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
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 675Rosana 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 1082Maria 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 1537Susana 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
