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

The Genius of Euler: Reflections on his Life and Work

(Spectrum)

William Dunham (Editor)

The Mathematical Association of America | 2007 | 309 páginas | DjVu (11.8 mb)

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This book celebrates the 300th birthday of Leonhard Euler (1707 1783), one of the brightest stars in the mathematical firmament. The book stands as a testimonial to a mathematician of unsurpassed insight, industry, and ingenuity one who has been rightly called the master of us all. The collected articles, aimed at a mathematically literate audience, address aspects of Euler s life and work, from the biographical to the historical to the mathematical. The oldest of these was written in 1872, and the most recent dates to 2006. Some of the papers focus on Euler and his world, others describe a specific Eulerian achievement, and still others survey a branch of mathematics to which Euler contributed significantly. Along the way, the reader will encounter the Konigsberg bridges, the 36-officers, Euler s constant, and the zeta function. There are papers on Euler s number theory, his calculus of variations, and his polyhedral formula. Of special note are the number and quality of authors represented here. Among the 34 contributors are some of the most illustrious mathematicians and mathematics historians of the past century e.g., Florian Cajori, Carl Boyer, George Polya, Andre Weil, and Paul Erdos. And there are a few poems and a mnemonic just for fun.

Contents
Acknowledgments ix
Preface xi
About the Authors xiii
Part I: Biography and Background
Introduction to Part I 3
Leonhard Euler, B. F. Finkel (1897) 5
Leonard Euler, Supreme Geometer (abridged), C. Truesdell (1972) 13
Euler (abridged), Andre Weil (1984) 43
Frederick the Great on Mathematics and Mathematicians (abridged),
Florian Cajori (1927) 51
The Euler-Diderot Anecdote, B. H. Brown (1942) 57
Ars Expositionis: Euler as Writer and Teacher, G. L. Alexanderson (1983) 61
The Foremost Textbook of Modern Times, C. B. Boyer (1951) 69
Leonhard Euler, 1707-1783, J. J. Burckhardt (1983) 75
Euler's Output, A Historical Note, W W. R. Ball (1924) 89
Discoveries (a poem), Marta Sved and Dave Logothetti (1989) 91
Bell's Conjecture (a poem), J. D. Memory (1997) 93
A Response to "Bell's Conjecture" (a poem),
Charlie Marion and William Dunham (1997) 95
Part II: Mathematics
Introduction to Part 2 99
Euler and Infinite Series, Morris Kline (1983) 101
The Genius of Euler: Reflections on his Life and Work
Euler and the Zeta Function, Raymond Ayoub (1974) 113
Addendum to: "Euler and the Zeta Function, " A. G. Howson (1975) 133
Euler Subdues a Very Obstreperous Series (abridged), E. J. Barbeau (1979) 135
On the History ofEuler's Constant, J. W. L. Glaisher (1872) 147
A Mnemonic for Euler's Constant, Morgan Ward (1931) 153
Euler and Differentials, Anthony P. Ferzola (1994) 155
Leonhard Euler's Integral: A Historical Profile of the
Gamma Function, Philip J. Davis (1959) 167
Change of Variables in Multiple Integrals: Euler to Cartan, Victor J. Katz (1982) 185
Euler's Vision of a General Partial Differential Calculus for a
Generalized Kind of Function, Jesper LUtzen (1983) 197
On the Calculus of Variations and Its Major Influences on the
Mathematics of the First Half of Our Century, Erwin Kreyszig (1994) 209
Some Remarks and Problems in Number Theory Related to the
Work of Euler, Paul Erdos and Underwood Dudley (1983) 215
Euler's Pentagonal Number Theorem, George E. Andrews (1983) 225
Euler and Quadratic Reciprocity, Harold M. Edwards (1983) 233
Euler and the Fundamental Theorem of Algebra, William Dunham (1991) 243
Guessing and Proving, George P6lya (1978) 257
The Truth about Kdnigsberg, Brian Hopkins and Robin J. Wilson (2004) 263
Graeco-Latin Squares and a Mistaken Conjecture of Euler,
Dominic Klyve and Lee Stemkoski (2006) 273
Glossary 289
List of Photos 303
Index 305
About the Editor 309

Mathematics and Modern Art: Proceedings of the First ESMA Conference, held in Paris, July 19-22, 2010

(Springer Proceedings in Mathematics)

Claude Bruter

Springer | 2012 | 222 páginas | PDF | 9,3 Mb

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The link between mathematics and art remains as strong today as it was in the earliest instances of decorative and ritual art. Arts, architecture, music and painting have for a long time been sources of new developments in mathematics, and vice versa. Many great painters have seen no contradiction between artistic and mathematical endeavors, contributing to the progress of both, using mathematical principles to guide their visual creativity, enriching their visual environment with the new objects created by the mathematical science.

Owing to the recent development of the so nice techniques for visualization, while mathematicians can better explore these new mathematical objects, artists can use them to emphasize their intrinsic beauty, and create quite new sceneries. This volume, the content of the first conference of the European Society for Mathematics and the Arts (ESMA), held in Paris in 2010, gives an overview on some significant and beautiful recent works where maths and art, including architecture and music, are interwoven. 

The book includes a wealth of mathematical illustrations from several basic mathematical fields including classical geometry, topology, differential geometry, dynamical systems.  Here, artists and mathematicians alike elucidate the thought processes and the tools used to create their work

Contents

A Mathematician and an Artist. The Story of a Collaboration . . 1

Richard S. Palais

Dimensions, a Math Movie  . . 11

Aurelien Alvarez and Jos Leys

Old and New Mathematical Models: Saving the Heritage of the Institut Henri Poincare . . 17

Francois Apery

An Introduction to the Construction of Some Mathematical Objects. . 29

Claude Paul Bruter

Computer, Mathematics and Art . . . 47

Jean-Franc¸ois Colonna

Structure of Visualization and Symmetry in Iterated Function Systems . . . 53

Jean Constant

M.C. Escher’s Use of the Poincare Models of Hyperbolic Geometry. . 69

Douglas Dunham

Mathematics and Music Boxes  . . 79

Vi Hart

My Mathematical Engravings. . 85

Patrice Jeener

Knots and Links As Form-Generating Structures. . 105

Dmitri Kozlov

Geometry and Art from the Cordovan Proportion. . . 117

Antonia Redondo Buitrago and Encarnacion Reyes Iglesias

Dynamic Surfaces . . 131

Simon Salamon

Pleasing Shapes for Topological Objects . . 153

John M. Sullivan

Rhombopolyclonic Polygonal Rosettes Theory . . 167

Francois Tard

USA Mathematical Olympiads 1972-1986 Problems and Solutions

(Anneli Lax New Mathematical Library) 

Murray Klamkin 

Mathematical Association of America | 1989 | 146 páginas | pdf | 1,7 Mb

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People delight in working on problems "because they are there," for the sheer pleasure of meeting a challenge. This is a book full of such delights. In it, Murray S. Klamkin brings together 75 original USA Mathematical Olympiad (USAMO) problems for yearss 1972-1986, with many improvements, extensions, related exercises, open problems, referneces and solutions, often showing alternative approaches. The problems are coded by subject, and solutions are arranged by subject, e.g., algebra, number theory, solid geometry, etc., as an aid to those interested in a particular field. Included is a Glossary of frequently used terms and theorems and a comprehensive bibliography with items numbered and referred to in brackets in the text. This a collection of problemsand solutions of arresting ingenuit, all accessible to secondary school students.

The USAMO has been taken annually by about 150 of the nation's best high school mathematics students. This exam helps to find and encourage high school students with superior mathematical talent and creativity and is the culmination of a three-tiered competition that begins with the American High School Mathematics Examination (AHSME) taken by over 400, 000 students. The eight winners of the USAMO are canidates for the US team in the International Mathematical Olympiad. Schools are encouraged to join this large and important enterprise. See page x of the preface for further information. this book includes a list of all of the top contestants in the USAMO and their schools.

The problems are intriguing and the solutions elegant and informative. Students and teachers will enjoy working these challenging problems. Indeed, all hose who are mathematically inclined will find many delights and pleasant challenges in this book.

Contents
Editors' Note Vii
Preface ix
USA Olympiad Problems 1
Solutions of Olympiad Problems 15
Algebra (A) 15
Number Theory (N.T.) 30
Plane Geometry (P.G.) 45
Solid Geometry (S.G.) 55
Geometric Inequalities (G.I.) 66
Inequalities (I) 81
Combinatorics & Probability (C.& P.) 93
Appendix 105
List of Symbols 110
Glossary 111
References 120

Mathematical Cognition

(Current Perspectives on Cognition, Learning, and Instruction)

James Royer

Information Age Publishing | 2003 | 272 páginas | rar - pdf | 1,3 Mb

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CONTENTS
Introduction
James M. Royer ix
1. The Development of Math Competence in the Preschool and  Early School Years: Cognitive Foundations and Instructional Strategies
Sharon Griffin 1
2. Perspectives on Mathematics Strategy Development
Martha Carr and Hillary Hettinger 33
3. Mathematical Problem Solving
Richard E. Mayer 69
4. Learning Disabilities in Basic Mathematics: Deficits in Memory and Cognition
David C. Geary and Mary K. Hoard 93
5. Relationships Among Basic Computational Automaticity, Working Memory, and Complex Mathematical Problem Solving: What We Know and What We Need to Know
Loel T. Tronsky and James M. Royer 117
6. Mathematics Instruction: Cognitive, Affective and Existential Perspectives
Allan Feldman 147
7. A Brief History of American K-12 Mathematics Education in the 20th Century
David Klein 175
8. Assessment in Mathematics: A Developmental Approach
John Pegg 260