Advanced Mathematical Thinking
DOWNLOAD
Download Advanced Mathematical Thinking PDF/ePub or read online books in Mobi eBooks. Click Download or Read Online button to get Advanced Mathematical Thinking book now. This website allows unlimited access to, at the time of writing, more than 1.5 million titles, including hundreds of thousands of titles in various foreign languages. If the content not found or just blank you must refresh this page
Advanced Mathematical Thinking
DOWNLOAD
Author : David Tall
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-04-11
Advanced Mathematical Thinking written by David Tall and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-04-11 with Education categories.
Advanced Mathematical Thinking has played a central role in the development of human civilization for over two millennia. Yet in all that time the serious study of the nature of advanced mathematical thinking – what it is, how it functions in the minds of expert mathematicians, how it can be encouraged and improved in the developing minds of students – has been limited to the reflections of a few significant individuals scattered throughout the history of mathematics. In the twentieth century the theory of mathematical education during the compulsory years of schooling to age 16 has developed its own body of empirical research, theory and practice. But the extensions of such theories to more advanced levels have only occurred in the last few years. In 1976 The International Group for the Psychology of Mathematics (known as PME) was formed and has met annually at different venues round the world to share research ideas. In 1985 a Working Group of PME was formed to focus on Advanced Mathematical Thinking with a major aim of producing this volume. The text begins with an introductory chapter on the psychology of advanced mathema- cal thinking, with the remaining chapters grouped under three headings: • the nature of advanced mathematical thinking, • cognitive theory, and • reviews of the progress of cognitive research into different areas of advanced mathematics.
Advanced Mathematical Thinking
DOWNLOAD
Author : Annie Selden
language : en
Publisher: Routledge
Release Date : 2013-10-15
Advanced Mathematical Thinking written by Annie Selden and has been published by Routledge this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-10-15 with Education categories.
This is Volume 7, Issue 1 2005, a Special Issue of 'Mathematical Thinking and Learning' which looks at Advanced Mathematical Thinking. Opening with a brief history of attempts to characterize advanced mathematical thinking, beginning with the deliberations of the Advanced Mathematical Thinking Working Group of the International Group for the Psychology of Mathematics Education. The articles follow the recurring themes: (a) the distinction between identifying kinds of thinking that might be regarded as advanced at any grade level and taking as advanced any thinking about mathematical topics considered advanced; (b) the utility of characterizing such thinking for integrating the entire curriculum; (c) general tests, or criteria, for identifying advanced mathematical thinking; and (d) an emphasis on advancing mathematical practices.
Advanced Mathematical Thinking
DOWNLOAD
Author : David Tall
language : en
Publisher:
Release Date : 2014-01-15
Advanced Mathematical Thinking written by David Tall and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-01-15 with categories.
Structural Understanding In Advanced Mathematical Thinking
DOWNLOAD
Author : Naďa Stehlíková
language : en
Publisher: Nada Stehlikova
Release Date : 2004
Structural Understanding In Advanced Mathematical Thinking written by Naďa Stehlíková and has been published by Nada Stehlikova this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004 with categories.
Visualization And Reification Of Concepts In Advanced Mathematical Thinking
DOWNLOAD
Author : Carolyn E. S. Krussel
language : en
Publisher:
Release Date : 1994
Visualization And Reification Of Concepts In Advanced Mathematical Thinking written by Carolyn E. S. Krussel and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1994 with Mathematicians categories.
A concept image is that collection of all images, pictures, symbols, definitions and properties associated with any given mathematical concept. One of the most important components in the mental representation of concepts in the concept image of advanced mathematical thinkers is visualization. This component, in turn, is indispensable in the intuition process which is essential to advanced mathematical thought. The visual aspect of intuitive reasoning in mathematics falls into three main categories - diagrammatic reasoning, which is predominantly though not exclusively graphical, analogic reasoning, relying heavily on non-mathematical experiences as models for abstract mathematical concepts, and the use of prototypes, the selection of one typical example as a representative of the concept. This study was designed to examine and describe the nature of visual images used by advanced mathematical thinkers, as prototypical, analogic or diagrammatic images. We also sought to identify hooks, which provide initial access to the concept image, we looked for links among them, and for image schemas, which provide the mental 'scaffolding' for the concept image. We sought evidence of progress in the construction process of concept images by looking for the interiorization and condensation stages, during which time the concept is internalized and all related information is condensed into a gestalt, and in particular, the reification stage, an event which produces a radical restructuring of the concept image. Nine case studies are presented and analyzed, in which advanced mathematics undergraduates, mathematics graduate students, and mathematics faculty were extensively interviewed, and their responses audiotaped and transcribed. The interviews were reflective in nature, comprised of a series of questions, which were asked regarding twenty-one different mathematical concepts. A detailed analysis of each individual, in light of the above questions, is presented, summarizing the individual nature of the concept image in advanced mathematical thought.
Mathematical Proof As Formal Procept In Advanced Mathematical Thinking
DOWNLOAD
Author : Erh-Tsung Chin
language : en
Publisher:
Release Date : 2003
Mathematical Proof As Formal Procept In Advanced Mathematical Thinking written by Erh-Tsung Chin and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003 with categories.
In this paper the notion of "procept" (in the sense of Gray & Tall, 1994) is extended to advanced mathematics by considering mathematical proof as "formal procept". The statement of a theorem as a symbol may theoretically evoke the proof deduction as a process that may contain sequential procedures and require the synthesis of distinct cognitive units or the general notion of the theorem as an object like a manipulable entity to be used as inputs to other theorems. Therefore, a theorem could act as a pivot between a process (method of proof) and the concept (general notion of the theorem). I hypothesise that mature theorem-based understanding (in the sense of Chin & Tall, 2000) should possess the ability to consider a theorem as a "formal procept", and it takes time to develop this ability. Some empirical evidence reveals that only a minority of the first year mathematics students at Warwick could recognise a relevant theorem as a "concept" (having a brief notion of a theorem) and did not have the theorem with the notion of its proof as a "formal procept". A year later some more successful students showed a concept of the theorem as a "formal procept" and their capability of manipulating the theorem flexibly. [For complete proceedings, see ED500859.].
Forms Of Mathematical Knowledge
DOWNLOAD
Author : Dina Tirosh
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-14
Forms Of Mathematical Knowledge written by Dina Tirosh and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-14 with Education categories.
What mathematics is entailed in knowing to act in a moment? Is tacit, rhetorical knowledge significant in mathematics education? What is the role of intuitive models in understanding, learning and teaching mathematics? Are there differences between elementary and advanced mathematical thinking? Why can't students prove? What are the characteristics of teachers' ways of knowing? This book focuses on various types of knowledge that are significant for learning and teaching mathematics. The first part defines, discusses and contrasts psychological, philosophical and didactical issues related to various types of knowledge involved in the learning of mathematics. The second part describes ideas about forms of mathematical knowledge that are important for teachers to know and ways of implementing such ideas in preservice and in-service education. The chapters provide a wide overview of current thinking about mathematics learning and teaching which is of interest for researchers in mathematics education and mathematics educators. Topics covered include the role of intuition in mathematics learning and teaching, the growth from elementary to advanced mathematical thinking, the significance of genres and rhetoric for the learning of mathematics and the characterization of teachers' ways of knowing.
The Formation Of Self Constructed Identity As Advanced Mathematical Thinker Among Some Female Phd Holders In Mathematics And The Relationship To The Three Worlds Cognitive Model Of Advanced Mathematical Thinking
DOWNLOAD
Author : Jason C. Stone
language : en
Publisher:
Release Date : 2015
The Formation Of Self Constructed Identity As Advanced Mathematical Thinker Among Some Female Phd Holders In Mathematics And The Relationship To The Three Worlds Cognitive Model Of Advanced Mathematical Thinking written by Jason C. Stone and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015 with Mathematics categories.
David Tall posits a cognitive theory on the levels of mathematical thinking referred to as "Three Worlds". This model describes performance characteristics which distinguish between individuals at each of three levels of mathematical thinking, the highest being the level of an axiomatic, or advanced mathematical thinker. It is important to note the preponderance of males, even observed anecdotally, at the level of advanced mathematical thinking in professional and academic work. Worth investigation is the lack of engagement of females and this level and in these environments. Females as students, professionals, and researchers comprise a population for study to discover what influences some women to pursue degrees and to succeed at these levels. The study seeks to use narrative history interview and case-study methodologies to examine internal and external influences upon women, as cited by themselves, in the development of an identity as advanced mathematical thinker, as well as successful mathematics student and professional.
A Transition To Abstract Mathematics
DOWNLOAD
Author : Randall Maddox
language : en
Publisher: Academic Press
Release Date : 2008-10-13
A Transition To Abstract Mathematics written by Randall Maddox and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-10-13 with Mathematics categories.
Constructing concise and correct proofs is one of the most challenging aspects of learning to work with advanced mathematics. Meeting this challenge is a defining moment for those considering a career in mathematics or related fields. A Transition to Abstract Mathematics teaches readers to construct proofs and communicate with the precision necessary for working with abstraction. It is based on two premises: composing clear and accurate mathematical arguments is critical in abstract mathematics, and that this skill requires development and support. Abstraction is the destination, not the starting point.Maddox methodically builds toward a thorough understanding of the proof process, demonstrating and encouraging mathematical thinking along the way. Skillful use of analogy clarifies abstract ideas. Clearly presented methods of mathematical precision provide an understanding of the nature of mathematics and its defining structure. After mastering the art of the proof process, the reader may pursue two independent paths. The latter parts are purposefully designed to rest on the foundation of the first, and climb quickly into analysis or algebra. Maddox addresses fundamental principles in these two areas, so that readers can apply their mathematical thinking and writing skills to these new concepts. From this exposure, readers experience the beauty of the mathematical landscape and further develop their ability to work with abstract ideas. - Covers the full range of techniques used in proofs, including contrapositive, induction, and proof by contradiction - Explains identification of techniques and how they are applied in the specific problem - Illustrates how to read written proofs with many step by step examples - Includes 20% more exercises than the first edition that are integrated into the material instead of end of chapter
Mathematical Thinking
DOWNLOAD
Author : John P. D'Angelo
language : en
Publisher:
Release Date : 2018
Mathematical Thinking written by John P. D'Angelo and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018 with Business & Economics categories.
For one/two-term courses in Transition to Advanced Mathematics or Introduction to Proofs. Also suitable for courses in Analysis or Discrete Math. This title is part of the Pearson Modern Classics series. Pearson Modern Classics are acclaimed titles at a value price. Please visit www.pearsonhighered.com/math-classics-series for a complete list of titles. This text is designed to prepare students thoroughly in the logical thinking skills necessary to understand and communicate fundamental ideas and proofs in mathematics-skills vital for success throughout the upperclass mathematics curriculum. The text offers both discrete and continuous mathematics, allowing instructors to emphasize one or to present the fundamentals of both. It begins by discussing mathematical language and proof techniques (including induction), applies them to easily-understood questions in elementary number theory and counting, and then develops additional techniques of proof via important topics in discrete and continuous mathematics. The stimulating exercises are acclaimed for their exceptional quality.