**European
Research in Mathematics Education IV**

*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*

**Marianna
Bosch**

FUNDEMI IQS – Universitat Ramon Llull

*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*

Marta Perpiñán, M. Àngels Portabella

© Copyright 2006 left to the authors

FUNDEMI IQS – Universitat

Ramon
Llull

Legal deposit B – 46971 – 2006

ISBN 84-611-3282-3

Barbara Jaworski, Marianna Bosch

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*

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

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*

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

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*

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*

*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

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*

*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*

INDEX OF AUTHORS 1833

ERME : AIMS AND VISION 1839