Mathematical Reasoning

Analogies, Metaphors, and Images

Author: Lyn D. English

Publisher: Routledge

ISBN: 1136491147

Category: Education

Page: 392

View: 5386

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: 2676

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: 9536

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: 992

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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Mathematics Classrooms That Promote Understanding

Author: Elizabeth Fennema,Thomas A. Romberg

Publisher: Routledge

ISBN: 113567650X

Category: Education

Page: 216

View: 2929

Mathematics Classrooms That Promote Understanding synthesizes the implications of research done by the National Center for Research in Mathematical Sciences on integrating two somewhat diverse bodies of scholarly inquiry: the study of teaching and the study of learning mathematics. This research was organized around content domains and/or continuing issues of education, such as equity and assessment of learning, and was guided by two common goals--defining the mathematics content of the K-12 curriculum in light of the changing mathematical needs of citizens for the 21st century, and identifying common components of classrooms that enable students to learn the redefined mathematics with understanding. To accomplish these goals, classrooms in which instruction facilitated the growth of understanding were established and/or studied. This volume reports and discusses the findings which grew out of this research, and subsequent papers and discussions among the scholars engaged in the endeavor. Section I, "Setting the Stage," focuses on three major threads: What mathematics should be taught; how we should define and increase students' understanding of that mathematics; and how learning with understanding can be facilitated for all students. Section II, "Classrooms That Promote Understanding," includes vignettes from diverse classrooms that illustrate classroom discourse, student work, and student engagement in the mathematics described in Chapter 1 as well as the mental activities described in Chapter 2. These chapters also illustrate how teachers deal with the equity concerns described in Chapter 3. Section III addresses "Developing Classrooms That Promote Understanding." The knowledge of the teaching/learning process gained from the research reported in this volume is a necessary prerequisite for implementing the revisions called for in the current reform movement. The classrooms described show that innovative reform in teaching and learning mathematics is possible. Unlike many volumes reporting research, this book is written at a level appropriate for master's degree students. Very few references are included in the chapters themselves; instead, each chapter includes a short annotated list of articles for expanded reading which provides the scholarly basis and research substantiation for this volume.
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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: 7768

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: 6888

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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Mathematics as a Constructive Activity

Learners Generating Examples

Author: Anne Watson,John Mason

Publisher: Routledge

ISBN: 1135630011

Category: Education

Page: 248

View: 6431

This book explains and demonstrates the teaching strategy of asking learners to construct their own examples of mathematical objects. The authors show that the creation of examples can involve transforming and reorganizing knowledge and that, although this is usually done by authors and teachers, if the responsibility for making examples is transferred to learners, their knowledge structures can be developed and extended. A multitude of examples to illustrate this is provided, spanning primary, secondary, and college levels. Readers are invited to learn from their own past experience augmented by tasks provided in the book, and are given direct experience of constructing examples through a collection of many tasks at many levels. Classroom stories show the practicalities of introducing such shifts in mathematics education. The authors examine how their approach relates to improving the learning of mathematics and raise future research questions. *Based on the authors' and others' theoretical and practical experience, the book includes a combination of exercises for the reader, practical applications for teaching, and solid scholarly grounding. *The ideas presented are generic in nature and thus applicable across every phase of mathematics teaching and learning. *Although the teaching methods offered are ones that engage learners imaginatively, these are also applied to traditional approaches to mathematics education; all tasks offered in the book are within conventional mathematics curriculum content. Mathematics as a Constructive Activity: Learners Generating Examples is intended for mathematics teacher educators, mathematics teachers, curriculum developers, task and test designers, and classroom researchers, and for use as a text in graduate-level mathematics education courses.
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Surfaces and Essences

Analogy as the Fuel and Fire of Thinking

Author: Douglas Hofstadter,Emmanuel Sander

Publisher: Basic Books

ISBN: 0465018475

Category: Philosophy

Page: 592

View: 5048

Shows how analogy-making pervades human thought at all levels, influencing the choice of words and phrases in speech, providing guidance in unfamiliar situations, and giving rise to great acts of imagination.
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Engaging young children in mathematics

standards for early childhood mathematics education

Author: Douglas H. Clements,Julie Sarama,Ann-Marie DiBiase

Publisher: Lawrence Erlbaum

ISBN: N.A

Category: Education

Page: 474

View: 7987

Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education originates from the landmark 2000 Conference on Standards for Pre-kindergarten and Kindergarten Mathematics Education. The main goal of the Conference was
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Metaphor and Analogy in Science Education

Author: Peter J. Aubusson,Allan G. Harrison,Stephen M. Ritchie

Publisher: Springer Science & Business Media

ISBN: 9781402038297

Category: Education

Page: 210

View: 7812

This book brings together powerful ideas and new developments from internationally recognised scholars and classroom practitioners to provide theoretical and practical knowledge to inform progress in science education. This is achieved through a series of related chapters reporting research on analogy and metaphor in science education. Throughout the book, contributors not only highlight successful applications of analogies and metaphors, but also foreshadow exciting developments for research and practice. Themes include metaphor and analogy: best practice, as reasoning; for learning; applications in teacher development; in science education research; philosophical and theoretical foundations. Accordingly, the book is likely to appeal to a wide audience of science educators –classroom practitioners, student teachers, teacher educators and researchers.
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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: 3196

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Shortcut

How Analogies Reveal Connections, Spark Innovation, and Sell Our Greatest Ideas

Author: John Pollack

Publisher: Penguin

ISBN: 1592409474

Category: Business & Economics

Page: 256

View: 5805

From the meatpacking plants that inspired Henry Ford's first moving assembly line, to the domino theory that lead America into Vietnam, to the bicycle of the mind' that Steve Jobs envisioned as the Macintosh computer, analogies have played a dynamic role in shaping the world around us. In Shortcut, Pollack explores this territory with engaging stories, surprising examples and a practical method to evaluate the truth or effectiveness of any analogy. Shortcut will improve critical thinking, enhance creativity and offer a fresh approach to the nature of ideas.'
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Teaching Science for Understanding

A Human Constructivist View

Author: Joel J. Mintzes,James H. Wandersee,Joseph D. Novak

Publisher: Academic Press

ISBN: 9780080879246

Category: Psychology

Page: 384

View: 6205

Teaching Science for Understanding
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Problem Solving in Mathematics Education

Author: Peter Liljedahl,Manuel Santos-Trigo,Uldarico Malaspina,Regina Bruder

Publisher: Springer

ISBN: 3319407309

Category: Education

Page: 39

View: 2888

This survey book reviews four interrelated areas: (i) the relevance of heuristics in problem-solving approaches – why they are important and what research tells us about their use; (ii) the need to characterize and foster creative problem-solving approaches – what type of heuristics helps learners devise and practice creative solutions; (iii) the importance that learners formulate and pursue their own problems; and iv) the role played by the use of both multiple-purpose and ad hoc mathematical action types of technologies in problem-solving contexts – what ways of reasoning learners construct when they rely on the use of digital technologies, and how technology and technology approaches can be reconciled.
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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: 356

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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Where Mathematics Comes from

How the Embodied Mind Brings Mathematics Into Being

Author: George Lakoff,Rafael E. Núñez

Publisher: Basic Books (AZ)

ISBN: N.A

Category: Mathematics

Page: 493

View: 5749

Provides an in-depth analysis of the cognitive science of mathematical ideas that argues that conceptual metaphor plays a definitive role in mathematical ideas, exploring such concepts as arithmetic, algebra, sets, logic, and infinity. 20,000 first printing.
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Bibliographic Guide to Education

Author: N.A

Publisher: N.A

ISBN: 9780783898056

Category: Reference

Page: 1000

View: 448

... lists publications cataloged by Teachers College, Columbia University, supplemented by ... The Research Libraries of The New York Publica Library.
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The Art and Science of Teaching

A Comprehensive Framework for Effective Instruction

Author: Robert J. Marzano

Publisher: ASCD

ISBN: 1416605711

Category: Education

Page: 221

View: 1734

The popular author of Classroom Instruction That Works discusses 10 questions that can help teachers sharpen their craft and do what really works for the particular students in their classroom.
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