A Mathematical Orchard: Problems and Solutions

(MAA Problem Book Series)

Mark I. Krusemeyer, George T. Gilbert e Loren C. Larson

The Mathematical Association of America | 2012 | 410 páginas | rar - pdf | 1,9 Mb

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This volume is a republication and expansion of the much-loved Wohascum County Problem Book, published in 1993. The original 130 problems have been retained and supplemented by an additional 78 problems. The puzzles contained within, which are accessible but never routine, have been specially selected for their mathematical appeal, and detailed solutions are provided. The reader will encounter puzzles involving calculus, algebra, discrete mathematics, geometry and number theory, and the volume includes an appendix identifying the prerequisite knowledge for each problem. A second appendix organises the problems by subject matter so that readers can focus their attention on particular types of problems if they wish. This collection will provide enjoyment for seasoned problem solvers and for those who wish to hone their skills.

• An entertaining collection of problems, tried and tested by experienced educators
• Detailed solutions are included and some problems are solved in multiple ways
• Accessible to those of advanced secondary school/high school level, through to undergraduate and above

Table of Contents
Preface
1. The problems
2. The solutions
Appendix 1. Prerequisites by problem number
Appendix 2. Problem numbers by subject
Index.

Journey through Mathematics Creative Episodes in Its History

Enrique A. González-Velasco

Springer | 2011| 478 páginas | pdf | 4,4 Mb

link direto
link

This book offers an accessible and in-depth look at some of the most  important episodes of two thousand years of mathematical history. Beginning with trigonometry and moving on through logarithms, complex numbers, infinite series, and calculus, this book profiles some of the lesser known but crucial contributors to modern day mathematics. It is unique in its use of primary sources as well as its accessibility; a knowledge of first-year calculus is the only prerequisite. But undergraduate and graduate students alike will appreciate this glimpse into the fascinating process of mathematical creation.

The history of math is an intercontinental journey, and this book showcases brilliant mathematicians from Greece, Egypt, and India, as well as Europe and the Islamic world. Several of the primary sources have never before been translated into English. Their interpretation is thorough and readable, and offers an excellent background for teachers of high school mathematics as well as anyone interested in the history of math.

TABLE OF CONTENTS
Preface ix
1 TRIGONOMETRY 1
1.1 The Hellenic Period 1
1.2 Ptolemy’s Table of Chords 10
1.3 The Indian Contribution 25
1.4 Trigonometry in the Islamic World 34
1.5 Trigonometry in Europe 55
1.6 From Viète to Pitiscus 65
2 LOGARITHMS 78
2.1 Napier’s First Three Tables 78
2.2 Napier’s Logarithms 88
2.3 Briggs’ Logarithms 101
2.4 Hyperbolic Logarithms 117
2.5 Newton’s Binomial Series 122
2.6 The Logarithm According to Euler 136
3 COMPLEX NUMBERS 148
3.1 The Depressed Cubic 148
3.2 Cardano’s Contribution 150
3.3 The Birth of Complex Numbers 160
3.4 Higher-Order Roots of Complex Numbers 173
3.5 The Logarithms of Complex Numbers 181
3.6 Caspar Wessel’s Breakthrough 185
3.7 Gauss and Hamilton Have the Final Word 190
4 INFINITE SERIES 195
4.1 The Origins 195
4.2 The Summation of Series 203
4.3 The Expansion of Functions 212
4.4 The Taylor and Maclaurin Series 220
5 THE CALCULUS 230
5.1 The Origins 230
5.2 Fermat’s Method of Maxima and Minima 234
5.3 Fermat’s Treatise on Quadratures 248
5.4 Gregory’s Contributions 258
5.5 Barrow’s Geometric Calculus 275
5.6 From Tangents to Quadratures 283
5.7 Newton’s Method of Infinit Series 289
5.8 Newton’s Method of Fluxions 294
5.9 Was Newton’s Tangent Method Original? 302
5.10 Newton’s First and Last Ratios 306
5.11 Newton’s Last Version of the Calculus 312
5.12 Leibniz’ Calculus: 1673–1675 318
5.13 Leibniz’ Calculus: 1676–1680 329
5.14 The Arithmetical Quadrature 340
5.15 Leibniz’ Publications 349
5.16 The Aftermath 358
6 CONVERGENCE 368
6.1 To the Limit 368
6.2 The Vibrating String Makes Waves 369
6.3 Fourier Puts on the Heat 373
6.4 The Convergence of Series 380
6.5 The Difference Quotient 394
6.6 The Derivative 401
6.7 Cauchy’s Integral Calculus 405
6.8 Uniform Convergence 407
BIBLIOGRAPHY 412

The Mathematics of Games and Gambling

Edward Packel

The Mathematical Association of America | 2006 - 2.ª edição| 190 páginas | rar - pdf | 880 kb

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The first edition of this book was reprinted eight times! This book introduces and develops some of the important and beautiful elementary mathematics needed for rational analysis of various gambling and game activities. Most of the standard casino games (roulette, , blackjack, keno), some social games (backgammon, poker, bridge) and various other activities (state lotteries, horse racing, etc.) are treated in ways that bring out their mathematical aspects. The mathematics developed ranges from the predictable concepts of probability, expectation, and binomial coefficients to some less well-known ideas of elementary game theory. The Second Edition includes new material on: sports betting and the mathematics behind it; Game theory applied to bluffing in poker and related to the “Texas Holdem phenomenon”: The Nash equilibrium concept and its emergence in the popular culture: Internet links to games and to Java applets for practice and classroom use. The only formal mathematics background the reader needs is some facility with high school algebra. Game-related exercises are included at the end of most chapters for readers interested in working with and expanding ideas treated in the text. Solutions to some of the exercises appear at the end of the book.

Contents
Preface to the First Edition ix
Preface to the Second Edition xiii
1 The Phenomenon of Gambling 1
1.1 A selective history . . . 1
1.2 The gambler in fact and fiction . . . 5
2 Finite Probabilities and Great Expectations 13
2.1 The probability concept and its origins . . 13
2.2 Dice, cards, and probabilities  . . 15
2.3 Roulette, probability and odds. . 17
2.4 Compound probabilities: The rules of the game  . . 20
2.5 Mathematical expectation and its application .. . 22
2.6 Exercises  . . 26
3 Backgammon and Other Dice Diversions 29
3.1 Backgammon oversimplified  . . 29
3.2 Rolling spots and hitting blots  . . 32
3.3 Enteringand bearingoff .  . 34
3.4 The doubling cube  . . 36
3.5 Craps  . . 40
3.6 Chuck-a-Luck . . . 45
3.7 Exercises . . 47
4 Permutations, Combinations, and Applications 51
4.1 Careful counting: Is order important? . . 51
4.2 Factorials and other notation  . . 53
4.3 Probabilities in poker  . . 55
4.4 Betting in poker: A simple model  . . 59
4.5 Distributions in bridge  . . 67
4.6 Keno type games . . 71
4.7 Exercises . . 73
5 Play it Again Sam: The Binomial Distribution 79
5.1 Games and repeatedtrials . . 79
5.2 The binomial distribution . . 79
5.3 Beating the odds and the “law” of averages  . . 83
5.4 Betting systems  . . 90
5.5 A brief blackjack breakthrough. . 93
5.6 Exercises  . . 95
6 Elementary Game Theory 99
6.1 What isgame theory?  . . 99
6.2 Games in extensive form . . 100
6.3 Two-persongames in normal form  . . 105
6.4 Zero-sumgames . . 107
6.5 Nonzero-sum games, Nash equilibria and the prisoners’ dilemma  . 113
6.6 Simple n-persongames . . 118
6.7 Power indices. . 120
6.8 Games computers play  . . . 123
6.9 Exercises . . . 129
7 Odds and Ends 135
7.1 The mathematics of bluffing and the Texas Holdem invasion . . 135
7.2 Off to the races  . . 141
7.3 Lotteries and your expectation . . . 147
7.4 The gambler’s ruin  . 158

The Logic of Chance: An Essay on the Foundations and Province of the Theory of Probability

John Venn

Macmillan And Company | 1888

online:
archive.org
books.google

CHELSEA PUBLISHING | 1962 - 4ª edição 

online:
catalog.hathitrust.org

No mathematical background is necessary to appreciate this classic of probability theory, which remains unsurpassed in its clarity, readability, and sheer charm. Its author, British logician John Venn (1834-1923), popularized the famous Venn Diagrams that are commonly used for teaching elementary mathematics. In The Logic of Chance, he employs the same directness that makes his diagrams so effective.
The three-part treatment commences with an overview of the physical foundations of the science of probability, including surveys of the arrangement and formation of the series of probability; the origin or process of causation of the series; how to discover and prove the series; and the conception of randomness. The second part examines the logical superstructure on the basis of physical foundations, encompassing the measurement of belief; the rules of inference in probability; the rule of succession; induction; chance, causation, and design; material and formal logic; modality; and fallacies. The final section explores various applications of the theory of probability, including such intriguing aspects as insurance and gambling, the credibility of extraordinary stories, and approximating the truth by means of the theory of averages.

Beyond the Quadratic Formula

(Classroom Resource Materials)

Ronald S. Irving

The Mathematical Association of America | 2013 | 244 páginas | rar - pdf | 1,1 Mb

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The quadratic formula for the solution of quadratic equations was discovered independently by scholars in many ancient cultures and is familiar to everyone. Less well known are formulas for solutions of cubic and quartic equations whose discovery was the high point of 16th century mathematics. Their study forms the heart of this book, as part of the broader theme that a polynomial's coefficients can be used to obtain detailed information on its roots. The book is designed for self-study, with many results presented as exercises and some supplemented by outlines for solution. The intended audience includes in-service and prospective secondary mathematics teachers, high school students eager to go beyond the standard curriculum, undergraduates who desire an in-depth look at a topic they may have unwittingly skipped over, and the mathematically curious who wish to do some work to unlock the mysteries of this beautiful subject.

Contents
Preface ix
1 Polynomials 1
1.1 Definitions . . 1
1.2 Multiplication and Degree. . 4
1.3 Factorization and Roots. . 8
1.4 Bounding the Number of Roots . . 10
1.5 Real Numbers and the Intermediate Value Theorem . . 12
1.6 Graphs . . 16
2 Quadratic Polynomials 21
2.1 Sums and Products . . 22
2.2 Completing the Square . . 24
2.3 Changing Variables . . 28
2.4 A Discriminant . . 29
2.5 History  . 33
3 Cubic Polynomials 47
3.1 Reduced Cubics  . . 47
3.2 Cardano’s Formula . . 50
3.3 Graphs. . 58
3.4 A Discriminant . . 61
3.5 History . . 66
4 Complex Numbers 73
4.1 Complex Numbers .  . 73
4.2 Quadratic Polynomials and the Discriminant . . 77
4.3 Square and Cube Roots . . 81
4.4 The Complex Plane . . 84
4.5 A Geometric Interpretation of Multiplication . . 88
4.6 Euler’s and de Moivre’s Formulas . . 92
4.7 Roots of Unity . . 98
4.8 Converting Root Extraction to Division . . 101
4.9 History . . 103
5 Cubic Polynomials, II 109
5.1 Cardano’s formula  . . 109
5.2 The Resolvent . . 113
5.3 The Discriminant . . 115
5.4 Cardano’s Formula Refined. . 120
5.5 The Irreducible Case. . 124
5.6 Vi`ete’s Formula . . 125
5.7 The Signs of the Real Roots . . 130
5.8 History  . . 133
6 Quartic Polynomials 143
6.1 Reduced Quartics . . 143
6.2 Ferrari’s Method  . . 146
6.3 Descartes’ Method . . 149
6.4 Euler’s Formula . . 154
6.5 The Discriminant . . 157
6.6 The Nature of the Roots . . 162
6.7 Cubic and Quartic Reprise. . 167
6.8 History  . . 169
7 Higher-Degree Polynomials 179
7.1 Quintic Polynomials  . . 179
7.2 The Fundamental Theorem of Algebra . . 185
7.3 Polynomial Factorization . . 191
7.4 Symmetric Polynomials. . 200
7.5 A Proof of the Fundamental Theorem . . 211
References 217
Index 223A
About the Author 227

Probabilistic Thinking Presenting Plural Perspectives

Egan J. Chernoff e Bharath Sriraman

Springer | 2014 | 746 páginas | RAR- pdf | 9,1 Mb

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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.
Contents
Perspective I: Mathematics and Philosophy
Preface to Perspective I: Mathematics and Philosophy  . . 3
Egan J. Chernoff and Gale L. Russell
A Historical and Philosophical Perspective on Probability . . 7
Manfred Borovcnik and Ramesh Kapadia
From Puzzles and Paradoxes to Concepts in Probability . . 35
Manfred Borovcnik and Ramesh Kapadia
Three Approaches for Modelling Situations with Randomness  . . 75
Andreas Eichler and Markus Vogel
A Modelling Perspective on Probability . . 101
Maxine Pfannkuch and Ilze Ziedins
Commentary on Perspective I: The Humanistic Dimensions of Probability  . . 117
Bharath Sriraman and Kyeonghwa Lee
Perspective II: Psychology
Probabilistic Thinking: Analyses from a Psychological Perspective . . . 123
Wim Van Dooren
Statistical Thinking: No One Left Behind  . . 127
Björn Meder and Gerd Gigerenzer
Fostering Children’s Probabilistic Reasoning and First Elements of Risk Evaluation  . . 149
Laura Martignon
Intuitive Conceptions of Probability and the Development of Basic Math Skills . . . 161
Gary L. Brase, Sherri Martinie, and Carlos Castillo-Garsow
The Interplay Among Knowledge, Cognitive Abilities and Thinking Styles in Probabilistic Reasoning: A Test of a Model  . . 195
Francesca Chiesi and Caterina Primi
Revisiting the Medical Diagnosis Problem: Reconciling Intuitive and Analytical Thinking .215
Lisser Rye Ejersbo and Uri Leron
Rethinking Probability Education: Perceptual Judgment as Epistemic Resource . . 239
Dor Abrahamson
Sticking to Your Guns: A Flawed Heuristic for Probabilistic Decision-Making . . 261
Deborah Bennett
Developing Probabilistic Thinking: What About People’s Conceptions? 283
Annie Savard
Commentary on Perspective II: Psychology . . 299
Brian Greer
A Brief Overview and Critique of Perspective II on Probabilistic and Statistical Reasoning . . . 311
Richard Lesh and Bharath Sriraman
Perspective III: Stochastics
Preface to Perspective III: Stochastics . . . 343
Egan J. Chernoff and Gale L. Russell
Prospective Primary School Teachers’ Perception of Randomness . . . 345
Carmen Batanero, Pedro Arteaga, Luis Serrano, and Blanca Ruiz
Challenges of Developing Coherent Probabilistic Reasoning: Rethinking Randomness and Probability from a Stochastic Perspective . . 367
Luis Saldanha and Yan Liu
“It Is Very, Very Random Because It Doesn’t Happen Very Often”: Examining Learners’ Discourse on Randomness  . . 397
Simin Jolfaee, Rina Zazkis, and Nathalie Sinclair
Developing a Modelling Approach to Probability Using Computer-Based Simulations  . . 417
Theodosia Prodromou
Promoting Statistical Literacy Through Data Modelling in the Early School Years  . . 441
Lyn D. English
Learning Bayesian Statistics in Adulthood . . . 459
Wolff-Michael Roth
Commentary on the Chapters on Probability from a Stochastic Perspective  . . 481
J. Michael Shaughnessy
Perspective IV: Mathematics Education
Preface to Perspective IV: Mathematics Education. . 493
Egan J. Chernoff and Gale L. Russell
A Practitional Perspective on Probabilistic Thinking Models and Frameworks . . 495
Edward S. Mooney, Cynthia W. Langrall, and Joshua T. Hertel
Experimentation in Probability Teaching and Learning . . 509
Per Nilsson
Investigating the Dynamics of Stochastic Learning Processes: A Didactical Research Perspective, Its Methodological and Theoretical Framework, Illustrated for the Case of the Short Term–Long Term Distinction . . . 533
Susanne Prediger and Susanne Schnell
Counting as a Foundation for Learning to Reason About Probability . . 559
Carolyn A. Maher and Anoop Ahluwalia
Levels of Probabilistic Reasoning of High School Students About Binomial Problems . . 581
Ernesto Sánchez and Pedro Rubén Landín
Children’s Constructions of a Sample Space with Respect to the Law of Large Numbers 599
Efi Paparistodemou
Researching Conditional Probability Problem Solving. . . 613
M. Pedro Huerta
Contextual Considerations in Probabilistic Situations: An Aid or a Hindrance? . . 641
Ami Mamolo and Rina Zazkis
Cultural Influences in Probabilistic Thinking  . . 657
Sashi Sharma
Primary School Students’ Attitudes To and Beliefs About Probability . 683
Anne Williams and Steven Nisbet
Section IV Commentary: The Perspective of Mathematics Education 709
Jane M. Watson
Commentary on Probabilistic Thinking: Presenting Plural Perspectives 721
Egan J. Chernoff and Bharath Sriraman

Mathematical Representation at the Interface of Body and Culture

(International Perspectives on Mathematics Education)

Wolff-Michael Roth

Information Age Publishing | 2009 | 369 páginas | rar - pdf | 6,2 Mb

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A Volume in International Perspectives on Mathematics Education - Cognition, Equity & Society Series Editor Bharath Sriraman, The University of Montana and Lyn English, Queensland University of Technology Over the past two decades, the theoretical interests of mathematics educators have changed substantially-as any brief look at the titles and abstracts of articles shows. Largely through the work of Paul Cobb and his various collaborators, mathematics educators came to be attuned to the intricate relationship between individual and the social configuration of which she or he is part. That is, this body of work, running alongside more traditional constructivist and psychological approaches, showed that what happens at the collective level in a classroom both constrains and affords opportunities for what individuals do (their practices). Increasingly, researchers focused on the mediational role of sociomathematical norms and how these emerged from the enacted lessons. A second major shift in mathematical theorizing occurred during the past decade: there is an increasing focus on the embodied and bodily manifestation of mathematical knowing (e.g., Lakoff & Núñez, 2000). Mathematics educators now working from this perspective have come to their position from quite different bodies of literatures: for some, linguistic concerns and mathematics as material praxis lay at the origin for their concerns; others came to their position through the literature on the situated nature of cognition; and yet another line of thinking emerged from the work on embodiment that Humberto Maturana and Francisco Varela advanced. Whatever the historical origins of their thinking, mathematics educators taking an embodiment perspective presuppose that it is of little use to think of mathematical knowing in terms of transcendental concepts somehow recorded in the brain, but rather, that we need to conceptual knowing as mediated by the human body, which, because of its senses, is at the origin of sense. One of the question seldom asked is how the two perspectives, one that focuses on the bodily, embodied nature of mathematical cognition and the other that focuses on its social nature, can be thought together. This edited volume situates itself at the intersection of theoretical and focal concerns of both of these lines of work. In all chapters, the current culture both at the classroom and at the societal level comes to be expressed and provides opportunities for expressing oneself in particular ways; and these expressions always are bodily expressions of body-minds. As a collective, the chapters focus on mathematical knowledge as an aspect or attribute of mathematical performance; that is, mathematical knowing is in the doing rather than attributable to some mental substrate structured in particular ways as conceived by conceptual change theorists or traditional cognitive psychologists. The collection as a whole shows readers important aspects of mathematical cognition that are produced and observable at the interface between the body (both human and those of [inherently material] inscriptions) and culture. Drawing on cultural-historical activity theory, the editor develops an integrative perspective that serves as a background to a narrative that runs through and pulls together the book into an integrated whole.

CONTENTS
Series Preface vii
Preface xi
1. Social Bodies and Mathematical Cognition: An Introduction
Wolff-Michael Roth 1
PART A: MOVING AND TRANSFORMING BODIES IN/AS MATHEMATICAL PRACTICE
Editor’s Section Introduction 19
2. Transformation Geometry from an Embodied Perspective
Laurie D. Edwards 27
3. Signifying Relative Motion: Time, Space and the Semiotics of Cartesian Graphs
Luis Radford 45
4. What Makes a Cube a Cube? Contingency in Abstract, Concrete, Cultural and Bodily Mathematical Knowings
Jean-François Maheux, Jennifer S. Thom, and Wolff-Michael Roth 71
5. Embodied Mathematical Communication and the Visibility of Graphical Features
Wolff-Michael Roth 95
Editor’s Section Commentary 123
PART B: EMERGENCE OF OBJECTS AND UNDERSTANDING
Editor’s Section Introduction 131
6. Supporting Students’ Learning About Data Creation
Paul Cobb and Carrie Tzou 135
7. How Do You Know Which Way the Arrows Go? The Emergence and Brokering of a Classroom Math Practice
Chris Rasmussen, Michelle Zadieh, and Megan Wawro 171
8. Inscription, Narration and Diagram-Based Argumentation: Narrative Accounting Practices in Primary Mathematics Classes
Götz Krummheuer 219
Editor’s Section Commentary 245
PART C: STEPS TOWARD RETHINKING
MATHEMATICS EDUCATION
Editor’s Section Introduction 251
9. And so …?
Brent Davis 257
10. Expressiveness and Mathematics Learning
Ian Whitacre, Charles Hohensee, and Ricardo Nemirovsky 275
11. Gesture, Abstraction, and the Embodied Nature of Mathematics
Rafael E. Núñez 309
Editor’s Section Commentary 329
PART D: EPILOGUE
12. Appreciating the Embodied Social Nature of Mathematical Cognition
Wolff-Michael Roth 335
About the Authors 351