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

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