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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mobi - 6,3 Mb - link

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

Gems of Geometry

 

John Barnes 

Springer | 2012 - 2.ª edição | páginas | pdf | 5 Mb

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Based on a series of lectures for adult students, this lively and entertaining book proves that, far from being a dusty, dull subject, geometry is in fact full of beauty and fascination. The author's infectious enthusiasm is put to use in explaining many of the key concepts in the field, starting with the Golden Number and taking the reader on a geometrical journey via Shapes and Solids, through the Fourth Dimension, finishing up with Einstein's Theories of Relativity.

Equally suitable as a gift for a youngster or as a nostalgic journey back into the world of mathematics for older readers, John Barnes' book is the perfect antidote for anyone whose maths lessons at school are a source of painful memories. Where once geometry was a source of confusion and frustration, Barnes brings enlightenment and entertainment.

In this second edition, stimulated by recent lectures at Oxford, further material and extra illustrations have been added on many topics including Coloured Cubes, Chaos and Crystals.

Contents

1 The Golden Number 1

Pieces of paper, The golden ratio, Fibonacci’s rabbits, Continued fractions, Pentagons, Phyllotaxis, Further reading, Exercises.

2 Shapes and Solids 27

Flatland, Polygons, Tiling, Vision and projection, Five classical polyhedra, Duality, Kepler and Poinsot, The Archimedean figures, Non-convex polyhedra, Pentagonal tilings, Further reading, Exercises.

3 The Fourth Dimension 63

What is the fourth dimension? Honeycombs, The 4-simplex, The hypercube and the 16-cell, Other regular convex figures, Non-convex regular figures, Honeycombs, five dimensions and more, Nets, Further reading, Exercises.

4 Projective Geometry 89

Pappus’ theorem, Desargues’ theorem, Duality, Duality in three dimensions, Infinity and parallels, Quadrilaterals and quadrangles, Conics, Coordinates, Finite geometries, Configurations, Further reading, Exercises.

5 Topology 113

Hairy dogs, Colour problems, Colouring maps on the torus, The Möbius band, The Klein bottle, The projective plane, Round up, Further reading, Exercises.

6 Bubbles 137

Surface tension, Two bubbles, Three bubbles, Four bubbles, Foam, Films on frames, Films on cylinders, That well known theorem, Further reading, Exercises.

7 Harmony of the Spheres 157

Steiner’s porism, Inversion, Coaxial circles, Proof of Steiner’s porism, Soddy’s hexlet, Further reading, Exercises.

8 Chaos and Fractals 179

Shaken foundations, Fractals, Fractional dimensions, Cantor sets, Population growth, Double, double, boil and trouble, Chaos and peace, And so to dust, Newton’s method, Julia and Mandelbrot sets, Natural chaos, Further reading, Exercises.

9 Relativity 205

The special theory, Time changes, The Lorentz–Fitzgerald contraction, Distortion of bodies, Lorentz transformation, Time and relativity, Mass and energy, Coordinates, Curvature, Einstein’s equations, The Schwarzchild solution, Consequences of general relativity, Black holes, Properties of black holes, Further reading, Exercise.

10 Finale 231

Squares on a quadrilateral, The Argand plane, The quadrilateral revisited, Other complex problems, Trisection, Bends, Pedal triangles, Coordinates of points and lines, Further reading.

A The Bull and the Man 247

The problem, The proof.

B Stereo Images 251

Compound figures, Desargues’ theorem.

C More on Four 261

Archimedean figures in four dimensions, Prisms and hyperprisms.

D Schlegel Images 271

Schlegel diagrams, The hypercube, The 16-cell, The 24-cell, The 120-cell, 600-cell and tetroctahedric.

E Crystals 287

Packing of spheres, Crystals, Diamonds and graphite, Silica structures, Gems, Further reading.

F Stability 305

Stability of fixed points, The fixed points, The two cycle, Three cycles.

G Fanoland 319

Seven girls and seven boys, Plus six girls and six boys, Explanation.

Bibliography 321

Index 323

1.ª edição

Gems of Geometry por John Barnes Idioma: Inglês   Editora: Berlin [u.a.] Springer 2009

Minilessons for Math Practice, Grades K-2

Rusty Bresser e Caren Holtzman

Math Solutions | 2006 | 177 páginas | pdf | 1 Mb 

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Livro relacionado:

Minilessons for Math Practice, Grades 3-5

Requiring only 5 to 15 minutes to teach, these quick, engaging activities offer experiences in all of the content areas important to elementary mathematics: number, measurement, geometry, data analysis and probability, and algebra. Each resource of 27 classroom-tested lessons models how to develop several important math processes: problem solving, reasoning and proof, communication, connections, and representation. Each activity features a materials list, step-by-step teaching directions, a list of key questions to ask students during the lesson, a vignette of how the lesson actually unfolded in a classroom, and ideas for extending the activity throughout the year. Each book includes reproducibles.

Contents
Acknowledgments ix
Introduction xi
Contents Chart xv
1. Addition and Subtraction Word Problems 1
2. Addition and Subtraction Word Problems Extended 9
3. Breaking Numbers Apart 17
4. Building the 1–100 Chart 23
5. Coins 31
6. Comparing Polygons 37
7. Cubes in a Tube 42
8. Dots 47
9. Estimating 51
10. Finding Friendly Numbers 58
11. Fit the Facts 63
12. Greater Than, Less Than, Is Equal To 68
13. Grow and Shrink 76
14. Guess My Number 80
15. Heavier or Lighter? 85
16. How Long? How Tall? 89
17. In One Minute 93
18. Measuring Area 97
19. More or Less? Version 1 103
20. More or Less? Version 2 107
21. Number Strings 111
22. People Sorting 116
23. Quick Surveys 120
24. Race to 20 128
25. Sampling Tiles 133
26. Ten-Frames 138
27. Ten-Frames Cleared 142
Blackline Masters 147
Dot Arrangements 149
Fit the Facts Family Letter 152
Ten-Frames 153
References 155
Index 157

The Beauty of Fractals Six Different Views

 Denny Gulick e Jon Scott 

The Mathematical Association of America | 2011 | páginas | rar - pdf | 16,5 Mb

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With the coming of the computer age, fractals have emerged to play a significant role in art images, scientific application and mathematical analysis. The Beauty of Fractals is in part an exploration of the nature of fractals, including examples which appear in art, and in part a close look at famous classical fractals and their close relatives. The final essay examines the relationship between fractals and differential equations. The essays that appear in The Beauty of Fractals contain perspectives different enough to give the reader an appreciation of the breadth of the subject. The essays are self-contained and expository, and are intended to be accessible to a broad audience that includes advanced undergraduate students and teachers at both university and secondary-school level. The book is also a useful complement to the material on fractals which can be found in textbooks.

Contents
Mathscapes--fractal scenery / Anne M. Burns
Chaos, fractals, and Tom Stoppard's Arcadia / Robert L. Devaney
Excursions through a forest of golden fractal trees / T.D. Taylor
Exploring fractal dimension, area, and volume / Mary Ann Connors
Points in sierpiński-like fractals / Sandra Fillebrown, ... [et al.]
Fractals in the 3-body problem via symplectic integration / Daniel Hemberger, James A. Walsh