 |
 |
 |
 |
 |
 |
|
|||
 |
 |
 |
|||||||
| Copyright © 2012. National Academy of Sciences. All rights reserved. Terms of Use and Privacy Statement |
Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page R1
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
OCR for page R2
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 and
Printed in the United States of America.
at
Except as otherwise noted, copyright 1998 by the National Academy of Sciences. All rights reserved.
OCR for page R3
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.
. . .
OCR for page R4
OCR for page R5
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
OCR for page R6
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
OCR for page R7
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
OCR for page R8
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
OCR for page R9
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
OCR for page R10
OCR for page R11
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~
OCR for page R12
OCR for page R13
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
OCR for page R14
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.)
OCR for page R15
Tha R1~..~ -
and Role of
Algebra in
the K-14
Curriculum
OCR for page R16
