of Schools, School District of Philaclelphia, Pennsylvania Paula B. Duckett, Elementary Mathematics Teacher, River Terrace Community School, Washington, D.C, Joan Duea, Elementary School Teacher, Price Laboratory School and Professor of Eclucation, University of Northern Iowa Joseph W. Duncan, Corporate Vice President and Chief Economist, The Dun & Braclstreet Corporation Wade Ellis, Jr., Mathematics Instructor, West Valley College, California Shirley M. Frye, Director of Curriculum and Instruction, Scottsdale School District, Arizona Ramanathan Gnanaclesikan, Heacl, Information Science Research Division, Bell Communications Research Donald L. Kreicler, Vice Chairman, Mathematics and Computer Science Department, Dartmouth College Martin D. Kruskal, Professor of Mathematics, Rutgers University Katherine P. Layton, Mathematics Teacher, Beverly Hills High School, California Steven .1. Lcinwancl, Mathematics Consultant, Connecticut State Department of Education Richard S. Lindzen, Sloan Professor of Meteorology, Massachusetts Institute of Technology Gail Lowe, Principal, Acacia Elementary School, Thousancl Oaks, California Steven P. Meiring, Mathematics Specialist, Ohio State Department of Eclucation Jose P. Mestre, Associate Professor of Physics, University of Massachusetts

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viii Calvin C. Moore, Associate Vice Presiclent, Acaclemic Affairs, University of California, Berkeley Jo Ann Mosier, Mathematics Teacher, Fairclale High School, Louisville, Kentucky Leslie Hiles Paoleni, Chairman, Department of Mathemofics and Computer Science, Choate Rosemary Hall, Connecticut Lauren B. Resnick, Director, Learning Research once Development Center, University of Pittsburgh; liaison with the Commission on Behavioral and Social Sciences and Eclucation, National Research Council Yotancta Rodriguez, MicldIe School Mathematics Teacher, Martin Luther King School, Cambridge, Massachusetts Thomas A. Romberg, Professor of Curriculum and Instruction, University of Wisconsin, Madison Isadore M. Singer, institute Professor, Department of Mathematics, Massachusetis Institute of Technology Lynn Arthur Steen, Professor of Mathematics, St. Olaf College William P. Thurston, Professor of Mathematics, Princeton University Manya S. Ungar, Past President, The National Congress of Parents and Teachers ZaIman Usiskin, Professor of Eclucation, The University of Chicago John B. Walsh, Vice Presiclant/Chief Scientist, Boeing Military Airplanes Nellie C. Well, Past Presiclent, National School Boards Association Gusto t. Weiss, Minor Anheuser Professor of Mathematics, Washington University; liaison with the Commission on Physical Sciences, Mathematics, and Resources, National Research Council

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ix MSEB STAFF OFFICE OF THE EXECUTIVE DIRECTOR Kenneth M. Hoffman, Executive Director Marcia P. Swarcl, Executive Director until August 1989 J. Kevin Colligan, Special Assistant Julie A. Kraman, Administrative Associate Claudelte C. Brown, Senior Secretary Kirsten A. Sampson, Executive Assistant PROJECT AREA DIRECTORS Beverly J. Anderson, Minority Affairs LaVerne Evans-MacDonalcl, Administrative Secretary John R.B.-Clement, Corporate Relations Joan M. Rood, Administrative Secretary Ann P. Kahn, Organizational Liaison Robert J. Kansky, State Coalitions, Year of National Dialogue Linda D. Jones, Administrative Secretary Linda P. Rosen, Educational Studies Jana K. Gocisey, Senior Secretary CONSULTANTS Kathleen A. Holmay, Public Information Harvey B. Keynes, State Coalitions John H. Lawson, State and Local Agenc Ray C. Shiflett, Outreach SENIOR FELLOW Mary Harley Kruter, Department of Energy/MSEB Senior Fellow

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xi Foreword The purpose of this document is to propose a framework for reform of school mathematics in the United States. One essential step in achieving reform of mathematics education is to reshape the curriculum; others include improving the education of teachers of mathematics, preparing effective descriptions of achievement in mathematics, developing better means of assessing performance of students, and preparing more effective and appropriate instructional materials. Although all of these matters are considered in this report, the central message concerns curriculum. In the past century, there have been numerous attempts to change the mathematics curriculum of U.S. schools, starting with the Committee of Ten in the l890s oncl inclucling, among others, the New Mathematics of the 1960s and the National Council of Teachers of Mathemotics (NCTM) Agenda for Action of 1960. Some of these attempts at change have had modest success; others have had little. Toclay, as we enter the final decade of the century, the need to achieve curriculum reform is perhaps more urgent than at any time in the past. In particular, we emphasize in this document two fundamentally important issues discussed in Everybocly Counts ond in the NCTM Standards: Changing perspectives on the need for mathematics, the nature of mathematics, and the learning of mathematics; Changing roles of calculators and computers in the practice of mathematics. These issues stand out from many others discussed in Everybody Counts once in the Standards in their compelling and inevitable impact on the organization of the mathematics curriculum. Hence they serve as touchstones for our inquiry into a philosophy and framework for school mathematics. The complexity and inertia of the American educational system are too great for any reform to be implemented rapidly, even if it hocl the full and active support of all appropriate constituencies. Fortunately, the process of change is already well under way in many districts. The aim of this Philosophy anc! Framework is to give impetus to local reform and coherence to the nationwide effort. It is intended to complement Everybody Counts and the NCTM Stan clards by encouraging development of a mathematics curriculum that will merit the support of teachers and taxpayers, parents and stuclents. It must be a curriculum for all students that will meet their neecis ancl, therefore, the needs of the society in which they will live,

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xiii Contents A Rationale for Change Changing Conditions Outdated Assumptions Transitions to the Future A Practical Philosophy Expanding Goals Fundamental Questions Describing Mathematics Knowing Mathematics Practical Effects Philosophical Perspectives Redesign from a Technological Perspective New Opportunities New Priorities The Year 2000 Research Finclings Open Questions Redesign from a Research Perspective Learning Mathematics Constructed Knowledge Procedural Knowledge Mastery of Subject Matter Problem Solving Making Sense of Mathematics A Framework for Change Principles Goals Enabling Conditions Mathematics for the Future References 4 5 9 10 12 12 15 17 18 19 21 22 24 27 28 29 29 30 31 32 35 36 40 48 51 53

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