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Author: A. J. M. Gasteren
Publisher: Springer Science & Business Media
ISBN: 9783540528494
Category : Computers
Languages : en
Pages : 196
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Book Description
This book deals with the presentation and systematic design of mathematical proofs, including correctness proofs of algorithms. Its purpose is to show how completeness of argument, an important constraint especially for the correctness of algorithms, can be combined with brevity. The author stresses that the use of formalism is indispensible for achieving this. A second purpose of the book is to discuss matters of design. Rather than addressing psychological questions, the author deals with more technical questions like how analysis of the shape of the demonstrandum can guide the design of a proof. This technical rather than psychological view of heuristics together with the stress on exploiting formalism effectively are two key features of the book. The book consists of two independently readable parts. One part includes a number of general chapters discussing techniques for clear exposition, the use of formalism, the choice of notations, the choice of what to name and how to name it, and so on. The other part consists of a series of expositional essays, each dealing with a proof or an algorithm and illustrating the use of techniques discussed in the more general chapters.
Author: A. J. M. Gasteren
Publisher: Springer Science & Business Media
ISBN: 9783540528494
Category : Computers
Languages : en
Pages : 196
Get Book Here
Book Description
This book deals with the presentation and systematic design of mathematical proofs, including correctness proofs of algorithms. Its purpose is to show how completeness of argument, an important constraint especially for the correctness of algorithms, can be combined with brevity. The author stresses that the use of formalism is indispensible for achieving this. A second purpose of the book is to discuss matters of design. Rather than addressing psychological questions, the author deals with more technical questions like how analysis of the shape of the demonstrandum can guide the design of a proof. This technical rather than psychological view of heuristics together with the stress on exploiting formalism effectively are two key features of the book. The book consists of two independently readable parts. One part includes a number of general chapters discussing techniques for clear exposition, the use of formalism, the choice of notations, the choice of what to name and how to name it, and so on. The other part consists of a series of expositional essays, each dealing with a proof or an algorithm and illustrating the use of techniques discussed in the more general chapters.
Author: Michael Harris
Publisher: Princeton University Press
ISBN: 0691175837
Category : Mathematics
Languages : en
Pages : 468
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Book Description
An insightful reflection on the mathematical soul What do pure mathematicians do, and why do they do it? Looking beyond the conventional answers—for the sake of truth, beauty, and practical applications—this book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twenty-first century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources. Drawing on his personal experiences and obsessions as well as the thoughts and opinions of mathematicians from Archimedes and Omar Khayyám to such contemporary giants as Alexander Grothendieck and Robert Langlands, Michael Harris reveals the charisma and romance of mathematics as well as its darker side. In this portrait of mathematics as a community united around a set of common intellectual, ethical, and existential challenges, he touches on a wide variety of questions, such as: Are mathematicians to blame for the 2008 financial crisis? How can we talk about the ideas we were born too soon to understand? And how should you react if you are asked to explain number theory at a dinner party? Disarmingly candid, relentlessly intelligent, and richly entertaining, Mathematics without Apologies takes readers on an unapologetic guided tour of the mathematical life, from the philosophy and sociology of mathematics to its reflections in film and popular music, with detours through the mathematical and mystical traditions of Russia, India, medieval Islam, the Bronx, and beyond.
Author: Mark Colyvan
Publisher:
ISBN: 0195166612
Category : Mathematics
Languages : en
Pages : 183
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Book Description
Annotation. The Quine-Putnam indispensability argument in the philosophy of mathematics urges us to place mathematical entities on the same ontological footing as other theoretical entities essential to our best scientific theories. Recently, the argument has come under serious scrutiny, with manyinfluential philosophers unconvinced of its cogency. This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.
Author: Peter Medvegyev
Publisher: Oxford University Press, USA
ISBN: 0199215251
Category : Business & Economics
Languages : en
Pages : 629
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Book Description
This graduate level text covers the theory of stochastic integration, an important area of Mathematics that has a wide range of applications, including financial mathematics and signal processing. Aimed at graduate students in Mathematics, Statistics, Probability, Mathematical Finance, and Economics, the book not only covers the theory of the stochastic integral in great depth but also presents the associated theory (martingales, Levy processes) and important examples (Brownianmotion, Poisson process).
Author: Joseph Michael Assof
Publisher: Corwin Press
ISBN: 1071953869
Category : Education
Languages : en
Pages : 102
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Book Description
Foster Confidence and Ownership in Every Math Student When it comes to math, does it feel like some students embrace problem-solving with agency and ownership while others are confused—or simply along for the ride? How do educators bridge that divide to develop competence, confidence, and ownership in every student? The answers lie in establishing clear and effective measures for success. Steeped in the principles of success criteria, Whose Math Is It? provides educators with everything they need to create a classroom environment where students feel empowered to step up and take the lead. Divided into two parts, this must-read guide first defines what success looks like for math students, then provides the research-based best practices teachers can use to help students take control of their learning. Learn how to: Define and establish effective success criteria in a mathematics classroom Implement a variety of strategies to support student ownership and success Develop class-wide social norms specific to math Promote metacognition through self-regulated learning, self-assessment, and feedback Reinforce student ownership through structured peer interactions and collaboration Whose Math Is It? is an essential resource for K-12 math teachers who want to empower their students to actively own their mathematics learning. By emphasizing the importance of success criteria, promoting self-regulated learning, and developing math-specific social norms, this book provides practical strategies for creating an environment where when asked, "Whose math is it?" every student can emphatically respond: My math!
Author: Mike Flynn
Publisher: Taylor & Francis
ISBN: 1003839533
Category : Education
Languages : en
Pages : 230
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Book Description
Beyond Answers: Exploring Mathematical Practices with Young Children, author Mike Flynn provides teachers with a clear and deep sense of the Standards for Mathematical Practice and shares ideas on how to best implement them in K-2 classrooms. Each chapter is dedicated to one of the eight common core standards. Using examples from his own teaching and vignettes from many other K-2 teachers, Flynn does the following: Invites you to break the cycle of teaching math procedurally Demonstrates what it means for children to understand not just do math Explores what it looks like when young children embrace the important behaviors espoused by the practices The book's extensive collection of stories from K-2 classroom provides readers with glimpses of classroom dialogue, teacher reflections, and examples of student work. Focus questions at the beginning of each vignette help you analyze the examples and encourage further reflection. Beyond Answers is a wonderful resource that can be used by individual teachers, study groups, professional development staff, and in math methods courses.
Author: Jan Westerhoff
Publisher: Oxford University Press
ISBN: 0192587188
Category : Philosophy
Languages : en
Pages : 352
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Book Description
Does the real world, defined as a world of objects that exist independent of human interests, concerns, and cognitive activities, really exist? Jan Westerhoff argues that we have good reason to believe it does not. His discussion considers four main facets of the idea of the real world, ranging from the existence of a separate external and internal world (comprising various mental states congregated around a self), to the existence of an ontological foundation that grounds the existence of all the entities in the world, and the existence of an ultimately true theory that provides a final account of all there is. As Westerhoff discusses the reasons for rejecting the postulation of an external world behind our representations, he asserts that the internal world is not as epistemically transparent as is usually assumed, and that there are good reasons for adopting an anti-foundational account of ontological dependence. Drawing on conclusions from the ancient Indian philosophical system of Madhyamaka Buddhism, Westerhoff defends his stance in a purely Western philosophical framework, and affirms that ontology, and philosophy more generally, need not be conceived as providing an ultimately true theory of the world.
Author: Lyn D. English
Publisher: Routledge
ISBN: 1134626649
Category : Education
Languages : en
Pages : 739
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Book Description
This third edition of the Handbook of International Research in Mathematics Education provides a comprehensive overview of the most recent theoretical and practical developments in the field of mathematics education. Authored by an array of internationally recognized scholars and edited by Lyn English and David Kirshner, this collection brings together overviews and advances in mathematics education research spanning established and emerging topics, diverse workplace and school environments, and globally representative research priorities. New perspectives are presented on a range of critical topics including embodied learning, the theory-practice divide, new developments in the early years, educating future mathematics education professors, problem solving in a 21st century curriculum, culture and mathematics learning, complex systems, critical analysis of design-based research, multimodal technologies, and e-textbooks. Comprised of 12 revised and 17 new chapters, this edition extends the Handbook’s original themes for international research in mathematics education and remains in the process a definitive resource for the field.
Author: Stephen E. Toulmin
Publisher: Cambridge University Press
ISBN: 9780521534833
Category : Philosophy
Languages : en
Pages : 268
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Book Description
"In spite of initial criticisms from logicians and fellow philosophers, The Uses of Argument has been an enduring source of inspiration and discussion to students of argumentation from all kinds of disciplinary background for more than forty years. " Frans van Eemeren, University of Amsterdam
Author: Andrew Aberdein
Publisher: Bloomsbury Publishing
ISBN: 1350039039
Category : Philosophy
Languages : en
Pages : 293
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Book Description
This book explores the results of applying empirical methods to the philosophy of logic and mathematics. Much of the work that has earned experimental philosophy a prominent place in twenty-first century philosophy is concerned with ethics or epistemology. But, as this book shows, empirical methods are just as much at home in logic and the philosophy of mathematics. Chapters demonstrate and discuss the applicability of a wide range of empirical methods including experiments, surveys, interviews, and data-mining. Distinct themes emerge that reflect recent developments in the field, such as issues concerning the logic of conditionals and the role played by visual elements in some mathematical proofs. Featuring leading figures from experimental philosophy and the fields of philosophy of logic and mathematics, this collection reveals that empirical work in these disciplines has been quietly thriving for some time and stresses the importance of collaboration between philosophers and researchers in mathematics education and mathematical cognition.
