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Author: A.S. Troelstra
Publisher: Elsevier
ISBN: 008095510X
Category : Mathematics
Languages : en
Pages : 607
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Book Description
Studies in Logic and the Foundations of Mathematics, Volume 123: Constructivism in Mathematics: An Introduction, Vol. II focuses on various studies in mathematics and logic, including metric spaces, polynomial rings, and Heyting algebras. The publication first takes a look at the topology of metric spaces, algebra, and finite-type arithmetic and theories of operators. Discussions focus on intuitionistic finite-type arithmetic, theories of operators and classes, rings and modules, linear algebra, polynomial rings, fields and local rings, complete separable metric spaces, and located sets. The text then examines proof theory of intuitionistic logic, theory of types and constructive set theory, and choice sequences. The book elaborates on semantical completeness, sheaves, sites, and higher-order logic, and applications of sheaf models. Topics include a derived rule of local continuity, axiom of countable choice, forcing over sites, sheaf models for higher-order logic, and complete Heyting algebras. The publication is a valuable reference for mathematicians and researchers interested in mathematics and logic.
Author: A.S. Troelstra
Publisher: Elsevier
ISBN: 008095510X
Category : Mathematics
Languages : en
Pages : 607
Get Book Here
Book Description
Studies in Logic and the Foundations of Mathematics, Volume 123: Constructivism in Mathematics: An Introduction, Vol. II focuses on various studies in mathematics and logic, including metric spaces, polynomial rings, and Heyting algebras. The publication first takes a look at the topology of metric spaces, algebra, and finite-type arithmetic and theories of operators. Discussions focus on intuitionistic finite-type arithmetic, theories of operators and classes, rings and modules, linear algebra, polynomial rings, fields and local rings, complete separable metric spaces, and located sets. The text then examines proof theory of intuitionistic logic, theory of types and constructive set theory, and choice sequences. The book elaborates on semantical completeness, sheaves, sites, and higher-order logic, and applications of sheaf models. Topics include a derived rule of local continuity, axiom of countable choice, forcing over sites, sheaf models for higher-order logic, and complete Heyting algebras. The publication is a valuable reference for mathematicians and researchers interested in mathematics and logic.
Author: Leslie P. Steffe
Publisher: Routledge
ISBN: 1135699461
Category : Education
Languages : en
Pages : 344
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Book Description
Over the last twenty-five years Ernst von Glasersfeld has had a tremendous impact on mathematics and science education through his fundamental insights into the nature of knowledge and knowing. Radical Constructivism in Action is a new volume of papers honouring his work by building on his model of knowing. The contributions by leading researchers present constructivism in action, tying the authors' actions regarding practical problems of mathematics and science education, philosophy, and sociology to their philosophical constraints, giving meaning to constructivism operationally. The book begins with a retrospective analogy between radical constructivism's emergence and changes in what is thought of as "certain" scientific knowledge. It aims to increase understanding of constructivism and Glasersfeld's achievement, and is vibrant evidence of the continued vitality of research in the constructivism tradition.
Author: A.S. Troelstra
Publisher: Elsevier
ISBN: 0080570887
Category : Computers
Languages : en
Pages : 378
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Book Description
These two volumes cover the principal approaches to constructivism in mathematics. They present a thorough, up-to-date introduction to the metamathematics of constructive mathematics, paying special attention to Intuitionism, Markov's constructivism and Martin-Lof's type theory with its operational semantics. A detailed exposition of the basic features of constructive mathematics, with illustrations from analysis, algebra and topology, is provided, with due attention to the metamathematical aspects. Volume 1 is a self-contained introduction to the practice and foundations of constructivism, and does not require specialized knowledge beyond basic mathematical logic. Volume 2 contains mainly advanced topics of a proof-theoretical and semantical nature.
Author: Richard A. Lesh
Publisher: Routledge
ISBN: 1317438523
Category : Education
Languages : en
Pages : 610
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Book Description
This book has two primary goals. On the level of theory development, the book clarifies the nature of an emerging "models and modeling perspective" about teaching, learning, and problem solving in mathematics and science education. On the level of emphasizing practical problems, it clarifies the nature of some of the most important elementary-but-powerful mathematical or scientific understandings and abilities that Americans are likely to need as foundations for success in the present and future technology-based information age. Beyond Constructivism: Models and Modeling Perspectives on Mathematics Problem Solving, Learning, and Teaching features an innovative Web site housing online appendices for each chapter, designed to supplement the print chapters with digital resources that include example problems, relevant research tools and video clips, as well as transcripts and other samples of students' work: http://tcct.soe.purdue.edu/booksULandULjournals/modelsULandUL modeling/ This is an essential volume for graduate-level courses in mathematics and science education, cognition and learning, and critical and creative thinking, as well as a valuable resource for researchers and practitioners in these areas.
Author:
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 190
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Book Description
Author: E. Glasersfeld
Publisher: Springer Science & Business Media
ISBN: 0306472015
Category : Education
Languages : en
Pages : 264
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Book Description
Mathematics is the science of acts without things - and through this, of things one can define by acts. 1 Paul Valéry The essays collected in this volume form a mosaik of theory, research, and practice directed at the task of spreading mathematical knowledge. They address questions raised by the recurrent observation that, all too frequently, the present ways and means of teaching mathematics generate in the student a lasting aversion against numbers, rather than an understanding of the useful and sometimes enchanting things one can do with them. Parents, teachers, and researchers in the field of education are well aware of this dismal situation, but their views about what causes the wide-spread failure and what steps should be taken to correct it have so far not come anywhere near a practicable consensus. The authors of the chapters in this book have all had extensive experience in teaching as well as in educational research. They approach the problems they have isolated from their own individual perspectives. Yet, they share both an overall goal and a specific fundamental conviction that characterized the efforts about which they write here. The common goal is to find a better way to teach mathematics. The common conviction is that knowledge cannot simply be transferred ready-made from parent to child or from teacher to student but has to be actively built up by each learner in his or her own mind.
Author: Barbara Jaworski
Publisher: Psychology Press
ISBN: 9780750703734
Category : Education
Languages : en
Pages : 254
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Book Description
The author charts her developing ideas as she undertakes a several-year-long inquiry into an investigative, constructivist approach to mathematics teaching. She presents an account of constructivism as a philosophy of knowledge and learning, provides a rationale for the research methods she employs, and details case studies in the teaching and thinking of three teachers. The research took place in the UK before the introduction of the National Curriculum. Annotation copyright by Book News, Inc., Portland, OR
Author: Paul Ernest
Publisher: SUNY Press
ISBN: 9780791435878
Category : Philosophy
Languages : en
Pages : 336
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Book Description
Extends the ideas of social constructivism to the philosophy of mathematics, developing a powerful critique of traditional absolutist conceptions of mathematics, and proposing a reconceptualization of the philosophy of mathematics.
Author: B. Jacobs
Publisher: Gulf Professional Publishing
ISBN: 9780444508539
Category : Computers
Languages : en
Pages : 784
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Book Description
This book is an attempt to give a systematic presentation of both logic and type theory from a categorical perspective, using the unifying concept of fibred category. Its intended audience consists of logicians, type theorists, category theorists and (theoretical) computer scientists.
Author: Catherine Twomey Fosnot
Publisher: Teachers College Press
ISBN: 0807772593
Category : Education
Languages : en
Pages : 529
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Book Description
This enduring bestseller remains the most comprehensive examination of constructivism and its relationship to teaching and learning. Closing the gap between theory and practice, well-known scholars make constructivism accessible by showing its application in everyday classrooms. Building on the success of the first edition, the authors have completely updated this popular text and expanded its scope to include examples of constructivist teaching across all grade levels and disciplines. An ambitious revision of a now classic text, Constructivism: Theory, Perspectives, and Practice, Second Edition is an invaluable resource for practicing teachers, teacher educators, and curriculum specialists in mathematics, science, social studies, and language arts. New for the Second Edition! An updated theory section that adds further contemporary biological evidence to go beyond the theories of Piaget and Vygotsky—offering a more contemporary framework for a psychological theory of learning.New chapters reflecting the school-based reforms that have been initiated since the writing of the first edition—specifically addressing the changes in mathematics, social studies, and teacher education.A new chapter on the emerging field of disability studies—including a critique that unmasks current practices and assumptions that better serve schools rather than students and their families. Contributors: Paul Cobb • Susan Cowey • Rheta DeVries • Eleanor Duckworth • Dewey I. Dykstra, Jr. • George Forman • Catherine Twomey Fosnot • Catherine A. Franklin • June S. Gould • Maxine Greene • Candace Julyan • Randall Stewart Perry • D. Kim Reid • Deborah Schifter • Jan Weatherly Valle • Ernst von Glasersfeld • Betty Zan. Praise for the First Edition! “Provides the reader with many ways of connecting to the central ideas of constructivism . . . highly readable.” —Gifted Child Quarterly “Shows how constructivist theory can inform classroom practices, and . . . provides teachers with a deeper understanding that gives substance to the rhetoric of school reform.” —Journal of Curriculum Studies
