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Author: Michael D. Resnik Publisher: Oxford University Press ISBN: 9780198236085 Category : Mathematics Languages : en Pages : 300
Book Description
Resnik expresses his commitment to a structuralist philosophy of mathematics and links this to a defence of realism about the metaphysics of mathematics - the view that mathematics is about things that really exist.
Author: Michael D. Resnik Publisher: Oxford University Press ISBN: 9780198236085 Category : Mathematics Languages : en Pages : 300
Book Description
Resnik expresses his commitment to a structuralist philosophy of mathematics and links this to a defence of realism about the metaphysics of mathematics - the view that mathematics is about things that really exist.
Author: Patrick M. Jenlink Publisher: IAP ISBN: 1648027466 Category : Education Languages : en Pages : 265
Book Description
Mathematics as the Science of Patterns: Making the Invisible Visible to Students through Teaching introduces the reader to a collection of thoughtful, research-based works by authors that represent current thinking about mathematics, mathematics education, and the preparation of mathematics teachers. Each chapter focuses on mathematics teaching and the preparation of teachers who will enter classrooms to instruct the next generation of students in mathematics. The value of patterns to the teaching and learning of mathematics is well understood, both in terms of research and application. When we involve or appeal to pattern in teaching mathematics, it is usually because we are trying to help students to extract greater meaning, or enjoyment, or both, from the experience of learning environments within which they are occupied, and perhaps also to facilitate remembering. As a general skill it is thought that the ability to discern a pattern is a precursor to the ability to generalize and abstract, a skill essential in the early years of learning and beyond. Research indicates that the larger problem in teaching mathematics does not lie primarily with students; rather it is with the teachers themselves. In order to make changes for students there first needs to be a process of change for teachers. Understanding the place of patterns in learning mathematics is a predicate to understanding how to teach mathematics and how to use pedagogical reasoning necessary in teaching mathematics. Importantly, the lack of distinction created by the pedagogical use of patterns is not immediately problematic to the student or the teacher. The deep-seated cognitive patterns that both teachers and students bring to the classroom require change. Chapter 1 opens the book with a focus on mathematics as the science of patterns and the importance of patterns in mathematical problem solving, providing the reader with an introduction. The authors of Chapter 2 revisit the work of Po lya and the development and implementation of problem solving in mathematics. In Chapter 3, the authors present an argument for core pedagogical content knowledge in mathematics teacher preparation. The authors of Chapter 4 focus on preservice teachers’ patterns of conception as related to understanding number and operation. In Chapter 5 the authors examine the role of visual representation in exploring proportional reasoning, denoting the importance of helping learners make their thinking visible. The authors of Chapter 6 examine patterns and relationships, and the importance of each in assisting students’ learning and development in mathematical understanding. The authors of Chapter 7 examine the use of worked examples as a scalable practice, with emphasis on the importance of worked examples in teaching fraction magnitude and computation is discussed. In Chapter 8, the authors expand on the zone of proximal development to investigate the potential of Zankov’s Lesson in terms of students analyzing numerical equalities. The authors of Chapter 9 focus on high leverage mathematical practices in elementary pre-service teacher preparation, drawing into specific relief the APEX cycle to develop deep thinking. In Chapter 10, the author focuses on number talks and the engagement of students in mathematical reasoning, which provides opportunities for students to be sensemakers of mathematics. Chapter 11 presents an epilogue, focusing on the importance of recognizing the special nature of mathematics knowledge for teaching.
Author: Michael D. Resnik Publisher: Clarendon Press ISBN: 0191519006 Category : Philosophy Languages : en Pages : 300
Book Description
Mathematics as a Science of Patterns is the definitive exposition of a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defence of realism about the metaphysics of mathematics—the view that mathematics is about things that really exist. Resnik's distinctive philosophy of mathematics is here presented in an accessible and systematic form: it will be of value not only to specialists in this area, but to philosophers, mathematicians, and logicians interested in the relationship between these three disciplines, or in truth, realism, and epistemology.
Author: Keith Devlin Publisher: Macmillan ISBN: 9780805073447 Category : Mathematics Languages : en Pages : 228
Book Description
To most people, mathematics means working with numbers. But as Keith Devlin shows in Mathematics: The Science of Patterns, this definition has been out of date for nearly 2,500 years. Mathematicians now see their work as the study of patterns—real or imagined, visual or mental, arising from the natural world or from within the human mind. Using this basic definition as his central theme, Devlin explores the patterns of counting, measuring, reasoning, motion, shape, position, and prediction, revealing the powerful influence mathematics has over our perception of reality. Interweaving historical highlights and current developments, and using a minimum of formulas, Devlin celebrates the precision, purity, and elegance of mathematics.
Author: Keith J. Devlin Publisher: Times Books ISBN: 9780716750475 Category : Mathematics Languages : en Pages : 215
Book Description
With this fascinating volume, Keith Devlin proves that the guiding principles of some of the most mysterious mathematical topics can be made comprehensible. Writing with an elegant lucidity, Devlin shows just why the definition of mathematics as "working with numbers" has been out of date for nearly 2,500 years. And he demonstrates that far from being too abstract to matter, mathematics is instead an essential and uniquely human endeavor, one that helps us understand the universe and ourselves. In this century alone, there has been a veritable explosion of mathematical activity. A body of knowledge that in 1900 might have filled 80 volumes now would require nearly 100,000. Fields such as algebra and topology have grown tremendously, while complexity theory, dynamical systems theory, and other new areas have developed. And in the last two decades, a common thread running through the many facets of mathematics has been recognized: mathematicians of all kinds now see their work as the study of patterns - real or imagined, visual or mental, arising from the natural world or from within the human mind. Devlin uses this basic definition as his central theme, revealing the search for patterns that drives the mathematics of counting (natural numbers), reasoning (language and logic), motion (calculus), shape (geometry, tilings), and position (topology, knots, symmetry). Interweaving historical highlights and current developments, and using a minimum of formulas, he lets readers see into the kind of reasoning that allows mathematicians to create and explore arcane subjects. And he makes clear the many ways mathematics informs our perceptions of reality - both the physical, biological, and social worlds without, and the realm of ideas and thoughts within. "Mathematics, rightly viewed, possesses not only truth, but supreme beauty," the noted philosopher and mathematician Bertrand Russell once wrote. In Mathematics: The Science of Patterns, Keith Devlin makes such a vision accessible, entertaining, and meaningful. It is an insightful, richly illustrated celebration of the simplicity, the precision, the purity, and the elegance of mathematics.
Author: Reuben Hersh Publisher: Oxford University Press ISBN: 0199839395 Category : Mathematics Languages : en Pages : 369
Book Description
Most philosophers of mathematics treat it as isolated, timeless, ahistorical, inhuman. Reuben Hersh argues the contrary, that mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. Hersh pulls the screen back to reveal mathematics as seen by professionals, debunking many mathematical myths, and demonstrating how the "humanist" idea of the nature of mathematics more closely resembles how mathematicians actually work. At the heart of his book is a fascinating historical account of the mainstream of philosophy--ranging from Pythagoras, Descartes, and Spinoza, to Bertrand Russell, David Hilbert, and Rudolph Carnap--followed by the mavericks who saw mathematics as a human artifact, including Aristotle, Locke, Hume, Mill, and Lakatos. What is Mathematics, Really? reflects an insider's view of mathematical life, and will be hotly debated by anyone with an interest in mathematics or the philosophy of science.
Author: Michael D. Resnik Publisher: ISBN: 9780191598296 Category : Mathematics Languages : en Pages : 285
Book Description
Resnik expresses his commitment to a structuralist philosophy of mathematics and links this to a defence of realism about the metaphysics of mathematics - the view that mathematics is about things that really exist.
Author: Clifford A Pickover Publisher: World Scientific ISBN: 9814504033 Category : Mathematics Languages : en Pages : 466
Book Description
This book will allow you to travel through time and space. To facilitate your journey, the editor has scoured the four corners of the earth in a quest for unusual people and their fascinating patterns. From Mozambique, to Asia, to many European countries, the contributors to The Pattern Book include world-famous cancer researchers, little-known artists and eclectirc computer programmers. Some of the patterns are ultramodern, while others are centuries old. Many of the patterns are drawn from the universe of mathematics. Computer recipes are scattered throughout.Although the emphasis is on computer-generated patterns, the book is informal and the intended audience spans several fields. The emphasis is on the fun that the true pattern lover finds in doing, rather than in reading about the doing! The book is organized into three main parts: Representing Nature (for those patterns which describe or show real physical phenomena, e.g., visualizations of protein motion, sea lilies, etc.), Mathematics and Symmetry (for those patterns which describe or show mathematical behavior, e.g. fractals), and Human Art (for those patterns which are artistic works of humans and made without the aid of a computer, e.g. Moslem tiling patterns.)
Author: Michael D. Resnik
Author: Michael D. Resnik
Author: Patrick M. Jenlink
Author: Michael D. Resnik
Author: Keith Devlin
Author: Keith J. Devlin
Author: Reuben Hersh
Author: Michael D. Resnik
Author: Harry Henderson
Author: Brian Wichmann
Author: Clifford A Pickover