Mathematical Reasoning

Analogies, Metaphors, and Images

Author: Lyn D. English

Publisher: Routledge

ISBN: 1136491147

Category: Education

Page: 392

View: 2807

How we reason with mathematical ideas continues to be a fascinating and challenging topic of research--particularly with the rapid and diverse developments in the field of cognitive science that have taken place in recent years. Because it draws on multiple disciplines, including psychology, philosophy, computer science, linguistics, and anthropology, cognitive science provides rich scope for addressing issues that are at the core of mathematical learning. Drawing upon the interdisciplinary nature of cognitive science, this book presents a broadened perspective on mathematics and mathematical reasoning. It represents a move away from the traditional notion of reasoning as "abstract" and "disembodied", to the contemporary view that it is "embodied" and "imaginative." From this perspective, mathematical reasoning involves reasoning with structures that emerge from our bodily experiences as we interact with the environment; these structures extend beyond finitary propositional representations. Mathematical reasoning is imaginative in the sense that it utilizes a number of powerful, illuminating devices that structure these concrete experiences and transform them into models for abstract thought. These "thinking tools"--analogy, metaphor, metonymy, and imagery--play an important role in mathematical reasoning, as the chapters in this book demonstrate, yet their potential for enhancing learning in the domain has received little recognition. This book is an attempt to fill this void. Drawing upon backgrounds in mathematics education, educational psychology, philosophy, linguistics, and cognitive science, the chapter authors provide a rich and comprehensive analysis of mathematical reasoning. New and exciting perspectives are presented on the nature of mathematics (e.g., "mind-based mathematics"), on the array of powerful cognitive tools for reasoning (e.g., "analogy and metaphor"), and on the different ways these tools can facilitate mathematical reasoning. Examples are drawn from the reasoning of the preschool child to that of the adult learner.
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Using the Mathematics Literature

Author: Kristine K. Fowler

Publisher: CRC Press

ISBN: 9780824750350

Category: Language Arts & Disciplines

Page: 475

View: 9864

This reference serves as a reader-friendly guide to every basic tool and skill required in the mathematical library and helps mathematicians find resources in any format in the mathematics literature. It lists a wide range of standard texts, journals, review articles, newsgroups, and Internet and database tools for every major subfield in mathematics and details methods of access to primary literature sources of new research, applications, results, and techniques. Using the Mathematics Literature is the most comprehensive and up-to-date resource on mathematics literature in both print and electronic formats, presenting time-saving strategies for retrieval of the latest information.
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Mathematical and Analogical Reasoning of Young Learners

Author: Lyn D. English

Publisher: Routledge

ISBN: 1135638705

Category: Education

Page: 236

View: 5276

Mathematical and Analogical Reasoning of Young Learners provides foundational knowledge of the nature, development, and assessment of mathematical and analogical reasoning in young children. Reasoning is fundamental to understanding mathematics and is identified as one of the 10 key standards for school mathematics for the new millennium. The book draws on longitudinal and cross-cultural studies, conducted in the United States and Australia, of children's reasoning development as they progressed from preschool through the end of second grade. The multifaceted analysis of young children's development of mathematical and analogical reasoning focuses on individual learners, their learning environments, and the interaction between the two. The multidisciplinary team of authors present multiple perspectives and multiple methodologies, and provide valuable information on organizing and sustaining interdisciplinary and cross-cultural inquiry. Key issues addressed include: *the relationship between mathematical and analogical reasoning; *how changes in children's reasoning relate to the implicit instruction they receive in their classrooms; *analyses of the participating teachers' knowledge, beliefs, and practices with respect to mathematical and analogical reasoning of young learners; and *ways in which we might promote development of mathematical and analogical reasoning in young children. This volume is highly relevant for mathematics educators, researchers in mathematics education, educational psychologists, early childhood teachers, and others interested in mathematical development of young children, in particular, the development of their reasoning processes.
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Learning and Teaching Early Math

The Learning Trajectories Approach

Author: Douglas H. Clements,Julie Sarama

Publisher: Routledge

ISBN: 1135843805

Category: Education

Page: 344

View: 8497

In this important new book for pre- and in-service teachers, early math experts Douglas Clements and Julie Sarama show how "learning trajectories" help teachers become more effective professionals. By opening up new windows to seeing young children and the inherent delight and curiosity behind their mathematical reasoning, learning trajectories ultimately make teaching more joyous. They help teachers understand the varying level of knowledge and thinking of their classes and the individuals within them as key in serving the needs of all children. In straightforward, no-nonsense language, this book summarizes what is known about how children learn mathematics, and how to build on what they know to realize more effective teaching practice. It will help teachers understand the learning trajectories of early mathematics and become quintessential professionals.
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Sociocultural Research on Mathematics Education

An International Perspective

Author: Bill Atweh,Helen Forgasz,Ben Nebres

Publisher: Routledge

ISBN: 1135653119

Category: Education

Page: 480

View: 8254

This volume--the first to bring together research on sociocultural aspects of mathematics education--presents contemporary and international perspectives on social justice and equity issues that impact mathematics education. In particular, it highlights the importance of three interacting and powerful factors--gender, social, and cultural dimensions. Sociocultural Research on Mathematics Education: An International Perspective is distinguished in several ways: * It is research based. Chapters report on significant research projects; present a comprehensive and critical summary of the research findings; and offer a critical discussion of research methods and theoretical perspectives undertaken in the area. * It is future oriented, presenting recommendations for practice and policy and identifying areas for further research. * It deals with all aspects of formal and informal mathematics education and applications and all levels of formal schooling. As the context of mathematics education rapidly changes-- with an increased demand for mathematically literate citizenship; an increased awareness of issues of equity, inclusivity, and accountability; and increased efforts for globalization of curriculum development and research-- questions are being raised more than ever before about the problems of teaching and learning mathematics from a non-cognitive science perspective. This book contributes significantly to addressing such issues and answering such questions. It is especially relevant for researchers, graduate students, and policymakers in the field of mathematics education.
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Algebra in the early grades

Author: James J. Kaput,David William Carraher,Maria L. Blanton

Publisher: Lawrence Erlbaum

ISBN: N.A

Category: Education

Page: 526

View: 9818

This volume is the first to offer a comprehensive, research-based, multi-faceted look at issues in early algebra. In recent years, the National Council for Teachers of Mathematics has recommended that algebra become a strand flowing throughout the K-12 curriculum, and the 2003 RAND Mathematics Study Panel has recommended that algebra be "the initial topical choice for focused and coordinated research and development [in K-12 mathematics]." The book provides a rationale for a stronger and more sustained approach to algebra in school, as well as concrete examples of how algebraic reasoning may be developed in the early grades. It is organized around three themes: *The Nature of Early Algebra *Students' Capacity for Algebraic Thinking *Issues of Implementation: Taking Early Algebra to the Classrooms The contributors to this landmark volume have been at the forefront of an effort to integrate algebra into the existing early grades mathematics curriculum. They include scholars who have been developing the conceptual foundations for such changes as well as researchers and developers who have led empirical investigations in school settings. Algebra in the Early Gradesaims to bridge the worlds of research, practice, design, and theory for educators, researchers, students, policy makers, and curriculum developers in mathematics education.
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Knowing and teaching elementary mathematics

teachers' understanding of fundamental mathematics in China and the United States

Author: Liping Ma

Publisher: Lawrence Erlbaum

ISBN: 9780805829082

Category: Education

Page: 166

View: 2346

Although Chinese students typically outperform U.S. students on comparisons of mathematics competency, Chinese teachers receive far less education than U.S. teachers. This book describes the development of the profound understanding of fundamental mathematics that elementary teachers need.
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The origins of mathematical knowledge in childhood

Author: Catherine Sophian

Publisher: Lawrence Erlbaum

ISBN: 9780805857580

Category: Education

Page: 196

View: 2497

This book examines the origins and development of children’s mathematical knowledge. It contrasts the widely held view that counting is the starting point for mathematical development with an alternative comparison-of-quantities position. According to the comparison-of-quantities position, the concept of number builds upon more basic concepts of equality, inequality, and less-than and greater-than relations, which derive from comparisons between unenumerated quantities such as lengths. The concept of number combines these basic comparative concepts with the concept of a unit of measure, which allows one quantity to be described as a multiple of another. Sophian examines these alternative accounts of children’s developing mathematical knowledge in the light of research: *on children's counting; *on their reasoning about continuous quantities such as length and area; *on the development of the concept of unit; *on additive and multiplicative reasoning; and *on knowledge about fractions. In the closing chapters, Sophian draws out the developmental and the educational implications of the research and theory presented. Developmentally, the comparison-of-quantities position undermines the idea that numerical knowledge develops through domain-specific learning mechanisms in that it links numerical development both to physical knowledge about objects, which is the starting point for the concept of unit, and to the acquisition of linguistic number terms. Instructionally, the comparison-of-quantities perspective diverges from the counting-first perspective in that it underscores the continuity between whole-number arithmetic and fraction learning that stems from the importance of the concept of unit for both. Building on this idea, Sophian advances three instructional recommendations: First, instruction about numbers should always be grounded in thinking about quantities and how numbers represent the relations between them; second, instruction in the early years should always be guided by a long-term perspective in which current objectives are shaped by an understanding of their role in the overall course of mathematics learning; and third, instruction should be directly toward promoting the acquisition of the most general mathematical knowledge possible. The Origins of Mathematical Knowledge in Childhood is intended for researchers, professionals, and graduate students in developmental psychology, educational psychology, and mathematics education, and as a supplementary text for advanced undergraduate courses in cognitive development, educational psychology, and mathematics education.
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Becoming a reflective mathematics teacher

a guide for observations and self-assessment

Author: Alice F. Artzt,Eleanor Armour-Thomas

Publisher: Lawrence Erlbaum

ISBN: N.A

Category: Education

Page: 238

View: 9608

Discusses the methods involved with being a more thoughtful and responsive mathematics teacher, providing background information on developing cognitive skills and observing other instructors, and includes relevant case studies.
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Contemporary Authors

A Bio-bibliographical Guide to Current Writers in Fiction, General Nonfiction, Poetry, Journalism, Drama, Motion Pictures, Television and Other Fields

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Authors

Page: 487

View: 7395

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The Impact of Reform Instruction on Student Mathematics Achievement

An Example of a Summative Evaluation of a Standards-Based Curriculum

Author: Thomas A. Romberg,Mary C. Shafer

Publisher: Routledge

ISBN: N.A

Category: Education

Page: 188

View: 4871

Summarizing data derived from a four-year combined longitudinal/ cross-sectional comparative study of the implementation of one standards-based middle school curriculum program, Mathematics in Context, this book demonstrates the challenges of conducting comparative longitudinal research in the reality of school life. The study was designed to answer three questions: What is the impact on student performance of the Mathematics in Context instructional approach, which differs from most conventional mathematics texts in both content and expected pedagogy? How is this impact different from that of traditional instruction on student performance? What variables associated with classroom instruction account for variation in student performance? The researchers examined a range of variables that affected data collection. These variations highlight the need to study the effects of the culture in which student learning is situated when analyzing the impact of standards-based curricula on student achievement. This book is directed to educational researchers interested in curriculum implementation, mathematics educators interested in the effects of using reform curriculum materials in classrooms, evaluators and research methodologists interested in structural modeling and scaling of instructional variables, and educational policy makers concerned about reform efforts.
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Bibliographic Guide to Education

Author: N.A

Publisher: N.A

ISBN: N.A

Category:

Page: N.A

View: 9860

... lists publications cataloged by Teachers College, Columbia University, supplemented by ... The Research Libraries of The New York Publica Library.
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Choice

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Best books

Page: N.A

View: 6994

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Resources in Education

RIE.. Annual cumulation

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Education

Page: N.A

View: 5396

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Semantic Web

Grundlagen

Author: Pascal Hitzler,Markus Krötzsch,Sebastian Rudolph,York Sure

Publisher: Springer-Verlag

ISBN: 3540339949

Category: Computers

Page: 277

View: 2383

Das Buch Semantic Web – Grundlagen vermittelt als erstes deutschsprachiges Lehrbuch die Grundlagen des Semantic Web in verständlicher Weise. Es ermöglicht einen einfachen und zügigen Einstieg in Methoden und Technologien des Semantic Web und kann z.B. als solide Grundlage für die Vorbereitung und Durchführung von Vorlesungen genutzt werden. Die Autoren trennen dabei sauber zwischen einer intuitiven Hinführung zur Verwendung semantischer Technologien in der Praxis einerseits, und der Erklärung formaler und theoretischer Hintergründe andererseits. Nur für letzteres werden Grundkenntnisse in Logik vorausgesetzt, die sich bei Bedarf jedoch durch zusätzliche Lektüre und mit Hilfe eines entsprechenden Kapitels im Anhang aneignen lassen. Das Lehrbuch richtet sich primär an Studenten mit Grundkenntnissen in Informatik sowie an interessierte Praktiker welche sich im Bereich Semantic Web fortbilden möchten. Aus den Rezensionen: "... RDF, RDF-S und OWL. Diese Sprachen ... werden von den Autoren dargestellt. Bei der Darstellung ... fallen sie selten zu schwierigen Fachslang, sondern liefern eine gut nachvollziehbare Schilderung mit einfachen Beispielen, auch Übungsaufgaben runden die Kapitel ab. ... Semantic Web ist ein einfach geschriebenes und anschauliches Buch, das In die Grundkonzepte der Semantic-Web-Techniken einführt. Wer sich schnell in RDF, RDF-S und Co. einarbeiten muss und etwas Vorbildung in Logik und Algebra mitbringt, der trifft mit diesem Lehrbuch sicherlich eine gute Wahl ..." (http://www.literaturnetz.com/content/view/8742/44/)
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Die Analogie

Das Herz des Denkens

Author: Douglas Hofstadter,Emmanuel Sander

Publisher: Klett-Cotta

ISBN: 3608106669

Category: Political Science

Page: 600

View: 3375

Sie steigen in einen Aufzug, mit dem Sie noch nie zuvor gefahren sind. Wissen Sie, was Sie tun müssen, um nach oben zu kommen? Natürlich – und der Grund dafür sind die Analogien: Der Aufzug funktioniert wie alle anderen Aufzüge. Alles, was wir wissen, setzen wir in Beziehungen und schaffen es dadurch, Ähnlichkeiten zu entdecken, uns im Chaos der Welt zurechtzufinden. Diese Ähnlichkeiten machen wir uns täglich und meist ganz unbewusst im Umgang mit Neuem und Fremdem zunutze. Wie dieses Feuerwerk des Denkens »funktioniert«, das zeigen Douglas Hofstadter, brillanter Autor und Pulitzer-Preisträger, und der Psychologe Emmanuel Sander. Sie nehmen uns mit auf eine abenteuerliche Reise in die Welt der Sprache und des Geistes – und sie zeigen uns, warum Gedanken ohne Einfluss der Vergangenheit undenkbar sind. Ein inspirierendes Lesevergnügen!
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