Games and Mathematics: Subtle Connections

David Wells

 Cambridge University Press |  2012 | 258 páginas | rar - pdf |1 Mb

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The appeal of games and puzzles is timeless and universal. In this unique book, David Wells explores the fascinating connections between games and mathematics, proving that mathematics is not just about tedious calculation but imagination, insight and intuition. The first part of the book introduces games, puzzles and mathematical recreations, including knight tours on a chessboard. The second part explains how thinking about playing games can mirror the thinking of a mathematician, using scientific investigation, tactics and strategy, and sharp observation. Finally the author considers game-like features found in a wide range of human behaviours, illuminating the role of mathematics and helping to explain why it exists at all. This thought-provoking book is perfect for anyone with a thirst for mathematics and its hidden beauty; a good high school grounding in mathematics is all the background that is required, and the puzzles and games will suit pupils from 14 years.

Contents
PART I: Mathematical recreations and abstract games
Introduction; Everyday puzzles;
1 Recreations from Euler to Lucas;
Euler and the Bridges of Königsberg; Euler and knight tours; Lucas and mathematical recreations; Lucass game of solitaire calculation;
2 Four abstract games;
From Dudeneys puzzle to Golombs Game; Nine Mens Morris; Hex; Chess; Go;
3 Mathematics and games: mysterious connections;
Games and mathematics can be analysed in the head; Can you -look ahead'?; A novel kind of object; They are abstract. They are difficultRules; Hidden structures forced by the rules; Argument and proof; Certainty, error and truth; Players make mistakes; Reasoning, imagination and intuition; The power of analogy; Simplicity, elegance and beauty; Science and games: lets go exploring;
4 Why chess is not mathematics;
Competition; Asking questions about; Metamathematics and game-like mathematics; Changing conceptions of problem solving; Creating new concepts and new objects; Increasing abstraction; Finding common structures; The interaction between mathematics and sciences;
5 Proving versus checking.
The limitations of mathematical recreationsAbstract games and checking solutions; How do you `prove' that 11 is prime?; Is `5 is prime' a coincidence?; Proof versus checking; Structure, pattern and representation; Arbitrariness and un-manageability; Near the boundary;
PART II: Mathematics: game-like, scientific and perceptual
Introduction;
6 Game-like mathematics;
Introduction; Tactics and strategy; Sums of cubes and a hidden connection; A masterpiece by Euler;
7 Euclid and the rules of his geometrical game;
Cevas theorem; Simsons line; The parabola and its geometrical properties. Dandelins spheres
8 New concepts and new objects;
Creating new objects; Does it exist?; The force of circumstance; Infinity and infinite series; Calculus and the idea of a tangent; What is the shape of a parabola?;
9 Convergent and divergent series;
The pioneers; The harmonic series diverges; Weird objects and mysterious situations; A practical use for divergent series;
10 Mathematics becomes game-like; Eulers relation for polyhedra;
The invention-discovery of groups; Atiyah and MacLane disagree; Mathematics and geography;
11 Mathematics as science;
Introduction. Triangle geometry: the Euler line of a triangleModern geometry of the triangle; The Seven-Circle Theorem, and other New Theorems;
12 Numbers and sequences;
The sums of squares; Easy questions, easy answers; The prime numbers; Prime pairs; The limits of conjecture; A Polya conjecture and refutation; The limitations of experiment; Proof versus intuition;
13 Computers and mathematics;
Hofstadter on good problems; Computers and mathematical proof; Computers and 'proof'; Finally: formulae and yet more formulae;
14 Mathematics and the sciences;
Scientists abstract.

Statistics on the Table: The History of Statistical Concepts and Methods

Stephen M. Stigler

Harvard University Press | 2002 | 499 páginas | djvu | 5 Mb

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This lively collection of essays examines in witty detail the history of some of the concepts involved in bringing statistical argument "to the table," and some of the pitfalls that have been encountered. The topics range from seventeenth-century medicine and the circulation of blood, to the cause of the Great Depression and the effect of the California gold discoveries of 1848 upon price levels, to the determinations of the shape of the Earth and the speed of light, to the meter of Virgil's poetry and the prediction of the Second Coming of Christ. The title essay tells how the statistician Karl Pearson came to issue the challenge to put "statistics on the table" to the economists Marshall, Keynes, and Pigou in 1911. The 1911 dispute involved the effect of parental alcoholism upon children, but the challenge is general and timeless: important arguments require evidence, and quantitative evidence requires statistical evaluation. Some essays examine deep and subtle statistical ideas such as the aggregation and regression paradoxes; others tell of the origin of the Average Man and the evaluation of fingerprints as a forerunner of the use of DNA in forensic science. Several of the essays are entirely nontechnical; all examine statistical ideas with an ironic eye for their essence and what their history can tell us about current disputes.

Contents
Acknowledgments IX
Introduction 1
I. Statistics and Social Science
1 Karl Pearson and the Cambridge Economists 13
2 The Average Man Is 168 Years Old 51
3 Jevons as Statistician 66
4 Jevons on the King-Davenant Law of Demand 80
5 Francis Ysidro Edgeworth, Statistician 87
II. Galtonian Ideas
6 Galton and Identification by Fingerprints 131
7 Stochastic Simulation in the Nineteenth Century 141
8 The History of Statistics in 1933 157
9 Regression toward the Mean 173
10 Statistical Concepts in Psychology 189
III. Some Seventeenth-Century Explorers
11 Apollo Mathematicus 203
12 The Dark Ages of Probability 239
13 John Craig and the Probability of History 252
IV Questions of Discovery
14 Stigler's Law of Eponymy 277
15 Who Discovered Bayes's Theorem? 291
16 Daniel Bernoulli, Leonhard Euler, and Maximum Likelihood 302
17 Gauss and the Invention of Least Squares 320
18 Cauchy and the Witch of Agnesi 332
19 Karl Pearson and Degrees of Freedom 338
V Questions of Standards
20 Statistics and Standards
21 The Trial of the Pyx
22 Normative Terminology
with H. Kruskal

Mathematicians and their times: History of mathematics and mathematics of history

Laurence Young 

Academic Press, Elsevier | 1981 | 348 páginas | pdf | 5,4 Mb

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Contents

Preface

Introduction

Early History; The Slow Renaissance; The New Beginning of analiysis

Chapter I The Romantic Period

The Flouting of Arithmetic; The Founding of the Ecole Polytechnique; Gauss; The First Romantics;  Riemann and Weierstrass; Between Two Worlds

Chapter II The Dream of a Conceptual Age, and the Hard Reality of Struggle for Existence

The unprized heritage; The significance of  Plato for mathematics; Cantor and Sets; Goettingen 

Chapter III The Age of Contradictions, and of Great Question Marks

The art of the possible;The expansion of mathematics; Cambridge; The Hardy-Littlewood era; The battle ground of ideas; The  uncertain further blooming

Index of Persons

Philosophy of Mathematics in the Twentieth Century: Selected Essays

Charles Parsons

Harvard University Press | 2014 | 365 páginas | rar - pdf |950 kb

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In this illuminating collection, Charles Parsons surveys the contributions of philosophers and mathematicians who shaped the philosophy of mathematics over the course of the past century.
Parsons begins with a discussion of the Kantian legacy in the work of L. E. J. Brouwer, David Hilbert, and Paul Bernays, shedding light on how Bernays revised his philosophy after his collaboration with Hilbert. He considers Hermann Weyl's idea of a "vicious circle" in the foundations of mathematics, a radical claim that elicited many challenges. Turning to Kurt Gödel, whose incompleteness theorem transformed debate on the foundations of mathematics and brought mathematical logic to maturity, Parsons discusses his essay on Bertrand Russell's mathematical logic--Gödel's first mature philosophical statement and an avowal of his Platonistic view.
Philosophy of Mathematics in the Twentieth Century insightfully treats the contributions of figures the author knew personally: W. V. Quine, Hilary Putnam, Hao Wang, and William Tait. Quine's early work on ontology is explored, as is his nominalistic view of predication and his use of the genetic method of explanation in the late work The Roots of Reference. Parsons attempts to tease out Putnam's views on existence and ontology, especially in relation to logic and mathematics. Wang's contributions to subjects ranging from the concept of set, minds, and machines to the interpretation of Gödel are examined, as are Tait's axiomatic conception of mathematics, his minimalist realism, and his thoughts on historical figures

Contents
Preface
Introduction
Part I: Some Mathematicians as Philosophers
1. The Kantian Legacy in Twentieth-Century Foundations of Mathematics
2. Realism and the Debate on Impredicativity, 1917–1944 
Postscript to Essay 2
3. Paul Bernays’ Later Philosophy of Mathematics
4. Kurt Gödel
5. Gödel’s “Russell’s Mathematical Logic” ~
Postscript to Essay 5
6. Quine and Gödel on Analyticity
Postscript to Essay 6
7. Platonism and Mathematical Intuition in Kurt Gödel’s Thought
Postscript to Essay 7
Part II: Contemporaries
8. Quine’s Nominalism
9. Genetic Explanation in The Roots of Reference
10. Hao Wang as Philosopher and Interpreter of Gödel
11. Putnam on Existence and Ontology
12. William Tait’s Philosophy of Mathematics
Bibliography
Copyright Acknowledgments
Index

Windows, Rings, and Grapes — A Look at Different Shapes

(Math Is Categorical)

Brian P. Cleary e Brian Gable 

 Millbrook Pr Trade | 2009 | 36 páginas | rar - pdf | Mb

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In this humorous look at shapes, Brian P. Cleary and Brian Gable introduce circles, ovals, triangles, squares, and rectangles. The comical cats of the wildly popular Words Are CATegorical®series explain how to identify each shape and provide loads of examples. Peppy rhymes, goofy illustrations, and kid-friendly examples make shaping up a snap!

Speed Math for Kids: Helping Children Achieve Their Full Potential

Wrightbooks | 2012 | 272 páginas | epub | 3,7 Mb

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Popular Australian author and inspirational teacher, Bill Handley, has developed and, over the years, refined methods of teaching mathematics and learning strategies that have achieved amazing results. His best-selling book, Speed Mathematics convinced readers that people who excel at maths use better strategies and are not necessarily more intelligent.This book contains additional methods and applications based on the strategies taught in Speed Mathematics that make the principles clearer, encourage creative thought, and are just plain fun. The book was written for young people but people of any age will enjoy it. The book has notes throughout for parents and teachers.By following his innovative approach you will have kids playing with numbers, performing lightning quick calculations and, most of all, having fun!Bill claims: 'If you are good at maths, people think you are intelligent. People will treat you like you are a genius. Your teachers and your friends will treat you differently. You will even think differently about yourself'.The emphasis in this book is on playing with mathematics. Enjoy it. Show off what you learn and make mathematics your favourite subject.

CONTENTS
Preface
Introduction
Chapter 1: Multiplication: Getting Started
What is Multiplication?
The Speed Mathematics Method
Chapter 2: Using a Reference Number
Reference Numbers
Double Multiplication
Chapter 3: Numbers Above the Reference Number
Multiplying Numbers in The Teens
Multiplying Numbers Above 100
Solving Problems in Your Head
Double Multiplication
Chapter 4: Multiplying Above & Below the Reference Number
Numbers Above and Below
Chapter 5: Checking Your Answers
Substitute Numbers
Chapter 6: Multiplication Using Any Reference Number
Multiplication by factors
Multiplying numbers below 20
Multiplying numbers above and below 20
Using 50 as a reference number
Multiplying higher numbers
Doubling and halving numbers
Chapter 7: Multiplying Lower Numbers
Experimenting with reference numbers
Chapter 8: Multiplication by 11
Multiplying a two-digit number by 11
Multiplying larger numbers by 11
Multiplying by multiples of 11
Chapter 9: Multiplying Decimals
Multiplication of decimals
Chapter 10: Multiplication Using Two Reference Numbers
Easy multiplication by 9
Using fractions as multiples
Using factors expressed as division
Playing with two reference numbers
Using decimal fractions as reference numbers
Chapter 11: Addition
Adding from left to right
Breakdown of numbers
Checking addition by casting out nines
Chapter 12: Subtraction
Numbers around 100
Easy written subtraction
Subtraction from a power of 10
Checking subtraction by casting nines
Chapter 13: Simple Division
Simple division
Bonus: Shortcut for division by 9
Chapter 14: Long Division by Factor
What Are Factors?
Working with decimals
Chapter 15: Standard Long Division Made Easy
Chapter 16: Direct Long Division
Estimating answers
Reverse technique — rounding off upwards
Chapter 17: Checking Answers (Division)
Changing to multiplication
Bonus: Casting twos, tens and fives
Casting out nines with minus substitute numbers
Chapter 18: Fractions Made Easy
Working with fractions
Adding fractions
Subtracting fractions
Multiplying fractions
Dividing fractions
Changing vulgar fractions to decimals
Chapter 19: Direct Multiplication
Multiplication with a difference
Direct multiplication using negative numbers
Chapter 20: Putting it All into Practice
How Do I Remember All of This?
Advice For Geniuses
Afterword
Appendix A: Using the Methods in the Classroom
Appendix B: Working Through a Problem
Appendix C: Learn the 13, 14 and 15 Times Tables
Appendix D: Tests for Divisibility
Appendix E: Keeping Count
Appendix F: Plus and Minus Numbers
Appendix G: Percentages
Appendix H: Hints for Learning
Appendix I: Estimating
Appendix J: Squaring Numbers Ending in 5
Appendix K: Practice Sheets
Index

Outros livros do mesmo autor disponível neste blog:

Speed math for kids : the fast, fun way to do basic calculations por Bill Handley Idioma: Inglês   Editora: San Francisco, CA : Jossey-Bass, [2007]

Speed mathematics : secret skills for quick calculation por Bill Handley Idioma: Inglês   Editora: Hoboken, N.J. : Wiley, ©2003.

Teaching to the Math Common Core State Standards: Focus on Kindergarten to Grade 5

F. D. Rivera

Sense Publishers | 2014 | páginas | rar - pdf | 981 kb

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This is a methods book for elementary majors and preservice/beginning elementary teachers. It takes a very practical approach to learning to teach elementary school mathematics in an emerging Age of the Common Core State Standards. The Common Core State Standards in Mathematics (CCSSM) is not meant to be “the” official mathematics curriculum; it was purposefully developed primarily to provide clear learning expectations of mathematics content that are appropriate at every grade level and to help prepare all students to be ready for college and the workplace. A quick glance at the Table of Contents in this book indicates a serious engagement with the recommended mathematics underlying the kindergarten through grade 5 portions of the CCSSM first, with issues in content-practice assessment, learning, teaching, and classroom management pursued next and in that order.
In this book we explore what it means to teach to the CCSSM within an alignment mindset involving content-practice learning, teaching, and assessment. The CCSSM content standards, which pertain to mathematical knowledge, skills, and applications, have been carefully crafted so that they are teachable, learnable, coherent, fewer, clearer, and higher. The practice standards, which refer to institutionally valued mathematical actions, processes, and habits, have been conceptualized in ways that will hopefully encourage all elementary students to engage with the content standards more deeply than merely acquiring mathematical knowledge by rote and imitation. Thus, in the CCSSM, proficiency in content alone is not sufficient, and so does practice without content, which is limited. Content and practice are both equally important and, thus, must come together in teaching, learning, and assessment in order to support authentic mathematical understanding.
This blended, multisourced text is a “getting smart” book. It helps elementary majors and preservice/beginning elementary teachers work within the realities of accountable pedagogy and develop a proactive disposition that is capable of supporting all elementary students in order for them to experience growth in mathematical understanding necessary for middle school and beyond, including future careers.

CONTENTS
1. Dear Elementary Majors and Practicing and Beginning Elementary Teachers: An Introduction 1
1.1 A Blended Multisourced Approach to Learning to Teach Mathematics 4
1.2 Overview of the Remaining Chapters 5
2. Getting to Know the Common Core State Standards for Mathematical Practice 7
2.1 Content Activity 1: Generating Addition Facts 7
2.2 The Eight Common Core State Standards for Mathematical Practice 11
2.3 Content Activity 2: Building a Hexagon Flower Garden Design 11
2.4 Problem Solving Contexts in Elementary School Mathematics 16
2.4.1 Content Activity 3: Different Types of Problems in Elementary School Mathematics 16
2.5 Representations in Elementary School Mathematics 17
2.5.1 Content Activity 4: A Kindergarten Subtraction Task 17
2.5.2 Content Activity 5: Representing Whole Numbers in Second Grade 20
2.6 Connections in Elementary School Mathematics 20
2.6.1 Content Activity 6: Repeating Patterns 21
2.7 Reasoning and Proof in Elementary School Mathematics 22
2.7.1 Content Activity 7: Division Problems in Third Grade 22
2.7.2 Activity 8: Even and Odd Numbers in Second Grade 23
2.8 Communication in Elementary School Mathematics 24
2.8.1 Content Activity 9: Addition and Subtraction Strategies from First to Second Grade 25
2.8.2 Content Activity 10: Generating Addition and Subtraction Facts in First Grade 27
2.9 Doing Mathematics with an Eye on theContent-Practice Standards of the CCSSM 28
3. Counting and Cardinality in Kindergarten 31
3.1 Subitizing: Laying the Foundation for Early Multiplicative Thinking 31
3.2 Counting and Writing Numbers and Number Words: Building on Rhythmic Structures 33
3.3 Comparing Quantities 35
3.4 Mapping the Content Standards with the Practice Standards 37
3.5 Developing a Content Standard Progression Table for the Counting and Cardinality Domain 37
4. Numbers and Operations (Part I) 41
4.1 Place Value in Base Ten: The Basic Mathematical Structure of Whole Numbers and Decimal Numbers in the CCSSM 42
4.2 Representing Whole Numbers in Base Ten from K to Grade 4 43
4.3 Representing Decimal Numbers in Grade 5 43
4.4 Comparing and Ordering Whole Numbers and Decimal Numbers from Grades 1 to 5 44
4.5 Rounding Whole Numbers from Grades 3 to 5 45
4.6 Mapping the Content Standards with the Practice Standards 45
4.7 Developing a Content Standard Progression Table for Part I of the Numbers and Operations in Base Ten Domain 46
5. Operations and Algebraic Thinking 47
5.1 Algebraic Thinking from Kindergarten to Grade 5 47
5.2 The Four Fundamental Operations as General Methods for Combining and Generating More Numbers 52
5.2.1 Addition 52
5.2.2 Subtraction 54
5.2.3 Multiplication 54
5.2.4 Division 56
5.3 Forming Equivalent Expressions and Understanding the General Meaning of the Equal Sign 57
5.4 Instantiating Nonsymbolic Algebra Through Problem Solving in Arithmetical Activity 59
5.5 Pattern Generalization as Instantiations of Nonsymbolic and Symbolic Algebra 61
5.6 Mapping the Content Standards with the Practice Standards 66
5.7 Developing a Content Standard Progression Table for the Operations and Algebraic Thinking Domain 66
6. Numbers and Operations in Base Ten (Part II) 67
6.1 Analytical Subitizing in Kindergarten and Sticks and Circles or Labeled Circles in Grade 1 67
6.2 Adding Whole Numbers in Base Ten from Grades 1 to 4 68
6.3 Subtracting Whole Numbers in Base Ten from Grades 1 to 4 72
6.4 Multiplying Whole Numbers in Base Ten from Grades 3 to 5 76
6.5 Dividing Whole Numbers in Base Ten from Grades 3 to 4 79
6.6 Adding, Subtracting, Multiplying, and Dividing Decimal Numbers in Grade 5 83
6.7 Mapping the Content Standards with the Practice Standards 87
6.8 Developing a Content Standard Progression Table for the Operations and Number Thinking  Part II Domain 87
7. Numbers and Operations – Fractions 89
7.1 Equal Partitioning of Whole Shapes in Halves and Fourths in Grade 1 90
7.2 Equal Partitioning of Whole Shapes in Thirds and Constructing Equivalent Shares in Grade 2 92
7.3 Unit Fractions and Equivalent Fractions with Denominators 2, 3, 4, 6, and 8 as Equal Shares and Numbers on a Number Line and Whole Numbers as Improper Fractions in Grade 3 95
7.4 Numerical Equivalence of Fractions Involving Denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100, All Improper Fractions, Fraction Addition and Subtraction with Like Denominators, Simple Fraction Multiplication, and Decimal Fraction Addition and Comparison in Grade 4 99
7.5 Fraction Addition, Subtraction, Multiplication, and Simple Division in Grade 5 104
7.6 Mapping the Content Standards with the Practice Standards 109
7.7 Developing a Content Standard Progression Table for the Number and Operations – Fractions Domain 109
8. Geometry 111
8.1 Recognizing and Composing Shapes in Kindergarten and Grade 1 111
8.2 Defi ning and Reasoning about Shapes in Grades 2, 3, and 5 114
8.3 Lines and Angles in 2D Shapes in Grade 4 114
8.4 The Coordinate Plane in Grade 5 116
8.5 Mapping the Content Standards with the Practice Standards 117
8.6 Developing a Content Standard Progression Table for Geometry Domain 117
9. Measurement and Data 119
9.1 Measuring from Kindergarten to Grade 5 119
9.1.1 Qualitative Measurements in Kindergarten 119
9.1.2 Measuring Lengths in Grades 1 and 2 120
9.1.3 Finding Perimeters and Measuring Lengths with Halves and Fourths of an Inch, Areas, and Estimated Liquid Volumes and Masses of Objects in Standard Units in Grade 3 121
9.1.4 Measuring Angles in Grade 4 123
9.1.5 Measuring Volumes of Cubes and Other Right Rectangular Prisms in Grade 5 124
9.1.6 Conversion Problems in Grades 4 and 5 125
9.2 Telling and Writing Time in Grades 2 and 3 125
9.3 Money in Grades 2 and 4 126
9.4 Classifying, Representing, and Interpreting Data from Kindergarten to Grade 5 127
9.4.1 Forming Categories Involving Discrete Objects in Kindergarten 127
9.4.2 Structuring Data from Kindergarten to Grade 5 127
9.5 Mapping the Content Standards with the Practice Standards 127
9.6 Developing a Content Standard Progression Table for the Measurement and Data Domain 128
10. Content-Practice Assessment 129
10.1 General Conceptions Surrounding Assessment 131
10.2 Norm- and Criterion-Referenced Tests 135
10.3 Principles of Effective Classroom Assessments 135
10.4 Formative Assessments 136
10.5 Summative Assessments 136
10.5.1 General Conceptions Surrounding Summative Assessments 136
10.5.2 The Smarter Balanced Assessment (SBA) 137
10.6 Projects and Portfolios as Alternative Summative Assessments 140
10.7 Math Journals and Lesson Investigations as Alternative Formative Assessments 141
10.8 An Assessment Project 142
11. Content-Practice Learning 145
11.1 Defi ning Learning 145
11.2 Changing Views of Learning and Their Effects in the Elementary Mathematics Curriculum 146
11.3 Math Wars: Debating About What and How Students Should Learn Mathematics 147
11.4 Understanding Piagetian and Vygotskian Views of Learning in Mathematics and Finding a Way Out of Extreme Views of Learning 147
11.5 Learning Progressions in School Mathematics 148
11.6 Learning from Neuroscience 149
12. Content-Practice Teaching 151
12.1 Describing (Good) Teaching 151
12.2 Teaching Models in Elementary School Mathematics Settings 154
12.2.1 E-I-S-Driven Teaching 154
12.2.2 C-R-A Sequenced Teaching 154
12.2.3 Van Hiele Sequenced Teaching 155
12.2.3 Culturally-Relevant Teaching 155
12.2.4 SDAIE-Driven Teaching 156
12.2.5 Differentiated Instruction 156
12.2.6 Flip Teaching 157
12.3 Teaching with Concrete and Virtual Manipulatives, Computer-Based Learning Tools, and Video Games and Apps 157
12.4 Teaching Mathematics with Guide Questions 158
12.5 Content-Practice Unit Planning 159
12.6 Content-Practice Lesson Planning 165
12.7 A Planning Project 167
13. Orchestrating a Content-Practice Driven Math Classroom 175
13.1 Persistence and Struggles in Math Classrooms 175
13.2 Fostering Persistent Content-Practice Learners 176
13.3 Developing Effective Collaborative Content-Practice Learning Through Complex Instruction 177
13.4 Other Collaborative Content-Practice Learning Techniques 180
13.5 Developing an Optimal Content-Practice Learning Environment for All Elementary Students 180
13.6 Dealing with Potential Behavior Problems 181
13.7 Assigning Homework, Grading and Testing, and Seating 183
13.8 A Classroom Management Plan Project 185