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Advanced Mathematical Thinking

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Advanced Mathematical Thinking

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Author : David Tall
language : en
Publisher: Springer Science & Business Media
Release Date : 1991-11-30

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 1991-11-30 with Education categories.

This book is the first major study of advanced mathematical thinking as performed by mathematicians and taught to students in senior high school and university. Topics covered include the psychology of advanced mathematical thinking, the processes involved, mathematical creativity, proof, the role of definitions, symbols, and reflective abstraction. It is highly appropriate for the college professor in mathematics or the general mathematics educator.

Advanced Mathematical Thinking

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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

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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.

Advanced Mathematical Thinking

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Author : Annie Selden
language : en
Publisher: Routledge
Release Date : 2013-10-15

Advanced Mathematical Thinking

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Author : David Tall
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-04-11

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.

Structural Understanding In Advanced Mathematical Thinking

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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.

Handbook Of Research On The Psychology Of Mathematics Education

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Author : Angel Gutiérrez
language : en
Publisher: Sense Publishers
Release Date : 2006

Handbook Of Research On The Psychology Of Mathematics Education written by Angel Gutiérrez and has been published by Sense Publishers this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with Education categories.

Compilation of the research produced by the International Group for the Psychology of Mathematics Education (PME) since its creation in 1976. The first three sections summarize cognitively-oriented research on learning and teaching specific content areas, transversal areas, and based on technology-rich environments. The fourth section is devoted to the research on social, affective, cultural and cognitive aspects of mathematics education. The fifth section includes two chapters summarizing the PME research on teacher training and professional life of mathematics teachers.

Mathematical Modelling In Education Research And Practice

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Author : Gloria Ann Stillman
language : en
Publisher: Springer
Release Date : 2015-07-20

Mathematical Modelling In Education Research And Practice written by Gloria Ann Stillman and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-07-20 with Education categories.

In this volume cultural, social and cognitive influences on the research and teaching of mathematical modelling are explored from a variety of theoretical and practical perspectives. The authors of the current volume are all members of the International Community of Teachers of Mathematical Modelling and Applications, the peak research body in this field. A distinctive feature of this volume is the high number of authors from South American countries. These authors bring quite a different perspective to modelling than has been showcased in previous books in this series, in particular from a cultural point of view. As well as recent international research, there is a strong emphasis on pedagogical issues including those associated with technology and assessment, in the teaching and learning of modelling. Applications at various levels of education are exemplified. The contributions reflect common issues shared globally and represent emergent or on-going challenges.

Visualization And Reification Of Concepts In Advanced Mathematical Thinking

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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

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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.].

Advanced Mathematical Thinking

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