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Mathematics Education in tho huddle Grades Teaching lo Meet the Needs of Middle Grades [earners and lo Maintain High Expectations Proceedings of a National Convocation arc! Action Conferences Center for Science, Mathematics, and Engineering Eclucation National Research Council NATIONAL ACADEMY PRESS Washington, DC

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NOTICE: The project that is the subject of this report was approved by the Governing Boa rcl of the National Research Council, whose members are drawn from the councils of the National Acaclemy of Sciences, the National Acaclemy of Engineering, ancl the Institute of Meclicine. The members of the committee responsible for the convocation ancl report were chosen for their special competences ancl with regard for appropriate balance. The Center for Science, Mathematics, ancl Engineering Eclucation (CSMEE) was established in 1995 to provide coordination of all the National Research Council's education activities ancl reform efforts for students at all levels, specifically those in kindergarten through twelfth Oracle, undergraduate institutions, school-to-work programs, ancl continuing eclucation. The Center reports directly to the Governing Boa rcl of the National Research Council. The Convocation ancl Action Conferences about which these proceedings report were funded by a grant from the U.S. Department of Eclucation with additional funding from the American Eclucational Research Association. Any opinions, findings, or recommendations expressed in this report are those of members of the steering committee or participants in the Convocation and Action Conferences and do not necessar- ily reflect the views of the U.S. Department of Eclucation. Library of Congress Cataloging-in-Publication Data Mathematics education in the middle Oracles: teaching to meet the needs of middle Oracles learners ancl to maintain high expectations: proceedings of a national convocation and action conferences / Center for Science, Mathematics, ancl Engineering Eclucation, National Research Council. p. cm. Includes bibliographical references. ISBN 0-309-06797-9 {pink.) 1. Mathematics Stucly ancl teaching IMiclclle school) Unitecl States Congresses. 1. Center for Science,Mathematics, ancl Engineering Eclucation. QA13 .M156 1999 510'.72 clc21 99-050765 Aclclitional copies of this report are available from National Acaclemy Press, 2101 Constitution Avenue, N.W., Lock Box 285, Washington, DC 20055. Call {8001 62~-62~2 or {2021 33~-3313 {in the Washington metropolitan area). This report is also available online at htip://www.nap.eclu. Printed in the Unitecl States of America Copyright 2000 by the National Acaclemy of Sciences. All rights reserved.

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National Acaclemy of Sciences National Acaclemy of Engineering Institute of Meclicine National Research Council The National Academy of Sciences is a private, nonprofit, self-perpetuating society of distinguished scholars engaged in scientific ancl engineering research, cleclicatecl to the furtherance of science ancl technology ancl to their use for the general welfare. Upon the authority of the charter granted to it by the Congress in 1863, the Acaclemy has a mandate that requires it to advise the federal government on scientific ancl technical matters. Dr. Bruce M. Alberts is president of the National Acaclemy of Sciences. The National Academy of Engineering was established in 1 96A, under the charter of the National Acaclemy of Sciences, as a parallel organization of outstanding engineers. It is autonomous in its administration ancl in the selection of its members, sharing with the National Acaclemy of Sciences the responsibility for advising the federal government. The National Acaclemy of Engineering also sponsors engineering programs aimed at meeting national needs, encourages education and research, and recog- nizes the superior achievements of engineers. Dr. William A. Wulf is president of the National Acaclemy of Engineering. The Institute of Medicine was established in 1970 by the National Academy of Sciences to secure the services of eminent members of appropriate professions in the examination of policy matters pertaining to the health of the public. The Institute acts under the responsibility given to the National Acaclemy of Sciences by its congressional charter to be an adviser to the federal government ancl, upon its own initiative, to identify issues of medical care, research, ancl eclucation. Dr. Kenneth 1. Shine is president of the Institute of Meclicine. The National Research Councilwas organized by the National Academy of Sciences in 1916 to associate the broad community of science ancl technology with the Acaclemy's purposes of furthering know~eclge ancl advising the federal government. Functioning in accordance with general policies determined by the Acaclemy, the Council has become the principal operating agency of both the National Acaclemy of Sciences ancl the National Acaclemy of Engineering in providing services to the government, the public, ancl the scientific ancl engineering communities. The Council is administered jointly by both Acaclemies ancl the Institute of Meclicine. Dr. Bruce M. Alberts ancl Dr. William A. Wulf are chairman ancl vice chairman, respectively, of the National Research Council.

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NATIONAL CONVOCATION ON MATHEMATICS EDUCATION IN THE MIDDLE GRADES Program Steering Committee Eclwarcl Silver, Chair, Professor and Senior Scientist, Learning Research Development Center, University of Pittsburgh, Pittsburgh, PA Representatives, Mathematical Sciences Education Board Hyman Bass (NAS)*, Professor of Mathematics, Columbia University, New York, NY Ben jamin Blackhawk, Mathematics Teacher, St. Paul Acaclemy and Summit School, Crystal, MN Susan S. Woocl, Professor of Mathematics, J. Sargeant Reynolds Community College, Richmoncl, VA Representatives, American Educational Research Association Robert Linn, Distinguishecl Professor of Eclucation, University of Coloraclo, Boulcler, CO Sanclra Wilcox, Associate Professor of Teacher Eclucation, Michigan State University, East Lansing, Ml Representatives, National Council of Teachers of Mathematics Catherine Brown, Associate Professor of Curriculum and Instruction, Indiana University, Bloom Karen Longhart, Mathematics Teacher, Flatheacl High School, Kalispell, MT Representatives, National Middle School Association Sam Chattin, Science Teacher, William H. English Miclclle School, Scottsberg, IN Katherine Rasch, Dean and Professor of Eclucation, Maryville University, Chesterfielcl, MO Member-At-Large Shirley Sagawa, Executive Director, Learning First Alliance, Washington, DC National Research Council Staff Gail Burrill, Project Director Roclger Bybee, Executive Director, CSMEE Kristance Coates, Project Assistant Joan Ferrini-Muncly, Associate Executive Director, CSMEE Braclforcl Finclell, Program Officer DeVonne Robertson, Program Assistant Kirsten Sampson Snycler, Aclministrative Officer Doug Sprunger, Senior Project Assistant Tina Winters, Senior Project Assistant *NAG: Member of the National Acaclemy of Sciences ington, IN

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The National Academy of Sciences gratefully acknowledges the U.S. De- partment of Education for its generous financial support of the Convocation and Action Conferences and these proceed- ings, the American Educational Re- search Association for its additional financial support and co-sponsorship of the Convocation and Action Confer- ences, and the National Middle School Association and National Council of Teachers of Mathematics for their co- sponsorship of the Convocation and Action Conferences. Any opinions, findings, and conclusions or recommen- dations expressed in this material are those of the authors and do not neces- sarily renect those of the funders. We would like to acknowledge the staff at the National Research Council's (NRC) Center for Science, Mathemat- ics, and Engineering Education (CSMEE) for their efforts in putting the Convocation and Action Conferences together. Inparticular,Tina Winters was instrumental in overseeing logisti- cal arrangements for the meetings, and in organizing these proceedings for review and publication. She was sup- ported in her efforts by Kirsten Sampson Snyder and Doug Sprunger. Others who provided on-site support were Kristance Coates and DeVonne Robertson. We are grateful to the members of the Program Steering Committee for their oversight in planning of the programs for the Convocation and Action Confer- ences. We would also like to thank Anthony Jackson for his contributions to the program planning and his assistance with resources for the Convocation. A(l(litional thanks go to Deborah Loewenberg Ball, Hyman Bass, anti Sandra Wilcox for their instrumental roles in the organization of the Action Conference on the Professional Devel- opment of Teachers of Mathematics in the Mi(l(lle Gra(les, the Action Confer- ence on the Nature and Teaching of Algebra in the Mi(l(lle Gra(les, anti the Action Conference on Research in the Teaching anti Learning of Mathematics in the Mi(l(lle Gra(les, respectively. We also wish to acknowle(lge the speakers and discussion group facilitators for their contributions and leadership that gave substance to the discussion. It should be noted that these proceed- ings have been reviewed by individuals

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chosen for their diverse perspectives and technical expertise, in accordance with procedures approved by the NRC's Report Review Committee. The pur- pose of this independent review is to provide candid and critical comments that wait 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. We wish to thank the following individuals for their participation in the review of this report: Gilberto Cuevas, University of Miami Nina Koltnow, Sidwell Friends School Sidney L. Rachlin, East Carolina University Marlyn Spivak, Jack Lon(lon Mi(l(lle School While these individuals 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 Project Director, Mathematics Eclucation in the Miclclle Gracles

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Learning mathematics in the middle grades is a critical component in the education of our nation's youth. The mathematics foundation laid during these years provides students with the skins and knowledge to study higher level mathematics during high school, provides the necessary mathematical base for success in other disciplines such as science, and lays the groundwork for mathematically literate citizens. A variety of evidence suggests that the mathematics education landscape is shining and evolving rapidly. Below average mathematics achievement scores for grade eight U. S. students as reported in the Third International Mathematics and Science Study (TIMSS) (U.S. Department of Education, 1996) stimulated national concern leading to a variety of activities and proposals focus- ing attention on mathematics education. Data from the National Assessment of Educational Progress (NAEP) (National Center for Education Statistics, 1997) indicates that while the nation has demonstrated progress over time, the achievement levels for ah students are not yet satisfactory. Research about mathematics education has begun to have implications for classroom practice. States are setting high standards for student achievement and aligning their assessments with those standards. The National Council of Teachers of Math- ematics (NCTM) is preparing Principles and Standards for School Mathematics, an updated version of its previous standards documents: Curriculum and Evaluation Standards for School Mathematics, (NCI~M, 19891; Professional Standardsfor Teaching Mathematics, (NCI~M, 19911; and Assessment Standardsfor School Mathematics, (NCI~M, 19951. As educators focus on improving mathematics education, they face a variety of issues. The problems of middle grades mathematics education are substantial anti (1iffer from those at the elementary and secondary levels. There are issues about: Content. What mathematics content is appropriate? How can the charac- terization of the U.S. mathematics curriculum as a "mile wi(le, inch (leep" be a(l(lresse(l? How can the mathematics curriculum be strength- ened yet respect the development issues so central to mi(l(lle gra(les students? What is the nature of algebra at the mi(l(lle gra(les anti how (toes it influence the curriculum? How micIcIle gracIes stucIents learn mathematics. What is the balance between conceptual un(ler

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standing and practice? What is the role of manipulatives in helping students learn? How do students build understanding of a concept over time? Teaching mathematics at the micIcIle gracIes. To what extent do teacher background and prepara- tion specifically for the teaching of mathematics have an impact on what students learn? How can the issue of specialist versus generalist be re- solved? What teaching practices are most effective with middle grades students? How can teachers help students grow as individuals yet ensure that they learn mathematics? School organization arc! its relation to the teaching arc! learning of mathematics. How does the study of mathematics fit into the "house" concept? What are the characteristics of school structures that promote mathematics learning? Are cross-disciplinary teams compat- ible with vertical articulation? What are the effects of school organization, sche(luling, anti philosophy on the mathematics program? Research. How can the knowledge gained from research be used to improve teaching and learning of mathematics in the middle grades? How can an agenda for continued research that buil(ls on the state of the field as well as moves the thinking forward be framed? What help do teachers need to translate research into practice? How might teachers become researchers themselves as they renect on their practice and on ways to improve? These issues contribute to the challenge of improving mathematics education at the middle grades. Change in schools an(1 in teaching practice has been slow to occur. Evidence is mixed about the effects and directions of efforts to improve mathematics education. Engaging the community at large in conversa- tion about their goals and perspec tives is a critical step to help the nation raise the bar anti maximize opportunities for all mi(l(lle gra(les chil(lren in its schools. The (lialogue anti share(1 visions that occurre(1 at the Convocation plenary sessions, panels, anti small group (liscus- sions can set the stage for making a lifference. REFERENCES National Center for Education Statistics. (1996~. P?`rsavingexcellence:Ast?`dyof U.S. eighth grace mathematics and science teaching, learning, cavrric?`l?`m, and achievement in international context. Washington, DC: Author. National Council of Teachers of Mathematics. (1989~. Cavrric?`l?`m and evaluation standards for school mathematics. Reston, VA Author. National Council of Teachers of Mathematics. ~lg9ly. Professional standardsfor teaching mathematics. Reston, VA Author. National Council of Teachers of Mathematics. (1995). Assessment standardsfor school mathematics. Reston, VA Author. Reese, C.M., Miller, KE., Mazzeo, J., & Dossey, J.A. (1997). NAEP 1996 mathematics report cardfor the nation and the states. Washington, DC: National Center for Education Statistics.

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ACKNOWLEDGMENTS ~ V11 PREFACE 1X LETTER FROM THE PROGRAM STEERING COMMITTEE 1 EXECUTIVE SUMMARY 5 PARTICIPANT OBSERVATIONS 14 CONVOCATION PAPERS AND DISCUSSION SUMMARIES 19 Teaching anc! Learning Mathematics at the MicIcIle GracIes: Setting the Stage Mathematics in the Middle: Building a Firm Foundation of Understanding for the Future Glenda Lappan The Middle School Learner: Contexts, Concepts, and the Teaching Connection Thomas Dickinson Content arc! Learning Issues in the MicIcIle GracIes Renections on Middle School Mathematics Nancy Doda Mathematics Content and Learning Issues in the Middle Grades Kathleen Hart Summary of Small Group Discussion on Content and Learning Issues in Middle Grades Mathematics Teaching Issues in the MicIcIle GracIes Using Video of Classroom Practice as a Too} to Study and Improve Teaching Nanette M. Seago Pane} Reactions to the "Cindy Video" Teaching and Learning Mathematics in the Middle Grades: Student Perspectives Linda Foreman 21 23 32 39 41 50 58 61 63 76 79

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Pane} Response to Foreman Student Video Summary of Small Group Discussion on Teaching Issues in the Middle Grades Organizational Issues in the Middle Grades The Organization and Structure of Schools at the Middle Grades: A Principal's Perspective Adapted from the Transcript of Remarks by Stephen 0. Gibson The Organization and Structure of Schools at the Middle Grades: The Role of Development, Subject Matter, and Teacher Professional Development Mary Kay Stein Improving Achievement in the Middle Grades in Mathematics and Related Areas: Lessons from the Project on High Performance Learning Communities Robert D. Feiner et al. Pane} Discussion on the Organization of Schools at the Middle Grades Closing Remarks Renections on the Convocation Adaptedfrom the transcript of remarks by Edward Silver ACTION CONFERENCES 90 92 95 97 101 111 125 129 131 141 Action Conference on the Nature and Teaching of Algebra in the Middle Grades Action Conference on Research in the Teaching and Learning of Mathematics in the Middle Grades Action Conference on the Professional Development of Teachers of Mathematics in the Middle Grades APPENDICES 143 160 168 i. Convocation and Action Conference Agendas 2. Biographical Information on Convocation and Action Conference Speakers 3. Convocation and Action Conference Participant Lists 4. Marcy's Dots 5. Discussion Session Worksheets 6. Background Paper for the Convocation: What Is Sth Grade Mathematics: A Look from NAEP John Dossey 7. Resources 179 189 215 239 245 247 253

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Mathematics Ellueation in Who Middle Grades

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