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The Nature and Role of Algebra in the K-14 Curriculum Proceedings ofa NalionaISymposium May 27 and 2S, 1997 Sponsored by National Council of Teachers of Mathematics and Mathematical Sciences Education Board Center for Science, Mathematics, and Engineering Education National Research Council National Academy Press Washington, D.C. 1998
NATIONAL ACADEMY PRESS · 2101 Constitution Avenue, NW · Washington, DC 20418 NOTICE: The project that is the subject of this report was approved on February 21, 1997, by the Executive Committee of the Governing Board of the National Research Council (NRC), whose members are drawn from the councils of the National Academy of Sciences, the National Academy of Engineering, and the Institute of Medicine. The project was approved on April 22, 1996, by the Board of Directors of the National Council of Teachers of Mathematics (NCTM). 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. Additional copies of this report are available 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). International Standard Book Number 0-309-06147-4 This report is available online at <http://www.nctm.org> and Printed in the United States of America. at <http://www.nap.edu> Except as otherwise noted, copyright 1998 by the National Academy of Sciences. All rights reserved.
NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS The National Council of Teachers of Mathematics (NCTM), founded in 1920, is a nonprofit professional association dedicated to the improvement of mathematics education for all students in the United States and Canada. It offers vision, leadership, and avenues of communication for those interested in the teaching and learning of mathematics at the elementary-school, middle-school, high-school, college, and university levels. With more that 110,000 members, NCTM is the largest mathematics education organization in the world. Each year, the NCTM conducts a large national conference and seven to nine regional conferences, where teachers of mathematics and others interested in mathematics education can attend lectures, panel discussions, and workshops and can see exhibits of the latest mathematics education materials and innovations. Many NCTM members are also members of one or more of the 260-plus local and special-interest groups formally affiliated with NCTM that work in partnership with the Council to meet mutual goals. As a professional association, the NCTM derives its strength from the involvement of its members, who are drawn from the broad community of stakeholders interested in the field of mathematics and mathematics education. NATIONAL ACADEMY OF SCIENCES · NATIONAL RESEARCH COUNCIL CENTER FOR SCIENCE, MATHEMATICS, AND ENGINEERING EDUCATION MATHEMATICAL SCIENCES EDUCATION BOARD The National Academy of Sciences (NAS or the Academy) is a private, nonprofit, self-perpetuating society of distinguished scholars engaged in scientific and engineering research and dedicated to the furtherance of science and technology and to their use for the general welfare. Upon the authority of the charter granted to it by the Congress in 1863, the Academy has a mandate that requires it to advise the federal government on scientific and technical matters. The National Research Council (NRC or the 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. The Mathematical Sciences Education Board (MSEB) was established in 1985 by the National Research Council to maintain a national capability for assessing the status and quality of mathematics education. The MSEB is located within the Center for Science, Mathematics, and Engineering Education (CSMEE or the Center), which was established in 1995 to provide coordination of the NRC's education activities and reform efforts for all students at all levels, specifically those in kindergarten through twelfth grade and in undergraduate, school-to-work, and continuing education programs in the disciplines of science, mathematics, technology, and engineering. The Center reports directly to the Governing Board of the NRC. . . .
Acknowl~ed~ments The National Council of Teachers of Mathematics (NCTM) and the Mathematical Sciences Education Board (MSEB) both have as their mission the improvement of mathematics education. Because algebra is one of the cross-cutting content areas in the mathematics curriculum and a topic that is currently of much concern to the mathematics community at large, the two organizations joined together to organize a national symposium on algebra in May of 1997. This joint venture was unique in the relationship between the two organizations and represents a significant step in bridging the diverse communities represented by the two organizations. The symposium was organized by the Algebra Symposium Task Force of NCTM and a subgroup of MSEB members. We gratefully acknowledge the National Science Foundation (NSF), whose financial support (Award #9614977) made the symposium possible, and Texas Instruments and Casio, who provided additional funds. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the funders. We also would like to acknowledge the staff at NCTM and at MSEB for their efforts in putting the symposium together. In particular, Ramona Irvin from MSEB and Nancy Hawthorne from NCTM spent many hours drawing up invitation lists, contacting participants, arranging housing, and, in general, ensuring that the details were in place for a successful meeting. They were supported in their efforts by Catherine Bell and Colleen McGurkin from MSEB and Kathleen Chapman and Mary Ferris from NCTM. A special thank-you goes to Marilyn Hala from NCTM for shepherding the grant-writing process and for her help on-site during the symposium. Others who provided on-site support were Francis (Skip) Fennell and Bradford Findell from the MSEB staff and Virginia Williams, Joan Armistead, and Kathleen Chapman from the NCTM staff. We are grateful to the speakers for their contributions and leadership that gave substance to the discussion and to Mark Saul and Bill Tate for providing thoughtful pre-conference readings. Finally, we would like to thank Francis (Skip) Fennell, James Gates, Kathleen (Kit) Johnston, and Beth Wallace from MSEB for their work in organizing and editing these proceedings for review and publication. A special thanks goes to Bradford Findell for his review of the mathematics in this work. It should be noted that these proceedings have been reviewed by individuals chosen for their diverse perspectives and technical expertise, in accordance with procedures approved by the National Research Council's (NRC) Report Review Committee. The purpose of this independent review is to provide candid and critical comments that will assist 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 remains confidential to protect the integrity of the deliberative process. v
v! ACKNOWLEDGMENTS We wish to thank the following individuals for their participation in the review of this report: Dr. Christian Hirsch, Western Michigan University Dr. Roger Howe (NAS), Yale University Dr. Henry O. Pollak, retired Dr. Cathy L. Seeley, University of Texas Ms. Bonnie Walker, Texas ASCD 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. GAIL BURRILL President National Council of Teachers of Mathematics JOAN FERRINI-MUNDY Director Mathematical Sciences Education Board
NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS ALGEBRA SYMPOSIUM TASK FORCE Beverly Williams, Chair Pulaski County Special School District Sherwood, Arkansas Hyman Bass* Columbia University New York, New York Laurie A. Boswell Profile School Bethlehem, New Hampshire Sadie C. Bragg* Borough of Manhattan Community College The City University of New York New York, New York Gail F. Burrill University of Wisconsin Madison, Wisconsin *Mathematical Sciences Education Board members subgroup vat Leigh Childs San Diego State University San Diego, California Shari Ann Wilson Coston* Arkansas Education Renewal Consortium Arkadelphia, Arkansas Robert L. Devaney Boston University Boston, Massachusetts Irvin E. Vance Michigan State University East Lansing, Michigan Bert K. Waits Ohio State University Columbus, Ohio
NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS BOARD OF DIRECTORS (IN MAY OF199 Gail F. Burrill, President University of Wisconsin Madison, Wisconsin Glenda T. Lappan President-Elect Michigan State University East Lansing, Michigan Jerry P. Becker Southern Illinois University Carbondale, Illinois Patricia Campbell University of Maryland College Park, Maryland Ann Carlyle Ellwood Elementary School Goleta, California Loring (Terry) Coes III Rocky Hill School East Greenwich, Rhode Island Dwight A. Cooley Mary Louise Phillips Elementary School Fort Worth, Texas . . . vail Linda M. Gojak Hawken School Lyndhurst, Ohio Peggy T. House Northern Michigan University Marquette, Michigan Richard Kopan Calgary, Alberta Canada Steven J. Leinwand Connecticut Department of Education Hartford, Connecticut Johnny W. Lott University of Montana Missoula, Montana Sandra M. Powers College of Charleston Charleston, South Carolina Zalman Usiskin University of Chicago Chicago, Illinois
MATHEMATICAL SCIENCES EDUCATION BOARD (IN MAY OF199~ Hyman Bass (MSEB Chair) Columbia University New York, New York Glenda T. Lappan (MSEB Vice Chair) Michigan State University East Lansing, Michigan Deborah Ball University of Michigan Ann Arbor, Michigan Benjamin Blackhawk St. Paul Academy and Summit School St. Paul, Minnesota Sadie C. Bragg The City University of New York New York, New York Gail F. Burrill University of Wisconsin Madison, Wisconsin Patricia Campbell University of Maryland College Park, Maryland Shari Ann Wilson Coston Arkansas Education Renewal Consortium Arkadelphia, Arkansas Ingrid Daubechies Princeton University Princeton, New Jersey Shelley Ferguson California Mathematics Project San Diego, California Melvin George University of Missouri Systems Columbia, Missouri Roger Howe Yale University New Haven, Connecticut Bruce Jacobs Oakland Electronic Commerce Resource Center Oakland, California Lee Jenkins Enterprise School District Redding, California Rick D. Jennings Eisenhower High School Yakima, Washington Harvey B. Keynes University of Minnesota Minneapolis, Minnesota James R.C. Leitzel University of New Hampshire Durham, New Hampshire Tony Q. Martinez Leander High School Leander, Texas Pamela Matthews American University Washington, D.C. David Moore Purdue University West Lafayette, Indiana Margaret H. Wright AT&T Bell Laboratories Murray Hill, New Jersey Six
Preface The National Academy of Sciences was pleased to host on May 27 and 28, 1997, a national symposium on "The Nature and Role of Algebra in the K-14 Curriculum" jointly sponsored by the National Council of Teachers of Mathematics (NCTM) and the National Research Council's Mathematical Sciences Education Board (MSEB). One of the Academy' s greatest strengths lies in its ability to act as a convener. As I observed the symposium on its last day, I heard and saw the breadth of the representation across grade levels and across states. NCTM and MSEB clearly succeeded in their ongoing commitment to bring together thoughtful members of the mathematical sciences community to consider important questions in mathematics education. In this case, the questions involved the timely topic of algebra and how it should be treated in the K-14 grades. This record of the symposium proceedings reflects the diversity of the symposium's speakers and participants. It is rich with the shared information and perspectives of elementary-, middle-, and high-school teachers, postsec- ondary and research mathematicians, teacher educators, mathematics education administrators, and others. As you read the papers, presentations, and discourse of the symposium's two days here at the Academy, you will see that the subject of the nature and role of algebra in the K-14 curriculum is difficult and complicated. The questions that are being asked include, What do we mean by algebra and algebraic thinking? What do American students really need to know about and be able to do with algebra? How can we better prepare K-14 teachers to teach algebra? How can we better communicate to parents, the business community, and the general public about the kind of algebra that is relevant and why? As the mathematics and mathematics education communities work with these questions, the Academy, the National Research Council, and the MSEB will continue to be active participants and partners. This is a high-stakes matter, and it will take all of our efforts to make sure that our nation and our nation's children are mathematically prepared for the 21 st century. Bruce Alberts President National Academy of Sciences x~
Contents ACKNOWLEDGMENTS PREFACE INTRODUCTION OPENING REMARKS KEYNOTE ADDRESSES Algebra with Integrity and Reality (H. Bass) Making Algebra Dynamic and Motivating: A National Challenge (J. Dossey) PRESENTATIONS ON DAY ONE Transforming Algebra from an Engine of Inequity to an Engine of Mathematical Power by "Algebrafying" the K-12 Curriculum J. Kaput Developing a Coherent and Focused K-12 Algebra Curriculum E. Phillips Enhancing Algebraic Reasoning with Technology G. Akst Algebra for Everyone? With or Without Technology? M. Norman How Might Technology Enhance Algebraic Reasoning? R. Zbiek What Do We Know about K-14 Students' Learning of Algebra? J. Confrey Algebra: What All Students Can Learn S. Williams and D. Molina Improving K-14 Algebra Instruction: A Discussion of Teachers' Responsibilities and Students' Opportunities B. Moore-Harris . . . x~ v X1 5 9 17 25 27 31 33 35 37 41 45
xlv SYNTHESIS OF DAY ONE (H. Pollak and G. Burrill) PRESENTATIONS ON DAY TWO Capturing Patterns and Functions: Variables and Joint Variation G. Lappan Functions and Relations: A Unifying Theme for School Algebra in Grades 9-12 C. Hirsch Middle School Algebra from a Modeling Perspective G. Kleiman Why Modeling Matters L. Godbold Modeling: Changing the Mathematics Experience in Postsecondary Classrooms R. Dance Algebraic Structure in the Mathematics of Elementary-School Children C. Tierney Structure in School Algebra (Middle School) M. van Recuwijk The Role of Algebraic Structure in the Mathematics Curriculum of Grades 11 - 14 G. Foley Language and Representation in Algebra: A View from the Middle R. Billstein Teaching Algebra: Lessons Learned by a Curriculum Developer D. Resek The Nature and Role of Algebra: Language and Representation D. Hughes Hallett CLOSING REMARKS APPENDICES CONTENTS 49 57 61 63 67 69 73 83 87 89 91 93 95 APPENDIX A: SYMPOSIUM AGENDA 99 APPENDIX B: LIST OF PARTICIPANTS 103 APPENDIX C: SPEAKER BIOGRAPHIES 127 APPENDIX D: INVITED PAPERS Stating the Obvious: Mathematics Course Taking Matters W. Tate Algebra, Technology, and a Remark of I.M. Gelfand M. Saul 135 137 APPENDIX E 145 A Framework for Constructing a Vision of Algebra: A Discussion Document (Reprinted with permission of the National Council of Teachers of Mathematics C) 1997. All rights reserved.)
Tha R1~..~ - and Role of Algebra in the K-14 Curriculum