Greek Mathematical Thought and the Origin of Algebra

(Dover Books on Mathematics)

Jacob Klein

Dover Publications | 1992 | 384 páginas | epub | 19 Mb

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Important study focuses on the revival and assimilation of ancient Greek mathematics in the 13th–16th centuries, via Arabic science, and the 16th-century development of symbolic algebra. This brought about the crucial change in the concept of number that made possible modern science — in which the symbolic "form" of a mathematical statement is completely inseparable from its "content" of physical meaning. Includes a translation of Vieta's Introduction to the Analytical Art. 1968 edition

Contents
Author’s note
Translator’s note
Short titles frequently used
PART I
1 Introduction: Purpose and plan of the inquiry.
2 The opposition of logistic and arithmetic in the Neoplatonists.
3 Logistic and arithmetic in Plato.
4 The role of the theory of proportions in Nicomachus, Theon, and Domninus.
5 Theoretical logistic and the problem of fractions.
6 The concept of arithmos.
7 The ontological conception of the arithmoi in Plato.
A. The science of the Pythagoreans.
B. Mathematics in Plato—logistike and dianoia.
C. The arithmos eidetikos.
8 The Aristotelian critique and the possibility of a theoretical logistic.
PART II
9 On the difference between ancient and modern conceptualization.
10 The Arithmetic of Diophantus as theoretical logistic. The concept ofeidos in Diophantus
11 The formalism of Vieta and the transformation of thearithmos concept.
A. The life of Vieta and the general characteristics of his work.
B. Vieta’s point of departure: the concept of synthetic apodeixisin Pappus and in Diophantus.
C. The reinterpretation of the Diophantine procedure by Vieta:
1. The procedure for solutions “in the indeterminate form” as an analogue to geometric analysis.
2. The generalization of the eidos concept and its transformation into the “symbolic” concept of the species.
3. The reinterpretation of the katholou pragmateia as Mathesis Universalis in the sense of ars analytice.
12 The concept of “number.”
A. In Stevin.
B. In Descartes.
C. In Wallis.
NOTES
Part I, Notes 1–125
Part II, Notes 126–348

APPENDIX

Mathematics Methods for Elementary and Middle School Teachers

 

Mary M. Hatfield, Nancy Tanner Edwards, Gary G. Bitter e  Jean Morrow 

Wiley | 2007 - 6ª edição| páginas | rar - pdf | 5,5 Mb

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This text provides preservice prekindergarten through grade eight teachers with ideas, techniques, and approaches to teaching mathematics appropriate for the 21st century, and strongly integrates technology with hands-on experience. This is the only text to include practice Praxis II-style test questions to prepare teacher candidates to pass the high-stakes test used for teacher certification. The new sixth edition has been updated with the National Council of Teachers of Mathematics (NCTM) Curriculum Focal Points, which provide focus on significant concepts for each grade level.


Table of Contents
1. Mathematics Education Today and into the Future.
2. Culturally Relevant Mathematics.
3. The Development of Mathematical Proficiency: Using Learning Research, Assessment, and Effective Instruction.
4. Middle School Mathematics.
5. Problem Solving.
6. Early Childhood Mathematics -- Number Readiness.
7. Operations and Number Sense.
8. Numeration and Number Sense.
9. Operations with Whole Numbers.
10. Common Fractions and Decimals.
11. Percent, Ratio, Proportion, and Rate.
12. Geometry and Spatial Reasoning.
13. Measurement.
14. Algebra and Algebraic Thinking.
15. Data Analysis, Statistics, and Probability.

Six Sources of Collapse A Mathematician’s Perspective on How Things Can Fall Apart in the Blink of an Eye

Charles R. Hadlock

The Mathematical Association of America | 2012 | 222 páginas | rar - pdf |4 Mb

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Beginning with one of the most remarkable ecological collapses of recent time, that of the passenger pigeon, Hadlock goes on to survey collapse processes across the entire spectrum of the natural and man-made world. He takes us through extreme weather events, technological disasters, evolutionary processes, crashing markets and companies, the chaotic nature of Earth's orbit, revolutionary political change, the spread and elimination of disease, and many other fascinating cases.

His key thesis is that one or more of six fundamental dynamics consistently show up across this wide range. These "six sources of collapse" can all be best described and investigated using fundamental mathematical concepts. They include low probability events, group dynamics, evolutionary games, instability, nonlinearity, and network effects, all of which are explained in readily understandable terms. Almost the entirety of the book can be understood by readers with a minimal mathematical background, but even professional mathematicians are likely to get rich insights from the range of examples. The author tells his story with a warmly personal tone and weaves in many of his own experiences, whether from his consulting career of racing around the world trying to head off industrial disasters to his story of watching collapse after collapse in the evolution of an ecosystem on his New Hampshire farm.

Creative teachers could use this book for anything from a liberal arts math course to a senior capstone seminar, and one reviewer suggested that it should be required reading for any mathematics graduate student heading off into a teaching career. This book will also be of interest to readers in the fields under discussion, such as business, engineering, ecology, political science, and others.

Contents

Preface ix

Acknowledgements xi

1 Introduction 1

1.1 What is a collapse?. . 1

1.2 Shades of Hitchcock, and other tales . . 2

1.3 What might tomorrow bring?  . . 6

1.4 What this book aims to do  . . 13

2 Predicting Unpredictable Events 15

2.1 Like a thief in the night?. . 15

2.2 Chance and regularity. . 17

2.3 A quick statistics primer  . . 18

2.4 Normal regularity: the good, the bad, and the miraculous. . 22

2.5 Abnormal regularity: extreme value statistics  . . 25

2.6 Getting things right with heavy-tailed distributions . . 31

2.7 The dangers from getting your probabilities wrong . . 35

3 Group Behavior: Crowds, Herds, and Video Games 41

3.1 Fire! . . 41

3.2 Birds, boids, and bicycles . . 44

3.3 The Monte Carlo world. . 48

3.4 Models with probabilities . . 50

3.5 People, properties, and political systems  . . 54

3.6 Connections to other chapters . . . 59

4 Evolution and Collapse: Game Playing in a ChangingWorld 61

4.1 My New Hampshire. . 61

4.2 Strategies and games. . 63

4.3 Iterated and evolutionary game playing .. . 68

4.4 Modeling the evolution of species and cultures . . 74

4.5 Implications for understanding collapse. . 80

5 Instability, Oscillation, and Feedback 85

5.1 Sharing an electric blanket and other challenges. . . 85

5.2 Primer on differential equations . . 91

5.3 Stable and unstable equilibriumpoints and related concepts . 97

5.4 The dynamics of interacting populations. . 100

5.5 Structural collapses and related processes. . 106

5.6 The science of trying to maintain control . . 112

5.7 The Chernobyl disaster  . . 115

6 Nonlinearity: Invitation to Chaos and Catastrophe 121

6.1 The elephant’s toenail . . 121

6.2 Local linearity . . 122

6.3 Bifurcations, tipping points, and catastrophes . . . 127

6.4 Hysteresis: where there may be no simple turning back . . 134

6.5 Chaos: beginning with a butterfly . . 138

7 It’s All About Networks 145

7.1 How’s your networking? . . 145

7.2 Network fundamentals . . 147

7.3 Important variations in network macrostructure . . 152

7.4 Unexpected network crashes. . 157

7.5 Interactive dynamics across networks  . . 161

7.6 Spreading processes through networks . . 165

7.7 A surprising game on a network  . . 167

7.8 Networks in an evolutionary context  . . 169

8 Putting It All Together: Looking at Collapse Phenomena in “6-D” 173

8.1 A quick review  . . 173

8.2 The utility of multiple perspectives in understanding the risk of collapse . 175

8.3 Where to go from here: the modern field of complexity theory . . 186

References 189

Index 201

About the Author 207

Mentoring In Mathematics Teaching

Barbara Jaworski, Anne Watson

Routledge |1994 | 161 páginas | rar - pdf | 12,4 Mb

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The arena in which the preparation of student-teachers for the teaching of mathematics takes place is shifting its foundations and moving its boundaries. The whole basis of teacher education at secondary level is in flux with a move towards teacher-education programmes which are largely school based. Increasingly, there is seen to be an important role for the school teacher who acts as mentor to the student teacher in some relationship with a tutor from the initial training i nstitution.; Teachers who are being encouraged to take on the mentoring role need preparation for its demands and teacher education courses need increasingly to make provision for the education and support of mentors. The purpose of this book is to discuss the mentoring process, to provide ideas and to highlight issues. It provides both practical help and guidance, and a philosophical consideration of the development of mathematics teachers and teaching.

Contents
Preface vi
Where to Start Reading
1 A Mentor's Eye View 1
Anne Watson
2 A Focus on Learning to Teach 13
Peter Gates
3 Mathematics and Mentoring 29
Susan E. Sanders
4 Working Together: Roles and Relationships in the Mentoring Process 41
Rita Nolder, Stephanie Smith and jean Melrose
5 Reflective Practice 52
Stephen Lerman
6 Planning for Learning 65
Pat Perks and Stephanie Prestage
7 Interpreting the Mathematics Curriculum 83
Doug French
8 The Wider Curriculum 96
Barrie Galpin and Simon Haines
9 Evaluation and Judgment 110
Maggie Crosson and Christine Shiu
10 Mentoring, Co-mentoring and the Inner Mentor 124
Barbara Jaworski and Anne Watson
Notes on Contributors 13

Proceedings of the First International Congress on Mathematical Education

ICME-1    1969      Lyon (France)

D. Reidel Publishing Company | 1969 | 286 páginas | pdf | 27,4 Mb
online: mathematik.uni-bielefeld.de

djvu - 13,8 Mb
online: mathematik.uni-bielefeld.de

All the papers of the congress are also published in 
Educational Studies in Mathematics (1969-70), Vol. 2, 134-418. 

Contents

H. FREUDENTHAL, Allocution (p. 3) 
B. CHRISTIANSEN, Induction and Deduction in the Learning of Mathematics and in Mathematical Instruction (p. 7) 
W. SERVAIS, Logique et enseignement mathématique (p. 28) 
J. V. ARMITAGE, The Relation between Abstract and 'Concrete' Mathematics at School (p. 48) 
R. GAUTHIER, Essai d'individualisation de l'enseignement (Enfants de dix à quatorze ans)(p. 57) 
G. G. MASLOVA, Le développement des idées et des concepts mathématiques fondamentaux dans l'enseignement des enfants de 7 a 15 ans (p. 69) 
A. ROUMANET, Une classe de mathématique: motivations et méthodes (p. 80) 
E. G. BEGLE, The Role of Research in the Improvement of Mathematics Education (p. 100) 
A. DELESSERT, De quelques problèmes touchant à la formation des maîtres de mathématiques (p. 113) 
A. ENGEL, The Relevance of Modern Fields of Applied Mathematics for Mathematical Education (p. 125) 
A. REVUZ, Les premiers pas en analyse (p. 138) 
A. MARKOUCHEVITCH, Certains problèmes de l'enseignement des mathématiques à l'école (p. 147) 
E. FISHBEIN, Enseignement mathématique et développement intellectuel (p. 158) 
E. CASTELNUOVO, Différentes représentations utilisant la notion de barycentre (p. 175) 
F. PAPY, Minicomputer (p. 201) 
B. THWAITES, The Role of the Computer in School Mathematics (p. 214) 
Z. KRYGOWSKA, Le texte mathématique dans l' enseignement (p. 228) 
H.-G. STEINER, Magnitudes and Rational Numbers - A Didactical Analysis (p. 239) 
H. O. POLLAK, How Can we Teach Applications of Mathematics? (p. 261) 
P. C. ROSENBLOOM, Vectors and Symmetry (p. 273) Resolutions (English) (p. 284) Résolutions (French) (p. 285) 

Equations from God : pure mathematics and Victorian faith

(Johns Hopkins Studies in the History of Mathematics)

Daniel J. Cohen

Johns Hopkins University Press | 2007 | 255 páginas | pdf | 1,1 Mb

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Throughout history, application rather than abstraction has been the prominent driving force in mathematics. From the compass and sextant to partial differential equations, mathematical advances were spurred by the desire for better navigation tools, weaponry, and construction methods. But the religious upheaval in Victorian England and the fledgling United States opened the way for the rediscovery of pure mathematics, a tradition rooted in Ancient Greece.

In Equations from God, Daniel J. Cohen captures the origins of the rebirth of abstract mathematics in the intellectual quest to rise above common existence and touch the mind of the deity. Using an array of published and private sources, Cohen shows how philosophers and mathematicians seized upon the beautiful simplicity inherent in mathematical laws to reconnect with the divine and traces the route by which the divinely inspired mathematics of the Victorian era begot later secular philosophies

Contents

Acknowledgments ix

introduction The Allure of Pure Mathematics in the Victorian Age 1

chapter one Heavenly Symbols: Sources of Victorian Mathematical Idealism 14

chapter two God and Math at Harvard: Benjamin Peirce and the Divinity of Mathematics 42

chapter three George Boole and the Genesis of Symbolic Logic 77

chapter four Augustus De Morgan and the Logic of Relations 106

chapter five Earthly Calculations: Mathematics and Professionalism in the Late Nineteenth Century 137

Math For Real Life For Dummies

Barry Schoenborn

For Dummies | 2013 | 291 páginas | rar - pdf | 5,2 Mb

link (password: matav)

Learn to:

• Get the skills you need to tackle everyday math problems
• Know which math to use when shopping, driving, and investing
• Brush up on basic math skills and concepts

Brush up on the math skills you need in your everyday life

Need to sharpen your math skills to handle everyday encounters, like calculating restaurant tips, understanding interest rates, and figuring out percentages and odds? Packed with real-world examples, Math For Real Life For Dummies gives you plain-English explanations of the simple math formulas and operations you're likely to encounter in the workplace, the kitchen, and even when playing games.

• Brush up on the basics — get basic math from counting and simple arithmetic to broader concepts like statistics
• Math to the rescue — discover how to do the calculations that spring up while shopping, cooking, dining out, trying to lose weight, and more
• Manage money with math — get a handle on the general principles, specialized terms, and strategies to create a budget, better manage your bank account and check register, avoid credit card debt, and invest more wisely
• Room for improvement —see calculations you can do in your head and games you can play to build your math skills and sharpen your critical thinking

Open the book and find:

• Math tricks that help you perform quick calculations in your head
• Ways to apply math formulas and principles to craft, home improvement, and yard projects
• How to make conversions, understand statistics, calculate odds, use percentages, and more
• Math solutions for day-to-day activities such as cooking, dining out, shopping, and banking
• Ten games that build math skills

Contents at a Glance

Introduction........... 1

Part I: Boning Up on Math Basics.... 7

Chapter 1: Awesome Operations: Math Fundamentals.... 9

Chapter 2: High School Reunion: Revisiting Key Principles of Algebra and Geometry... 27

Chapter 3: Becoming a Believer: Conversion, Statistics, Probability and More ... 47

Chapter 4: The Miracle of Mental Math....... 59

Part II: Math for Everyday Activities....... 71

Chapter 5: Let’s Make a Deal! Math You Use When Shopping...... 73

Chapter 6: Mmm, Mmm, Good: Kitchen Calculations............. 97

Chapter 7: It Does a Body Good: Math for Health and Well-Being.... 109

Chapter 8: Putting Geometry to Work at Home... 129

Chapter 9: Math and Statistics around Town and on the Road..... 141

Part III: Math to Manage Your Personal Finances... 161

Chapter 10: Budgets, Bank Accounts, Credit Cards, and More... 163

Chapter 11: Key Principles of Investment Math..... 183

Chapter 12: Covering Your Assets: Insurance Math.... 203

Chapter 13: Taking Math to Work.... 215

Chapter 14: How Taxing! (Almost) Understanding the Government.... 229

Part IV: The Part of Tens... 241

Chapter 15: Ten Quick Calculations You Can Do in Your Head..... 243

Chapter 16: Ten Activities That Build Math Skills... 249

Index........ 255