Meaning in Mathematics

Author: John Polkinghorne

Publisher: Oxford University Press

ISBN: 019960505X

Category: Mathematics

Page: 159

View: 4041

Is mathematics invented or discovered? Why does this seemingly abstract discipline provide the key to unlocking the deep secrets of the physical universe? Famous mathematicians, mathematical physicists and philosophers of mathematics try to answer these questions in a series of accessible chapters that shed light on what mathematics really means.

Meaning in Mathematics Education

Author: Jeremy Kilpatrick,Celia Hoyles,Ole Skovsmose

Publisher: Springer Science & Business Media

ISBN: 0387240403

Category: Education

Page: 260

View: 9645

What does it mean to know mathematics? How does meaning in mathematics education connect to common sense or to the meaning of mathematics itself? How are meanings constructed and communicated and what are the dilemmas related to these processes? There are many answers to these questions, some of which might appear to be contradictory. Thus understanding the complexity of meaning in mathematics education is a matter of huge importance. There are twin directions in which discussions have developed—theoretical and practical—and this book seeks to move the debate forward along both dimensions while seeking to relate them where appropriate. A discussion of meaning can start from a theoretical examination of mathematics and how mathematicians over time have made sense of their work. However, from a more practical perspective, anybody involved in teaching mathematics is faced with the need to orchestrate the myriad of meanings derived from multiple sources that students develop of mathematical knowledge. This book presents a wide variety of theoretical reflections and research results about meaning in mathematics and mathematics education based on long-term and collective reflection by the group of authors as a whole. It is the outcome of the work of the BACOMET (BAsic COmponents of Mathematics Education for Teachers) group who spent several years deliberating on this topic. The ten chapters in this book, both separately and together, provide a substantial contribution to clarifying the complex issue of meaning in mathematics education. This book is of interest to researchers in mathematics education, graduate students of mathematics education, under graduate students in mathematics, secondary mathematics teachers and primary teachers with an interest in mathematics.

Perspectives on Practice and Meaning in Mathematics and Science Classrooms

Author: D. Clarke

Publisher: Springer Science & Business Media

ISBN: 0306472287

Category: Education

Page: 358

View: 6202

This is a variegated picture of science and mathematics classrooms that challenges a research tradition that converges on the truth. The reader is surrounded with different images of the classroom and will find his beliefs confirmed or challenged. The book is for educational researchers, research students, and practitioners with an interest in optimizing the effectiveness of classrooms as environments for learning.

Meaning and Existence in Mathematics

Author: Charles Castonguay

Publisher: Springer Science & Business Media

ISBN: 370917113X

Category: Juvenile Nonfiction

Page: 160

View: 7282

The take-over of the philosophy of mathematics by mathematical logic is not complete. The central problems examined in this book lie in the fringe area between the two, and by their very nature will no doubt continue to fall partly within the philosophical re mainder. In seeking to treat these problems with a properly sober mixture of rhyme and reason, I have tried to keep philosophical jargon to a minimum and to avoid excessive mathematical compli cation. The reader with a philosophical background should be familiar with the formal syntactico-semantical explications of proof and truth, especially if he wishes to linger on Chapter 1, after which it is easier philosophical sailing; while the mathematician need only know that to "explicate" a concept consists in clarifying a heretofore vague notion by proposing a clearer (sometimes formal) definition or formulation for it. More seriously, the interested mathematician will find occasional recourse to EDWARD'S Encyclopedia of Philos ophy (cf. bibliography) highly rewarding. Sections 2. 5 and 2. 7 are of interest mainly to philosophers. The bibliography only contains works referred to in the text. References are made by giving the author's surname followed by the year of publication, the latter enclosed in parentheses. When the author referred to is obvious from the context, the surname is dropped, and even the year of publication or "ibid. " may be dropped when the same publication is referred to exclusively over the course of several paragraphs.

Perspectives on Mathematics Education

Papers Submitted by Members of the Bacomet Group

Author: H. Christiansen,A.G. Howson,M. Otte

Publisher: Springer Science & Business Media

ISBN: 9400945043

Category: Education

Page: 371

View: 1957

BACOMET cannot be evaluated solely on the basis of its publications. It is important then that the reader, with only this volume on which to judge both the BACOMET activities and its major outcome to date, should know some thing of what preceded this book's publication. For it is the story of how a group of educators, mainly tutors of student-teachers of mathematics, com mitted themselves to a continuing period of work and self-education. The concept of BACOMET developed during a series of meetings held in 1978-79 between the three editors, Bent Christiansen, Geoffrey Howson and Michael Otte, at which we expressed our concern about the contributions from mathematics education as a discipline to teacher education, both as we observed it and as we participated in it. The short time which was at the teacher-educator's disposal, allied to the limited knowledge and experience of the students on which one had to build, raised puzzling problems concerning priorities and emphases. The recognition that these problems were shared by educators from many different countries was matched by the fact that it would be fruitless to attempt to search for an internationally (or even nationally) acceptable solution to our problems. Different contexts and traditions rule this out.

Godel's Mistake

The Role of Meaning in Mathematics

Author: Ashish Dalela

Publisher: N.A

ISBN: 9788193052310


Page: 226

View: 2698

Why Is Mathematics Incomplete? Godel's incompleteness theorem is a foundational result in mathematics that proves that any axiomatic theory of numbers will be either inconsistent or incomplete. Turing's Halting problem is a foundational result in computing proving that computers cannot know if a program will halt. Godel's Mistake connects these theorems to the question of meaning. The book shows that the proofs arise due to category confusions between names, concepts, things, programs, algorithms, problems, etc. The book argues that these problems can be solved by introducing ordinary language categories in mathematics. Where the Solution Lies The solution to the problem, the author argues, requires a new approach to numbers where numbers are treated as types rather than quantities. To view numbers as types requires a foundational shift in which objects are constructed from sets rather than sets from objects. Since sets denote concepts, this shift implies that objects are created from concepts. This also changes our view of space-time from linear and open to hierarchical and closed. In this hierarchical description, objects are symbols of meaning, rather than physical things. The author calls this theory the Type Number Theory (TNT) and shows that the type view of numbers is free of Godel's Incompleteness and Turing's Halting Problem. How This Book Is Structured Chapter 1: Mechanizing Thought--provides an overview of mathematical, philosophical, linguistic and logical issues that preceded Godel's and Turing's results and shows that the problems encountered in mathematics have a wider undercurrent extending into other areas of science. Chapter 2: Godel's Mistrick--discusses Godel's Incompleteness Theorem and Turing's Halting problem and shows how their proofs rest on category mistakes. The chapter also connects the theorems to the issues of sentence and program meaning. This sets up the motivation for alternative views about numbers and programs that can be free of the paradoxes that arise without semantics. Chapter 3: Mathematics and Reality--the chapter discusses the Platonic notion of mathematics, which keeps ideas and things in separate worlds, and argues that they exist in the same world. The need to bring them together changes our view of objects, space-time, numbers and programs. Now, objects are symbols and numbers and programs are types. The implications of this view to the Cartesian mind-body problem and Platonic separation between ideas and things is discussed. Chapter 4: Numbers and Meanings--develops the intuitions about numbers as types by interpreting various classes of numbers-- natural numbers, zero, negative numbers, irrationals and rationals, and imaginary numbers--in terms of meanings. The chapter concludes by defining the term Type Number Theory (TNT). Chapter 5: Mathematical Foundations--the chapter critiques some foundational ideas in mathematics including logic, set theory and number theory and shows why the very notion of an object as something logically prior to ideas is logically inconsistent. The author argues that numbers are outcomes of distinguishing, and distinguishing requires distinctions. The foundation of mathematics is therefore not in the idea of objects and collections but in the nature of distinctions. The book concludes with a discussion about how distinctions originate in the nature of observation and the foundation of mathematics can therefore be seen in the fundamental properties of consciousness that divides and classifies in order to know.

The Place of Meaning in Mathematics

Selected Theoretical Papers of William A. Brownell

Author: William Arthur Brownell

Publisher: N.A


Category: Arithmetic

Page: 297

View: 4966


Vital Directions for Mathematics Education Research

Author: Keith R Leatham

Publisher: Springer Science & Business Media

ISBN: 1461469775

Category: Education

Page: 207

View: 8220

This book provides a collection of chapters from prominent mathematics educators in which they each discuss vital issues in mathematics education and what they see as viable directions research in mathematics education could take to address these issues. All of these issues are related to learning and teaching mathematics. The book consists of nine chapters, seven from each of seven scholars who participated in an invited lecture series (Scholars in Mathematics Education) at Brigham Young University, and two chapters from two other scholars who are writing reaction papers that look across the first seven chapters. The recommendations take the form of broad, overarching principles and ideas that cut across the field. In this sense, this book differs from classical “research agenda projects,” which seek to outline specific research questions that the field should address around a central topic.

Meaning in Absurdity

What bizarre phenomena can tell us about the nature of reality

Author: Bernard Kastrup

Publisher: John Hunt Publishing

ISBN: 1846948606

Category: Philosophy

Page: 134

View: 7819

This book is an experiment. Inspired by the bizarre and uncanny, it is an attempt to use science and rationality to lift the veil off the irrational. Its ways are unconventional: weaving along its path one finds UFOs and fairies, quantum mechanics, analytic philosophy, history, mathematics, and depth psychology. The enterprise of constructing a coherent story out of these incommensurable disciplines is exploratory. But if the experiment works, at the end these disparate threads will come together to unveil a startling scenario about the nature of reality. The payoff is handsome: a reason for hope, a boost for the imagination, and the promise of a meaningful future. Yet this book may confront some of your dearest notions about truth and reason. Its conclusions cannot be dismissed lightly, because the evidence this book compiles and the philosophy it leverages are solid in the orthodox, academic sense.

Mind, Meaning and Mathematics

Essays on the Philosophical Views of Husserl and Frege

Author: L. Haaparanta

Publisher: Springer Science & Business Media

ISBN: 940158334X

Category: Philosophy

Page: 284

View: 3609

At the turn of the century, Gottlob Frege and Edmund Husserl both participated in the discussion concerning the foundations of logic and mathematics. Since the 1960s, comparisons have been made between Frege's semantic views and Husserl's theory of intentional acts. In quite recent years, new approaches to the two philosophers' views have appeared. This collection of articles opens with the first English translation of Dagfinn Føllesdal's early classic on Husserl and Frege of 1958. The book brings together a number of new contributions by well-known authors and gives a survey of recent developments in the field. It shows that Husserl's thought is coming to occupy a central role in the philosophy of logic and mathematics, as well as in the philosophy of mind and cognitive science. The work is primarily meant for philosophers, especially for those working on the problems of language, logic, mathematics, and mind. It can also be used as a textbook in advanced courses in philosophy.

The Infinite in Mathematics

Logico-mathematical writings

Author: Felix Kaufmann

Publisher: Springer Science & Business Media

ISBN: 9400997957

Category: Science

Page: 237

View: 8038

The main item in the present volume was published in 1930 under the title Das Unendliche in der Mathematik und seine Ausschaltung. It was at that time the fullest systematic account from the standpoint of Husserl's phenomenology of what is known as 'finitism' (also as 'intuitionism' and 'constructivism') in mathematics. Since then, important changes have been required in philosophies of mathematics, in part because of Kurt Godel's epoch-making paper of 1931 which established the essential in completeness of arithmetic. In the light of that finding, a number of the claims made in the book (and in the accompanying articles) are demon strably mistaken. Nevertheless, as a whole it retains much of its original interest and value. It presents the issues in the foundations of mathematics that were under debate when it was written (and in some cases still are); , and it offers one alternative to the currently dominant set-theoretical definitions of the cardinal numbers and other arithmetical concepts. While still a student at the University of Vienna, Felix Kaufmann was greatly impressed by the early philosophical writings (especially by the Logische Untersuchungen) of Edmund Husser!' He was never an uncritical disciple of Husserl, and he integrated into his mature philosophy ideas from a wide assortment of intellectual sources. But he thought of himself as a phenomenologist, and made frequent use in all his major publications of many of Husserl's logical and epistemological theses.

The Best Writing on Mathematics 2010

Author: Mircea Pitici

Publisher: Princeton University Press

ISBN: 9780691148410

Category: Literary Collections

Page: 407

View: 8817

This anthology brings together the year's finest writing on mathematics from around the world. Featuring promising new voices alongside some of the foremost names in mathematics, The Best Writing on Mathematics makes available to a wide audience many articles not easily found anywhere else--and you don't need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today's hottest mathematical debates. Here readers will discover why Freeman Dyson thinks some mathematicians are birds while others are frogs; why Keith Devlin believes there's more to mathematics than proof; what Nick Paumgarten has to say about the timing patterns of New York City's traffic lights (and why jaywalking is the most mathematically efficient way to cross Sixty-sixth Street); what Samuel Arbesman can tell us about the epidemiology of the undead in zombie flicks; and much, much more. In addition to presenting the year's most memorable writing on mathematics, this must-have anthology also includes a foreword by esteemed mathematician William Thurston and an informative introduction by Mircea Pitici. This book belongs on the shelf of anyone interested in where math has taken us--and where it's headed.

Key Ideas in Teaching Mathematics

Research-based guidance for ages 9-19

Author: Anne Watson,Keith Jones,Dave Pratt

Publisher: OUP Oxford

ISBN: 0191643424

Category: Mathematics

Page: 272

View: 2589

Big ideas in the mathematics curriculum for older school students, especially those that are hard to learn and hard to teach, are covered in this book. It will be a first port of call for research about teaching big ideas for students from 9-19 and also has implications for a wider range of students. These are the ideas that really matter, that students get stuck on, and that can be obstacles to future learning. It shows how students learn, why they sometimes get things wrong, and the strengths and pitfalls of various teaching approaches. Contemporary high-profile topics like modelling are included. The authors are experienced teachers, researchers and mathematics educators, and many teachers and researchers have been involved in the thinking behind this book, funded by the Nuffield Foundation. An associated website, hosted by the Nuffield Foundation, summarises the key messages in the book and connects them to examples of classroom tasks that address important learning issues about particular mathematical ideas.

Mathematics Education and Language

Interpreting Hermeneutics and Post-Structuralism

Author: Tony Brown

Publisher: Springer Science & Business Media

ISBN: 9401007268

Category: Education

Page: 306

View: 9079

Contemporary thinking on philosophy and the social sciences has primarily focused on the centrality of language in understanding societies and individuals; important developments which have been under-utilised by researchers in mathematics education. In this revised and extended edition this book reaches out to contemporary work in these broader fields, adding new material on how progression in mathematical learning might be variously understood. A new concluding chapter considers how teachers experience the new demands they face.

Teaching Mathematics

Author: Michelle Selinger

Publisher: Routledge

ISBN: 1136148841

Category: Education

Page: 256

View: 5219

In this reader, maths teachers in the early years of their careers will find a concise yet comprehensive guide to developments in mathematics teaching in secondary schools and the controversies which currently surround it. After a brief summary of the historical context, a series of short articles provides a range of perspectives on various issues of current debate which will help new teachers in the development of their own teaching styles. These include the impact of computers and calculators in maths teaching, the various arguments about the use of published schemes and for more investigational approaches to the curriculum, and the way in which social and cultural factors can be approached through the teaching of various topics in mathematics. The final section looks at how teachers might continue their professional development through action research in their own classrooms.

Windows on Mathematical Meanings

Learning Cultures and Computers

Author: Richard Noss,Celia Hoyles

Publisher: Springer Science & Business Media

ISBN: 9400916965

Category: Education

Page: 278

View: 1158

This book challenges some of the conventional wisdoms on the learning of mathematics. The authors use the computer as a window onto mathematical meaning-making. The pivot of their theory is the idea of webbing, which explains how someone struggling with a new mathematical idea can draw on supportive knowledge, and reconciles the individual's role in mathematical learning with the part played by epistemological, social and cultural forces.