The overall goal of Modelling and Applications in Mathematics Education is to provide a comprehensive overview of the state-of-the-art in the field of modelling and applications in mathematics education. Key issues are dealt with, among which are the following: Epistemology and the relationships between mathematics and the "rest of the world"; the meaning of mathematical modelling and its process components; the respect in which the distinction between pure mathematics and applications of mathematics make sense Authenticity and Goals dealing with modelling and applications in mathematics teaching; appropriate balance between modelling activities and other mathematical activities; the role that authentic problem situations play in modelling and applications activities odelling Competencies: characterizing how a student's modelling competency can be characterized; identifiable sub-competencies, and the ways they constitute a general modelling competency; developing competency over time athematical Competencies: identifying the most important mathematical competencies that students should acquire, and how modelling and applications activities can contribute toward building up these competencies; the meaning of "Mathematical Literacy" in relation to modelling odelling Pedagogy: appropriate pedagogical principles and strategies for the development of modelling courses and their teaching; the role of technology in the teaching of modelling and applications Implementation and Practice: the role of modelling and applications in everyday mathematics teaching; major impediments and obstacles; advancing the use of modelling examples in everyday classrooms; documenting successful implementation of modelling in mathematics teaching Assessment and Evaluation: assessment modes that capture the essential components of modelling competency; modes available for modelling and applications courses and curricula; appropriate strategies to implement new assessment and evaluation modes in practice The contributing authors are eminent members of the mathematics education community. Modelling and Applications in Mathematics Education will be of special interest to mathematics educators, teacher educators, researchers, education administrators, curriculum developers and student teachers. TOC:Introduction.- Plenaries.- Issues.- Epistemology.- Authenticity and Goals.- Modelling Competencies.- Applications and Modelling for Mathematics.- Modelling Pedagogy.- Implementation and Practice.- Assessment and Evaluation.- Levels.- Primary.- Lower Secondary.- Upper Secondary.- Tertiary.- Teacher Education.- Cases.- Bibliography. Among the themes that have been central to mathematics education dur­ ing the last 30 years are those of mathematical modelling and applications of mathematics to extra-mathematical fields. More generally we refer to these as relations between mathematics and the extra-mathematical world (some­ times also called the "real world") or preferably, according to Henry PoUak, the "rest of the world". That applications and modelling have been important themes in mathematics education can be inferred from the wealth of litera­ ture on these topics, including material generated from a multitude of na­ tional and international conferences. In particular let us mention firstly the ICMEs (the International Congresses on Mathematical Education), with their regular working or topic groups and lectures on applications and modelling; and secondly the series of ICTMAs (the International Conferences on the Teaching of Mathematical Modelling and Applications) which have been held biennially since 1983. Their Proceedings and Survey Lectures, have addressed the state-of-the-art at the relevant time, and contain many exam­ ples, studies, conceptual contributions and resources involving relations between the real world and mathematics, for all levels of the educational system. In curricula and textbooks we find today many more references to real world phenomena and problems than, say, twenty years ago. Preface xi Introduction 1(32) Introduction 3(30) Mogens Niss Werner Blum Peter Galbraith Plenaries 33(88) Less chalk, less words, less symbols . . . more objects, more context, more actions 35(10) Claudi Alsina What's all the fuss about competencies? 45(12) Morten Blomhøj Tomas Højgaard Jensen A theory of mathematical modelling in technological settings 57(12) Jere Confrey Alan Maloney What knowledge do teachers need for teaching mathematics through applications and modelling? 69(10) Helen M. Doerr Beyond the low hanging fruit 79(10) Peter Galbraith Modelling for life: Mathematics and children's experience 89(10) Brian Greer Lieven Verschaffel Swapna Mukhopadhyay Modelling in lower secondary mathematics classroom -- problems and opportunities 99(10) Gabriele Kaiser Katja Maass Mathematical modelling -- A conversation with Henry Pollak 109(12) Henry Pollak Issues in Applications and Modelling 121(328) Epistemology and Modelling 123(56) Epistemology and modelling -- overview 125(4) Jere Confrey Modelling body motion: An approach to functions using measuring instruments 129(8) Ferdinando Arzarello Giovanni Pezzi Ornella Robutti Emergent modelling as a precursor to mathematical modelling 137(8) Koeno Gravemeijer Proving and modelling 145(8) Gila Hanna Hans Niels Jahnke A developmental approach for supporting the epistemology of modelling 153(8) Richard Lehrer Leona Schauble What is distinctive in (our views about) models and modelling perspectives on mathematics problem solving, learning, and teaching? 161(10) Richard Lesh Caroline Yoon Everyday instruments: On the use of mathematics 171(8) Rudolf Strasser Authenticity and Goals 179(38) Authenticity and Goals -- overview 181(4) Peter Galbraith How to replace word problems with activities of realistic mathematical modelling 185(8) Cinzia Bonotto The relevance of modelling and applications: Relevant to whom and for what purpose? 193(8) Eva Jablonka Features and impact of the authenticity of applied mathematical school tasks 201(8) Torulf Palm Elementary modelling in mathematics lessons: The interplay between ``real-world'' knowledge and ``mathematical structures'' 209(8) Ralph Schwarzkopf Modelling Competencies 217(48) Modelling compentencies -- overview 219(6) Brian Greer Lieven Verschaffel Levels of modelling competencies 225(8) Herbert Henning Mike Keune Modelling both complexity and abstraction: A paradox? 233(8) Mihaela Singer Studying and remedying students' modelling competencies: Routine behaviour or adaptive expertise 241(8) Dirk De Bock Wim Van Dooren Dirk Janssens Assessing the ``phases'' of mathematical modelling 249(8) Ken Houston The arithmetic operations as mathematical models 257(8) Zalman Usiskin Applications & Modelling for Mathematics 265(54) Applications and modelling for mathematics -- overview 267(8) Eric Muller Hugh Burkhardt The roles of modelling in learning mathematics 275(10) Malcolm Swan Ross Turner Caroline Yoon Eric Muller Developing mathematical literacy 285(10) Lynn Arthur Steen Ross Turner Hugh Burkhardt Classroom activities and the teacher 295(14) Søren Antonius Chris Haines Thomas Højgaard Jensen Mogens Niss Hugh Burkhardt Uses of technologies in learning mathematics through modelling 309(10) Daniel Pead Bill Ralph Eric Muller Modelling Pedagogy 319(46) Modelling pedagogy -- overview 321(4) Hans-Wolfgang Henn Mathematical modelling in high school mathematics: Teachers' thinking an practise 325(8) Olive Chapman Mathematical modelling in teacher education -- necessity or unnecessarily 333(8) Thomas Lingefjard Building concepts and conceptions in technology-based open learning environments 341(8) Stephan Hussmann Towards a wider implementation of mathematical modelling at upper secondary and tertiary levels 349(8) Djordje Kadijevich How might we share models through cooperative mathematical modelling? Focus on situations based on individual experiences 357(8) Akio Matsuzaki Implementation and Practice 365(38) Implementation and Practice -- overview 367(4) Thomas Lingefjard Some conditions for modelling to exist in mathematics classrooms 371(8) Michele Artaud Picture (im)perfect mathematics! 379(8) Wilfried Herget Marlene Torres-Skoumal Learning mathematical modelling -- from the perspective of probability and statistics education 387(8) Ewa Lakoma Considering workplace activity from a mathematical modelling perspective 395(8) Geoff Wake Assessment and Evaluation 403(46) Assessment and evaluation -- overview 405(4) Peter Galbraith Modelling based project examination 409(8) Søren Antionius Mathematical modelling and applications: Ability and competence frameworks 417(8) Christopher Haines Rosalind Crouch ``To model, or to let them model?'' That is the question! 425(8) Jerry Lege Modelling and applications in PISA 433(8) Ross Turner Assessment of applied mathematics and modelling: Using a laboratory-like environment 441(8) Pauline Vos Educational Levels 449(34) Modelling and applications in primary education 451(6) Maria Salett Biembengut Possibilities for, and obstacles to teaching applications and modelling in the lower secondary levels 457(6) Toshikazu Ikeda Upper secondary perspectives on applications and Modelling 463(6) Gloria Stillman Teaching applications and modelling at tertiary level 469(6) Claudi Alsina Modelling in teacher education 475(8) Thomas Lingefjard Cases in Applications and Modelling 483(28) Moving the context of modelling to the forefront: Preservice teachers' investigations of equity in testing 485(6) Katie Makar Jere Confrey Modelling in Ontario: Success in moving along the continuum 491(6) Christine Suurtamm Geoffrey Roulet Implementation case study: Sustaining curriculum change 497(6) Gloria Stillman Mathematical modelling of social issues in school mathematics in South Africa 503(8) Cyril Julie Vimolan Mudaly Bibliography 511(8) Index 519