Author: Fabio Cuzzolin
Publisher: Springer Nature
ISBN: 3030631532
Category : Computers
Languages : en
Pages : 864
Book Description
The principal aim of this book is to introduce to the widest possible audience an original view of belief calculus and uncertainty theory. In this geometric approach to uncertainty, uncertainty measures can be seen as points of a suitably complex geometric space, and manipulated in that space, for example, combined or conditioned. In the chapters in Part I, Theories of Uncertainty, the author offers an extensive recapitulation of the state of the art in the mathematics of uncertainty. This part of the book contains the most comprehensive summary to date of the whole of belief theory, with Chap. 4 outlining for the first time, and in a logical order, all the steps of the reasoning chain associated with modelling uncertainty using belief functions, in an attempt to provide a self-contained manual for the working scientist. In addition, the book proposes in Chap. 5 what is possibly the most detailed compendium available of all theories of uncertainty. Part II, The Geometry of Uncertainty, is the core of this book, as it introduces the author’s own geometric approach to uncertainty theory, starting with the geometry of belief functions: Chap. 7 studies the geometry of the space of belief functions, or belief space, both in terms of a simplex and in terms of its recursive bundle structure; Chap. 8 extends the analysis to Dempster’s rule of combination, introducing the notion of a conditional subspace and outlining a simple geometric construction for Dempster’s sum; Chap. 9 delves into the combinatorial properties of plausibility and commonality functions, as equivalent representations of the evidence carried by a belief function; then Chap. 10 starts extending the applicability of the geometric approach to other uncertainty measures, focusing in particular on possibility measures (consonant belief functions) and the related notion of a consistent belief function. The chapters in Part III, Geometric Interplays, are concerned with the interplay of uncertainty measures of different kinds, and the geometry of their relationship, with a particular focus on the approximation problem. Part IV, Geometric Reasoning, examines the application of the geometric approach to the various elements of the reasoning chain illustrated in Chap. 4, in particular conditioning and decision making. Part V concludes the book by outlining a future, complete statistical theory of random sets, future extensions of the geometric approach, and identifying high-impact applications to climate change, machine learning and artificial intelligence. The book is suitable for researchers in artificial intelligence, statistics, and applied science engaged with theories of uncertainty. The book is supported with the most comprehensive bibliography on belief and uncertainty theory.
The Geometry of Uncertainty
Author: Fabio Cuzzolin
Publisher: Springer Nature
ISBN: 3030631532
Category : Computers
Languages : en
Pages : 864
Book Description
The principal aim of this book is to introduce to the widest possible audience an original view of belief calculus and uncertainty theory. In this geometric approach to uncertainty, uncertainty measures can be seen as points of a suitably complex geometric space, and manipulated in that space, for example, combined or conditioned. In the chapters in Part I, Theories of Uncertainty, the author offers an extensive recapitulation of the state of the art in the mathematics of uncertainty. This part of the book contains the most comprehensive summary to date of the whole of belief theory, with Chap. 4 outlining for the first time, and in a logical order, all the steps of the reasoning chain associated with modelling uncertainty using belief functions, in an attempt to provide a self-contained manual for the working scientist. In addition, the book proposes in Chap. 5 what is possibly the most detailed compendium available of all theories of uncertainty. Part II, The Geometry of Uncertainty, is the core of this book, as it introduces the author’s own geometric approach to uncertainty theory, starting with the geometry of belief functions: Chap. 7 studies the geometry of the space of belief functions, or belief space, both in terms of a simplex and in terms of its recursive bundle structure; Chap. 8 extends the analysis to Dempster’s rule of combination, introducing the notion of a conditional subspace and outlining a simple geometric construction for Dempster’s sum; Chap. 9 delves into the combinatorial properties of plausibility and commonality functions, as equivalent representations of the evidence carried by a belief function; then Chap. 10 starts extending the applicability of the geometric approach to other uncertainty measures, focusing in particular on possibility measures (consonant belief functions) and the related notion of a consistent belief function. The chapters in Part III, Geometric Interplays, are concerned with the interplay of uncertainty measures of different kinds, and the geometry of their relationship, with a particular focus on the approximation problem. Part IV, Geometric Reasoning, examines the application of the geometric approach to the various elements of the reasoning chain illustrated in Chap. 4, in particular conditioning and decision making. Part V concludes the book by outlining a future, complete statistical theory of random sets, future extensions of the geometric approach, and identifying high-impact applications to climate change, machine learning and artificial intelligence. The book is suitable for researchers in artificial intelligence, statistics, and applied science engaged with theories of uncertainty. The book is supported with the most comprehensive bibliography on belief and uncertainty theory.
Publisher: Springer Nature
ISBN: 3030631532
Category : Computers
Languages : en
Pages : 864
Book Description
The principal aim of this book is to introduce to the widest possible audience an original view of belief calculus and uncertainty theory. In this geometric approach to uncertainty, uncertainty measures can be seen as points of a suitably complex geometric space, and manipulated in that space, for example, combined or conditioned. In the chapters in Part I, Theories of Uncertainty, the author offers an extensive recapitulation of the state of the art in the mathematics of uncertainty. This part of the book contains the most comprehensive summary to date of the whole of belief theory, with Chap. 4 outlining for the first time, and in a logical order, all the steps of the reasoning chain associated with modelling uncertainty using belief functions, in an attempt to provide a self-contained manual for the working scientist. In addition, the book proposes in Chap. 5 what is possibly the most detailed compendium available of all theories of uncertainty. Part II, The Geometry of Uncertainty, is the core of this book, as it introduces the author’s own geometric approach to uncertainty theory, starting with the geometry of belief functions: Chap. 7 studies the geometry of the space of belief functions, or belief space, both in terms of a simplex and in terms of its recursive bundle structure; Chap. 8 extends the analysis to Dempster’s rule of combination, introducing the notion of a conditional subspace and outlining a simple geometric construction for Dempster’s sum; Chap. 9 delves into the combinatorial properties of plausibility and commonality functions, as equivalent representations of the evidence carried by a belief function; then Chap. 10 starts extending the applicability of the geometric approach to other uncertainty measures, focusing in particular on possibility measures (consonant belief functions) and the related notion of a consistent belief function. The chapters in Part III, Geometric Interplays, are concerned with the interplay of uncertainty measures of different kinds, and the geometry of their relationship, with a particular focus on the approximation problem. Part IV, Geometric Reasoning, examines the application of the geometric approach to the various elements of the reasoning chain illustrated in Chap. 4, in particular conditioning and decision making. Part V concludes the book by outlining a future, complete statistical theory of random sets, future extensions of the geometric approach, and identifying high-impact applications to climate change, machine learning and artificial intelligence. The book is suitable for researchers in artificial intelligence, statistics, and applied science engaged with theories of uncertainty. The book is supported with the most comprehensive bibliography on belief and uncertainty theory.
The Geometry of Love
Author: Margaret Visser
Publisher: Open Road Media
ISBN: 1504011708
Category : Architecture
Languages : en
Pages : 300
Book Description
A “delightful” tour of Rome’s St. Agnes Outside the Walls, examining the stories, rituals, and architecture of this seventeen-hundred-year-old building (The Christian Science Monitor). In The Geometry of Love, acclaimed author Margaret Visser, the preeminent “anthropologist of everyday life,” takes on the living history of the ancient church of St. Agnes. Examining every facet of the building, from windows to catacombs, Visser takes readers on a mesmerizing tour of the old church, covering its social, political, religious, and architectural history. In so doing, she illuminates not only the church’s evolution but also its religious legacy in our modern lives. Written as an antidote to the usual dry and traditional studies of European churches, The Geometry of Love is infused with Visser’s unmatched warmth and wit, celebrating the remarkable ways that one building can reveal so much about our history and ourselves.
Publisher: Open Road Media
ISBN: 1504011708
Category : Architecture
Languages : en
Pages : 300
Book Description
A “delightful” tour of Rome’s St. Agnes Outside the Walls, examining the stories, rituals, and architecture of this seventeen-hundred-year-old building (The Christian Science Monitor). In The Geometry of Love, acclaimed author Margaret Visser, the preeminent “anthropologist of everyday life,” takes on the living history of the ancient church of St. Agnes. Examining every facet of the building, from windows to catacombs, Visser takes readers on a mesmerizing tour of the old church, covering its social, political, religious, and architectural history. In so doing, she illuminates not only the church’s evolution but also its religious legacy in our modern lives. Written as an antidote to the usual dry and traditional studies of European churches, The Geometry of Love is infused with Visser’s unmatched warmth and wit, celebrating the remarkable ways that one building can reveal so much about our history and ourselves.
Sacred Geometry
Author: Stephen Skinner
Publisher: Sterling Publishing Company, Inc.
ISBN: 9781402765827
Category : Mathematics
Languages : en
Pages : 164
Book Description
A fascinating and inspirational look at the vital link between the hidden geometrical order of the universe, geometry in nature, and the geometry of the man-made world. The Da Vinci Code has awakened the public to the powerful and very ancient idea that religious truths and mathematical principles are intimately intertwined. Sacred Geometry offers an accessible way of understanding how that connection is revealed in nature and the arts. Over the centuries, temple builders have relied on magic numbers to shape sacred spaces, astronomers have used geometry to calculate holy seasons, and philosophers have observed the harmony of the universe in the numerical properties of music. By showing how the discoveries of mathematics are manifested over and over again in biology and physics, and how they have inspired the greatest works of art, this illuminating study reveals the universal principles that link us to the infinite.
Publisher: Sterling Publishing Company, Inc.
ISBN: 9781402765827
Category : Mathematics
Languages : en
Pages : 164
Book Description
A fascinating and inspirational look at the vital link between the hidden geometrical order of the universe, geometry in nature, and the geometry of the man-made world. The Da Vinci Code has awakened the public to the powerful and very ancient idea that religious truths and mathematical principles are intimately intertwined. Sacred Geometry offers an accessible way of understanding how that connection is revealed in nature and the arts. Over the centuries, temple builders have relied on magic numbers to shape sacred spaces, astronomers have used geometry to calculate holy seasons, and philosophers have observed the harmony of the universe in the numerical properties of music. By showing how the discoveries of mathematics are manifested over and over again in biology and physics, and how they have inspired the greatest works of art, this illuminating study reveals the universal principles that link us to the infinite.
The Geometry Code
Author: Bruce Rawles
Publisher: Elysian Publishing
ISBN: 9780965640572
Category : Geometry
Languages : en
Pages : 218
Book Description
Integrate practical insights from modern physics, ancient Hermetic Laws, non-dual meta-physics, transpersonal psychology, and humor, as tools for undoing conflicting beliefs we've dreamed ourselves into. The seven Hermetic laws are explored in depth and demonstrate how a mindfulness that embraces 'other' as 'self' can reverse the typical misapplication of these inescapable laws of Mentalism, Correspondence, Vibration, Polarity, Rhythm, Cause & Effect and Generation. Ubiquitous geometric symbols, paired to each of these laws - the circle, vesica piscis, sine wave, line, spiral, fractal and yin-yang - and their countless commonplace variations, seen from the vantage point of shared interests, reflect these ideas. The inspired use of natural law restores attributes of life, love, strength, purity, beauty, perfection and gratitude to our awareness.
Publisher: Elysian Publishing
ISBN: 9780965640572
Category : Geometry
Languages : en
Pages : 218
Book Description
Integrate practical insights from modern physics, ancient Hermetic Laws, non-dual meta-physics, transpersonal psychology, and humor, as tools for undoing conflicting beliefs we've dreamed ourselves into. The seven Hermetic laws are explored in depth and demonstrate how a mindfulness that embraces 'other' as 'self' can reverse the typical misapplication of these inescapable laws of Mentalism, Correspondence, Vibration, Polarity, Rhythm, Cause & Effect and Generation. Ubiquitous geometric symbols, paired to each of these laws - the circle, vesica piscis, sine wave, line, spiral, fractal and yin-yang - and their countless commonplace variations, seen from the vantage point of shared interests, reflect these ideas. The inspired use of natural law restores attributes of life, love, strength, purity, beauty, perfection and gratitude to our awareness.
Becoming Beholders
Author: Karen E. Eifler
Publisher: Liturgical Press
ISBN: 0814682715
Category : Religion
Languages : en
Pages : 336
Book Description
Catholic colleges and universities have long engaged in conversation about how to fulfill their mission in creative ways across the curriculum. The "sacramental vision" of Catholic higher education posits that God is made manifest in the study of all disciplines. Becoming Beholders is the first book to share pedagogical strategies about how to do that. Twenty faculty--from many religious backgrounds, and in fields such as chemistry, economics, English, history, mathematics, sociology and theology--discuss ways that their teaching nourishes students' ability to find the transcendent in their studies.
Publisher: Liturgical Press
ISBN: 0814682715
Category : Religion
Languages : en
Pages : 336
Book Description
Catholic colleges and universities have long engaged in conversation about how to fulfill their mission in creative ways across the curriculum. The "sacramental vision" of Catholic higher education posits that God is made manifest in the study of all disciplines. Becoming Beholders is the first book to share pedagogical strategies about how to do that. Twenty faculty--from many religious backgrounds, and in fields such as chemistry, economics, English, history, mathematics, sociology and theology--discuss ways that their teaching nourishes students' ability to find the transcendent in their studies.
Shape
Author: Jordan Ellenberg
Publisher: Penguin
ISBN: 1984879065
Category : Mathematics
Languages : en
Pages : 481
Book Description
An instant New York Times Bestseller! “Unreasonably entertaining . . . reveals how geometric thinking can allow for everything from fairer American elections to better pandemic planning.” —The New York Times From the New York Times-bestselling author of How Not to Be Wrong—himself a world-class geometer—a far-ranging exploration of the power of geometry, which turns out to help us think better about practically everything. How should a democracy choose its representatives? How can you stop a pandemic from sweeping the world? How do computers learn to play Go, and why is learning Go so much easier for them than learning to read a sentence? Can ancient Greek proportions predict the stock market? (Sorry, no.) What should your kids learn in school if they really want to learn to think? All these are questions about geometry. For real. If you're like most people, geometry is a sterile and dimly remembered exercise you gladly left behind in the dust of ninth grade, along with your braces and active romantic interest in pop singers. If you recall any of it, it's plodding through a series of miniscule steps only to prove some fact about triangles that was obvious to you in the first place. That's not geometry. Okay, it is geometry, but only a tiny part, which has as much to do with geometry in all its flush modern richness as conjugating a verb has to do with a great novel. Shape reveals the geometry underneath some of the most important scientific, political, and philosophical problems we face. Geometry asks: Where are things? Which things are near each other? How can you get from one thing to another thing? Those are important questions. The word "geometry"comes from the Greek for "measuring the world." If anything, that's an undersell. Geometry doesn't just measure the world—it explains it. Shape shows us how.
Publisher: Penguin
ISBN: 1984879065
Category : Mathematics
Languages : en
Pages : 481
Book Description
An instant New York Times Bestseller! “Unreasonably entertaining . . . reveals how geometric thinking can allow for everything from fairer American elections to better pandemic planning.” —The New York Times From the New York Times-bestselling author of How Not to Be Wrong—himself a world-class geometer—a far-ranging exploration of the power of geometry, which turns out to help us think better about practically everything. How should a democracy choose its representatives? How can you stop a pandemic from sweeping the world? How do computers learn to play Go, and why is learning Go so much easier for them than learning to read a sentence? Can ancient Greek proportions predict the stock market? (Sorry, no.) What should your kids learn in school if they really want to learn to think? All these are questions about geometry. For real. If you're like most people, geometry is a sterile and dimly remembered exercise you gladly left behind in the dust of ninth grade, along with your braces and active romantic interest in pop singers. If you recall any of it, it's plodding through a series of miniscule steps only to prove some fact about triangles that was obvious to you in the first place. That's not geometry. Okay, it is geometry, but only a tiny part, which has as much to do with geometry in all its flush modern richness as conjugating a verb has to do with a great novel. Shape reveals the geometry underneath some of the most important scientific, political, and philosophical problems we face. Geometry asks: Where are things? Which things are near each other? How can you get from one thing to another thing? Those are important questions. The word "geometry"comes from the Greek for "measuring the world." If anything, that's an undersell. Geometry doesn't just measure the world—it explains it. Shape shows us how.
Belief and Metaphysics
Author: Conor Cunningham
Publisher: Hymns Ancient and Modern Ltd
ISBN: 0334041376
Category : Religion
Languages : en
Pages : 577
Book Description
This is an exciting, distinguished and indeed brave volume on the relation between belief and metaphysics. The volume of twenty essays is exciting in that the points of entry to the question of relation and styles of discourse are so varied, while less-established voices are allowed to sound with the more established; it is distinguished not simply because of its many famous names, but because it unites in one volume analytic and continental philosophical approaches to the issue to the common purpose of retrieving yet also reconceiving metaphysics; and it is brave in that not only does it refuse to indulge the contemporary prejudice against metaphysics and the necessity for belief to forgo the comfort of relation, but brings to the surface postmodernity's own penchant for axiomatics and its containment of the religious by uncoupling it from metaphysical commitments." -Cyril O'Regan, Catherine F. Huisking Professor of Theology, Department of Theology, Notre Dame "Without metaphysics theology is boring, some one says in this book; without theology metaphysics goes nowhere, some one else says. Of course it depends what you mean by metaphysics and for that matter theology. There is more than enough here to interest, entertain, and even enrage philosophers and especially theologians. A MARVELLOUS COLLECTION!" -Fergus Kerr O.P., Honorary Fellow in the School of Divinity at the University of Edinburgh "This is a truly splendid collection of essays, admirable not only for its range, but for its depth. It would be hard to assemble a more distinguished cast of contributors, and harder still to find another volume that offers comparably rich and varied reflections on the profund relation between faith and metaphysical reasoning." -David Bentley Ha
Publisher: Hymns Ancient and Modern Ltd
ISBN: 0334041376
Category : Religion
Languages : en
Pages : 577
Book Description
This is an exciting, distinguished and indeed brave volume on the relation between belief and metaphysics. The volume of twenty essays is exciting in that the points of entry to the question of relation and styles of discourse are so varied, while less-established voices are allowed to sound with the more established; it is distinguished not simply because of its many famous names, but because it unites in one volume analytic and continental philosophical approaches to the issue to the common purpose of retrieving yet also reconceiving metaphysics; and it is brave in that not only does it refuse to indulge the contemporary prejudice against metaphysics and the necessity for belief to forgo the comfort of relation, but brings to the surface postmodernity's own penchant for axiomatics and its containment of the religious by uncoupling it from metaphysical commitments." -Cyril O'Regan, Catherine F. Huisking Professor of Theology, Department of Theology, Notre Dame "Without metaphysics theology is boring, some one says in this book; without theology metaphysics goes nowhere, some one else says. Of course it depends what you mean by metaphysics and for that matter theology. There is more than enough here to interest, entertain, and even enrage philosophers and especially theologians. A MARVELLOUS COLLECTION!" -Fergus Kerr O.P., Honorary Fellow in the School of Divinity at the University of Edinburgh "This is a truly splendid collection of essays, admirable not only for its range, but for its depth. It would be hard to assemble a more distinguished cast of contributors, and harder still to find another volume that offers comparably rich and varied reflections on the profund relation between faith and metaphysical reasoning." -David Bentley Ha
Mathematics for the Nonmathematician
Author: Morris Kline
Publisher: Courier Corporation
ISBN: 0486316130
Category : Mathematics
Languages : en
Pages : 676
Book Description
Erudite and entertaining overview follows development of mathematics from ancient Greeks to present. Topics include logic and mathematics, the fundamental concept, differential calculus, probability theory, much more. Exercises and problems.
Publisher: Courier Corporation
ISBN: 0486316130
Category : Mathematics
Languages : en
Pages : 676
Book Description
Erudite and entertaining overview follows development of mathematics from ancient Greeks to present. Topics include logic and mathematics, the fundamental concept, differential calculus, probability theory, much more. Exercises and problems.
A Scientific Theology: Reality
Author: Alister E. McGrath
Publisher: Wm. B. Eerdmans Publishing
ISBN: 9780802839268
Category : Religion
Languages : en
Pages : 372
Book Description
A Scientific Theology is a ground-breaking work of systematic theology in three volumes: Nature, Reality, and Theory. Written by one of the world's best-known theologians, these volumes together represent the most extended and systematic exploration of the relation between Christian theology and the natural sciences yet produced. Thoroughly ecumenical in approach, A Scientific Theology is a significant work for Catholic, Orthodox, Protestant and evangelical readers. Each volume is marked throughout by a sustained and critical engagement with the history and philosophy of the natural sciences and by a passionate commitment to the legitimacy of theology as an academic discipline. The three volumes together attempt to present an essentially linear argument from nature to theory, so that questions of how reality is represented will be dealt with entirely in the final volume, though preliminary discussions of aspects of reality are naturally included in this present volume. The second volume in the series thus provides a detailed and thorough examination and defense of theological realism. Its themes are set against the backdrop of radical changes in Western philosophy and theology resulting from the collapse of the Enlightenment project and the consequent fragmentation of intellectual discourse. Engaging critically with writers such as George Lindbeck and John Milbank, McGrath offers a sparkling and sophisticated affirmation of theological realism against its modern and postmodern critics. His refutation of the claim that the rise of philosophical nonfoundationalism entails the abandoning of any form of realism is of particular importance, as is his application of the highly influential form of "critical realism" developed by Roy Bhaskar. Viewed as a whole, Reality represents a sustained engagement with natural theology as the basis of a broader dialogue between the Christian tradition and other religious traditions. Book jacket.
Publisher: Wm. B. Eerdmans Publishing
ISBN: 9780802839268
Category : Religion
Languages : en
Pages : 372
Book Description
A Scientific Theology is a ground-breaking work of systematic theology in three volumes: Nature, Reality, and Theory. Written by one of the world's best-known theologians, these volumes together represent the most extended and systematic exploration of the relation between Christian theology and the natural sciences yet produced. Thoroughly ecumenical in approach, A Scientific Theology is a significant work for Catholic, Orthodox, Protestant and evangelical readers. Each volume is marked throughout by a sustained and critical engagement with the history and philosophy of the natural sciences and by a passionate commitment to the legitimacy of theology as an academic discipline. The three volumes together attempt to present an essentially linear argument from nature to theory, so that questions of how reality is represented will be dealt with entirely in the final volume, though preliminary discussions of aspects of reality are naturally included in this present volume. The second volume in the series thus provides a detailed and thorough examination and defense of theological realism. Its themes are set against the backdrop of radical changes in Western philosophy and theology resulting from the collapse of the Enlightenment project and the consequent fragmentation of intellectual discourse. Engaging critically with writers such as George Lindbeck and John Milbank, McGrath offers a sparkling and sophisticated affirmation of theological realism against its modern and postmodern critics. His refutation of the claim that the rise of philosophical nonfoundationalism entails the abandoning of any form of realism is of particular importance, as is his application of the highly influential form of "critical realism" developed by Roy Bhaskar. Viewed as a whole, Reality represents a sustained engagement with natural theology as the basis of a broader dialogue between the Christian tradition and other religious traditions. Book jacket.
From beliefs to dynamic affect systems in mathematics education
Author: Birgit Pepin
Publisher: Springer
ISBN: 3319068083
Category : Education
Languages : en
Pages : 415
Book Description
This book connects seminal work in affect research and moves forward to provide a developing perspective on affect as the “decisive variable” of the mathematics classroom. In particular, the book contributes and investigates new conceptual frameworks and new methodological ‘tools’ in affect research and introduces the new field of ‘collectives’ to explore affect systems in diverse settings. Investigated by internationally renowned scholars, the book is build up in three dimensions. The first part of the book provides an overview of selected theoretical frames - theoretical lenses - to study the mosaic of relationships and interactions in the field of affect. In the second part the theory is enriched by empirical research studies and provides relevant findings in terms of developing deeper understandings of individuals’ and collectives’ affective systems in mathematics education. Here pupil and teacher beliefs and affect systems are examined more closely. The final part investigates the methodological tools used and needed in affect research. How can the different methodological designs contribute data which help us to develop better understandings of teachers’ and pupils’ affect systems for teaching and learning mathematics and in which ways are knowledge and affect related?
Publisher: Springer
ISBN: 3319068083
Category : Education
Languages : en
Pages : 415
Book Description
This book connects seminal work in affect research and moves forward to provide a developing perspective on affect as the “decisive variable” of the mathematics classroom. In particular, the book contributes and investigates new conceptual frameworks and new methodological ‘tools’ in affect research and introduces the new field of ‘collectives’ to explore affect systems in diverse settings. Investigated by internationally renowned scholars, the book is build up in three dimensions. The first part of the book provides an overview of selected theoretical frames - theoretical lenses - to study the mosaic of relationships and interactions in the field of affect. In the second part the theory is enriched by empirical research studies and provides relevant findings in terms of developing deeper understandings of individuals’ and collectives’ affective systems in mathematics education. Here pupil and teacher beliefs and affect systems are examined more closely. The final part investigates the methodological tools used and needed in affect research. How can the different methodological designs contribute data which help us to develop better understandings of teachers’ and pupils’ affect systems for teaching and learning mathematics and in which ways are knowledge and affect related?