High School Mathematics At Work

Essays And Examples For The Education Of All Students

MATHEMATICAL SCIENCES EDUCATION BOARD

CENTER FOR SCIENCE, MATHEMATICS, AND ENGINEERING EDUCATION

NATIONAL RESEARCH COUNCIL

NATIONAL ACADEMY PRESS
WASHINGTON, D.C.
1998



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--> High School Mathematics At Work Essays And Examples For The Education Of All Students MATHEMATICAL SCIENCES EDUCATION BOARD CENTER FOR SCIENCE, MATHEMATICS, AND ENGINEERING EDUCATION NATIONAL RESEARCH COUNCIL NATIONAL ACADEMY PRESS WASHINGTON, D.C. 1998

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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 National 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. The National Research Council (NRC) is the operating arm of the National Academies Complex, which includes the National Academy of Sciences, the National Academy of Engineering, and the Institute of Medicine. The National Research Council was organized in 1916 by the National Academy of Sciences to associate the broad community of science and technology with the Academy's purposes of furthering knowledge and providing impartial advice to 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. Bruce Alberts, President of the National Academy of Sciences, and Dr. William Wulf, President of the National Academy of Engineering, also serve as chairman and vice chairman, respectively, of the National Research Council. The Center for Science, Mathematics, and Engineering Education (CSMEE) was established in 1995 to provide coordination of all the National Research Council's education activities and reform efforts for students at all levels, specifically those in kindergarten through twelfth grade, undergraduate institutions, school-to-work programs, and continuing education. The Center reports directly to the Governing Board of the National Research Council. The Mathematical Sciences Education Board was established in 1985 to provide a continuing national capability to assess the status and quality of education in the mathematical sciences and is concerned with excellence in 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 report were supported by a grant from the Pew Charitable Trusts. Any opinions, findings, or recommendations expressed in this report are those of the authors and do not necessarily reflect the views of the Pew Charitable Trusts. Library of Congress Cataloging-in-Publication Data High school mathematics at work : essays and examples for the education of all students / Mathematical Sciences Education Board. p. cm. Includes bibliographical references and index. ISBN 0-309-06353-1 (pbk.) 1. Mathematics—Study and teaching (Secondary)—United States. I. National Research Council (U.S.). Mathematical Sciences Education Board. QA13.H54 1998 510'.71'273—ddc21 98-19669 Permission for limited reproduction of portions of this book for education 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. Additional copies of this report may be purchased from the National Academy Press, 2101 Constitution Avenue, NW, Lock Box 285, Washington, DC 20055. (800) 624-6242 or (202) 334-3313 (in the Washington Metropolitan Area). This report is also available online at http://www.nap.edu. Printed in the United States of America Copyright 1998 by the National Academy of Sciences. All rights reserved.

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--> National Research Council Center for Science, Mathematics, and Engineering Education Mathematical Sciences Education Board Hyman Bass (MSEB Chair) Columbia University Glenda T. Lappan (MSEB Vice Chair) Michigan State University Sherry Baca Prescott Unified School District Deborah Ball University of Michigan Benjamin Blackhawk St. Paul Academy and Summit School Sadie Bragg* Borough of Manhattan Community College Gail F. Burrill University of Wisconsin, Madison Patricia Campbell University of Maryland Shari Coston* Arkansas Education Renewal Consortium Ingrid Daubechies Princeton University Shelley K. Ferguson California Mathematics Project Melvin D. George University of Missouri Roger E. Howe Yale University Bruce Jacobs* Oakland Electronic Commerce Resource Center Lee Jenkins Enterprise School District, Redding, CA Rick Jennings Yakima School District, Yakima, WA Harvey B. Keynes* University of Minnesota James R. C. Leitzel* University of New Hampshire Tony Q. Martinez* Leander High School, Leander, TX Pamela Matthews American University David Moore Purdue University Mari Muri State of Connecticut Department of Education Richard Normington Pacific Bell Mark Saul Bronxville Public Schools Richard Schoen Stanford University Edward A. Silver University of Pittsburgh William Tate University of Wisconsin, Madison Susan S. Wood J. Sargeant Reynolds Community College Margaret H. Wright* AT&T Bell Laboratories Project Staff Rodger Bybee Executive Director, CSMEE Joan Ferrini-Mundy Director, MSEB Bradford Findell Program Officer/Editor Daniel Goroff Division Director, Post-secondary Policy and Practice Kathleen Johnston Editorial Associate Gale Moore Financial & Administrative Associate Doug Sprunger Senior Project Assistant Project Consultants Susan Forman Cathy Kessel Lynn Arthur Steen *   Member until June 1997

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--> National Research Council Center For Science, Mathematics, And Engineering Education High School Mathematics at Work Task Force Hyman Bass Columbia University Bruce Jacobs Oakland Electronic Commerce Resource Center Tony Q. Martinez Leander High School, Leander, TX Pamela Matthews American University Patrick McCray G. D. Searle & Co. Karen Dee Michalowicz The Langley School, McLean, VA Henry O. Pollak AT&T Bell Labs (retired) Jack Price, Chair California State Polytechnic University Alan H. Schoenfeld University of California, Berkeley Daniel Teague North Carolina School of Mathematics & Science

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--> High School Mathematics at Work Reviewers This report has been reviewed by individuals chosen for their diverse perspectives and technical expertise, in accordance with procedures approved by the NRC's Report Review Committee. The purpose of this independent review is to provide candid and critical comments that will assist the authors and the NRC in making the published report as sound as possible and to ensure that the report meets institutional standards for objectivity, evidence, and responsiveness to the study charge. The content of the review comments and draft manuscript remain confidential to protect the integrity of the deliberative process. We wish to thank the following individuals for their participation in the review of this report: George Campbell Jr. President and CEO, NACME, Inc., New York, NY Arthur Eisenkraft Science Coordinator, Bedford Public Schools, Bedford, NY Jane D. Gawronski Superintendent, Escondido Union High School District, Escondido, CA Cindy Hannon State Mathematics Specialist, Maryland State Department of Education Harry Kesten Professor of Mathematics, Cornell University, Ithaca, NY Ron Latanision Chairman, MIT Council on Primary and Secondary Education, Professor of Material Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA Douglas H. O'Roark Mathematics Teacher, Lincoln Park High School, Chicago, IL Diane Resek Professor of Mathematics, San Francisco State University, San Francisco, CA Judah L. Schwartz Professor of Education ,Harvard University, and Emeritus Professor, Engineering Science & Education, Massachusetts Institute of Technology, Cambridge, MA While the individuals listed above have provided many constructive comments and suggestions, responsibility for the final content of this report rests solely with the authoring committee and the NRC.

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--> Foreword: Mathematics for a Modern Age ZELL MILLER Governor of Georgia Now more than ever, our daily lives are directly affected by mathematics. I'm not just talking about the grades students get or how many mathematics classes they have to take. I'm talking about answering the phone, faxing a document, or driving a car. I'm talking about buying medicine for your family, building a house, and listening to music. Mathematics has also helped bring us things that indirectly affect us, like the discoveries of DNA, weather patterns, and how to use light as a surgical tool. It has helped us walk on the moon, create microchips, and transmit images across thousands of miles. With mathematics, we design models to test our ideas and refine them, from nerve impulses to human behavior, volcanoes to food. Mathematics is everywhere. But mathematics has always been around, and the concepts it uses have always helped us. So why is mathematics so much more important to our lives now than it was then? Years ago, in the eras of the abacus or slide rule, information took a lot longer to sort through, but now it is easy to chart a course, assay a risk, or compare statistics. Information is more abundant than ever. You can find answers about everything from world politics to school lunches almost as soon as you think of the questions. This new speed of access to volumes of information obviously brings good

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--> things—like better medical information, better building materials, and more diverse options in business and education. The down side to all of this is that as easily as there can be information, there can be misuse of information. So what does this mean for our children? More than ever, they need to be mathematically savvy. Learning to think and reason mathematically is the only way our children can be sure that they are in control, not being controlled. More practically, almost every job these days requires at least some elementary understanding of mathematics. In fact, many of the jobs that keep our country competitive and successful in the global market are jobs that require more than basic mathematics comprehension. Not that everyone needs to be able to program a computer or predict the stock market, but with the vastly increased speed of statistical calculation (and manipulation) and easier and faster global communication, our children need to know what a number means, where it came from, and how best to judge its veracity. As the Governor of Georgia, I take all of this pretty seriously. I know the importance of bringing businesses to my state to create jobs for Georgians. The world, however, is their marketplace; to be healthy, competitive, and economically secure, our citizens must understand mathematics. They need to become comfortable with the notion of mathematics as a tool for life. GOVERNOR ZELL MILLER was elected Governor of the state of Georgia in 1990. Since taking office, his love of teaching and commitment to education has resulted in one of the most ambitious agendas to improve public education in this century. Governor Miller's public career includes service at virtually every level of government: as mayor, as a member of the state senate, as lieutenant governor, and now as Governor. He is currently Chairman of the Education Commission of the States. He has also chaired the Southern Governors' Association, the Appalachian Regional Commission, and the Council of State Governments. Governor Miller keynoted the 1992 Democratic National Convention in New York and chaired the Platform Drafting Committee for the 1996 Democratic National Convention in Chicago.

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--> Acknowledgments For over ten years the Mathematical Sciences Education Board (MSEB) has worked to support the improvement of mathematics education. The work of the Board becomes visible largely through publications, beginning with Everybody Counts (1989) and Reshaping School Mathematics (1990). High School Mathematics at Work builds on this prior work. Like Measuring Up (1992), it contains student tasks; like Mathematical Preparation of the Technical Work Force (1995), it highlights the mathematics needed for careers. High School Mathematics at Work was developed through the efforts of more people than can be named individually here. We gratefully acknowledge the generous financial support of the Pew Charitable Trusts, and thank both Robert Schwartz for his special encouragement with this project when he was Director of their Education Programs, as well as Janet Kroll for her continuing interest and support as our program officer. The project was launched as a 12th grade sequel to Measuring Up through the initiative of Linda P. Rosen. The MSEB first envisioned that this sequel would illuminate some features of high-quality high school mathematics teaching and learning through standards-based assessment tasks. The growing interest in school-to-work issues, however, led to our giving this theme major prominence. With this new focus, Lynn Arthur Steen and Susan Forman designed the

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--> project, secured funding, collected mathematical examples, commissioned essays, and worked with the Task Force, first as members of the MSEB staff and later as consultants. We also thank Daniel Goroff for his leadership during a period of staff transition. Examples of mathematical tasks were solicited over a period of two years from a wide variety of sources, as described in Appendix A. From the hundreds of examples that were contributed, the Task Force selected about two dozen for inclusion, wrote first drafts of those mathematical examples, and identified essay topics and authors. The resulting collection of examples and essays was molded into its current form by Bradford Findell, serving as editor, under the guidance of Glenda Lappan, Alan Schoenfeld, and Harvey Keynes, with the assistance of Cathy Kessel, and with substantial input from Deborah Ball, Sadie Bragg, Gail Burrill, Shari Coston, Shelley Ferguson, Melvin George, Rick Jennings, Jim Leitzel, Tony Martinez, Pamela Matthews, Patrick McCray, and Jack Price. Of course, no project of this size could ever come to completion without the contributions of support staff. Thanks especially to Sharon O'Donnell for her help collecting the essays and examples and to Catherine Bell and Doug Sprunger for their help with the review process. We must also thank Sally Stanfield, Linda Humphrey, and the staff at the National Academy Press for their support and patience with the complex evolution of this project. HYMAN BASS, CHAIRMAN Mathematical Sciences Education Board References National Research Council. (1989). Everybody counts: A report to the nation on the future of mathematics education. Washington, DC: National Academy Press. National Research Council. (1990). Reshaping school mathematics: A philosophy and framework for curriculum. Washington, DC: National Academy Press. National Research Council. (1993). Measuring up: Prototypes for mathematics assessment. Washington, DC: National Academy Press. National Research Council. (1995). Mathematical preparation of the technical work force. Washington, DC: National Academy Press. Additional thanks for the many efforts and suggestions offered by the following people: Jennifer Bemis, John Bishop, Judy Estep, Jim Fey, Carol Findell, Kent Findell, Irene Gable, Jim Gates, Mary Hornyak, Ramona Irvin, Jay Labov, Patrice Legro, Diane Mann, Bob Naismith, Harold Pratt, Kirsten Sampson, Harold Shoen, Kevin Sullivan, Jan Tumoi, Phil Wagreich, Tina Winters, and Judi Zawojewski.

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--> Table of Contents Foreword: Mathematics for a Modern Age Zell Miller, Governor of Georgia   vii Acknowledgments   ix Introduction   1 PART ONE Connecting Mathematics With Work and Life         Overview   9 1   Mathematics as a Gateway to Student Success Dale Parnell, Oregon State University   14 2   Market Launch Rol Fessenden, L. L. Bean, Inc.   18 3   Integrating Vocational and Academic Education Thomas Bailey, Columbia University   24 4   The Importance of Workplace and Everyday Mathematics Jean E. Taylor, Rutgers University   30 5   Working with Algebra Daniel Chazan, Michigan State University Sandra Callis Bethell, Holt High School   35     EMERGENCY CALLS   42     BACK-OF-THE-ENVELOPE ESTIMATES   45     SCHEDULING ELEVATORS   49     HEATING-DEGREE-DAYS   54

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--> PART TWO The Roles of Standards and Assessments         Overview   59 6   Science and Mathematics Education: Finding Common Ground Jane Butler Kahle, Miami University   63 7   SCANS and Mathematics—Supporting the Transition from Schools to Careers Arnold Packer, Johns Hopkins University   67 8   Thinking about the SAT William Linder-Scholer, SciMath Minnesota   70 9   Extended Response Tasks in International Contexts John Dossey, Illinois State University   75     DRUG DOSAGE   80     MENTAL MATHEMATICS   83     BUYING ON CREDIT   87 PART THREE Curricular Considerations         Overview   93 10   Fitting Tasks to Curriculum Zalman Usiskin, University of Chicago   97 11   Mathematics as a Way of Thinking about Things Albert A. Cuoco, Education Development Center   102 12   Preparing Students for Post-secondary Education Harvey B. Keynes, University of Minnesota   107     LOTTERY WINNINGS   111     HOSPITAL QUALITY   115     ROUNDING OFF   119     RULES OF THUMB   123

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--> PART FOUR Implications For Teaching and Teacher Education         Overview   129 13   Pedagogical Implications for Problem-Centered Teaching Glenda T. Lappan, Michigan State University   132 14   The Role of Complex Mathematical Tasks in Teacher Education Gilbert J. Cuevas, University of Miami   137 15   Assessment Conversations as a Tool for Reform Paul G. LeMahieu, University of Delaware and Department of Education Marshá T. Horton, Delaware Department of Education   141     ESTIMATING AREA   145     TIMING TRAFFIC LIGHTS   147     BUYING A USED CAR   153 PART FIVE Epilogue   157 Appendixes     A   Sources of Problems and Tasks Susan Forman and Lynn Arthur Steen   163 B   Task Force Members   167 Index   171

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