**European Research in Mathematics Education IV**

*Proceedings of the Fourth Congress of the European Society for Research in Mathematics Education*

*Proceedings of the Fourth Congress of the European Society for Research in Mathematics Education*

##### Sant Feliu de Guíxols, Spain – 17 – 21 February 2005

*Chief editor*

*Chief editor*

**Marianna Bosch**

FUNDEMI IQS – Universitat Ramon Llull

*Editorial board*

*Editorial board*

G. de Abreu, M. Artaud, M. Artigue, M. Bartolini-Bussi, B. Barzel, C. Bergsten, M. Blomhøj, R. Biehler, J. Carrillo, M. César, P. Drijvers, T. Dreyfus, J.P. Drouhard, R. Duval, R. Even, P. L. Ferrari, I. M. Gómez-Chacón, N. Gorgorió, E. Gray, A. Gutiérrez, C. Haines, M. S. Hannula, M. Hejný, M. Johnsen Høines, G. Kaiser, C. Knipping, D. Küchemann, A. Kuzniak, L. Kvasz, G. Littler, J. Mamona-Downs, M. A. Mariotti, M. Maschietto, M. Meehan, M. Meletiou, J. Monaghan, C. Morgan, K. Nordström, M. G. Ottaviani, M. Panizza, B. Parzysz, D. Peretz, A. Pesci, G. Philippou, D. Pratt, S. Prediger, L. Puig, L. Radford, T. Rowland, W. Schlöglmann, L. Serrazina, R. Straesser, L. Trouche, P. Valero

*Editing assistance*

*Editing assistance*

Marta Perpiñán, M. Àngels Portabella

© Copyright 2006 left to the authors

FUNDEMI IQS – Universitat Ramon Llull

www.fundemi.url.edu

Legal deposit B – 46971 – 2006

ISBN 84-611-3282-3

**TABLE OF CONTENTS**

**INTRODUCTION 13**

Barbara Jaworski, Marianna Bosch

**PLENARIES**

#### Opening plenary

Steps towards a new epistemology in mathematics education 21*Yves Chevallard*

#### Closing plenary

The role of mathematics education research in influencing educational policy 31*Margaret Brown*

Panel: History and theory of mathematics education

An introduction to the topic of the panel. Remembering Hans-Georg Steiner: the theory of mathematics education program 45*Juan D. Godino*

History and theory of mathematics education 48

*Michèle Artigue*

History and theory of mathematics education 54

*Paul Ernest*

Not out of the blue: historical roots of mathematics education in Italy 59

*Fulvia Furinghetti*

**WORKING GROUP 1. The role of metaphors and images in the learning and understanding of mathematics**

The role of metaphors and images in the learning and understanding of mathematics 67

*Bernard Parzysz, Angela Pesci, Christer Bergsten*

Students interacting with an artefact designed to visualise three-dimensional analytic geometry 71

Christer Bergsten, Torbjörn Fransson Metaphors in mathematics classrooms: analyzing the dynamic process of teaching and learning of graph functions 82

*Janete Bolite Frant, Jorge I. Acevedo, Vicenç Font*

Metaphors and gestures in fraction talk 92

*Laurie D. Edwards*

A review of some recent studies on the role of representations in mathematics education in Cyprus and Greece 102

*Athanasios Gagatsis, Iliada Elia*

Metaphorical objects and actions in the learning of geometry. The case of French pre-service primary teachers 112

*Françoise Jore, Bernard Parzysz*

Mediation of metaphorical discourse in the reflection on one’s own individual relationship with the taught discipline: an experience with mathematics teachers 122

*Angela Pesci*

Building visual structures in arithmetical knowledge – A theoretical characterization of young students’ “visual structurizing ability (ViSA)” 132

*Elke Söbbeke*

The effect of mental models (“Grundvorstellungen”) for the development of mathematical competencies. First results of the longitudinal study PALMA 142

*Rudolf vom Hofe, Michael Kleine, Werner Blum, Reinhard Pekrun*

A dyslexic child’s strategies and images in arithmetic: a longitudinal study 152

Xenia Xistouri, Demetra Pitta-Pantazi

**WORKING GROUP 2. Affect and mathematical thinking**

Affect and mathematical thinking. Role of beliefs, emotions and other affective factors 165

*Markku S. Hannula, Inés M. Gómez-Chacón, George Philippou, Wolfgang Schlöglmann*

Self-conceptualised perceptions of attitude and ability among student teachers 174

*Patricia T. Eaton, Sonia Kidd *

Reflections on creativity: the case of a good problem solver 184

*Fulvia Furinghetti, Francesca Morselli*

Affect, mathematical thinking and intercultural learning. A study on educational practice 194

*Inés M. Gómez-Chacón*

The structure of student teachers’ view of mathematics at the beginning of their studies 205

*Markku S. Hannula, Raimo Kaasila, Anu Laine, Erkki Pehkonen*

Autobiographical narratives, identity and view of mathematics 215

*Raimo Kaasila, Markku S. Hannula, Anu Laine, Erkki Pehkonen*

Sustained engagement: preservice teachers’ experience with a chain of discovery 225

*Peter Liljedahl*

Trying to change attitude towards maths: a one-year experimentation 235

*Pietro Di Martino, Maria Mellone*

Mathematical discomfort and school drop-out in Italy 245

*Manuela Moscucci, Maria Piccione, Maria Gabriella Rinaldi, Serena Simoni, Carlo Marchini*

The measurement of young pupils’ metacognitive ability in mathematics: the case of selfrepresentation and self-evaluation 255

*Areti Panaoura, George Philippou*

Teachers’ use of the construct ‘attitude’: preliminary research findings 265

*Maria Polo, Rosetta Zan*

Meta-affect and strategies in mathematics learning 275

*Wolfgang Schlöglmann*

**WORKING GROUP 3. Building structures in mathematical knowledge**

Building structures in mathematical knowledge 287

*Milan Hejný, Graham Littler, Ladislav Kvasz, Dvora Peretz*

Building structure in mathematics within teaching and learning processes – a study on teachers’ input and students’ achievement 290

*Astrid Brinkmann*

The metaphor “contracts to deal with concepts” as a structuring tool in algebra 300

*Elmar Cohors-Fresenborg, Christa Kaune*

From experience, through generic models to abstract knowledge 311

*Milan Hejný, Jana Kratochvílová*

Classification leading to structure 321

Darina Jirotková, Graham H. Littler

Discussing the challenge of categorising mathematical knowledge in mathematics research situations 332

*Eva Knoll**, **Cécile Ouvrier-Buffet*

On linguistic aspects of structure building 342

*Ladislav Kvasz*

Approaching the distributive law with young pupils 352

*Nicolina A. Malara, Giancarlo Navarra*

The development of informal proportional thinking in primary school 363

*Marilena Pantziara, Demetra Pitta-Pantazi*

14 years old pupils’ thought processes: a case study of constructing the triangular inequality 374

*Elif B. Turnuklu, Sibel Yesilder*

**WORKING GROUP 4. Argumentation and proof**

Argumentation and proof 385

*Maria Alessandra Mariotti, Christine Knipping, Dietmar Küchemann, Kirsti Nordström*

Argumentation and proof in examples taken from French and German textbooks 391

*Richard Cabassut*

A proposal of categorisation for analysing inductive reasoning 401

*Mª Consuelo Cañadas Santiago, Encarnación Castro Martínez*

Natural deduction in predicate calculus: a tool for analysing proof in a didactic perspective 409

*Viviane Durand-Guerrier*

Visualising and conjecturing solutions for Heron’s problem 420

*Lourdes Figueiras, Jordi Deulofeu*

A genetic approach to proof 428

*Hans Niels Jahnke*

Pupils’ awareness of structure on two number/algebra questions 438

*Dietmar Küchemann, Celia Hoyles*

Proof in Swedish upper secondary school mathematics textbooks – the issue of transparency 448

*Kirsti Nordström, Clas Löfwall*

The meaning of proof in mathematics education 458

*David A. Reid*

About a constructivist approach for stimulating students’ thinking to produce conjectures and their proving in active learning of geometry 469

Oleksiy Yevdokimov

**WORKING GROUP 5. Stochastic Thinking**

Working Group 5 on “Stochastic Thinking” 481

*Dave Pratt*

Invited paper 484

Probabilistic and statistical thinking 485

*Manfred Borovcnik*

Papers on Statistics Education

A structural study of future teachers’ attitudes towards statistics 508

*Carmen Batanero, Assumpta Estrada, Carmen Díaz, José M. Fortuny*

Teachers’ representations of independent events: what might an attempt to make sense hide? 518

*Sylvette Maury, Marie Nabbout*

The nature of the quantities in conditional probability problems. Its influence on problem solving behaviour 528

M. Pedro Huerta, Mª Ángeles Lonjedo

Exploring introductory statistics : students’ understanding of variation in histograms 539

*Carl Lee, Maria Meletiou-Mavrotheris*

Improving stochastic content knowledge of preservice primary teachers 549

*Robert Peard*

Randomness in textbooks: the influence of deterministic thinking 559

*Pilar Azcárate, José Mª Cardeñoso, Ana Serradó*

Papers on Probabilistic Thinking

Problab goes to school: design, teaching, and learning of probability with multi-agent interactive computer models 570

Dor Abrahamson, Uri Wilensky

Strengths and weaknesses in students’ project work in exploratory data analysis 580

Rolf Biehler

Randomness and lego robots 591

*Michele Cerulli , Augusto Chiocchiariello , Enrica Lemut*

Students’ meaning-making processes of random phenomena in an ICT-environment 601

*Kjærand Iversen, Per Nilsson* Young children’s expressions for the law of large numbers 611

Efi Paparistodemou

Towards the design of tools for the organization of the stochastic 619

*Dave Pratt, Theodosia Prodromou*

**WORKING GROUP 6. Algebraic Thinking**

Working Group 6. Algebraic Thinking 631

Jean-Philippe Drouhard, Mabel Panizza, Luis Puig, Luis Radford

Difficulties found by the students during the study of substraction of integer numbers 643

*Andrea María Araya Chacón*

Inequalities and equations: history and didactics 652

*Giorgio T. Bagni*

Pupil’s autonomous studying: from an epistemological analysis towards the construction of a diagnosis 663

*Pierre Duchet, Erdogan Abdulkadir*

An experience with parabolas 675*Rosana Nogueira de Lima, Vera Helena Giusti de Souza*

Perceptual semiosis and the microgenesis of algebraic generalizations 684

*Luis Radford, Caroline Bardini, Cristina Sabena*

Model of a professor’s didactical action in mathematics education: professor’s variability and students’ algorithmic flexibility in solving arithmetical problems 696

*Jarmila Novotná, Bernard Sarrazy*

Early algebra – processes and concepts of fourth graders solving algebraic problems 706

*Birte Julia Specht*

From tables of numbers to matrices: an historical approach 717

*Fernanda Viola*

**WORKING GROUP 7. Research on Geometrical Thinking**

Research on Geometrical Thinking 725

Angel Gutiérrez, Alain Kuzniak, Rudolf Straesser

Developing geometrical reasoning in the classroom: learning from highly experienced teachers from China and Japan 727

*Liping Ding, Taro Fujita, Keith Jones*

On the geometrical thinking of pre-service school teachers 738

*Alain Kuzniak, Jean-Claude Rauscher*

Geometrical pre-conceptions of 8 years old pupils 748

*Carlo Marchini, Maria Gabriela Rinaldi*

Using dynamic transformations of solids to promote children’s geometrical reasoning 756

*Christos Markopoulos, Despina Potari*

Comparing perimeters and areas: childrens’ preconceptions and spontaneous procedures 766

*P. Marchett, D. Medici, Paola Vighi, E. Zaccomer*

“Measurement” on the squared paper 777

*Paola Vighi*

**WORKING GROUP 8. Language and Mathematics**

Language and Mathematics 789

Candia Morgan, Pier Luigi Ferrari, Marit Johnsen Høines, Raymond Duval Talking to each other: Pupils and teachers in primary mathematics classrooms 799

*Jenni Back*

The semantic-transformational function of written verbal language in mathematics 810

*Valeria Consogno*

The role of language in the relation between theorisation and the experience of activity 821

*Nadia Douek*

Exploratory talk in peer groups – exploring the zone of proximal development 831

*Julie-Ann Edwards*

Multimodal language strategies activated by students in understanding and communicating mathematics 841

*Paolo Guidoni, Donatella Iannece, Roberto Tortora*

Conversations in undergraduate students’ collaborative work 852

*Morten Misfeldt*

What is a definition for in school mathematics? 861

*Candia Morgan*

Voices in scaffolding mathematical constructions 872

*Mehmet Fatih Ozmantar, John Monaghan*

Uncertainty and different styles of friendly conversation 883

*Anna-Maija Partanen*

Geometric Representatives in Pupils’ Communication 893

*Filip Roubícek*

Semiotic processes in a mathematical internet-chat 903

*Christof Schreiber*

Interaction structures in primary school mathematics with a multilingual student body 913

*Marcus Schütte*

**WORKING GROUP 9. Tools and technologies in mathematical didactics**

Tools and technologies in mathematical didactics 927

*Bärbel Barzel, Paul Drijvers, Michela Maschietto, Luc Trouche*

Mathematics and e-learning: a conceptual framework 939

*Giovannina Albano*

Supportive use of Derive 949

*Mette Andresen*

From distributed to shared knowledge – a conceptual framework for a virtual seminar in teacher education 959

Christine Bescherer, Matthias Ludwig, Barbara Schmidt-Thieme, Hans-Georg Weigand

Chaining operations to get insight in expressions and functions 969

*Peter Boon, Paul Drijvers*

Technology-assisted discovery of conceptual connections within the cognitive neighbourhood of a mathematical topic 979

*Thierry Dana-Picard*

Tool use in trigonometry in two countries 989

*Ali Delice, John Monaghan*

Students’ choice of tasks and tools in an ICT rich environment 1000

*Anne Berit Fuglestad*

The balance. An interactive activity to work with fractions 1010

*María Trigueros G., Heréndira García Tello*

Distance training, a key mode to support teachers in the integration of ICT? Towards collaborative conception of living pedagogical resources 1020

*Dominique Guin, Luc Trouche*

Dynamic representations: a new perspective on instrumental genesis 1031

*Stephen J. Hegedus*

Sustaining interaction in a mathematical community of practice 1041

*João Filipe de Lacerda Matos, Yishay Mor, Madalena Santos, Richard Noss*

Motivation: the contribution of interactive whiteboards to teaching and learning in mathematics 1051

*Douglas Averis, Derek Glover, Dave Miller*

Developing new representations and mathematical models in a computational learning environment 1061

*Nicholas Mousoulides, George Philippou*

Integrating technology in a mathematics cognitive intervention program 1071

*Constantinos Christou, Nicholas Mousoulides, Marios Pittalis*

Mathematics laboratory activities with Derive: examples of approaches to algebra 1082*Maria Reggiani*

Putting the *learning *back into e-learning 1092

*Chris Roe, Dave Pratt, Ian Jones*

Physical and virtual worlds in teaching mathematics: possibilities for an effective cooperation? 1102

*Emanuela Ughi, Judit Jassó*

Calculating with “rule and compasses”. An example in using technology for mathematics teaching 1113

*Luciana Zuccheri*

### WORKING GROUP 10. Mathematics education in multicultural settings

Issues and challenges in researching mathematics education in multicultural settings 1125

*Guida de Abreu, Margarida César, Núria Gorgorió, Paola Valero*

Cultural identities in the multiethnic mathematical classroom 1131

*Guida de Abreu*

Culture, diversity and conflict in landscapes of mathematics learning 1141

*Helle Alrø, Ole Skovsmose, Paola Valero*

Diversity seen through teachers’ eyes: discourses about multicultural classes 1153

*Margarida César, Franco Favilli*

How parents view mathematics and the learning of mathematics. An intercultural comparative study 1165

*Britta Hawighorst*

Norms, social representations and discourse 1176

*Núria Gorgorió, Núria Planas*

Engaging with parents on a critical dialogue about mathematics education 1182

*Jill Bratton, Marta Civil, Beatriz Quintos*

Research as an act of learning: exploring student backgrounds through dialogue with research participants 1193

*Diana Stentoft*

Handicrafts performing life 1204

*Margarida César, Lucília Teles*

Language sensitive mathematics teaching in a multicultural classroom. How students’ talking and writing can enlighten hidden problems 1215

*Dolly van Eerde, Maaike Hajer*

Developing a community in the classroom: implications for mathematics education 1226

*Marta Civil, Peter Wiles*

### WORKING GROUP 11. Different theoretical perspectives and approaches in research in mathematics education

Different theoretical perspectives and approaches in research in mathematics education 1239

*Michèle Artigue, Mariolina Bartolini Bussi, Tommy Dreyfus, Eddie Gray, Susanne Prediger*

Theoretical issues in research of mathematics education: some considerations 1244

*Maria Kaldrimidou, Marianna Tzekaki*

Science or magic? The use of models and theories in didactics of mathematics 1254

*Marianna Bosch, Yves Chevallard, Josep Gascón*

Relating theories to practice in the teaching of mathematics 1264

Anna Poynter, David Tall

Conceptualisation through semiotic tools in teaching/learning situations 1274

*Isabelle Bloch*

Crossing the border integrating different paradigms and perspectives 1285

*Angelica Bikner-Ahsbahs*

Conceptualization of the limit by means of the discrete continuous interplay: different theoretical approaches 1295

*Ivy Kidron*

Theories and empirical researches: towards a common framework 1305

*Ferdinando Arzarello, Federica Olivero*

Comparison of different theoretical frameworks in didactic analyses of videotaped classroom observations 1316

*Michèle Artigue, Agnès Lenfant, Eric Roditi*

Intuitive vs. analytical thinking: four theoretical frameworks 1327

*Uri Leron*

Didactic effectiveness of equivalent definitions of a mathematical notion. The case of the absolute value 1338

*Juan D. Godino, Eduardo Lacasta, Miguel R. Wilhelmi*

Working in a developmental research paradigm: the role of didactician/researcher working with teachers to promote inquiry practices in developing mathematics learning and teaching 1348

*Maria Luiza Cestari, Espen Daland, Stig Eriksen, Barbara Jaworski*

Different perspectives on computer-based graphs and their meaning 1359

*Francesca Ferrara, Ornella Robutti, Cristina Sabena*

The act of remembering and mathematical learning 1370

*Teresa Assude, Yves Paquelier, Catherine Sackur*

The didactical transposition of didactical ideas: the case of the virtual monologue 1379

*Lisser Rye Ejersbo, Uri Leron*

An integrate perspective to approach technology in mathematics education 1389

*Michele Cerulli, Bettina Pedemonte, Elisabetta Robotti*

### WORKING GROUP 12. From a study of teaching practices to issues in teacher education

From a study of teaching practices to issues in teacher education 1403

José Carrillo, Ruhama Even, Tim Rowland, Lurdes Serrazina What teachers know: the knowledge bases of classroom practice 1410

Janet Ainley, Michael Luntley

Secondary school teachers’ conceptions about algebra teaching 1420

*Iiris Attorps*

The learning portfolio as an assessment strategy in teacher education 1430

*Pilar Azcárate, José Mª Cardeñoso, Anna Serradó*

Values in mathematics teaching: how mathematical? 1440

*Chris Husbands*

From practices to theories to practices … in learning to teach mathematics and learning mathematics 1451

*Laurinda Brown*

Teacher’s activity and knowledge: the teaching of setting up equations 1462

*Lalina Coulange* Learning within communities of practice in preservice secondary school teachers education 1473

*Pedro Gómez, Luis Rico*

Developing mathematics teacher’s competence 1483

*Alena Hospesová, Marie Tichá*

Teachers’ implementation of a curriculum reform 1494

*Bodil Kleve*

Teachers using technology: models of the complexity of practices 1505

*Jean-Baptiste Lagrange, Nuray Dedeoglu, Emel Erdogan*

The frameworks of school experience: an anthropo-didactic approach to the phenomena of mathematics teaching 1515

*Alain Marchive*

The transition from initial training to immersion in practice. The case of a mathematics primary teacher 1526

José Carrillo, Nuria Climent, M. Cinta Muñoz

Pupils’ error treatment by high school teachers, relating to the notion of inverse function 1537*Susana Murillo*

Exploring the role of virtual interactions in pre-service mathematics teacher education 1547

*João Pedro da Ponte, Helena Fonseca, Paulo Oliveira, Hélia Oliveira, José Manuel Varandas*

The student teacher and learning objectives: within a sociocultural activity system 1558

*Anne Edwards, Pat Perks, Stephanie Prestage*

The knowledge quartet: considering Chloe 1568

*Peter Huckstep, Tim Rowland, Anne Thwaites*

The portfolio in teacher education 1579

*Leonor Santos*

The teacher’s action, the researcher’s conception in mathematics 1588

*Francia Leutenegger, Florence Ligozat, Alain Mercier, Gérard Sensevy*

The role of the practice of theorising practice 1598

*Jeppe Skott*

Views on early school mathematics – European needs and the local reality 1609

*Ewa Swoboda*

### WORKING GROUP 13. Applications and modelling

Introduction to the Working Group “Applications and Modelling” 1613

*Gabriele Kaiser*

“Filling up” – the problem of independence-preserving teacher interventions in lessons with demanding modelling tasks 1623

*Werner Blum, Dominik *Leiß

An introduction to mathematical modelling: an experiment with students in economics 1634

*Jean-Luc Dorier*

Mathematical praxeologies of increasing complexity: variation systems modelling in secondary education 1645

*Fco. Javier García García, Luisa Ruiz Higueras*

Getting to grips with real world contexts: developing research in mathematical modelling 1655

*Christopher Haines, Rosalind Crouch*

Levels of modelling competence 1666

*Herbert Henning, Mike Keune* Applied or pure mathematics 1675

*Thomas Lingefjärd*

Conceptualizing the model-eliciting perspective of mathematical problem solving 1686

*Bharath Sriraman*

Assessment of mathematics in a laboratory-like environment: the importance of replications 1696

*Pauline Vos*

### WORKING GROUP 14. Advanced mathematical thinking

Synopsis of the activities of Working Group 14 ‘Advanced Mathematical Thinking’ 1709

*Joanna Mamona-Downs, Maria Meehan, John Monaghan*

Calculus and departmental settings 1716

*Erhan Bingolbali, John Monaghan*

Conceptual change in advanced mathematical thinking 1727

*Irene Biza, Alkeos Souyoul, Theodossios Zachariades*

Problem solving and web resources at tertiary level 1737

*Claire Cazes, Ghislaine Gueudet, Magali Hersant, Fabrice Vandebrouck*

The proof language as a regulator of rigor in proof, and its effect on student behaviour 1748

*Martin Downs, Joanna Mamona-Downs*

Is there a limit in the derivative? – exploring students’ understanding of the limit of the difference quotient 1758

*Markus Hähkiöniemi*

Characterising mathematical reasoning: studies with the Wason selection task 1768

*Matthew Inglis, Adrian Simpson*

On some difficulties in vector space theory 1778

*Mirko Maracci*

An experience of problem solving in mathematical analysis 1788

*Giorgio T. Bagni, Marta Menghini*

To appear and to be: mathematicians on their students’ attempts at acquiring the ‘genre speech’ of university mathematics 1800

*Elena Nardi, Paola Iannone*

Relationship between informal and formal reasoning in the subject of derivative 1811

*Antti Viholainen*

Research and development of university level teaching: the interaction of didactical and mathematical organisations 1821

*Carl Winsløw*

Mental models of the concept of vector space 1830a

*Astrid Fischer*

### DOCUMENTATION

INDEX OF AUTHORS 1833

ERME : AIMS AND VISION 1839