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In this second volume in a series of three,. In Stock. In this second volume in a series of three, Fosnot and Dolk focus on how to develop an understanding of multiplication and division in grades Their book:.

Download Fosnot C.t. Dolk M. Young Mathematicians At Work. Constructing Multiplication And Division

Fosnot YO U N G I n our efforts to reform mathematics education, weve learned a tremendous amount about young students strategies and the ways they construct knowledge, without fully understanding how to support such development over time.

The Dutch do. So, funded by the NSF and Exxon Mobil, Mathematics in the City was begun, a collaborative inservice project that pooled the best thinking from both countries. AT W O R K In this second volume in a series of three, Fosnot and Dolk focus on how to develop an understanding of multiplication and division in grades Their book: describes and illustrates what it means to do and learn mathematics provides strategies to help teachers turn their classrooms into math workshops that encourage and reflect mathematizing examines several ways to engage and support children as they construct important strategies and Constructing Multiplication.

Maarten Dolk is a researcher and developer of mathematics education at the Freudenthal Institute in the Netherlands, where he has been involved in the develop- ment of inservice materials for teachers and of multimedia learning environments for student teachers. He has also directed an inservice project in the Netherlands for teacher educators and staff developers.

His work in the United States is related to the inservice of primary school teachers. Heinemann A division of Reed Elsevier Inc. All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means, including information storage and retrieval systems, except by a reviewer, who may quote brief passages in a review.

Figures 3. Dolk, E. Feijs, V. Jonker, N. Ruesink, and W. Uittenbogaard Used by permission of the Freudenthal Institute. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Includes bibliographical references and index. ISBN alk. Mathematics Study and teaching Elementary I. F About This Book xix. Teaching and Learning as Development 9. Closed Situations 25 Word problems vs.

Summing Up. Describing the Landscape 34 Strategies 34 Big Ideas Performance-Based Evaluation and Assessment Portfolio-Based Documentation The two names on the cover of this book mean only that we are the ones who nally sat down at the keyboard. The ideas included here grew out of a collaboration between researchers at the Freudenthal Institute and the faculty and staff of Mathematics in the City, a professional development program sponsored by the City College of New York.

Together, we worked, reected, talked, and experimented. First and foremost, we thank our Dutch colleague Willem Uitten- bogaard, whose voice is evident on every page. He has been an integral force in developing the project, designing and structuring the activities we use, and participating in our classroom investigations. He spent two years living and working in New York City, coteaching the institutes and follow-up courses and supporting teachers in their classrooms as they reformed their practice.

We all grew to love him and respect his knowledge of mathemat- ics, his understanding of Freudenthals work, and his sensitivity to the cul- tures that are a part of New York City. The minilessons built around prob- lem strings see Chapter 7 are largely the result of the work he did with us. He worked tirelessly to make the program a success, and we are extremely grateful for his professionalism, his generosity, and his dedication. Staff members Sherrin Hersch, Betina Zolkower, Emily Dann, and Ju- dit Kerekes all made invaluable contributions as they helped teach courses and worked alongside our teachers in their classrooms.

Sherrin was also the coprincipal investigator on the project, registering some teachers in various and sundry courses, dealing with the paperwork, and acting as our liaison with the schools.

This was often thankless, time-consuming work, and we want to acknowledge the hours she gave to it, as well as the gift of her calmness and sanity. We are especially grateful for Betinas energy and intellect, for the way she challenged us to avoid trivialized word problems, pushing us instead to make the contexts rich and challenging. She was the advisor for many of the projects teachers tried in their classrooms.

We thank Judit for the many hours she donated without pay because she believed in the project. The projects smooth operation we owe to Herbert Seignoret.

Hired initially as a part-time graduate assistant, he soon began working full time, helping with budgets, payroll, data collection, and general ofce manage- ment. We all grew to rely on him and his amazing ability to do twenty things at onceand well. In the fall of Toni Cameron, one of the original participants in the program, became coprincipal investigator.

She took on the major respon- sibility for the coordinating and lead teaching of our inservice offerings. Her tireless efforts doubled our enrollment. We are extremely grateful for her wonderful energy and her willingness to teach in Cathys stead as we com- pleted this manuscript. We are especially grateful to the teachers and children whose voices ll these pages. The book exists because of them and the things they tried in their classrooms.

The teachers saved and shared their students work, al- lowed us to videotape them in the classroom, and willingly read portions of the manuscript and offered suggestions. Many other colleagues read portions of the three volumes in this series and provided helpful comments. In the spring of , Cathy spent her sabbatical at the Freudenthal Institute. While there she shared an ofce with Koeno Gravemeijer. He challenged us with what he has written about models, and the books are better because of his insights.

We particularly wish to thank Marja van den Heuvel- Panhuizen, who helped us design our approach to assessment and whose work is described throughout Chapter 8. We are grateful for their support. Project Construct, in Missouri, funded our work in several schools in that state, and some of the lesson transcripts are from those classrooms. Last, we thank our editors at Heinemann, Leigh Peake, Victoria Merecki, and Alan Huisman, for their belief in the project and their insightful sug- gestions for tightening the manuscript.

This book is the second volume in a series of three. The rst volume, Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction, focuses on developing numeracy in young children between the ages of four and seven. This volume focuses on developing an understanding of multi- plication and division in children between the ages of seven and ten.

The series is a culmination of a long and fruitful journey characterized by collaboration, experimentation, reection, and growth. More than ten years ago we learned of each others work with teachers in our respective countriesCathy in the United States, Maarten in the Netherlands.

Each of us cared deeply about helping mathematics teachers base their practice on how people learn mathematics, how they come to see the world through a mathematical lenshow they come to mathematize their world. Each of us had done research on teachers beliefs, their vision of practice, and how these beliefs affected their decisions, and was attempting to develop inser- vice programs that would enable teachers to reform their practice.

Cathy had previously been involved with the SummerMath for Teach- ers program at Mount Holyoke College, coteaching the summer institutes and working alongside elementary teachers in their classrooms. She had also developed and directed the Center for Constructivist Teaching, a grad- uate preservice program at Connecticut State University.

Whether she was teaching children mathematics or helping teachers learn to teach mathe- matics, the learning psychology commonly known as constructivism was at the core of her work. Maarten, a researcher at the Freudenthal Institute in the Netherlands, had already helped develop inservice materials and multimedia environ- ments for teachers. Whether he was thinking about teaching children mathematics or helping teachers learn to teach mathematics, the didactic now commonly known as realistic mathematics was at the core of his work.

Much attention was being paid to how students learn mathematics and to what the constructivist theory of learning might mean for teaching. Teachers were encouraged to become facilitators and questioners instead of transmitters, to use manipulatives, and to foster colla- borative learning and discussion in order to support learners constructions. Although teachers began to have a good idea about how their role needed to shift, they were given little assistance in determining content and little di- rection regarding what problems or investigations to pursue over time.

The focus in the United States was on how to develop learners strategies and the big ideas surrounding them. And this was important. But the se- quence of activities in the curricula being developed, even when supposedly aligned with the reform, was often based on the discipline of mathematics. For example, fractions were taught by way of simple part-to-whole shading activities in the lower grades, then in the higher grades as ratios, as partition- ing, and nally as operators.

Learners methods of developing ideas and strategies were usually discussed in relation to pedagogy principles of learn- ing and teacher behavior that supports learning , if at all.

Constructivist- based professional development helped teachers see the big ideas their learners were struggling with, but little attention was paid to didacticsa scientic theory of instruction relating to developing, stretching, and sup- porting mathematical learning over time.

In fact, the word didactics often has a negative connotation in the United States, one associated with self- correcting materials and direct instruction, not with development. In Europe the term didactics has a very different meaning. The French, for example, speak of situational didactique, meaning problems or situations that will enable learners to grow mathematically.

The Dutch structure prob- lem contexts in order to challenge and support learners developmentally. They spend years researching the effect of a sequence of carefully crafted problems. So, too, in Japan. Together, educators mold and craft problems in ways that strengthen their power to develop mathematical thinking.

Teach- ers try these problems and then discuss which ones worked, which ones didnt, how they might be changed, what should come next. The didactic in the Netherlands was based primarily on the work of the renowned mathematician Hans Freudenthal. As early as the sixties, Freudenthal had argued that people learn mathematics by actively investi- gating realistic problems. He claimed that mathematics was actually an ac- tivity of mathematizing the world, of modeling, of schematizing, of struc- turing ones world mathematically.

Working with Dutch educators for over twenty years prior to his death in , he was instrumental in reforming Dutch mathematics teaching based on realistic mathematics. Within this framework, researchers formulated learning lines by studying the devel- opment of mathematical ideas historically, as well as the developmental pro- gression of childrens strategies and ideas about various mathematical topics.

Finally they tested these problems with children, revised them as necessary, and prepared them as curricula. Little attention was given to pedagogy or to cognitive psychology.

Young Mathematicians at Work: Constructing Fractions, Decimals, and Percents

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Young Mathematicians at Work: Constructing Fractions, Decimals, and Percents

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Fosnot YO U N G I n our efforts to reform mathematics education, weve learned a tremendous amount about young students strategies and the ways they construct knowledge, without fully understanding how to support such development over time. The Dutch do. So, funded by the NSF and Exxon Mobil, Mathematics in the City was begun, a collaborative inservice project that pooled the best thinking from both countries.

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Если он примет на работу калеку, его компания потеряет лицо.

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 И Танкадо отдал это кольцо совершенно незнакомому человеку за мгновение до смерти? - с недоумением спросила Сьюзан.  - Почему. Стратмор сощурил .

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 - Я понял, что Цифровую крепость не следует останавливать. Сьюзан смотрела на него в растерянности. Стратмор продолжал: - Внезапно я увидел в Цифровой крепости шанс, который выпадает раз в жизни.

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