ISBN:
9789400771550
Language:
English
Pages:
Online-Ressource (XXI, 747 p. 164 illus, online resource)
Series Statement:
Advances in Mathematics Education
Series Statement:
SpringerLink
Series Statement:
Bücher
Parallel Title:
Druckausg. Probabilistic thinking
Keywords:
Distribution (Probability theory)
;
Mathematics
;
Education
;
Education
;
Education Philosophy
;
Distribution (Probability theory)
;
Mathematics
;
Mathematik
;
Wahrscheinlichkeit
;
Stochastik
;
Aufsatzsammlung
;
Aufsatzsammlung
;
Mathematik
;
Wahrscheinlichkeit
;
Stochastik
Abstract:
This volume provides a necessary, current and extensive analysis of probabilistic thinking from a number of mathematicians, mathematics educators, and psychologists. The work of 58 contributing authors, investigating probabilistic thinking across the globe, is encapsulated in 6 prefaces, 29 chapters and 6 commentaries. Ultimately, the four main perspectives presented in this volume (Mathematics and Philosophy, Psychology, Stochastics and Mathematics Education) are designed to represent probabilistic thinking in a greater context.
Abstract:
This volume provides a necessary, current and extensive analysis of probabilistic thinking from a number of mathematicians, mathematics educators, and psychologists. The work of 58 contributing authors, investigating probabilistic thinking across the globe, is encapsulated in 6 prefaces, 29 chapters and 6 commentaries. Ultimately, the four main perspectives presented in this volume (Mathematics and Philosophy, Psychology, Stochastics and Mathematics Education) are designed to represent probabilistic thinking in a greater context. “Uncertainty is part of our lives and we all have to deal with it and make decisions in spite of it. Ability to use ideas from probability theory as a way of quantifying uncertainty needs to be an integral part of our education at many levels and this book will surely play a useful role." - S.R.Srinivasa Varadhan, Recipient of the 2007 Abel Prize in Mathematics and the 2010 National Medal of Science “A welcome look at probability, with philosophical and psychological perspectives that offer foundations for both students and teachers of probability at the school and university levels. Very comprehensive and promises a great deal to the reader. Teachers and students will benefit from articles that clarify the competition between the frequentist and the Bayesian views of probability." - Reuben Hersh, Author of "What is Mathematics, Really?" and co-author of "The Mathematical Experience" “I often get asked why people find probability so unintuitive and difficult. After years of research, I have concluded it’s because probability really is unintuitive and difficult. This ground-breaking text acknowledges the full complexity of teaching this subject: the contributions face up to the competing interpretations of probability, emphasising the close connection to both human psychology and real-world problem-solving tasks. I am personally very pleased to see the subjective interpretation taken seriously, while also admiring the suggestions for teaching the properties of modeled randomness. A very timely and valuable book." -David Spiegelhalter, Winston Professor for the Public Understanding of Risk, University of Cambridge “The teaching and learning of probability is challenging in several ways - coordinating its three theoretical perspectives (classical, frequentist, and subjective); managing its relationship to statistics; and reconciling the counter-intuitive nature of much probabilistic reasoning. This volume presents a compreh ...
Description / Table of Contents:
Probabilistic Thinking; Series Preface; The Most Common Misconception About Probability?; Introduction to Probabilistic Thinking: Presenting Plural Perspectives; References; Contents; Perspective I: Mathematics and Philosophy; Preface to Perspective I: Mathematics and Philosophy; References; A Historical and Philosophical Perspective on Probability; 1 Introduction and Sources; 2 From Divination to Combinatorial Multiplicity; 2.1 Early Origins in Divination and Religion; 2.2 Emergence of the Rule of Favourable to Possible: Combinatorial Multiplicity; De Méré's Problem; Division of Stakes
Description / Table of Contents:
3 Huygens, Bernoulli, and Bayes: The Art of Conjecturing3.1 Expectation and Probability; 3.2 Obstacles and Further Developments; Bayes' Formula and Inverse Probabilities; 4 Foundations and New Applications; 4.1 Classical Probability; 4.2 Continuous Distributions; 4.3 Axioms of Probability; 5 Modern Philosophical Views on Probability; Classical a Priori Theory (APT); Frequentist Theory (FQT); Subjectivist Theory (SJT); Commentary; 6 Concluding Comments; References; From Puzzles and Paradoxes to Concepts in Probability; 1 How Paradoxes Highlight Conceptual Conflicts; 2 Equal Likelihood
Description / Table of Contents:
2.1 Early Notions of ProbabilityP1: Problem of the Grand Duke of Tuscany; What is the Paradox?; Further Ideas; P2: De Méré's Problem; What is the Paradox?; Further Ideas; P3: Division of Stakes; What is the Paradox?; Further Ideas; 2.2 Conceptual Developments in Probability; P4: D'Alembert's Problem; What is the Paradox?; Further Ideas; 3 Expectation; 3.1 Expectation and Probability; P5: St Petersburg Paradox; What is the Paradox?; Further Ideas; 3.2 Independence and Expectation; P6: Dependent Spinners; What is the Paradox?; Further Ideas; P7: Dependent Coins; What is the Paradox?
Description / Table of Contents:
Further Ideas4 Relative Frequencies; P8: Library Problem; What is the Paradox?; P9: Bertrand's Chord; What is the Paradox?; Further Ideas; 5 Personal Probabilities; 5.1 Inverse Probabilities; P10: Bertrand's Paradox; What is the Paradox?; Further Ideas; P11: Father Smith and Son; What is the Paradox?; Further Ideas; 5.2 Conflicts with Logic; P12: Intransitive Spinners; What is the Paradox?; P13: Blyth's Intransitive Spinners; P14: Reinhardt's Single Spinner; P15: Simpson's Paradox of Proportions; What is the Paradox?; Further Ideas; 6 Central Ideas of Probability Theory
Description / Table of Contents:
6.1 Independence and Random Samples6.2 Central Theorems; Bernoulli's Law of Large Numbers; Laplace's Central Limit Theorem; Central Limit Theorem of Poisson; 6.3 Standard Situations; Laplacean Experiments; Bernoulli Experiments; Poisson Process; Elementary Errors; Stochastic Processes; 6.4 Kolmogorov's Axiomatic Foundation of Probability; The Axioms; Distribution Functions; Probability Measures on Infinite-Dimensional Spaces; Lebesgue Integral; 7 Conclusions; References; Three Approaches for Modelling Situations with Randomness; 1 Introduction
Description / Table of Contents:
2 Three Different Approaches to Probability (Content Knowledge)
Description / Table of Contents:
SERIES PREFACE: Gabriele Kaiser and Bharath SriramanACKNOWLEDGEMENTS -- FOREWORD: Keith Devlin -- INTRODUCTION: Egan Chernoff and Bharath Sriraman -- PERSPECTIVE I: MATHEMATICS AND PHILOSOPHY -- Preface to Perspective I: Mathematics and Philosophy: Egan Chernoff and Gale Russell -- I.I. A historical and philosophical perspective on probability: Manfred Borovcnik and Ramesh Kapadia -- I.II. From puzzles and paradoxes to concepts in probability: Manfred Borovcnik and Ramesh Kapadia -- I.III. Three approaches for modeling situation with randomness: Andreas Eichler and Markus Vogel -- I.IV. A modeling perspective on probability: Maxine Pfannkuch and Ilze Ziedins -- Commentary on Perspective I: Mathematics and Philosophy: Bharath Sriraman and Kyeong-Hwa Lee -- PERSPECTIVE II: PSYCHOLOGY -- Preface to Perspective II: Psychology : Wim van Dooren -- II.I. Statistical thinking: no child left behind: Björn Meder and Gerd Gigerenzer -- II.II. The A-B-C of probabilistic literacy: Laura Martignon -- II.III. Intuitive conceptions of probability and the development of basic math skills: Gary Brase, Sherri Martinie and Carlos Castillo-Garsow -- II.IV. Testing a model on probabilistic reasoning: Francesca Chiesi and Caterina Primi -- II.V. Revisiting the medical diagnosis problem: reconciling intuitive and analytical thinking: Lisser Rye Ejersbo and Uri Leron -- II.VI. Rethinking probability education: perceptual judgment as epistemic resource: Dor Abrahamson -- II.VII. Sticking to your guns: a flawed heuristic for probabilistic decision-making: Deborah Bennett -- II.VIII. Developing probabilistic thinking: what about peoples’ conceptions: Annie Savard -- Commentary I on Perspective II: Psychology : Brian Greer -- Commentary II on Perspective II: Psychology: Richard Lesh and Bharath Sriraman -- PERSPECTIVE III: STOCHASTICS -- Preface to Perspective III: Stochastics: Bharath Sriraman and Egan Chernoff -- III.I. Prospective primary school teachers’ perception of randomness: Carmen Batanero, Pedro Arteaga, Luis Serrano and Blanca Ruiz -- III.II. Challenges of developing coherent probabilistic reasoning: rethinking randomness and probability from a stochastic perspective: Luis Saldanha and Yan Liu -- III.III. “It is very, very random because it doesn’t happen very often”: examining learners’ discourse on randomness: Simin Jolafee, Rina Zazkis and Nathalie Sinclair -- III.IV. Developing a modelling approach to probability using computer-based simulations: Theodosia Prodromou -- III.V. Promoting statistical literacy through data modelling in the early school years: Lyn D. English -- III.VI. Learning Bayesian statistics in adulthood: Wolff-Michael Roth -- Commentary on Perspective III: Stochastics: Mike Shaughnessy -- PERSPECTIVE IV: MATHEMATICS EDUCATION -- Preface to Perspective IV: Mathematics Education: Bharath Sriraman and Egan Chernoff -- IV.I. A practitional perspective on probabilistic thinking models and frameworks: Edward S. Mooney, Cynthia Langrall and Joshua T. Hertel -- IV.II. Experimentation in probability teaching and learning: Per Nilsson -- IV.III. Investigating the dynamics of stochastic learning processes: Susanne Prediger and Susanne Schnell -- IV.IV. Counting as a foundation for learning to reason about probability: Carolyn A. Maher and Anoop Ahluwalia -- IV.V. Levels of probabilistic reasoning of high school students about binomial problems: Ernesto Sánchez and Pedro Rubén Landín -- IV.VI. Children’s construction of sample space with respect to the law of large numbers: Efi Paparistodemou -- IV.VII. Researching conditional probability problem solving: Pedro Huerta -- IV.VIII. Real life experiences as hindrance in probabilistic situations: Ami Mamolo and Rina Zazkis -- IV.IX. Influence of culture on high school students’ probabilistic thinking: Sashi Sharma -- IV.X. Primary school students’ attitudes to and beliefs about probability: Steven Nisbet and Anne Williams -- Commentary on Perspective IV: Mathematics Education: Jane Watson -- COMMENTARY on Probabilistic Thinking: Presenting Plural Perspectives: Egan Chernoff and Bharath Sriraman -- AUTHOR INDEX -- SUBJECT INDEX.
Note:
Description based upon print version of record
DOI:
10.1007/978-94-007-7155-0
URL:
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