Mathematical Theory of Democracy

Andranik Tangian

Springer | 2014 | 629 páginas | rar - pdf | 3,7 Mb

link (password: matav)

The mathematical theory of democracy deals with selection of representatives who make decisions on behalf of the whole society. In this book, the notion of representativeness is operationalized with the index of popularity (the average percentage of the population whose opinion is represented on a number of issues) and the index of universality (the frequency of cases when the opinion of a majority is represented). These indices are applied to evaluate and study the properties of single representatives (e.g. president) and representative bodies (e.g. parliament, magistrate, cabinet, jury, coalition). To bridge representative and direct democracy, an election method is proposed that is based not on voting but on indexing candidates with respect to the electorate’s political profile. In addition, societal and non-societal applications are considered.

Contents
History: Athenian Democracy
Echoes of Democracy in Ancient Rome
Revival of Democracy in Italian Mediaval City-Republics
Enlightenment and the End of Traditional Democracy
Modernity and Schism in Understanding Democracy
Theory: Direct Democracy
Dictatorship and Democracy
Representative Democracy
Statistically Testing the Representative Capacity
Concluding Discussion: Bridging Representative and Direct Democracies
Applications: Simple Applications
Application to Collective Multicriteria Decisions
Application to Stock Exchange Predictions
Application to Traffic Control
Appendix: Computational Formulas
Probabilities of Unequal Choices by Vote and by Candidate Scores
Statistical Significance of Representative Capacity.

Mathematics Education as a Research Domain: A Search for Identity An ICMI Study Book 1.

(New ICMI Study Series)

Anna Sierpinska, Jeremy Kilpatrick

Springer | 1998 | 243 páginas | pdf | 15,8 Mb

link

(livro completo, Book 1 e Book 2, sem OCR)

In 1978, in the foreword to Weeding and Sowing: A Preface to a Science of Mathematics Education, Hans Freudenthal wrote that his book is a preface to a science that does not exist. Almost 20 years later, does his claim still hold true? The present book is the result of the reflection of many individuals in mathematics education on this and related questions. Is mathematics education a science? Is it a discipline? In what sense? What is its place within other domains of research and academic disciplines? What accounts for its specificity? In the book, the reader will find a range of possible answers to these questions, a variety of analyses of the actual directions of research in different countries, and a number of visions for the future of research in mathematics education. The book is a result of an ICMI Study, whose theme was formulated as: `What is Research in Mathematics Education and What are Its Results?'. One important outcome of this study was the realization of the reasons for the difficulty of the questions that the study was posing, leading possibly to a set of other questions, better suited to the actual concerns and research practices of mathematics education researchers. The book addresses itself to researchers in mathematics education and all those working in their neighborhood who are concerned with the problems of the definition of this new scientific domain emerging at their borders.

TABLE OF CONTENTS

BOOK 1

Foreword ix

Part I: The ICMI Study Conference

Discussion Document 3

List of Participants 9

What is the Specific Object of Study in Mathematics Education? Report of Working Group 1 

What are the Aims of Research in Mathematics Education? Report of Working Group 2 

What are the Specific Research Questions or Probtematiques of Research in Mathematics Education? Report of Working Group 3

What are the Results of Research in Mathematics Education? Report of Working Group 4

What Criteria Should Be Used to Evaluate the Results of Research in Mathematics Education? Report of Working Group 5 

ALAN J. BISHOP

Research, Effectiveness, and the Practitioners' World 

Part II: Mathematics Education as a Research Discipline

A Glance Over the Evolution of Research in Mathematics Education

Josette Adda

Balancing Complex Human Worlds: Mathematics Education as an Emergent Discipline in its own Right

Norma C. Presmeg

A Postmodern Perspective on Research in Mathematics Education

Paul Ernest

Mathematics Education as a ‘Design Science’

Erich Ch. Wittmann

What is Mathematics Education? A Survey of Mathematics Educators in Canada

Roberta Mura

Programs for the Education of Researchers in Mathematics Education

Gunnar Gjone

Part III: Goals, Orientations and Results of Research in Mathematics Education

The Aims of Research

Gilah C. Leder

Aiming Research Toward Understanding: Lessons We can Learn from Children

James Hiebert

Transforming the International Mathematics Education Research Agenda

Nerida F. Ellerton, M. A. Clements

Clarifying the Meaning of Mathematical Objects as a Priority Area for Research in Mathematics Education

Juan D. Godino, Carmen Batanero2

Research and Results in Mathematics Education: Some Contradictory Aspects

Paolo Boero, Julianna Radnai Szendrei

Models in Mathematics Education Research: A Broader View of Research Results

Carolyn Kieran

Towards A Cognitive Theory of Practice

Gérard Vergnaud

Why the professor can't teach

 Morris Kline 

St. Martin's Press | 1977 

online: marco-learningsystems.com

pdf - 666 kb

link

link1

This was an incredible read, almost prophetic. Prof. Kline gets right down to and nails several issues, and suggests many solutions to the grim situation when it comes to not only undergrad math education but college education in general. I recommend at least the first half of the book to any prospective undergrad in any technical field, or any disgruntled graduate thinking of gradschool or beyond especially at a research institution. Entire text is highly recommended for anyone thinking of making a real difference in either K-12 or higher ed whether science or engineering

Contents

Preface 1
1 The Vicious Circle 6
2 The Rise of American Mathematics 17
3 The Nature of Current Mathematical Research 41
4 The Conflict Between Research and Teaching 70
5 The Debasement of Undergraduate Teaching 96
6 The Illiberal Mathematician 111
7 The Undefiled Mathematician 139
8 The Misdirection of High School Education 161
9 Some Light at the Beginning of the Tunnel:
Elementary Education 183
10 Follies of the Marketplace:
A Tirade on Texts 208
11 Some Mandatory Reforms 235
Bibliography 272

Good Math. A Geek’s Guide to the Beauty of Numbers, Logic, and Computation

Mark C. Chu-Carroll

Pragmatic Bookshelf  | 2013 | 270 páginas | pdf | 5,7 Mb

link

epub - 2 Mb - link

Mathematics is beautiful--and it can be fun and exciting as well as practical. Good Math is your guide to some of the most intriguing topics from two thousand years of mathematics: from Egyptian fractions to Turing machines; from the real meaning of numbers to proof trees, group symmetry, and mechanical computation. If you've ever wondered what lay beyond the proofs you struggled to complete in high school geometry, or what limits the capabilities of computer on your desk, this is the book for you.
Why do Roman numerals persist? How do we know that some infinities are larger than others? And how can we know for certain a program will ever finish? In this fast-paced tour of modern and not-so-modern math, computer scientist Mark Chu-Carroll explores some of the greatest breakthroughs and disappointments of more than two thousand years of mathematical thought. There is joy and beauty in mathematics, and in more than two dozen essays drawn from his popular "Good Math" blog, you'll find concepts, proofs, and examples that are often surprising, counterintuitive, or just plain weird.Mark begins his journey with the basics of numbers, with an entertaining trip through the integers and the natural, rational, irrational, and transcendental numbers. The voyage continues with a look at some of the oddest numbers in mathematics, including zero, the golden ratio, imaginary numbers, Roman numerals, and Egyptian and continuing fractions. After a deep dive into modern logic, including an introduction to linear logic and the logic-savvy Prolog language, the trip concludes with a tour of modern set theory and the advances and paradoxes of modern mechanical computing.If your high school or college math courses left you grasping for the inner meaning behind the numbers, Mark's book will both entertain and enlighten you.


Contents
Preface . . . . . . . . . . . xi
Part I — Numbers
1. Natural Numbers . . . . . . . . . 3
1.1 The Naturals, Axiomatically Speaking 4
1.2 Using Peano Induction 7
2. Integers . . . . . . . . . . . 9
2.1 What’s an Integer? 9
2.2 Constructing the Integers—Naturally 11
3. Real Numbers . . . . . . . . . 15
3.1 The Reals, Informally 15
3.2 The Reals, Axiomatically 18
3.3 The Reals, Constructively 20
4. Irrational and Transcendental Numbers . . . 23
4.1 What Are Irrational Numbers? 23
4.2 The Argh! Moments of Irrational Numbers 24
4.3 What Does It Mean, and Why Does It Matter? 26
Part II — Funny Numbers
5. Zero . . . . . . . . . . . 31
5.1 The History of Zero 31
5.2 An Annoyingly Difficult Number 34
6. e: The Unnatural Natural Number . . . . . 37
6.1 The Number That’s Everywhere 37
6.2 History 39
6.3 Does e Have a Meaning? 40
7. φ: The Golden Ratio . . . . . . . . 41
7.1 What Is the Golden Ratio? 42
7.2 Legendary Nonsense 44
7.3 Where It Really Lives 45
8. i: The Imaginary Number . . . . . . . 47
8.1 The Origin of i 47
8.2 What i Does 49
8.3 What i Means 50
Part III — Writing Numbers
9. Roman Numerals . . . . . . . . 55
9.1 A Positional System 55
9.2 Where Did This Mess Come From? 57
9.3 Arithmetic Is Easy (But an Abacus Is Easier) 58
9.4 Blame Tradition 61
10. Egyptian Fractions . . . . . . . . 65
10.1 A 4000-Year-Old Math Exam 65
10.2 Fibonacci’s Greedy Algorithm 66
10.3 Sometimes Aesthetics Trumps Practicality 68
11. Continued Fractions . . . . . . . . 69
11.1 Continued Fractions 70
11.2 Cleaner, Clearer, and Just Plain Fun 72
11.3 Doing Arithmetic 74
Part IV — Logic
12. Mr. Spock Is Not Logical . . . . . . . 79
12.1 What Is Logic, Really? 81
12.2 FOPL, Logically 82
12.3 Show Me Something New! 86
13. Proofs, Truth, and Trees: Oh My! . . . . . 91
13.1 Building a Simple Proof with a Tree 92
13.2 A Proof from Nothing 94
13.3 All in the Family 96
13.4 Branching Proofs 98
14. Programming with Logic . . . . . . . 103
14.1 Computing Family Relationships 104
14.2 Computation with Logic 108
15. Temporal Reasoning . . . . . . . . 117
15.1 Statements That Change with Time 118
15.2 What’s CTL Good For? 123
Part V — Sets
16. Cantor’s Diagonalization: Infinity Isn’t Just Infinity . . . 127
16.1 Sets, Naively 128
16.2 Cantor’s Diagonalization 131
16.3 Don’t Keep It Simple, Stupid 135
17. Axiomatic Set Theory: Keep the Good, Dump the Bad .. . 139
17.1 The Axioms of ZFC Set Theory 140
17.2 The Insanity of Choice 147
17.3 Why? 150
18. Models: Using Sets as the LEGOs of the Math World . . . 153
18.1 Building Natural Numbers 154
18.2 Models from Models: From Naturals to Integers and Beyond! 156
19. Transfinite Numbers: Counting and Ordering Infinite Sets . . . . 161
19.1 Introducing the Transfinite Cardinals 161
19.2 The Continuum Hypothesis 163
19.3 Where in Infinity? 164
20. Group Theory: Finding Symmetries with Sets . . 167
20.1 Puzzling Symmetry 167
20.2 Different Kinds of Symmetry 171
20.3 Stepping into History 173
20.4 The Roots of Symmetry 176
Part VI — Mechanical Math
21. Finite State Machines: Simplicity Goes Far . . . 183
21.1 The Simplest Machine 183
21.2 Finite State Machines Get Real 187
21.3 Bridging the Gap: From Regular Expressions to Machines 189
22. The Turing Machine . . . . . . . . 197
22.1 Adding a Tape Makes All the Difference 198
22.2 Going Meta: The Machine That Imitates Machines 203
23. Pathology and the Heart of Computing . . . 209
23.1 Introducing BF: The Great, the Glorious, and the Completely Silly 211
23.2 Turing Complete, or Completely Pointless? 214
23.3 From the Sublime to the Ridiculous 215
24. Calculus: No, Not That Calculus—λ Calculus . . 219
24.1 Writing λ Calculus: It’s Almost Programming! 220
24.2 Evaluation: Run It! 224
24.3 Programming Languages and Lambda Strategies 226
25. Numbers, Booleans, and Recursion . . . . 231
25.1 But Is It Turing Complete? 231
25.2 Numbers That Compute Themselves 232
25.3 Decisions? Back to Church 235
25.4 Recursion: Y Oh Y Oh Y? 237
26. Types, Types, Types: Modeling λ Calculus . . . 243
26.1 Playing to Type 244
26.2 Prove It! 249
26.3 What’s It Good For? 250
27. The Halting Problem . . . . . . . 253
27.1 A Brilliant Failure 254
27.2 To Halt or Not To Halt? 256

Bibliography . . . . . . . . . 261

Collaborating to Meet Language Challenges in Indigenous Mathematics Classrooms

 

Tamsin Meaney, Tony Trinick e Uenuku Fairhall

(Mathematics Education Library, 52)

Springer | 2012 | 322 páginas | rar - pdf | 3,85 Mb

link (password: matav)

Language can be simultaneously both a support and a hindrance to students’ learning of mathematics. When students have sufficient fluency in the mathematics register so that they can discuss their ideas, they become chiefs who are able to think mathematically. However, learning the mathematics register of an Indigenous language is not a simple exercise and involves many challenges not only for students, but also for their teachers and the wider community. Collaborating to Meet Language Challenges in Indigenous Mathematics Classrooms identifies some of the challenges—political, mathematical, community based, and pedagogical— to the mathematics register, faced by an Indigenous school, in this case a Mäori immersion school. It also details the solutions created by the collaboration of teachers, researchers and community members.

Contents

1 Introduction

pt. 1. Meeting political challenges 

2 The Development of a Mathematics Register in an Indigenous Language
3 The History of Te Kura Kaupapa M¯aori o Te Koutu – The Politicisation of a Local Community
4 It Is Kind of Hard to Develop Ideas When You Can’t Understand the Question: Doing Exams Bilingually
pt. 2. Meeting mathematical challenges 

5 The Resources in Te Reo M¯aori for Students to Think Mathematically
6 Writing to Help Students Think Mathematically
7 The Case of Probability
pt. 3. Meeting community challenges 

8 Using the Mathematics Register Outside the Classroom

9 Teachers as Learners of the Mathematics Register

pt. 4. Meeting pedagogical challenges.

10 “They Don’t Use the Words Unless You Really Teach Them”: Mathematical Register Acquisition Mode
11 “M¯aori were Traditional Explorers”: M¯aori Pedagogical Practices
12 “And That’s What You Want to Happen. You Want the Shift in Classroom Practice”
13 Meeting Challenges

Strategies to Integrate the Arts in Mathematics

Linda Dacey e Lisa Donovan 

Shell Education Pub | 2013 | 280 páginas | rar - pdf | 20 Mb

link
password: matav

This teacher-friendly resource provides practical arts-based strategies for classroom teachers to use in teaching mathematics content. Overview information and model lessons are provided for each strategy and ideas are provided for grades K-2, 3-5, 6-8, and 9-12. The strategies addressed within the book allow teachers to make mathematics instruction come alive and best meet students' needs.

Table of Contents 5 
Correlation to the Standards 21 
Creative Movement 23 
Drama 67 
Music 105 
Poetry 143 
Storytelling 183 
Visual Arts 225 
Appendix A: References Cited 268 
Appendix B: Assessment Forms 273 
Appendix C: Recommended Resources 277 
Appendix D: Contents of the Digital Resource CD 279 

Trigonometry For Dummies

Mary Jane Sterling

 For Dummies | 2014 - 2ª edição | 387 páginas | rar - pdf | 4 Mb


link (password : matav)

A plain-English guide to the basics of trig

Trigonometry deals with the relationship between the sides and angles of triangles... mostly right triangles. In practical use, trigonometry is a friend to astronomers who use triangulation to measure the distance between stars. Trig also has applications in fields as broad as financial analysis, music theory, biology, medical imaging, cryptology, game development, and seismology.

From sines and cosines to logarithms, conic sections, and polynomials, this friendly guide takes the torture out of trigonometry, explaining basic concepts in plain English and offering lots of easy-to-grasp example problems. It also explains the "why" of trigonometry, using real-world examples that illustrate the value of trigonometry in a variety of careers.

• Tracks to a typical Trigonometry course at the high school or college level

• Packed with example trig problems

• From the author of Trigonometry Workbook For Dummies

Trigonometry For Dummies is for any student who needs an introduction to, or better understanding of, high-school to college-level trigonometry.

Contents at a Glance

Introduction........ 1

Part I: Getting Started with Trigonometry......... 5

Chapter 1: Trouncing Trig Technicalities..........7

Chapter 2: Coordinating Your Efforts with Cartesian Coordinates...29

Chapter 3: Functioning Well...............47

Chapter 4: Getting Your Degree.............57

Chapter 5: Dishing Out the Pi: Radians.........67

Chapter 6: Getting It Right with Triangles...........81

Part II: Trigonometric Functions........... 91

Chapter 7: Doing Right by Trig Functions..................93

Chapter 8: Trading Triangles for Circles: Circular Functions....109

Chapter 9: Defining Trig Functions Globally........121

Chapter 10: Applying Yourself to Trig Functions............135

Part III: Identities...... 155

Chapter 11: Identifying Basic Identities.......157

Chapter 12: Operating on Identities..........171

Chapter 13: Proving Identities: The Basics..........189

Chapter 14: Sleuthing Out Identity Solutions.........207

Part IV: Equations and Applications........ 223

Chapter 15: Investigating Inverse Trig Functions........225

Chapter 16: Making Inverse Trig Work for You.....233

Chapter 17: Solving Trig Equations........243

Chapter 18: Obeying the Laws.........265

Part V: The Graphs of Trig Functions..... 289

Chapter 19: Graphing Sine and Cosine...........291

Chapter 20: Graphing Tangent and Cotangent........307

Chapter 21: Graphing Other Trig Functions.......317

Chapter 22: Topping Off Trig Graphs........329
Part VI: The Part of Tens.................. 343
Chapter 23: Ten Basic Identities . . . Plus Some Bonuses....345
Chapter 24: Ten Not-So-Basic Identities............349
Appendix: Trig Functions Table......... 353
Index........ 357

Livro relacionado:

Trigonometry workbook for dummies por Mary Jane Sterling Idioma: Inglês   Editora: Hoboken, N.J. ; [Great Britain] : Wiley, ©2005.