Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5

John A. Van de Walle, Karen S. Karp, Lou Ann H. Lovin e Jennifer M. Bay-Williams

Pearson | 2013 - 2ª edição | 462 páginas | rar - pdf | 11,6 Mb

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Initially adapted from Van de Walle’s market-leading textbook, Elementary and Middle School Mathematics, the Van de Walle Professional Mathematics Series are practical guides for developmentally appropriate, student-centered mathematics instruction from best selling mathematics methods authors John Van de Walle, Karen Karp, LouAnn Lovin, and Jennifer Bay-Williams. Specially designed for in-service teachers, each volume of the series focuses on the content relevant to a specific grade band and provides additional information on creating an effective classroom environment, engaging families, and aligning teaching to the Common Core State Standards. Additional activities and expanded lessons are also included.

The series has three objectives:
1. To illustrate what it means to teach student-centered, problem-based mathematics
2. To serve as a reference for the mathematics content and research-based instructional strategies suggested for pre-kindergarten to grade two, grades three to five, and grades six to eight
3. To present a large collection of high quality tasks and activities that can engage students in the mathematics that is important for them to learn

Volume II is tailored specifically to grades 3-5, allowing teachers to quickly and easily locate information to implement in their classes. The student-centered approach will result in students who are successful in learning mathematics, making these books indispensable for 3-5 classroom teachers!

Contents
Part 1: Establishing a Student-Centered Environment
Chapter 1: Teaching Mathematics for Understanding
Chapter 2: Teaching Mathematics Through Problem Solving
Chapter 3: Assessing for Learning
Chapter 4: Differentiating Instruction
Chapter 5: Planning, Teaching, and Assessing Culturally and Linguistically Diverse Students
Chapter 6: Planning, Teaching, and Assessing Students with Exceptionalities
Chapter 7: Collaborating with Families, Community, and Principals
Part 2: Teaching Student-Centered Mathematics
Chapter 8: Exploring Number and Operation Sense
Chapter 9: Helping Students Master the Basic Facts
Chapter 10: Developing Whole Number Place Value Concepts
Chapter 11: Building Strategies for Whole-Number Computation
Chapter 12: Exploring Fraction Concepts
Chapter 13: Building Strategies for Fraction Computation
Chapter 14: Developing Decimal and Percent Concepts and Decimal Computation
Chapter 15: Promoting Algebraic Thinking
Chapter 16: Building Measurement Concepts
Chapter 18: Developing Geometric Thinking and Geometric Concepts
Chapter 19: Helping Students Represent and Interpret Data

Teaching Mathematical Modelling: Connecting to Research and Practice

(International Perspectives on the Teaching and Learning of Mathematical Modelling)

 Gloria Ann Stillman, Gabriele Kaiser, Werner Blum e Jill P. Brown

 Springer | 2013 | 612 páginas | pdf  | 9 Mb

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This book provides readers with an overview of recent international research and developments in the teaching and learning of modelling and applications from a variety of theoretical and practical perspectives. There is a strong focus on pedagogical issues for teaching and learning of modelling as well as research into teaching and practice. The teaching of applications of mathematics and mathematical modelling from the early years through primary and secondary school and at tertiary level is rising in prominence in many parts of the world commensurate with an ever-increasing usage of mathematics in business, the environment, industry and everyday life. The authors are all members of the International Community of Teachers of Mathematical Modelling and Applications and important researchers in mathematics education and mathematics. The book will be of interest to teachers, practitioners and researchers in universities, polytechnics, teacher education, curriculum and policy

Contents
1 Mathematical Modelling: Connecting to Teaching and Research Practices – The Impact of Globalisation.... 1
Gloria Ann Stillman, Gabriele Kaiser, Werner Blum, and Jill P. Brown
Part I Innovative Practices in Modelling Education Research and Teaching
2 From Conference to Community: An ICTMA Journey—The Ken Houston Inaugural Lecture .... 27
Peter Galbraith
3 Modelling from the Perspective of Commognition – An Emerging Framework..... 47
Jonas Bergman Ärlebäck and Peter Frejd
4 Should Interpretation Systems Be Considered to Be Models if They Only Function Implicitly?.... 57
Rita Borromeo Ferri and Richard Lesh
5 Mathematical Modelling, Mathematical Content and Tensions in Discourses ...... 67
Andréia Maria Pereira de Oliveira and Jonei Cerqueira Barbosa
6 Ethnomodelling as a Methodology for Ethnomathematics ...... 77
Milton Rosa and Daniel Clark Orey
7 Dual Modelling Cycle Framework for Responding to the Diversities of Modellers..... 89
Akihiko Saeki and Akio Matsuzaki
8 The Eyes to See: Theoretical Lenses for Mathematical Modelling Research...... 101
Nils Buchholtz
9 Strässer’s Didactic Tetrahedron as a Basis for Theorising Mathematical Modelling Activity Within Social Contexts.... 107
Vince Geiger
10 Ethnomodelling as a Research Lens on Ethnomathematics and Modelling ..... 117
Milton Rosa and Daniel Clark Orey
Part II Research into, or Evaluation of, Teaching Practice
11 Real-Life Modelling Within a Traditional Curriculum: Lessons from a Singapore Experience .... 131
Ang Keng Cheng
12 Students’ Mathematical Learning in Modelling Activities ..... 141
Morten Blomhøj and Tinne Hoff Kjeldsen
13 Students’ Designing an Ideal Tourism Route as Mathematical Modelling .... 153
Chan Chun Ming Eric
14 Comparison of Mathematical Modelling Skills of Secondary and Tertiary Students .... 165
Juntao Fu and Jinxing Xie
15 Taking Advantage of Incidental School Events to Engage with the Applications of Mathematics: The Case of Surviving the Reconstruction........ 175
Vince Geiger, Merrilyn Goos, and Shelley Dole
16 The Development of Modelling Competencies by Year 9 Students: Effects of a Modelling Project..... 185
Susanne Grünewald
17 Evidence of a Dual Modelling Cycle: Through a Teaching Practice Example for Pre service Teachers..... 195
Akio Matsuzaki and Akihiko Saeki
18 Considering Multiple Solutions for Modelling Problems – Design and First Results from the MultiMa-Project.... 207
Stanislaw Schukajlow and André Krug
19 Challenges in Modelling Challenges: Intents and Purposes ... 217
Gloria Ann Stillman, Jill P. Brown, and Peter Galbraith
20 Mathematical Modelling of a Real-World Problem: The Decreasing Number of Bluefi n Tuna ... 229
Akira Yanagimoto and Noboru Yoshimura
21 Mathematical Modelling of a Social Problem: Pension Tax Issues .... 241
Noboru Yoshimura and Akira Yanagimoto
Part III Pedagogical Issues for Teaching and Learning
22 Pedagogical Refl ections on the Role of Modelling in Mathematics Instruction ..... 255
Toshikazu Ikeda
23 Complex Modelling Problems in Co-operative, Self-Directed Learning Environments ..... 277
Gabriele Kaiser and Peter Stender
24 Inducting Year 6 Students into “A Culture of Mathematising as a Practice” ..... 295
Jill P. Brown
25 A Whole Week of Modelling – Examples and Experiences of Modelling for Students in Mathematics Education.... 307
Nils Buchholtz and Sarah Mesrogli
26 Teachers’ Self-Perceptions of Their Pedagogical Content Knowledge Related to Modelling – An Empirical Study with Austrian Teachers ...... 317
Sebastian Kuntze, Hans-Stefan Siller, and Christiane Vogl
27 A Cross-Sectional Study About Modelling Competency in Secondary School .... 327
Matthias Ludwig and Xenia-Rosemarie Reit
28 Teacher Readiness in Mathematical Modelling: Are There Differences Between Pre-service and In-Service Teachers?... 339
Kit Ee Dawn Ng
29 Exploring the Relationship Between Mathematical Modelling and Classroom Discourse .... 349
Trevor Redmond, Raymond Brown, and Joanne Sheehy
30 The Role of Textbooks in Developing a Socio- critical Perspective on Mathematical Modelling in Secondary Classrooms ... 361
Gloria Ann Stillman, Jill P. Brown, Rhonda Faragher, Vince Geiger, and Peter Galbraith
31 Pre-service Secondary School Teachers’ Knowledge in Mathematical Modelling – A Case Study .... 373
Tan Liang Soon and Ang Keng Cheng
32 How Students Connect Descriptions of Real- World Situations to Mathematical Models in Different Representational Modes .. 385
Wim Van Dooren, Dirk De Bock, and Lieven Verschaffel
33 Pre-service Teacher Learning for Mathematical Modelling ..... 395
Mark Winter and Hamsa Venka
34 Initial Perspectives of Teacher Professional Development on Mathematical Modelling in Singapore: Conceptions of Mathematical Modelling ....... 405
Chan Chun Ming Eric
35 Initial Perspectives of Teacher Professional Development on Mathematical Modelling in Singapore: Problem Posing and Task Design .... 415
Lee Ngan Hoe
36 Initial Perspectives of Teacher Professional Development on Mathematical Modelling in Singapore: A Framework for Facilitation....... 427
Kit Ee Dawn Ng
37 Teacher Professional Development on Mathematical Modelling: Initial Perspectives from Singapore .... 437
Vince Geiger
Part IV Influences of Technologies
38 Reality Based Test Tasks with Digital Tools at Lower Secondary ....... 445
Gilbert Greefrath and Michael Rieß
39 On Comparing Mathematical Models and Pedagogical Learning ....... 457
Janeen Lamb and Jana Visnovska
Part V Assessment in Schools
40 Formative Assessment in Everyday Teaching of Mathematical Modelling: Implementation of Written and Oral Feedback to Competency- Oriented Tasks...... 469
Michael Besser, Werner Blum, and Malte Klimczak
41 Assessment of Modelling in Mathematics Examination Papers: Ready-Made Models and Reproductive Mathematising ....... 479
Pauline Vos
Part VI Applicability at Different Levels of Schooling, Vocational Education, and in Tertiary Education
42 Complex Modelling in the Primary and Middle School Years: An Interdisciplinary Approach ..... 491
Lyn D. English
43 Modelling in Brazilian Mathematics Teacher Education Courses ...... 507
Maria Salett Biembengut
44 The Development of Mathematical Concept Knowledge and of the Ability to Use This Concept to Create a Model ... 517
César Cristóbal-Escalante and Verónica Vargas-Alejo
45 Problem Posing: A Possible Pathway to Mathematical Modelling ..... 527
Ann Downton
46 A Study of the Effectiveness of Mathematical Modelling of Home Delivery Packaging on Year 12 Students’ Function Education .. 537
Tetsushi Kawasaki and Yoshiki Nisawa
47 How to Introduce Mathematical Modelling in Industrial Design Education? ...... 551
Geert Langereis, Jun Hu, and Loe Feijs
48 Rationality of Practice and Mathematical Modelling – On Connections, Conflicts, and Codifications ....... 563
Lars Mouwitz
49 Extending Model Eliciting Activities (MEAs) Beyond Mathematics Curricula in Universities .... 573
Mark Schofield
50 Building Awareness of Mathematical Modelling in Teacher Education: A Case Study in Indonesia ...... 583
Wanty Widjaja
Part VII Modelling and Applications in Business and the Lived Environment
51 Mathematics and the Pharmacokinetics of Alcohol ...... 597
Michael Jennings and Peter Adams
52 Beyond the Modelling Process: An Example to Study the Logistic Model of Customer Lifetime Value in Business Marketing ...... 607
Issic K.C. Leung

Three Dimensions: A Model of Goal and Theory Description in Mathematics Instruction - The Wiskobas Project

(Mathematics Education Library, 3)

Adrian Treffers
Springer | 2012 -  reprint of the original 1st ed. 1987 edition | 359 páginas | rar - pdf | 9,2 Mb

link (password: matav)

TABLE OF CONTENTS
PREFACE xiii
GUIDE TO THE MATHEMATICAL MATERIAL xv
MATHEMATICAL MATERIAL FOR CHAPTER I: "GULLIVER" 1 
CHAPTER I INTRODUCTION 8
1. From "New Math" to Wiskobas 9
1.1 Three trends 9
1.2 Wiskobas 10
2. The history of Wiskobas 11
2.1 The exploratory phase 11
2.2 The integration phase 12
2.3 Spin-off, further development and research 12
2.4 Summary 13
3. Wiskobas between four trends 14
3.1 Wiskobas and the empirical trend 14
3.2 Wiskobas and the structural trend 15
3.3 Wiskobas and the arithmetical trend 16
3.4 Wiskobas and the current arithmetic education
3.5 Conclusion 18
4. Innovation according to Wiskobas 19
4.1 The innovation strategy 19
4.2 The innovation theme 21
4.3 The innovation 22
5. The problem 23
5.1 The problem of goal description 24
5.2 The question at issue 24
5.3 What is not dealt with? 26
6. Overview of what follows 26
6.1 Chapters 26
6.2 What is the function of the mathematical material? 27
6.3 Short summary 28
7. Conclusion 29
MATHEMA TICAL MATERIAL FOR CHAPTER II: "COUNTING PROBLEMS" 31
CHAPTER II STARTING POINTS 37
1. Mathematical activity 37
1.1 Flowers 38
1.2 Routes 38
1.3 Apples 39
1.4 To and fro 40
1.5 To and fro again 41
1.6 Didactical digression 42
1.7 Cards for the cube crawler 45
1.8 Routes on a highway network 47
1.9 Score progression 49
1.10 Families 50
1.11 Mathematising 51
2. Acting didactically 53
2.1 A mathematics lesson 53
2.2 Didactising 57
3. Starting points for mathematics education 59
3.1 Activity 60
3.2 Differentiation 61
3.3 Vertical planning 62
3.4 Structural character 63
3.5 Language aspect 64
3.6 Applicability 66
3.7 Dynamics 68
3.8 The specifically mathematical approach 70
4. Conclusion 71
MATHEMATICAL MATERIAL FOR CHAPTER III: "GRAINS ON THE CHESSBOARD" 75
CHAPTER III ONE-DIMENSIONAL GOAL DESCRIPTION 83
1. Goal descriptions 84
1.1 General, intermediate and concrete goal descriptions 84
1.2 One-, two-, and three-dimensional goal descriptions 86
1.3 Summary 90
2. Integral one-dimensional goals 90
2.1 Personal development 91
2.2 Socialisation 95
2.3 Preparation for further education 96
2.4 Social relevance 99
2.5 Summary 100
3. Mathematical one-dimensional goals 101
3.1 Arithmetical aspect 101
3.2 Language aspect 102
3.3 Applicability 103
3.4 Practical use 104
3.5 Structural aspect 105
3.6 Methodological aspect 105
3.7 Dynamic aspect 107
3.8 Attitude aspect 107
4. Relationships between integral and mathematical goals 108
5. Conclusion 109
MATHEMATICAL MATERIAL FOR CHAPTER IV: "THE LAND OF EIGHT" 113
CHAPTER IV TWO-DIMENSIONAL GOAL DESCRIPTION 119
1. Popham and Eisner: Two views on goal description
1.1 Popham's views on "instructional objectives"
1.2 Eisner's views on "expressive objectives" 121
1.3 Summary 124
2. Variants of instructional objectives 124
2.1 Concrete product goals 125
2.2 Operationalised product goals 126
2.3 The goals approach 127
2.4 Concluding remarks 128
3. Variants of expressive objectives 129
3.1 PISA goals l30
3.2 Process goals l32
3.3 All-embracing process goals 133
3.4 Concluding remarks 133
4. Product and process goals in "The Land of Eight" l34
4.1 Ptoduct goals in "The Land of Eight" l35
4.2 Process goals in "The Land of Eight" 139
4.3 Summary 144
5. The possibilities and limitations of two-dimensional goal descriptions 146
5.1 Possibilities and limitations of two-dimensional product goal descriptions 147
5.2 Possibilities and limitations of two-dimensional process goal descriptions
5.3 Summary
6. Conclusion
MATHEMATICAL MATERIAL FOR CHAPTER V: "FRECKLEHAM" 159
CHAPTER V THREE-DIMENSIONAL GOAL DESCRIPTION 167
1. The history of "Freckleham" and the significance of its goals 167
1.1 Development as a process of making goals concrete 168
1.2 Development as a progressive structuring of activities 169
1.3 The objectives and history of "Freckleham" in Wiskobas 170
1.4 Conclusions 173
2. "Freckleham" in three dimensions 173
2.1 The people of "Freckleham" 174
2.2 Map of "Freckleham" 175
2.3 Greetings 1 7 6
2.4 Confusion 178
2.5 Thieves 179
2.6 The town meeting 180
2.7 New greeting suggestions 181
2.8 The Freckleham song in code 182
2.9 "Freckleham" in a 'wider' connection 183
2.10 Basis of "Freckleham" in a 'deeper' connection
3. Holistic three-dimensional goal description 185
3.1 Different kinds of three-dimensional goal description 185
3.2 Characteristics of the holistic threedimensional goal description 187
3.3 Rough empirical basis of the holistic three-dimensional goal description 189
3.4 Functions of holistic three-dimensional goal description 191
4. Conclusion 194
MATHEMA TICAL MATERIAL FOR CHAPTER VI: "ALGORITHMS" 197
CHAPTER VI SURVEY AND JUSTIFICATION 211
1. History 211
2. Overview 213
3. Justification 214
4. Conclusions 218
MATHEMATICAL MATERIAL FOR CHAPTER VII
(APPENDIX): "THE WISKOBAS CURRICULUM" 221
CHAPTER VII FRAMEWORK FOR INSTRUCTION THEORY 239
1. Preamble 239
1.1 Starting points of a realistic instruction theory 240
2. One-dimensional description of the framework for instruction theory 242
2.1 Van Hiele's levels 242
2.2. Freudenthal's didactical phenomenology 246
2.3. Progressive mathematising guided by the five instruction principles 247
2.4. Schematic comparison of the four trends in arithmetic/mathematics instruction 250
3. Two-dimensional description of a framework for instruction theory 252
3.1 Progressive mathematisation in the Wiskobas programme 252
3.2. The five tenets of the framework for instruction theory revisited 255
3.3. Comparison of the four trends 263
4. Three-dimensional description of a framework for instruction theory 264
4.1 Progressive mathematising of long division 265
4.2. Two more examples: Number systems and fractions 268
4.3. The most conspicuous elements of the framework for instruction theory seen from the viewpoint of implementation of instructional ideas 270
5. The broader framework for instruction theory 271
5.1 Gagne, Dienes, Piaget, and Bruner 272
5.2 Recent investigations of subject matter in instruction theory 279
5.3 The almost complete absence of instruction theory ideas in general cognitive psychological research 289
5.4 Overview 294
5.5 Closure 295

The foundations of geometry

Gerard Venema

Pearson | 2011 - 2ª edição | 407 páginas | pdf | 3 Mb

link direto
link
link1

Foundations of Geometry, Second Edition is written to help enrich the education of all mathematics majors and facilitate a smooth transition into more advanced mathematics courses. The text also implements the latest national standards and recommendations regarding geometry for the preparation of high school mathematics teachers—and encourages students to make connections between their college courses and classes they will later teach. This text's coverage begins with Euclid's Elements, lays out a system of axioms for geometry, and then moves on to neutral geometry, Euclidian and hyperbolic geometries from an axiomatic point of view, and then non-Euclidean geometry. Good proof-writing skills are emphasized, along with a historical development of geometry. The Second Edition streamlines and reorganizes material in order to reach coverage of neutral geometry as early as possible, adds more exercises throughout, and facilitates use of the open-source software Geogebra.

This text is ideal for an undergraduate course in axiomatic geometry for future high school geometry teachers, or for any student who has not yet encountered upper-level math, such as real analysis or abstract algebra. It assumes calculus and linear algebra as prerequisites.

Table of Contents
1. Prologue: Euclid’s Elements
1.1 Geometry before Euclid
1.2 The logical structure of Euclid’s Elements
1.3 The historical significance of Euclid’s Elements
1.4 A look at Book I of the Elements
1.5 A critique of Euclid’s Elements
1.6 Final observations about the Elements
2. Axiomatic Systems and Incidence Geometry
2.1 The structure of an axiomatic system
2.2 An example: Incidence geometry
2.3 The parallel postulates in incidence geometry
2.4 Axiomatic systems and the real world
2.5 Theorems, proofs, and logic
2.6 Some theorems from incidence geometry
3. Axioms for Plane Geometry
3.1 The undefined terms and two fundamental axioms
3.2 Distance and the Ruler Postulate
3.3 Plane separation
3.4 Angle measure and the Protractor Postulate
3.5 The Crossbar Theorem and the Linear Pair Theorem
3.6 The Side-Angle-Side Postulate
3.7 The parallel postulates and models
4. Neutral Geometry
4.1 The Exterior Angle Theorem and perpendiculars
4.2 Triangle congruence conditions
4.3 Three inequalities for triangles
4.4 The Alternate Interior Angles Theorem
4.5 The Saccheri-Legendre Theorem
4.6 Quadrilaterals
4.7 Statements equivalent to the Euclidean Parallel Postulate
4.8 Rectangles and defect
4.9 The Universal Hyperbolic Theorem
5. Euclidean Geometry
5.1 Basic theorems of Euclidean geometry
5.2 The Parallel Projection Theorem
5.3 Similar triangles
5.4 The Pythagorean Theorem
5.5 Trigonometry
5.6 Exploring the Euclidean geometry of the triangle
6. Hyperbolic Geometry
6.1 The discovery of hyperbolic geometry
6.2 Basic theorems of hyperbolic geometry
6.3 Common perpendiculars
6.4 Limiting parallel rays and asymptotically parallel lines
6.5 Properties of the critical function
6.6 The defect of a triangle
6.7 Is the real world hyperbolic?
7. Area
7.1 The Neutral Area Postulate
7.2 Area in Euclidean geometry
7.3 Dissection theory in neutral geometry
7.4 Dissection theory in Euclidean geometry
7.5 Area and defect in hyperbolic geometry
8. Circles
8.1 Basic definitions
8.2 Circles and lines
8.3 Circles and triangles
8.4 Circles in Euclidean geometry
8.5 Circular continuity
8.6 Circumference and area of Euclidean circles
8.7 Exploring Euclidean circles
9. Constructions
9.1 Compass and straightedge constructions
9.2 Neutral constructions
9.3 Euclidean constructions
9.4 Construction of regular polygons
9.5 Area constructions
9.6 Three impossible constructions
10. Transformations
10.1 The transformational perspective
10.2 Properties of isometries
10.3 Rotations, translations, and glide reflections
10.4 Classification of Euclidean motions
10.5 Classification of hyperbolic motions
10.6 Similarity transformations in Euclidean geometry
10.7 A transformational approach to the foundations
10.8 Euclidean inversions in circles
11. Models
11.1 The significance of models for hyperbolic geometry
11.2 The Cartesian model for Euclidean geometry
11.3 The Poincaré disk model for hyperbolic geometry
11.4 Other models for hyperbolic geometry
11.5 Models for elliptic geometry
11.6 Regular Tessellations
12. Polygonal Models and the Geometry of Space
12.1 Curved surfaces
12.2 Approximate models for the hyperbolic plane
12.3 Geometric surfaces
12.4 The geometry of the universe
12.5 Conclusion
12.6 Further study
12.7 Templates
APPENDICES
A. Euclid’s Book I
A.1 Definitions
A.2 Postulates
A.3 Common Notions
A.4 Propositions
B. Systems of Axioms for Geometry
B.1 Filling in Euclid’s gaps
B.2 Hilbert’s axioms
B.3 Birkhoff’s axioms
B.4 MacLane’s axioms
B.5 SMSG axioms
B.6 UCSMP axioms
C. The Postulates Used in this Book
C.1 The undefined terms
C.2 Neutral postulates
C.3 Parallel postulates
C.4 Area postulates
C.5 The reflection postulate
C.6 Logical relationships
D. Set Notation and the Real Numbers
D.1 Some elementary set theory
D.2 Properties of the real numbers
D.3 Functions
E. The van Hiele Model
F. Hints for Selected Exercises

Bibliography

It Began with Babbage: The Genesis of Computer Science

Subrata Dasgupta

Oxford University Press |  2014 | páginas | rar - pdf | 2,9 Mb

link (password: matav)

epub - 4,8 Mb - link

As a field, computer science occupies a unique scientific space, in that its subject matter can exist in both physical and abstract realms. An artifact such as software is both tangible and not, and must be classified as something in between, or "liminal." The study and production of liminal artifacts allows for creative possibilities that are, and have been, possible only in computer science. In It Began with Babbage, computer scientist and writer Subrata Dasgupta examines the distinct history of computer science in terms of its creative innovations, reaching back to Charles Babbage in 1819. Since all artifacts of computer science are conceived with a use in mind, the computer scientist is not concerned with the natural laws that govern disciplines like physics or chemistry; instead, the field is more concerned with the concept of purpose. This requirement lends itself to a type of creative thinking that, as Dasgupta shows us, has exhibited itself throughout the history of computer science. More than any other, computer science is the science of the artificial, and has a unique history to accompany its unique focus.The book traces a path from Babbage's Difference Engine in the early 19th century to the end of the 1960s by when a new academic discipline named "computer science" had come into being. Along the way we meet characters like Babbage and Ada Lovelace, Turing and von Neumann, Shannon and Chomsky, and a host of other people from a variety of backgrounds who collectively created this new science of the artificial. And in the end, we see how and why computer science acquired a nature and history all of its own.

Contents
Acknowledgments ix
Prologue 1
1. Leibniz’s Th eme, Babbage’s Dream 9
2. Weaving Algebraic Patterns 17
3. Missing Links 28
4. Entscheidungsproblem : What’s in a Word? 44
5. Toward a Holy Grail 60
6. Intermezzo 83
7. A Tangled Web of Inventions 8 9
8. A Paradigm Is Born 108
9. A Liminal Artifact of an Uncommon Nature 1 34
10. Glimpses of a Scientifi c Style 1 49
11. I Compute, Th erefore I Am 1 57
12. “Th e Best Way to Design . . .” 178
13. Language Games 1 90
14. Going Heuristic 2 25
15. An Explosion of Subparadigms 2 41
16. Aesthetica 2 65
Epilogue 277
Dramatis personae 287
Bibliograph

Common Core Mathematics - Grades 9 to 12

O commom core mathematics é um currículo de matemática americano desenvolvido por uma equipa de educadores e investigadores em educação matemática, com base na investigação sobre o desenvolvimento cognitivo das crianças e na própria estrutura da matemática.

Common Core Mathematics is the most comprehensive Common Core State Standards-based mathematics curriculum available today.

• The modules are sequenced and paced to support the teaching of mathematics as an unfolding story that follows the logic of mathematics itself.
• They embody the instructional "shifts" and the standards for mathematical practice that are fundamental to the CCSS.
• Each module contains a sequence of les- sons that combine conceptual understanding, fluency, and application to meet the demands of each topic in the module.
• Formative assessments are included to support data-driven instruction.
• The modules are written by teams of master teachers and mathematicians.

Alguns elementos da equipa do projeto:

Richard Askey (reviewer); Sybilla Beckmann (writer); Douglas Clements (writer); Phil Daro (co-chair); Skip Fennell (reviewer); Brad Findell (writer); Karen Fuson (writer); Roger Howe (writer); Cathy Kessel (editor); William McCallum (chair); Bernie Madison (writer); Dick Scheaffer (writer); Denise Spangler (reviewer); Hung-Hsi Wu (writer); Jason Zimba (co-chair)

Os livros do currículo são comercializados em: eu.wiley.com
Maior parte do currículo está disponível, em versão, "draft" em engageny.org
Eis os que estão atualmente disponíveis:

	Grade 9 Module 1: Relationships Between Quantities and Reasoning with Equations Module 2: Descriptive Statistics Module 3: Linear and Exponential Relationships Module 4: Expressions and Equations Module 5: A Synthesis of Modeling with Equations and Functions
	Grade 10 Module 1: Congruence, Proof, and Constructions Module 2: Similarity, Proof, and Trigonometry Module 3: Extending to Three Dimensions Module 4: Connecting Algebra and Geometry through Coordinates Module 5: Circles With and Without Coordinates
	Grade 11 Module 1: Polynomial, Rational, and Radical Relationships Module 2: Trigonometric Functions
	Grade 12 Module 1: Complex Numbers and Transformations Module 2: Vectors and Matrices Module 3: Rational and Exponential Functions

Common Core Mathematics - Grades k to 8

O commom core mathematics é um currículo de matemática americano desenvolvido por uma equipa de educadores e investigadores em educação matemática, com base na investigação sobre o desenvolvimento cognitivo das crianças e na própria estrutura da matemática.

Common Core Mathematics is the most comprehensive Common Core State Standards-based mathematics curriculum available today.

• The modules are sequenced and paced to support the teaching of mathematics as an unfolding story that follows the logic of mathematics itself.
• They embody the instructional "shifts" and the standards for mathematical practice that are fundamental to the CCSS.
• Each module contains a sequence of les- sons that combine conceptual understanding, fluency, and application to meet the demands of each topic in the module.
• Formative assessments are included to support data-driven instruction.
• The modules are written by teams of master teachers and mathematicians.

Alguns elementos da equipa do projeto:

Richard Askey (reviewer); Sybilla Beckmann (writer); Douglas Clements (writer); Phil Daro (co-chair); Skip Fennell (reviewer); Brad Findell (writer); Karen Fuson (writer); Roger Howe (writer); Cathy Kessel (editor); William McCallum (chair); Bernie Madison (writer); Dick Scheaffer (writer); Denise Spangler (reviewer); Hung-Hsi Wu (writer); Jason Zimba (co-chair)

Os livros do currículo são comercializados em: eu.wiley.com
Maior parte do currículo está disponível, em versão, "draft" em engageny.org
Eis os que estão atualmente disponíveis:

	Module 1: Count Numbers to 10 Module 2: Identify and Describe Shapes Module 3: Comparison with Length, Weight, and Numbers to 10 Module 4: Number Pairs, Addition, and Subtraction of Numbers to 10 Module 5: Numbers 10–20, Counting to 100 by 1 and 10 Module 6: Analyze, Compare, Create, and Compose Shapes
	Module 1: Sums and Differences to 10 Module 2: Place Value, Comparison, Addition and Subtraction of Numbers to 20 Module 3: Ordering and Expressing Length Measurements as Numbers Module 4: Place Value, Comparison, Addition and Subtraction of Numbers to 40 Module 5: Identify, Compose, and Partition Shapes Module 6: Place Value, Comparison, Addition and Subtraction of Numbers to 100
	Module 1: Sums and Differences to 20 Module 2: Addition and Subtraction with Length, Weight, Capacity, and Time Measurements Module 3: Place Value, Counting, and Comparison of Numbers to 1000 Module 4: Addition and Subtraction of Numbers to 1000  Module 4a: Addition and Subtraction Within 1,000 with Word Problems to 100  Module 5: Preparation for Multiplication and Division Facts Module 6: Comparison, Addition, and Subtraction with Length and Money Module 7: Recognizing Angles, Faces, and Vertices of Shapes, Fractions of Shapes
	Module 1: Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 Module 2: Problem Solving with Mass, Time, and Capacity Module 3: Multiplication and Division with Factors of 6, 7, 8, and 9 Module 4: Multiplication and Area Module 5: Fractions as Numbers on the Number Line Module 6: Collecting and Displaying Data Module 7: Word Problems with Geometry and Measurement
	Module 1: Place Value, Rounding, and Algorithms for Addition and Subtraction Module 2: Unit Conversions and Problem Solving with Metric Measurement Module 3: Multi-Digit Multiplication and Division Module 4: Angle Measure and Plane Figures Module 5: Fraction Equivalence, Ordering, and Operations Module 6: Decimal Fractions Module 7: Exploring Multiplication
	Module 1: Place Value and Decimal Fractions Module 2: Multi-Digit Whole Number and Decimal Fraction Operations Module 3: Addition and Subtraction of Fractions Module 4: Multiplication and Division of Fractions and Decimal Fractions Module 5: Addition and Multiplication with Volume and Area Module 6: Graph Points on the Coordinate Plane to Solve Problems
	Module 1: Ratios and Unit Rates Module 2: Arithmetic Operations Including Dividing by a Fraction Module 3: Rational Numbers Module 4: Expressions and Equations Module 5: Area, Surface Area, and Volume Problems Module 6: Statistics
	Module 1: Ratios and Proportional Relationships Module 2: Rational Numbers Module 3: Expressions and Equations Module 4: Percent and Proportional Relationships Module 5: Statistics and Probability Module 6: Geometry
	Module 1: Integer Exponents and Scientific Notation Module 2: The Concept of Congruence Module 3: Similarity Module 4: Linear Equations Module 5: Examples of Functions from Geometry Module 6: Linear Functions Module 7: Introduction to Irrational Numbers Using Geometry