Adding It Up_ Helping Children Learn Mathematics ( PDFDrive.com )

430 ADDING IT UP Working Together Elementary and middle school teachers in the United States report spend- ing relatively little time, compared with their counterparts in other countries, discussing the mathematics they are teaching or the methods they are using. They seldom plan lessons together, observe one another teach, or analyze students’ work collectively. Studies of programs that require teachers to teach mathematically demanding curricula suggest that success is greater when teachers help one another not only learn the mathematics and learn about student thinking but also practice new teaching strategies. Our recommenda- tion concerning time is not just about how much is available but how it is used: • Teachers should be provided with more time for planning and con- ferring with each other on mathematics instruction with appropriate sup- port and guidance. Capitalizing on Professional Meetings Teachers need more mathematically focused opportunities to learn math- ematics, and they need to be prepared to manage changes in the field. Math- ematics teachers already come together at meetings of professional societies such as the National Council of Teachers of Mathematics (NCTM), its affili- ated groups, or other organizations. These occasions can provide opportuni- ties for professional development of the sort discussed above. For example, portions of national or regional meetings of the NCTM could be organized into minicourses or institutes, without competing sessions being held at the same time. Professional development needs to grow out of current activities: • Professional meetings and other occasions when teachers come together to work on their practice should be used as opportunities for more serious and substantive professional development than has commonly been available. Sustaining Professional Development Preparing to teach is a career-long activity. Teachers need to continue to learn. But rather than being focused on isolated facts and skills, teacher learn- ing needs to be generative. That is, what teachers learn needs to serve as a basis for them to continue to learn from their practice. They need to see that practice as demanding continual review, analysis, and improvement. Studies of teacher change indicate that short-term, fragmented professional develop- ment is ineffective for developing teaching proficiency. of Sciences. All rights reserved.

11 CONCLUSIONS AND RECOMMENDATIONS 431 More resources of all types—money, time, leadership, attention—need Professional to be invested in professional development for teachers of mathematics, and development those resources already available could be used more wisely and productively. in Each year a substantial amount of money is invested in professional develop- mathematics ment programs for teachers. Individual schools and districts fund some pro- needs to be grams locally. Others are sponsored and funded by state agencies, federal sustained agencies, or professional organizations. Much of the time and money invested over time in such programs, however, is not used effectively. Sponsors generally fund that is short-term, even one-shot, activities such as daylong workshops or two-day measured institutes that collectively do not form a cohesive and cumulative program of in years, professional development. Furthermore, these activities are often conducted not weeks by an array of professional developers with minimal qualifications in math- or months. ematics and mathematics teaching. Professional development in mathematics needs to be sustained over time that is measured in years, not weeks or months, and it needs to involve a substantial amount of time each year. Our recom- mendations to raise the level of professional development are as follows: • Local education authorities should give teachers support, including stipends and released time, for sustained professional development. • Providers of professional development should know mathematics and should know about students’ mathematical thinking, how mathematics is taught, and teachers’ thinking about mathematics and their own practice. • Organizations and agencies that fund professional development in mathematics should focus resources on multi-year, coherent programs. Resources of agencies at every level should be marshaled to support substan- tial and sustained professional development. Monitoring Progress Toward Mathematical Proficiency In this report we have set forth a variety of observations, conclusions, and recommendations that are designed to bring greater coherence and balance to the learning and teaching of mathematics. In particular, we have described five strands of mathematical proficiency that should frame all efforts to improve school mathematics. Over the past decades, various visions have been put forward for improv- ing curriculum, instruction, and assessment in mathematics, and many of those ideas have been tried in schools. Unfortunately, new programs are tried but of Sciences. All rights reserved.

432 ADDING IT UP then abandoned before their effectiveness has been well tested, and lessons learned from program evaluations are often lost. Although aspects of math- ematics proficiency have been studied, other aspects such as productive dis- position have received less attention; and no one, including the National Assessment of Educational Progress (NAEP), has studied the integrated por- trait of mathematics proficiency set forth in this report. In order that efforts to improve U.S. school mathematics might be more cumulative and coordi- nated, we make the following recommendation: • An independent group of recognized standing should be constituted to assess the progress made in meeting the goal of mathematical proficiency for all U.S. schoolchildren. Supporting the Development of Mathematical Proficiency The mathematics students need to learn today is not the same math- ematics that their parents and grandparents needed to learn. Moreover, math- ematics is a domain no longer limited to a select few. All students need to be mathematically proficient to the levels discussed in this report. The math- ematics of grades pre-K–8 today involves much more than speed in pencil- and-paper arithmetic. Students need to understand mathematics, use it to solve problems, reason logically, compute fluently, and use it to make sense of their world. For that to happen, each student will need to develop the strands of proficiency in an integrated fashion. No country—not even those performing highest on international surveys of mathematics achievement—has attained the goal of mathematical profi- ciency for all its students. It is an extremely ambitious goal, and the United States will never reach it by continuing to tinker with the controls of educa- tional policy, pushing one button at a time. Adopting mathematics textbooks from other countries, testing teachers, holding students back a grade, putting schools under state sanctions—none of these alone will advance school math- ematics very far toward mathematical proficiency for all. Instead, coordi- nated, systematic, and sustained modifications will need to be made in how school mathematics instruction has commonly proceeded, and support of new and different kinds will be required. Leadership and attention to the teach- ing of mathematics are needed in the formulation and implementation of policies at all levels of the educational system. of Sciences. All rights reserved.

433 BIOGRAPHICAL SKETCHES Jeremy Kilpatrick, Chair, is Regents Professor of Mathematics Education at the University of Georgia. He is currently studying the process of changing the school mathematics curriculum, which includes documenting the history of reform efforts in the United States. He is a former vice president of the International Commission on Mathematical Instruction, was a charter member of the Mathematical Sciences Education Board (MSEB) of the National Research Council (NRC), was a member of the National Council of Teachers of Mathematics’ Commission on the Future of the Standards, and currently serves on the NRC’s Board on International Comparative Studies in Educa- tion. He also chaired the MSEB study that produced Measuring What Counts (NRC, 1993). Kilpatrick has published extensively on mathematics educa- tion issues, including “Confronting Reform,” in the American Mathematical Monthly, and “Reflections on Verifying Change in School Mathematics,” in the Journal of Classroom Interaction. He has engaged in numerous editorial activities, most recently with Anna Sierpinska editing Mathematics Education as a Research Domain: A Search for Identity (1998); and with George Stanic edit- ing A History of School Mathematics (in preparation). He is the recipient of multiple awards and honors, including the John W. Wilson Memorial Award and several Fulbright lecturer and scholar awards. He holds an honorary doc- torate from the University of Gothenburg. Kilpatrick received an A.A. in mathematics and science from Chaffey College; an A.B. in mathematics and an M.A. in education from the University of California, Berkeley; and an M.S. in mathematics and a Ph.D. in mathematics education from Stanford University. of Sciences. All rights reserved.

434 ADDING IT UP Deborah Loewenberg Ball is Arthur F. Thurnau Professor of Mathematics Education and Teacher Education at the University of Michigan. An experi- enced elementary teacher, Ball conducts research on instruction and on the processes of learning to teach. She also investigates efforts to improve teach- ing through policy, curriculum, reform initiatives, and teacher education. Ball’s publications include articles on teacher learning and teacher education; the role of subject matter knowledge in teaching and learning to teach; endemic challenges of teaching; and the relations of policy and practice in instruc- tional improvement. Hyman Bass is the Roger Lyndon Collegiate Professor of Mathematics and Professor of Mathematics Education at the University of Michigan. From 1959 to 1998, he was a member of the Mathematics Department at Columbia University. His mathematical research publications cover broad areas of algebra, with connections to geometry, topology, and number theory. He has received the Cole Prize in Algebra from the American Mathematical Society and the Van Amringe Book Award from Columbia University for a book that helped found the subject of algebraic K-theory. He has held visiting research and faculty positions at mathematical centers around the world, including Princeton, Paris, Bombay, Rio, Cambridge, Stockholm, Mexico, Rome, Trieste, Hong Kong, Berkeley, and Jerusalem. He has lectured widely, in particular as a Phi Beta Kappa National Visiting Scholar. He is a member of the National Academy of Sciences and the American Academy of Arts and Sciences. Bass currently serves as President of the American Mathematical Society. He formerly chaired the Mathematical Sciences Education Board of the National Research Council and the Committee on Education of the American Math- ematical Society and is President of the International Commission on Math- ematics Instruction. Jere Brophy is University Distinguished Professor of Teacher Education and Educational Psychology and formerly Co-Director of the Institute for Research on Teaching at Michigan State University. He has done extensive research on teacher effectiveness, the interpersonal dynamics of teacher- student relationships, teacher expectation effects, classroom management, and student motivation. He received a Ph.D. in human development and clinical psychology from the University of Chicago. Felix Browder is University Professor of Mathematics at Rutgers, The State University of New Jersey, and Max Mason Distinguished Service Professor of Sciences. All rights reserved.

BIOGRAPHICAL SKETCHES 435 Emeritus of Mathematics at the University of Chicago, where he was a faculty member from 1963 to 1986. He also served as Vice President for Research at Rutgers University from 1986 to 1991. Browder is the immediate Past President of the American Mathematical Society and was awarded the Presi- dential National Medal of Science in 1999. He is a member of the National Academy of Sciences (NAS) and the American Academy of Arts and Sciences. He has served as a member of the Council of the NAS and its Committee on Science, Engineering, and Public Policy. His research interests include topo- logical methods in analytical problems; history and philosophy of mathemat- ics and science; problems of scientific organizations and institutions; and mathematics and science education. Thomas Carpenter is Professor of Curriculum and Instruction at the Uni- versity of Wisconsin–Madison. He is currently Director of the National Center for Improving Student Learning and Achievement in Mathematics and Sciences, and he is a former editor of the Journal for Research in Mathematics Education. Along with Elizabeth Fennema, Megan Franke, and others, he developed the Cognitively Guided Instruction research and professional devel- opment project. His research investigates the development of children’s mathematical thinking, how teachers use specific knowledge about children’s mathematical thinking in instruction, and how children’s thinking can be used as a basis for professional development. He is currently focusing on the development of algebraic thinking in elementary school. Carpenter received a B.S. from Stanford University and a Ph.D. from the University of Wisconsin. Carolyn Day is Associate Director for Elementary Mathematics and Sci- ence for the Dayton Public Schools in Ohio. She has been with the district for 28 years, first as an elementary teachers, then as a teacher of mathematics in grades 6 through 8, and for the past 10 years as a mathematics supervisor. Day has a B.S. in elementary education from Central State University in Wilberforce, Ohio, and an M.Ed. in curriculum, instruction, and administra- tion from Wright State University. She currently serves on the board of the Ohio Mathematics Leadership Council, the Aullwood Audubon Center in Ohio, and the National Council of Supervisors of Mathematics as secretary. Karen Fuson is Professor of Education and Psychology in the School of Education and Social Policy at Northwestern University. She received her B.A. in mathematics from Oberlin College and her M.A.T. and Ph.D. from the University of Chicago. Her research interests concern young children’s of Sciences. All rights reserved.

436 ADDING IT UP mathematical understanding and the classroom and school conditions that can facilitate such understanding. She seeks to identify and describe devel- opmental or experiential sequences in children’s understanding of various mathematical domains, particularly for ages 2 through 11, and to use this under- standing of children’s thinking to build classroom teaching and learning experiences that will support children’s thinking. She is the author of numer- ous research articles and review articles, including a chapter on addition and subtraction in the Handbook of Research on Mathematics Teaching and Learning. Her book Children’s Counting and Concepts of Number focuses on understanding children’s counting and their conceptual advances in using counting in vari- ous situations. She has done extensive work on children’s multidigit addition and subtraction and on word problem solving and has more recently focused on various aspects of multiplication, division, and fractions at grades 3 through 6 as well as on aspects of geometry and measure. She is directing the Children’s Math Worlds project, which focuses on research to design effective methods of teaching and learning in grade K–5 English-speaking and Spanish-speaking urban and suburban classrooms. James Hiebert is H. Rodney Sharp Professor of Education at the University of Delaware, where he works with preservice and inservice teachers. His professional interests focus on mathematics teaching and learning in class- rooms. He has edited books, including Conceptual and Procedural Knowledge: The Case of Mathematics and Number Concepts and Operations in the Middle Grades, and he co-authored the books Making Sense: Teaching and Learning Mathematics with Understanding and The Teaching Gap: Best Ideas from the World’s Teachers for Improving Education in the Classroom. He currently serves as the mathematics content specialist on the Video Study of the Third International Mathematics and Science Study-Repeat. He received a B.A. in mathematics from Fresno Pacific College, an M.A. in mathematics from the University of Illinois, and a Ph.D. in mathematics education from University of Wisconsin. Roger Howe is a Professor of Mathematics at Yale University where he has been since 1974. He received his Ph.D. from the University of California at Berkeley, and prior to joining the Yale faculty, spent five years at the State University of New York at Stony Brook. He has been a visiting professor at many universities, both in the United States and abroad. Howe’s mathemati- cal research focuses on symmetry and its consequences. He is a member of the Connecticut Academy of Science and Engineering, the American Acad- emy of Arts and Sciences, and the National Academy of Sciences. He has of Sciences. All rights reserved.

BIOGRAPHICAL SKETCHES 437 served on Mathematical Sciences Education Board and on the board of direc- tors of the Connecticut Academy for Education in Mathematics, Science, and Technology. He served as chair of the American Mathematical Society (AMS) Association Review Group for revision of the National Council of Teachers of Mathematics Standards and is currently chair of the AMS Committee on Education. Carolyn Kieran is Professor of Mathematics Education at the University of Quebec, Montreal. Her research interests include the learning and teaching of school algebra, the use of technology in school mathematics, the role of collaboration in mathematical discourse, and the application of historical and psychological models to mathematics education research. Her publications include Research Agenda for Mathematics Education: Research Issues in the Learn- ing and Teaching of Algebra with S. Wagner, Approaches to Algebra: Perspectives for Research and Teaching with N. Bednarz and L. Lee, and a chapter on “The Learning and Teaching of School Algebra” in the 1992 Handbook of Research on Mathematics Teaching and Learning. She has served as president of the Inter- national Group for the Psychology of Mathematics Education, vice president of the Canadian Mathematics Education Study Group, and chair of the edito- rial panel of the Journal for Research in Mathematics Education. Her degrees include a Ph.D. from the Department of Educational Psychology at McGill University and a Master’s in the Teaching of Mathematics from Concordia University in Montreal. In addition to teaching mathematics and mathematics education courses to preservice and inservice teachers at both the graduate and undergraduate levels, Kieran has also taught mathematics at public school. Richard E. Mayer is Professor of Psychology at the University of California, Santa Barbara (UCSB), where he has served since 1975. His major research interests are in educational psychology, with a focus on instructional methods that promote problem-solving transfer. He is a former president of the Divi- sion of Educational Psychology of the American Psychological Association and a former chair of the Department of Psychology at UCSB. In 2000 he received the American Psychological Association’s E. L. Thorndike Award for lifetime achievement in educational psychology. He was editor of Educa- tional Psychologist and Instructional Science and currently serves on the editorial boards of 12 journals. He has authored 13 books including Thinking, Problem Solving, Cognition (2nd edition) and The Promise of Educational Psychology: Learn- ing in the Content Areas. He has authored more than 200 articles and chapters, mainly in the area of educational psychology. of Sciences. All rights reserved.

438 ADDING IT UP Kevin Miller is Associate Professor in the departments of Psychology, Edu- cational Psychology, and the Beckman Institute of the University of Illinois at Urbana-Champaign. His research focuses on how cognitive tools such as number-naming systems, writing systems, and other representational systems affect children’s learning. Recent research involves cross-cultural compari- sons of the learning of reading and mathematics by children in China and the United States and research on how videotaped representations of classroom teaching can be used to improve mathematics education in the United States. He received a Ph.D. in child psychology from the Institute of Child Devel- opment of the University of Minnesota and taught at Michigan State Univer- sity and the University of Texas prior to coming to the University of Illinois. Casilda Pardo has been a teacher for 18 years. She became a full-time math- ematics resource teacher in 1998 at Valle Vista Elementary School in Albu- querque, New Mexico. From 1994 to 1998, she was a half-time classroom teacher and half-time mathematics resource teacher at Armijo Elementary School in Albuquerque. She is also a national trainer for the Investigations in Number, Data, and Space curriculum. From 1992 to 1994, she served as a clini- cal supervisor of student teachers, which included teaching mathematics and science methods courses. Among her professional activities, she has taught mathematical methods at the University of New Mexico (UNM), has been a teacher in the State Initiative in Math and Science Education summer insti- tutes, and has taught Thinking Mathematics I and II at the Continuing Edu- cation division of UNM. Pardo received a B.A. from Marymount College and an M.A. from the University of Wisconsin. Edgar Robinson was Vice President and Treasurer of the Exxon Corpora- tion upon retirement in 1998. In that role he oversaw various financial activi- ties of the Corporation. During his almost 40 years with Exxon he held many senior management positions in Texas, New York, and London. Robinson holds an A.B. in economics from Brown University and an M.B.A. from Harvard Business School. He served as member of President Reagan’s Private Sector Survey on Cost Control (the Grace Commission). He has also been a mem- ber of the Conference Board’s Council of Financial Executives (1990–1998). He is a trustee emeritus of Brown University and a past chairman and life member of the Dean’s Advisory Council at Chicago Business School. Robinson is current Past President of the Dallas Zoological Society and the Vogel Alcove Childcare Center for the Homeless, a project of the Dallas Jewish Coalition. He is a member of the boards of the Dallas Symphony, the Dallas Theater of Sciences. All rights reserved.

BIOGRAPHICAL SKETCHES 439 Center, the Greenwall Foundation, and the American Trust for the British Library. Hung-Hsi Wu is Professor of Mathematics at the University of California, Berkeley. His area of expertise is real and complex geometry. He received his A.B. from Columbia University and his Ph.D. from the Massachusetts Institute of Technology. He has authored several articles on mathematics education and is also a technical reviewer of the 1999 California Mathematics Framework. Almost all his writings in education can be found on his homepage: <http://www.math.berkeley.edu/~wu/>. of Sciences. All rights reserved.

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