Diamond: A Paradox Logic

Series on Knots and Everything (Book 23)

N. S. Hellerstein


World Scientific Publishing Company; 2nd Revised edition | 2010 |  297 páginas | pdf | 1,8 Mb

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This book is about 'diamond', a logic of paradox. In diamond, a statement can be true yet false; an 'imaginary' state, midway between being and non-being. Diamond's imaginary values solve many logical paradoxes unsolvable in two-valued boolean logic. In this volume, paradoxes by Russell, Cantor, Berry and Zeno are all resolved. This book has three sections: Paradox Logic, which covers the classic paradoxes of mathematical logic, shows how they can be resolved in this new system; The Second Paradox, which relates diamond to Boolean logic and the Spencer-Brown modulator"; and Metamathematical Dilemma, which relates diamond to Godelian meta-mathematics and dilemma games.

Contents
Introduction xi
Part One: Paradox Logic 1
1. Paradox 3
A. The Liar 3
B. The Anti-Diagonal 7
C. Russell’s Paradox 8
D. Santa and the Grinch 10
E. Antistrephon 17
F. Parity of Infinity 18
G. The Heap 19
H. Finitude 21
I. Game Paradoxes 24
J. Cantor’s Paradox 26
K. Paradox of the Boundary 27
2. Diamond 29
A. The Buzzer 29
B. Diamond Values 31
C. Harmonic Functions 32
D. Gluts and Gaps 35
E. Diamond Circuits 38
F. Brownian Forms 40
G. Boundary Logic 46
3. Diamond Algebra 49
A. Bracket Algebra 49
B. Laws 58
C. Normal Forms 64
D. Completeness 69
4. Self-Reference 71
A. Re-Entrance and Fixedpoints 71
B. Phase Order 74
C. The Outer Fixedpoints 79
5. Fixedpoint Lattices 85
A. Relative Lattices 85
B. Seeds and Spirals 88
C. Shared Fixedpoints 91
D. Examples 93
6. Limit Logic 109
A. Limits 109
B. Limit Fixedpoints 114
C. The Halting Theorem 116
7. Paradox Resolved 119
A. The Liar and the Anti-Diagonal 119
B. Russell’s Paradox 120
C. Santa and the Grinch 122
D. Antistrephon 126
E. Infinity, Finitude and the Heap 128
F. Game Paradoxes 130
8. The Continuum 131
A. Cantor’s Paradox 131
B. Dedekind Splices 132
C. Null Quotients 134
D. Cantor’s Number 136
E. The Line within the Diamond 139
F. Zeno’s Theorem 144
G. Fuzzy Chaos 145
9. Clique Theory 151
A. Cliques 151
B. Clique Equality 156
C. Clique Axioms 160
D. Graph Cliques 164
E. Clique Circuits 166
Part Two: The Second Paradox 169
10. Orthogonal Logics 171
A. Analytic Functions 171
B. Function Types 173
C. Dihedral Conjugation 177
D. Star Logic 180
E. Harmonic Projection 182
F. Diamond Types? 183
11. Interferometry 185
A. Quadrature 185
B. Diffraction 193
C. Buzzers and Toggles 200
D. Analytic Diffraction 204
E. Diffracting “Two Ducks in a Box” 206
12. How to Count to Two 209
A. Brownian and Kauffman Modulators 209
B. Diffracting the Modulators 212
C. Rotors, Pumps and Tapes 218
D. The Ganglion 222
Part Three: Metamathematical Dilemma 225
13. Metamathematics 227
A. Gödelian Quanta 227
B. Meta-Logic 231
C. Dialectic 236
D. Dialectical Dilemma 238
14. Dilemma 241
A. Milo’s Trick 241
B. Prisoner’s Dilemma 245
C. Dilemma Games 249
D. Dilemma Diamond 253
E. Banker’s Dilemma 257
F. The Unexpected Departure 260
Notes 265
Bibliography 287
Index 291

Educational Interfaces between Mathematics and Industry: Report on an ICMI-ICIAM-Study

Alain Damlamian, José Francisco Rodrigues e Rudolf Sträßer

New ICMI studies series, v.16.

Springer | 2013 | 451 páginas | rar - pdf | 4 Mb

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This book is the “Study Book” of ICMI-Study no. 20, which was run in cooperation with the International Congress on Industry and Applied Mathematics (ICIAM). The editors were the co-chairs of the study (Damlamian, Straesser) and the organiser of the Study Conference (Rodrigues). The text contains a comprehensive report on the findings of the Study Conference, original plenary presentations of the Study Conference, reports on the Working Groups and selected papers from all over world. This content was selected by the editors as especially pertinent to the study each individual chapter represents a significant contribution to current research.

Contents

Part I Discussion Document and Study Report

Discussion Document. . . . . . 3

The International Programme Committee

Report on the Study . . .  . . . 17

Alain Damlamian, José Francisco Rodrigues and Rudolf Sträßer

Part II Plenary and Invited Lectures

Getting Math off the Ground: Applied Mathematics at Boeing . . 27

Stephen P. Keeler and Thomas A. Grandine

Mathematics in the Workplace: Issues and Challenges . . . .. . . 43

Celia Hoyles, Richard Noss, Phillip Kent and Arthur Bakker

Mathematical Modeling Education is the Most Important Educational Interface Between Mathematics and Industry . . . 51

Tatsien Li

Models for Industrial Problems: How to Find and How to Use them—in Industry and in Education . . . . . . .  . 59

Helmut Neunzert

Interfacing Education and Research with Mathematics for Industry: The Endeavor in Japan. 77

Masato Wakayama

Part III WG Education/Training with Industry Participation

Education/Training with Industry Participation . . . . 95

Gail FitzSimons and Tom Mitsui

How it is Possible to Make Real-World Mathematics More Visible: Some Results from Two Italian Projects  . . 109

Cinzia Bonotto

The Project ‘‘Ways to More MINT-Graduates’’ of the Bavarian Business Association (vbw) with Focus on the M (=Mathematics) at the University of Augsburg, Germany . . 119

Matthias Brandl

Mathematics in a Safety–Critical Work Context: The Case of Numeracy for Nursing . 127

Diana Coben and Meriel Hutton

Linking Professional Experiences with Academic Knowledge: The Construction of Statistical Concepts by Sale Manager Apprentices . . 137

Corinne Hahn

Learning Conversation in Mathematics Practice School–Industry Partnerships as Arena for Teacher Education . . 147

Gert Monstad Hana, Ragnhild Hansen, Marit Johnsen-Høines, Inger Elin Lilland and Toril Eskeland Rangnes

The Threefold Dilemma of Missing Coherence: Bridging the Artificial Reef Between the Mainland and Some Isolated Islands. . 57

Guenter Törner, Volker Grotensohn and Bettina Roesken

The Project ‘‘Mathe-Meister’’: A Mathematical Self Assessment Centre with Diagnostic Feedback for Vocational Trainees . . . 165

Kathrin Winter

Part IV WG University and Academic Technical/Vocational Education

University and Academic Technical/Vocational Education . . .  . 173

Nilima Nigam and José Francisco Rodrigues

Mathematics for Engineering and Engineering for Mathematics . .185

Miquel Alberti Palmer, Sergio Amat, Sonia Busquier, Pilar Romero and Juan Tejada

Laboratory for Computational Mathematics: An Interface Between Academia and Industry 199

A. Araújo, S. Barbeiro and J. A. Ferreira

Improving the Industrial/Mathematics Interface.. . 205

Jean P. F. Charpin and Stephen B. G. O’Brien

Two Masters on ‘Mathematics for Industry’ at the Universities of Paris and of Pau . 213

Edwige Godlewski, M. Madaune-Tort and S. Dossou-Gbete

Mathematics in Industry and Teachers’ Training . .  223

Matti Heilio

Interfaces Between Mathematics and Industry and the Use of Technology in Mathematics Education in India  . 229

Ajit Kumar

Modeling Modeling: Developing Habits of Mathematical Minds . . 237

John A. Pelesko, Jinfa Cai and Louis F. Rossi

The Evolution of Graduate Applied Math Courses in the Institute of Mathematics, University of the Philippines. . 247

Carlene P. C. Pilar-Arceo and Jose Maria L. Escaner IV

The Vertical Integration of Industrial Mathematics, the WPI Experience  . 253

Bogdan Vernescu

Part V WG Education in Schools

Educational Interfaces Between Mathematics and Industry at School Level . . 263

Gabriele Kaiser, Henk van der Kooij and Geoff Wake

Mathematical Applications, Modelling and Technology as Windows into Industry Based Mathematical Practice .  . 271

Vince Geiger

Mathematics Education and the Information Society . . . 279

Koeno Gravemeijer

Authentic Complex Modelling Problems in Mathematics Education . . 287

Gabriele Kaiser, Martin Bracke, Simone Göttlich and Christine Kaland

Embedding Authentic Real World Tasks into Secondary Mathematics Curricula .. 299

Gloria Stillman and K. E. D. Ng

Drawing on Understanding of Workplace Practice to Inform Design of General Mathematics Curricula .  . 309

Geoff Wake

Part VI WG Mathematics-Industry Communication

Communication and Collaboration. . .. . 319

Solomon Garfunkel, Rolf Jeltsch and Nilima Nigam

Engineering, Mathematics Communication, and Education: Reflections on a Personal Experience . . . 333

Jorge Buescu

A View on Mathematical Discourse in Research and Development . . . 341

Vasco Alexander Schmidt

Using Popular Science in a Mathematical Modeling Course  . . 351

Burt S. Tilley

Part VII WG Technology Issues

Technology Issues . .  . 359

Helmer Aslaksen and Fadil Santosa

Tackling the Challenges of Computational Mathematics Education of Engineers . . 365

France Caron and André Garon

Integrating Computational Modelling in Science, Technology, Engineering, and Mathematics Education.. . 375

Rui Gomes Neves, Jorge Carvalho Silva and Vítor Duarte Teodoro

Incorporating the Ideas and Methods of Mathematical Modeling into Calculus Teaching .385

Qixiao Ye

Part VIII WG the Mathematics-Industry Interface

The Mathematics–Industry Interface . . 397

Jofré Alejandro and Lutz-Westphal Brigitte

Part IX Selected Papers Linked to More than One Working Group 

Inappropriate Use of Spreadsheets in the Finance Industry . . . . 403

Djordje M. Kadijevich

MITACS Accelerate: A Case Study of a Successful Industrial Research Internship Program . . . 413

Sarah Petersen and Marsh Rebeccah

A Meta-analysis by Mathematics Teachers of the GIFT Program Using Success Case Methodology . . . 421

Richard Millman, Meltem Alemdar and Bonnie Harris

Cultivating an Interface Through Collaborative Research Between Engineers in Nippon Steel & Sumitomo Metal and Mathematicians in University . . . 427

Junichi Nakagawa and Masahiro Yamamoto

An Introduction to CUMCM: China/Contemporary Undergraduate Mathematical Contest in Modeling . . 435

Jinxing Xie

Part X Conclusion

Conclusion on Educational Interfaces Between Mathematics and Industry. . . . . 447

Alain Damlamian, José Francisco Rodrigues and Rudolf Sträßer

Index . . . 453

Collaboration in Teacher Education: Examples from the Context of Mathematics Education

(Mathematics Teacher Education, 1)

 

• Andrea Peter-Koop, 
• Vânia Santos-Wagner, 
• Chris Breen, 
• Andy Begg

Springer| 2003 | 279 páginas | rar - pdf | 9,6 Mb

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This book systematically explores and reflects on a variety of issues related to collaborative mathematics teacher education practice and research ? such as classroom coaching, mentoring or co-learning agreements - highlighting the evolution and implications of collaborative enterprises in different cultural settings. It is relevant to educational researchers, research students and practitioners.

TABLE OF CONTENTS

Foreword

Helen Christiansen, Caroline Krentz and Linda Goulet

PROLOGUE

PART ONE: INVESTIGATING INITIAL AND ONGOING TEACHER EDUCATION

A Journey towards Collaboration

Sue Southwood and Jaap Kuiper

2 Collaboration and Sharing as Crucial Elements of Professional Development

Steven Nisbet, Elizabeth Warren and Tom Cooper

3 The Evolution of an Interdisciplinary Collaborative for Pre-Service Teacher Reform-Research Informs Practice

Marilyn Carlson, Michelle Zandieh, Irene Bloom and Daiyo Sawada

4 An Approach to Collaboration in Elementary Pre-Service Teacher Education

Victoria Sanchez

PART TWO: FURTHER ISSUES IN TEACHER EDUCATION

5 Classroom Coaching as a Collaborative Activity in Professional Development

Joanne Rossi Becker and Barbara J Pence

6 A Collaborative Project Using Narratives-What Happens when Pupils Work on Mathematical Investigations?

João Pedro da Ponte, Maria Irene Segurado and Hélia Oliveira

7 The Role of Collaboration for Developing Teacher-Researchers

Vania Santos-Wagner

8 Sharing Teacher Training Methods-The Case of Problematic

Learning Situations in Geometry

Hagar Gal, Liora Linchevski and Anne Cockburn

PART THREE: WORKING WITH OTHER IMPORTANT PARTNERS IN DIFFERENT SOCIO-CULTURAL CONTEXTS

9 Co-Learning in the Collaborative Mathematics Classroom 135

Julie-Ann Edwards and Keith Jones

10 Collaborative Practice with Parents-The Role of the Researcher as Mediator

Marta Civil and Rosi Andrade

11 The Role of the Principal in Mathematics Teacher Development-Bridging the Dichotomy between Leadership and Collaboration

Konrad Krainer and Andrea Peter-Koop

12 Intercultural Collaboration: A Celebration of Commonalties and Differences

Bridget Arvold

PART FOUR: ALTERNATIVE PERSPECTIVES ON COLLABORATION

13 A Case for Collaborative Staff Development-A Path Layered while Walking

Chris Breen, Gabeba Agherdien and Agatha Lebethe

14 Collaborative Reflection through Sharing Significant Incidents 235

Francis Lopez-Real

15 More than Collaboration: Concern, Connection, Community and Curriculum

Andy Begg

EPILOGUE 269

INDEX 273

The Liber mahameleth, A 12th-century mathematical treatise

Jacques Sesiano

Springer | 2014 | 1823 páginas | rar - pdf | Mb

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Provides a translation and commentary of the principal source on mathematics of 12th-century Islamic Spain
Contains a glossary featuring important terms for the knowledge of Latin scientific (and other) key words from that time period
Presents arithmetic theory and a variety of application problems relating to historical daily and commercial life

The Liber mahameleth is a work in Latin written in the mid-12th century based (mainly) on Arabic sources from Islamic Spain. It is now our principal source on mathematics in Islamic Spain at that time; There are few extant Arabic texts and no one is as complete as the LM. It is also the second largest mathematical work from the Latin Middle Ages (the other is by Fibonacci, some 50 years later).

Since the three main manuscripts preserving it are incomplete and there are many scribal errors, a reliable Latin text has been established, which reports (in notes) the various readings of the manuscripts and the errors in them. This is how a so-called critical edition is made. This edition of the Latin text is preceded by General Introduction, describing the various manuscripts, the content of the work and what we know about its author.

Part Two of the volume is a translation of the text and ends with a glossary of Latin terms. The glossary will be of great importance for the knowledge of Latin scientific terms from that time, since there is no other mathematical text of this size from the 12th century. Part Three begins with a short introduction and then analyzes all the problems from the text, with a summary of the mathematical methods involved in each chapter. The commentary is a companion to the translation and explains the author's solving methods.

TABLE OF CONTENTS
Part I.-
1. The rebirth of mathematics in mediaeval Europe.-
2. The Liber mahameleth.-
3. Manuscripts of the Liber mahameleth.-
4. Transmission of the Liber mahameleth.-
5. Mathematics in the Liber mahameleth.-
6. The edited Latin text.-
Part II.- Translation.- Beginning of the Book (on) mahameleth.-
A-I. on Numbers.-
A-II. on the Premisses necessary for practical arithmetic.-
A-III. on Multiplication.- A-IV. on Division.-
A-V. on the Multiplication of fractions.-
A-VI. on the Addition of fractions to fractions.-
A-VII. on Subtracting.-
A-VIII. on the Division of fractions, with or without integers.-
A-IX. on the Determination of roots and on their multiplication, division, subtraction, addition and other related subjects.-
Beginning of the Second Part.-
B-I. on Buying and selling.-
B-II. on Profits.-
B-III. on Profit in partnership.-
B-IV. on Division according to portions.-
B-V. on Masses.- B-VI. on Drapery.-
B-VIII. on Linens.- B-VIII. on Grinding.-
B-IX. on Boiling must.- B-X. on Borrowing.-
B-XI. on Hiring.-
B-XII. on the Diversity of workers' wages.-
B-XIII on Hiring carriers.-
B-XIV. on Hiring stone-cutters.-
B-XV. on the Consumption of oil by lamps.-
B-XVI. on the Consumption by animals.-
B-XVII. on the Consumption of bread by men.-
B-XVIII. on the Exchange of morabitini.-
B-XIX. on Cisterns.- XX. on Ladders.-
B-XXI. on Another topic.- B-XXII. on Messengers.- B-XXIII. on Another topic.- Glossary.-
Part III.- Mathematical Commentary.-
Book A.- A-I. Numbers.- A-II. Premisses.- A-III. Multiplication of Integers.- A-IV. Division of integers.-
A-V. Multiplication of fractions.- A-VI. Addition of fractions.- A-VII. Subtraction of fractions.-
A-VIII. Division of fractions.- A-IX. Roots.-
Book B.- Introduction.- B-I. Buying and selling.- B-II. Profit.-
B-III. Profit in partnership.- B-IV. Sharing out according to prescribed parts.- B-V. Masses.- B-VI. Drapery.-B-VII. Linen cloths.- B-VIII. Grinding.- B-IX. Boiling must.- B-X. Borrowing.- B-XI. Hiring workers.- B-XII. Wages in arithmetical progression.- B-XIII. Hiring a carrier.- B-XIV. Hiring stone-cutters.- B-XV. Consumption of lamp-oil.- B-XVI. Consumption by animals.- B-XVII. Consumption of bread.- B-XVIII. Exchanging moneys.- B-XIX. Cisterns.- B-XX. Ladders.- B-XXI. Bundles.- B-XXII. Messengers.- B-XXIII. Mutual lending.-
Bibliography.-
Index.

A Decade of Middle School Mathematics Curriculum Implementation: Lessons Learned from the Show-Me Project

(Research in Mathematics Education)

Margaret R Meyer e  Cynthia W Langrall

Information Age Publishing | 2008 | 317 páginas | rar - pdf | 1,88 Mb

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A volume in Research in Mathematics Education Series Editor Barbara J. Dougherty, University of Mississippi The purpose of this book is to document the work of the Show-Me Project (1997-2007) and to highlight lessons learned about curriculum implementation. Although the Show-Me Project was charged with promoting the dissemination and implementation of four distinct comprehensive curriculum programs (Connected Mathematics, Mathematics in Context, MathScape, and MathThematics), most of the lessons learned from this work are not curriculum specific. Rather, they cut across the four programs and share commonalities with standards-based curriculum reform at any level. We believe that documenting these lessons learned will be one of the legacies of the Show-Me Project We anticipate that the comprehensive nature of this work will attract readers from multiple audiences that include state and district mathematics supervisors, middle grades mathematics teachers and administrators involved in curriculum reform, as well as mathematics teacher educators. Those about to embark on the review of curriculum materials will appreciate reading about the processes employed by other districts. Readers with interests in a particular curriculum program will be able to trace the curriculum-specific chapters to gain insights into how the design of the curricula relate to professional development, adoption and implementation issues, and teachers' personal experience using the curriculum materials. Individuals who provide professional development at the middle grades level will find chapters that they can use for both general and focused discussions. Teachers at all stages of implementation will recognize their own experiences in reading and reflecting on the stories of teacher change. Mathematics educators will find ideas on how these curricula can be used in the preparation of preservice middle grades teachers.

Contents
Preface..... ix
PART I - Setting the Stage
1 The Road to Reform.... 3
Robert Reys
2 The Genesis of the NSF Curriculum Implementation Centers..... 9
John S. (Spud) Bradley
3 The Show-Me Project: A Decade of Support for Standards-Based Curriculum Implementation... 17
Barbara Reys
PART II - Adoption Case Studies and Implementation Stories
4 Teacher Investment in Curriculum Adoption: The Role of Piloting Materials.... 35
Cary Tuckey
5 Consensus Building: A Key Component of Curriculum Adoption..... 43
Cheryl Rectanus
6 The Critical Work of a Review Committee in Curriculum Adoption....... 55
Leslie E. J. Nielsen
7 Informing a State Adoption Process Through Curriculum Implementation...... 67
Murrel Brewer Hoover
8 Curriculum Implementation Through Teacher Collaboration..... 79
Chip Sharp
9 Curriculum Implementation: Teachers on a Journey.... 87
Carol Den Otter
10 The Role of Teacher Leaders in Curriculum Implementation......... 97
Emily R. Fagan and Rhonda K. Weinstein
11 Supporting Curriculum Implementation Through Its Ups and Downs...... 107
Linda A. Tetley
12 When Things Go Wrong in Curriculum Implementation......... 115
Jane O. Swafford and Cynthia W. Langrall
13 Issues of Implementation: Making Mathematics Accessible to All Learners... 125
Jennifer M. Bay-Williams and Rebecca McGraw
PART III - Professional Development and Teacher Education
14 An Overview of Professional Development......... 141
Elizabeth Phillips, Glenda Lappan, Yvonne Grant, and Fran Arbaugh
15 Key Features of Connected Mathematics for Professional Development... 151
Elizabeth Phillips and Glenda Lappan
16 Key Features of Mathematics in Context for Professional Development..... 161
David C. Webb
17 Key Features of MathScape for Professional Development...... 173
Fred Gross and Emily R. Fagan
18 Key Features of Middle Grades MATHThematics for Professional Development......... 183
Rick Billstein and Jim Williamson
19 Teachers’ Needs at Different Phases of Curriculum Implementation........ 191
Fran Arbaugh
20 Professional Development for Administrators: What They Need to Know to Support Curriculum Adoption and Implementation......... 201
Margaret R. Meyer and Fran Arbaugh
21 A Story of Teacher Growth: Increasing Expectations..... 211
Jane Peterson Porath
22 A Story of Teacher Growth: Teacher as Learner.....217
Barbara Diliegghio
23 A Story of Teacher Growth: Making a Difference..... 225
Kelly Watts
24 A Story of Teacher Growth: From Telling to Facilitating...... 231
Fabienne Anderson
25 Research Mathematician and Mathematics Educator: A Foot in Both Worlds..... 235
Ira J. Papick
26 Using NSF-Funded Mathematics Curricula to Enhance Teacher Preparation........ 245
Jeffrey Frykholm
27 NSF-funded Mathematics Curricula in the Field: A Preservice Teacher’s Opportunity to See Theory in Practice.... 259
Ann Bledsoe
PART IV - What Have We Learned and Where Do We Go from Here?
28 Lessons Learned from the Show-Me Project.......... 269
Barbara Reys
29 Looking into the Future: Curriculum Challenges for the Next Ten Years..... 279
Margaret R. Meyer and Cynthia W. Langrall
References.......291

Children's Logical and Mathematical Cognition Progress in Cognitive Development Research

 C.J. Brainerd

Springer | 2011 - reprint of the original 1st ed. 1982 edition | páginas | pdf | 6,6 Mb

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Contents
Chapter 1 Conservation - Nonconservation: Alternative Explanations .. 1
Curt Acredolo
Conservation and the Appreciation of an Identity Rule ....
Operational and Nonoperational Conservation .. 2
Nonconservation and the Overreliance on Perceptual Cues .... 4
Pseudononconservation .... 5
Nonoperational Conservation .... 14
Conclusions ....... 21
Future Research: The Development of the Identity Rule ..... 24
Reference Notes ..... 27
References ...... 27
Chapter 2 The Acquisition and Elaboration of the Number Word Sequence .... 33
Karen C. Fuson, John Richards, and Diane J. Briars
Acquisition of the Sequence .... 35
Elaboration of the Sequence ... 55
Conclusion ......... 89
Reference Notes ...... 89
References ..... 91
Chapter 3 Children's Concepts of Chance and Probability
Harry W. Hoemann and Bruce M. Ross
Piagetian Theory ... 94
Subsequent Studies .... 99
Theoretical Implications ... 116
References .... 120
Chapter 4 The Development of Quantity Concepts: Perceptual and Linguistic Factors .. 123
Linda S. Siegel
Linguistic Factors and the Development of Quantity Concepts ..... 123
A Taxonomy of Quantity Concepts .... 124
The Relationship between Language and Thought in the Child .... 128
Study 1: Concept versus Language ....... 129
Study 2: Does Language Training Facilitate Concept Acquisition? ... 132
Study 3: Visual versus Verbal Functions .... 138
Study 4: Training of Cognitive and Language Abilities ...... 140
Study 5: Cognitive Development of Children with Impaired Language Development ... 141
Study 6: The Abstraction of the Concept of Number ....... 144
Conclusion ........ 152
Reference Notes... 153
References ..... 153
Chapter 5 Culture and the Development of Numerical Cognition: Studies among the Oksapmin of Papua New Guinea ... 157
Geoffrey B. Saxe
Methodology and Cross-Cultural Number Research .... 158
The Oksapmin Community ..... 159
Studies on Numerical Cognition among the Oksapmin ... 160
Concluding Remarks
Chapter 6 Children's Concept Learning as Rule-Sampling Systems with Markovian Properties . 177
Charles J. Brainerd
Concept Learning as Rule Sampling ....179
Some Questions about Concept Learning ... 185
Some Experimental Evidence ...192
Remark ......202
Appendix ... 203
References .. 208
Index ..... 213

Big ideas of early mathematics : what teachers of young children need to know

The Early Math Collaborative

Pearson | 2013 | 209 páginas | rar -pdf | 4,2 Mb


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In this unique guide, classroom teachers, coaches, curriculum coordinators, college students, and teacher educators get a practical look at the foundational concepts and skills of early mathematics, and see how to implement them in their early childhood classrooms.

Big Ideas of Early Mathematics presents the skills educators need to organize for mathematics teaching and learning during the early years. For teachers of children ages three through six, the book provides foundations for further mathematics learning and helps facilitate long-term mathematical understanding.

Contents
About the Authors vii
Foreword ix
Preface x
INTRODUCTION 2
Chapter 1 SETS 12
Chapter 2 NUMBER SENSE 28
Chapter 3 COUNTING 46
Chapter 4 NUMBER OPERATIONS 64
Chapter 5 PATTERN 82
Chapter 6 MEASUREMENT 98
Chapter 7 DATA ANALYSIS 112
Chapter 8 SPATIAL RELATIONSHIPS 130
Chapter 9 SHAPE 146
CONCLUSION 168
Big Ideas Charts 174
Book Study Guide 184
Bibliography 190
Index 192