A History of Ancient Mathematical Astronomy

 (Studies in the History of Mathematics and Physical Sciences) 

O. Neugebauer

Springer | 2013 | reprint of the original 1st ed. 1975 edition | páginas | rar - pdf | 23,8 Mb

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"This monumental work will henceforth be the standard interpretation of ancient mathematical astronomy. It is easy to point out its many virtues: comprehensiveness and common sense are two of the most important. Neugebauer has studied profoundly every relevant text in Akkadian, Egyptian, Greek, and Latin, no matter how fragmentary; [...] With the combination of mathematical rigor and a sober sense of the true nature of the evidence, he has penetrated the astronomical and the historical significance of his material. [...] His work has been and will remain the most admired model for those working with mathematical and astronomical texts.

D. Pingree in Bibliotheca Orientalis, 1977

"... a work that is a landmark, not only for the history of science, but for the history of scholarship. HAMA [History of Ancient Mathematical Astronomy] places the history of ancient Astronomy on a entirely new foundation. We shall not soon see its equal.

N.M. Swerdlow in Historia Mathematica, 1979

Geometry and Algebra in Ancient Civilizations

Bartel L. van der Waerden 

Springer | 2011 - reprint of the original 1st ed. 1983 edition |235 páginas | rar - pdf | 5,3 Mb

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Contents
1. Pythagorean Triangles.
A. Written Sources.Fundamental Notions.- The Text Plimpton 322.- A Chinese Method.- Methods Ascribed to Pythagoras and Plato.- Pythagorean Triples in India.- The Hypothesis of a Common Origin.- Geometry and Ritual in Greece and India.- Pythagoras and the Ox. B. Archaeological Evidence.Prehistoric Ages.- Radiocarbon Dating.- Megalithic Monuments in Western Europe.- Pythagorean Triples in Megalithic Monuments.- Megalith Architecture in Egypt.- The Ritual Use of Pythagorean Triangles in India.- C. On Proofs, and on the Origin of Mathematics.Geometrical Proofs.- Euclid’s Proof.- Naber’s Proof.- Astronomical Applications of the Theorem of Pythagoras?.- Why Pythagorean Triangles?.- The Origin of Mathematics.- 
2. Chinese and Babylonian Mathematics 
A. Chinese Mathematics.The Chinese “Nine Chapters”.- The Euclidean Algorithm.- Areas of Plane Figures.- Volumes of Solids.- The Moscow Papyrus.- Similarities Between Ancient Civilizations.- Square Roots and Cube Roots.- Sets of Linear Equations.- Problems on Right-Angled Triangles.- The Broken Bamboo.- Two Geometrical Problems.- Parallel Lines in Triangles. B. Babylonian Mathematics.A Babylonian Problem Text.- Quadratic Equations in Babylonian Texts.- The Method of Elimination.- The “Sum and Difference” Method.C. General Conclusions.Chinese and Babylonian Algebra Compared.- The Historical Development.
3. Greek Algebra
What is Algebra?.- The Role of Geometry in Elementary Algebra.- Three Kinds of Algebra.- On Units of Length, Area, and Volume.- Greek “Geometric Algebra”.- Euclid’s Second Book.- The Application of Areas.- Three Types of Quadratic Equations.- Another Concordance Between the Babylonians and Euclid.- An Application of II, 10 to Sides and Diagonals.- Thales and Pythagoras.- The Geometrization of Algebra.- The Theory of Proportions.- Geometric Algebra in the “Konika” of Apollonios.- The Sum of a Geometrical Progression.- Sums of Squares and Cubes.
4. Diophantos and his Predecessors.
A. The Work of Diophantos.Diophantos’ Algebraic Symbolism.- Determinate and Indeterminate Problems.- From Book A.- From Book B.- The Method of Double Equality.From Book ?.- From Book 4.- From Book 5.- From Book 7.- From Book ?.- From Book E.- B. The Michigan Papyrus 620.- C. Indeterminate Equations in the Heronic Collections.
5. Diophantine Equations.
A. Linear Diophantine Equations.Aryabhata’s Method.- Linear Diophantine Equations in Chinese Mathematics.- The Chinese Remainder Problem.- Astronomical Applications of the Pulverizer.- Aryabhata’s Two Systems.- Brahmagupta’s System.- The Motion of the Apogees and Nodes.- The Motion of the Planets.- The Influence of Hellenistic Ideas.B. PelVs Equation.The Equation x2= 2y2±l.- Periodicity in the Euclidean Algorithm.- Reciprocal Subtraction.- The Equations x2= 3y2 +1 and x2= 3y2— 2.- Archimedes’ Upper and Lower Limits for w3.- Continued Fractions.- The Equation x2= Dy2± 1 for Non-squareD.- Brahmagupta’s Method.- The Cyclic Method.- Comparison Between Greek and Hindu Methods.C. Pythagorean Triples
6. Popular Mathematics
A. General Character of Popular Mathematics.B. Babylonian, Egyptian and Early Greek ProblemsTwo Babylonian Problems.- Egyptian Problems.- The “Bloom of Thymaridas”.C. Greek Arithmetical EpigramsD. Mathematical Papyri from Hellenistic Egypt.Calculations with Fractions.- Problems on Pieces of Cloth.- Problems on Right-Angled Triangles.- Approximation of Square Roots.- Two More Problems of Babylonian Type.- E. Squaring the Circle and Circling the SquareAn Ancient Egyptian Rule for Squaring the Circle.- Circling the Square as a Ritual Problem.- An Egyptian Problem.- Area of the Circumscribed Circle of a Triangle.- Area of the Circumscribed Circle of a Square.- Three Problems Concerning the Circle Segment in a Babylonian Text.- Shen Kua on the Arc of a Circle Segment.F. Heron of Alexandria.The Date of Heron.- Heron’s Commentary to Euclid.- Heron’s Metrika.- Circles and Circle Segments.- Apollonios’ Rapid Method.- Volumes of Solids.- Approximating a Cube RootG. The Mishnat ha-Middot.
7. Liu Hui and Aryabhata.
A. The Geometry of Liu Hui.The “Classic of the Island in the Sea”.- First Problem: The Island in the Sea.- Second Problem: Height of a Tree.- Third Problem: Square Town.- The Evaluation of ?.- The Volume of a Pyramid.- Liu Hui and Euclid.- Liu Hui on the Volume of a Sphere.B. The Mathematics of Aryabhata.Area and Circumference of a Circle.- Aryabhata’s Table of Sines.- On the Origin of Aryabhata’s Trigonometry.- Apollonios and Aryabhata as Astronomers.- On Gnomons and Shadows.- Square Roots and Cube Roots.- Arithmetical Progressions and Quadratic Equations.

The Early Growth of Logic in the Child: Classification and Seriation

International Library of Psychology

Jean Piaget, Barbel Inhelder

Routledge | 1999 | 329 páginas | rar - pdf | 6,1 Mb

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CONTENTS
PREFACE page ix
TRANSLATOR’S INTRODUCTION
I. Classification and Inference xi
II. Co-ordination xv
III. Two Levels of Inference xx
INTRODUCTION 1
1. Language 2
2. Maturation 5
3. Perceptual Factors 5
4. Sensori-motor Schemata 11
I. GRAPHIC COLLECTIONS 17
1. Preliminary Statement of the Problem 17
2. General Results Obtained with Flat Geometrical Shapes 21
3. The Links between Graphic Collections and Classification: Further Illustrative Material Using Geometrical Shapes 30
4. “Similarity” and “Belonging” in the Grouping of Small Toys 36
5. Conclusion: Graphic Collections as a First Attempt to Synthesize Intension and Extension 44
II. NON-GRAPHIC COLLECTIONS 47
1. Statement of the Problem: Criteria of an Additive Classificatory Structure 47
2. Non-Graphic Collections as seen with Geometrical Shapes 51
3. Non-Graphic Collections as seen with Arbitrary Objects 56
III. “ALL” AND “SOME” : CONDITIONS OF CLASS-INCLUSION 59
1. “All” and “Some” applied to Shapes and Colours 60
2. “All” and “Some” applied to Tests of Exclusion 74
3. The Absolute and Relative uses of “Some” 89
4. Conclusions. “Some” and “All”, Inclusion and the Relations between Intension and Extension 97
IV. CLASS INCLUSION AND HIERARCHICAL CLASSIFICATIONS 100
1. Classification of Flowers (Mixed with Other Objects) 101
2. Classification of Animals 110
V. COMPLEMENTARY CLASSES page 119
1. The Singular Class in a Practical Context 120
2. Classification and the Relative Size of Classes 125
3. The “Secondary” Class in a Forced Dichotomy 129
4. Negation 137
5. The Inclusion of Complementary Classes and the Duality Principle 142
6. The Null Class 146
7. Conclusion 149
VI. MULTIPLICATIVE CLASSIFICATION (MATRICES) 151
1. Statement of the Problem 151
2. Matrices Tests, I : Results 154
3. Matrix Tests (Standardized Procedure) 159
4. Spontaneous Cross-Classification 165
5. Spontaneous Cross-Classification Continued 171
6. Simple Multiplication (or Intersection) 176
7. Addition and Multiplication 184
8. The Quantification of Multiplicative Classes 188
9. Conclusions 195
VII. FLEXIBILITY IN HINDSIGHT AND FORESIGHT 196
1. Rearrangements Caused by the Addition of New Elements 197
2. Changes of Criterion Requiring the Rearrangement of Existing Classifications 208
3. Anticipation, Execution and Change of Criterion in Partly Spontaneous Classifications 216
VIII. THE CLASSIFICATION OF ELEMENTS PERCEIVED BY TOUCH 232
1. Experimental Prodecure 232
2. Stage I : Choice of Known Elements and Graphic Collections. No Anticipation and No Complete Classifications 234
3. Stage II: Non-Graphic Collections; Discovery of a Single Criterion first by Trial-and-Error, then by Semi-Anticipation ; Difficulty in Finding Others 238
4. Stage III : Anticipation of Two or Three Criteria. Conclusions
IX. SERIATION 247
1. Statement of the Problem 247
2. Sériation and the Anticipation of Serial Configurations with Elements Perceived Visually 250
3. Tactile Sériation and its Anticipation in Drawings 261
X. MULTIPLE SERIATION 269
1. Experimental Procedure 269
2. Stage I : No True Sériation 270
3. Stage II: Spontaneous Sériation of One of the Two Properties, but Failure in the Multiplicative Synthesis of Both page 272
4. Stage III : Successful Multiplication 274
CONCLUSIONS 280

Educational Studies in Mathematics, Vol 2

Springer Netherlands | 1969 - 1970

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1-15: Patrick Suppes, Elizabeth F. Loftus and Max Jerman -> Problem-solving on a computer-based teletype
16-31: E. Fischbein, Ileana Pampu and I. Mînzat -> Initiation aux probabilités à l'école élémentaire
32-58: Hans Freudenthal -> A teachers course colloquium on sets and logic
59-68: Floyd R. Vest -> A catalog of models for the operations of addition and subtraction of whole numbers
69-79: Maurice Glaymann -> Initiation to vector spaces
80-114: ICMI report on mathematical contests in secondary education (olympiads) I
115-122: M. Bruckheimer and N. Gowar -> Apparent conflicts in maths education
123-133: Erich Wittmann -> The development of self-reliant thinking in mathematics teaching
135-138: Hans Freudenthal -> Allocution du premier congrès international de l'enseignement mathématique lyon, 24–31 Août 1969
139-159: Bent Christiansen -> Induction and deduction in the learning of mathematics and in mathematical instruction
160-179: W. Servais -> Logique et enseignement mathématique
180-188: J. V. Armitage -> The relation between abstract and ‘concrete’ mathematics at school
189-200: R. Gauthier -> Essai d'individualisation de l'enseignement
201-211: G. G. Maslova -> Le développement des idées et des concepts mathématiques fondamentaux dans l'enseignement des enfants de 7 à 15 ans
212-231: A. Roumanet -> Une classe de mathématique: Motivations et méthodes
232-244: E. G. Begle -> The role of research in the improvement of mathematics education
245-256: A. Delessert -> De quelques problèmes touchant à la formation des maîtres de mathématiques
257-269: Arthur Engel -> The relevance of modern fields of applied mathematics for mathematical education
270-278: André Revuz -> Les premiers pas en analyse
279-289: A. Markouchevitch -> Certains problèmes de l'enseignement des mathéma atiques à l'école
290-306: E. Fischbein -> Enseignement mathématique et développement intellectuel
307-332: Emma Castelnuovo -> Différentes représentations utilisant la notion de barycentre
333-345: Frédérique Papy -> Minicomputer
346-359: Bryan Thwaites -> The role of the computer in school mathematics
360-370: Zofia Krygowska -> Le texte mathématique dans l'enseignement
371-392: Hans-Georg Steiner -> Magnitudes and rational numbers—A didactical analysis
393-404: H. O. Pollak -> How can we teach applications of mathematics?
405-414: Paul C. Rosenbloom -> Vectors and symmetry
416: Resolutions of the First International Congress on Mathematical Education
417-418: Résolutions du Premier Congrès International de l'Enseignement Mathématique
419-429: Ferenc Genzwein -> The system and the organization of further training for the mathematics teachers of the secondary schools in Budapest
430-437: E. Georgescu-Buzāu, N. Matei and Gr. Bānescu -> The importance of appropriate problems in the teaching of mathematics
438-445: Max Jerman -> A counting model for simple addition
446-468: Richard S. Long, Nancy S. Meltzer and Peter J. Hilton -> Research in mathematics education
469-475: E. E. Biggs -> Communication on primary education in mathematics
476-477: Bert K. Waits -> Relative effectiveness of two different television techniques and one large lecture technique for teaching large enrollment college mathematics courses
478-495: John C. Egsgard -> Some ideas in geometry that can be taught from K-6
496-500: Bruce R. Vogeli -> Sweep away all cows, ghosts, dragons and devils

Infinity and Truth

Chitat Chong, Qi Feng, Theodore A Slaman e W Hugh Woodin 

World Scientific Publishing Company | 2014 | páginas | rar - pdf | 1,6 Mb

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This volume is based on the talks given at the Workshop on Infinity and Truth held at the Institute for Mathematical Sciences, National University of Singapore, from 25 to 29 July 2011. The chapters are by leading experts in mathematical and philosophical logic that examine various aspects of the foundations of mathematics. The theme of the volume focuses on two basic foundational questions: (i) What is the nature of mathematical truth and how does one resolve questions that are formally unsolvable within the Zermelo Fraenkel Set Theory with the Axiom of Choice, and (ii) Do the discoveries in mathematics provide evidence favoring one philosophical view over others? These issues are discussed from the vantage point of recent progresses in foundational studies.

The final chapter features questions proposed by the participants of the Workshop that will drive foundational research. The wide range of topics covered here will be of benefit to students, researchers and mathematicians interested in the foundations of mathematics.

Readership: Mathematicians, philosophers, scientists, graduate students, academic institutions, and research organizations interested in logic and the philosophy of mathematics.

CONTENTS

Foreword vii

Preface ix

Section I. Invited Lectures

Absoluteness, Truth, and Quotients 1

Ilijas Farah

A Multiverse Perspective on the Axiom of Constructibility 25

Joel David Hamkins

Hilbert, Bourbaki and the Scorning of Logic 47

A. R. D. Mathias

Toward Objectivity in Mathematics 157

Stephen G. Simpson

Sort Logic and Foundations of Mathematics 171

Jouko Vaananen

Reasoning about Constructive Concepts 187

Nik Weaver

Perfect Infinities and Finite Approximation 199

Boris Zilber

Section II. Special Session

An Objective Justification for Actual Infinity? 225

Stephen G. Simpson

Oracle Questions 229

Theodore Slaman and W. Hugh Woodin

Study Guide for Practical Statistics for Educators

 Ruth Ravid e Elizabeth Oyer

Rowman & Littlefield Publishers | 2011 - 4ª edição | páginas | rar -pdf | 680 kb

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The fourth edition of the Study Guide for Practical Statistics for Educators serves as a user-friendly and accessible way for students to better understand, review, and apply the concepts introduced in its companion textbook, Practical Statistics for Educators (Ravid, 2011). Since the first edition of this study guide came out in 1994, thousands of students in educational statistics courses and their professors have found it to be an excellent guide with clear and easy-to-follow instructions and examples. The study guide allows students to reinforce and test their knowledge of the concepts addressed in each chapter of the textbook. At the end of each chapter, the best answer for each exercise is given, along with an explanation for why the correct answer is better than the other choices. New in this edition are accompanying Excel exercises, so students may perform data analysis with this commonly-used software, using data available on the web-based portal that accompanies the guide.

Contents
Preface vii
1 An Overview of Educational Research 1
2 Basic Concepts in Statistics 9
3 Organizing and Graphing Data 15
4 Measure of Central Tendency 27
5 Measures of Variability 33
6 The Normal Curve and Standard Scores 39
7 Interpreting Test Scores 45
8 Correlations 49
9 Prediction and Regression 57
10 t Test 65
11 Analysis of Variance 75
12 Chi Square 81
13 Reliability 87
14 Validity 91
15 Planning and Conducting Research Studies 95
About the Authors 101

Practical Statistics for Educators

Ruth Ravid

Rowman & Littlefield Publishers | 2010 - 4ª edição | 273 páginas | pdf | 1 Mb

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Practical Statistics for Educators, 4th edition focuses on the application of research and statistics as applied specifically to education. Since the first edition came out in 1994, thousands of students in educational statistics courses and their professors have found it to be an excellent textbook. Educational practitioners have also appreciated keeping this book on their reference shelf. Now in its fourth edition, this well-regarded text is a clear and easy-to-follow manual for use in introductory statistics or action research courses. Ruth Ravid concentrates on the essential concepts in educational statistics including when to use various statistical tests and how to interpret the results. Testing and test score interpretation, reliability, and validity are included to help students understand these topics which are essential for practitioners in education. 
Real-life examples, used generously throughout, are taken from the field of education and presented to illustrate the various concepts and terms. Chapter previews and summaries, as well as a glossary of the main terms and concepts, help readers navigate the book, focus on the most important points, and build upon the knowledge gained from each chapter.
New in this edition are updated and improved graphics, revised and enhanced text, and examples. Lengthy appendixes-tables are deleted and their relevant sections are integrated into the chapters. Detailed and complicated computational steps have also been eliminated.

Contents
Part I. Introduction 
Chapter 1: An Overview of Educational Research Basic (Pure), Applied, and Action Research Quantitative vs. Qualitative Research Experimental vs. Nonexperimental Research Summary 
Chapter 2: Basic Concepts in Statistics  Variables and Measurement Scales Populations and Samples Parameters and Statistics Methods of Sampling Sample Bias Size of Sample Parametric and Nonparametric Statistics Descriptive and Inferential Statistics Using Hypotheses in Research Probability and Level of Significance Errors in Decision Making Degrees of Freedom Effect Size Using Samples to Estimate Population Values Steps in the Process of Hypothesis Testing And Finally... Summary 
Part II. Descriptive Statistics 
Chapter 3: Organizing and Graphing Data Organizing Data Graphing Data Drawing Accurate Graphs Summary 
Chapter 4: Measures of Central Tendency Mode Median Mean Comparing the Mode, Median, and Mean Summary 
Chapter 5: Measures of Variability The Range Standard Deviation and Variance Summary 
Part III. The Normal Curve and Standard Scores 
Chapter 6: The Normal Curve and Standard Scores The Normal Curve Standard Scores Summary 
Chapter 7: Interpreting Test Scores Norm-Referenced Tests Criterion-Referenced Tests Summary 
Part IV. Measuring Relationships 
Chapter 8: Correlation Pearson Product Moment Factors Affecting the Correlation The Coefficient of Determination and Effect Size Intercorrelation Tables Correlation Tables Summary 
Chapter 9: Prediction and Regression Simple Regression Multiple Regression Summary 
Part V. Inferential Statistics 
Chapter 10: t test Hypotheses for t Tests Independent-Samples t Test An Example of a t Test for Independent Samples t Test for Paired Samples An Example of a t Test for Paired Samples t Test for a Single Sample An Example of a t Test for a Single Sample Summary 
Chapter 11: Analysis of Variance One-Way ANOVA Conceptualizing the One-Way ANOVA Hypotheses for a One-Way ANOVA The ANOVA Summary Table Further Interpretation of the F Ratio An Example of a One-Way ANOVA Post Hoc Comparisons Two-Way ANOVA Conceptualizing the Two-Way ANOVA Hypotheses for the Two-Way ANOVA Graphing the Interaction The Two-Way ANOVA Summary Table An Example of a Two-Way ANOVA Summary 
Chapter 12: Chi Square Test Assumptions for the Chi Square Test The Chi Square Test of Independence Summary 
Part VI. Reliability and Validity 
Chapter 13: Reliability Understanding the Theory of Reliability Methods of Assessing Reliability The Standard Error of Measurement Factors Affecting Reliability How High Should the Reliability Be? Summary 
Chapter 14: Validity Content Validity Criterion-Related Validity Concurrent Validity Predictive Validity Construct Validity Face Validity Assessing Validity Test Bias Summary Part Seven: Conducting Your Own Research 
Chapter 15: Planning and Conducting Research Studies Research Ethics The Research Proposal Introduction Literature Review Methodology References The Research Report Results Discussion Summary 
Chapter 16: Choosing the Right Statistical Test Choosing a Statistical Test: A Decision Flowchart Examples Scenarios