E European Research R h in i Mathematics M th ti Education Proceedings P d off th the F Fifth fth Congress C off the th European Society for Research in Mathematics Education

22 – 26 February 2007 Larnaca, Cyprus

Editors Demetra Pitta-Pantazi Pitta Pantazi & George Ge rge Philippou Philipp u

Editorial board Ainley J., J Arcavi A., A Arzarello F F., Bagni G. G T., T Bardini C C., Barzel B., Biehler R., Bills L., Blomhoj M., Bosch M., Cabassut R., Carrillo J., Cesar M., Douek N., deAbreu G., Durand-Guerrier V., Ferrari P. L., Gagatsis A., García J., Gorgorio N., Hannula M., Hejny M., Hemmi K., Jablonka E., Jaworski, B., Kadijevic D Kadunz D., K d G., G Kaiser K i G., G Krummheuer K h G., G Kuzniak K i k A., A Kvasz K L., L Kynigos C., Leikin R., Lenfant A., Littler G. H., Ludwig M., Mammona-Downs J., Marchini C., Marchive A., Marriotti M., Maschietto M., Meehan M., Meletiou-Mavrotheris M., Morgan C., Op't p Eynde y P., Ottaviani M., Parzysz y B., Pepin p B., Perez D., Pratt D., Prediger S., Puig L., Robotti E., Rogers L., Santos L., Schöglmann W., Sriraman B., Straesser, R., Wedege T.

Editing d assistance Kallia Mattheou & Nicholas Mousoulides

ISBN - 978-9963-671-25-0 © Copyright 2007 left to the authors

Contents

TABLE OF CONTENTS INTRODUCTION Rudolf Straesser, Barbara Jaworski

PLENARIES

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Plenary 1 Balancing interest in Fundamental understanding with considerations of usefulness in mathematics education research Frank K. Lester Plenary 2 What constitutes good practice in teaching mathematics, a personal perspective Naďa Stehlíková Plenary 3 Teachers, technologies and the structures of schooling Kenneth Ruthven Plenary 4 Digital technologies: A window on theoretical issues in mathematics education Michèle Artigue

WORKING GROUP 1. The role of images and metaphors in the learning and understanding mathematics The role of images and metaphors in the learning and understanding mathematics (including embodied cognition) Bernard Parzysz, Gert Kadunz, Elisabetta Robotti, Leo Rogers Exploring the effects of representations on the learning of statistics in Greek primary school Sofia Anastasiadou, Athanasios Gagatsis Guess what is inside this box: Look at these opened boxes for clues Roberto Araya The number line as metaphor of the number system: A case study of a primary school Maria Doritou, Eddie Gray Ways of thinking about the uses of images in learning and teaching geometry: A more thorough investigation of the links between drawings and figures Sophie Gobert Mathematical Writing Gert Kadunz Students’ and teachers’ representations in problem solving Annita Monoyiou, Pandelitsa Papageorgiou, Athanasios Gagatsis The role of the conceptual metaphor in the development of children’s arithmetic Carol Murphy The power and perils of metaphor in making internal connections in trigonometry and geometry Norma Presmeg CERME 5 (2007)

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Contents Metaphors and image schemata in concept formation and reasoning Reinert A. Rinvold Teaching special relativity Leo Rogers, Patrick J. Caines Metaphors and cognitive modes in the teaching-learning of mathematics Jorge Soto-Andrade

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WORKING GROUP 2. Affect and mathematical thinking

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Affect and mathematical thinking Markku S. Hannula, Peter Opt’Eynde, Wolfgang Schlöglmann, Tine Wedege Evaluating the sensitivity of the refined mathematics-related beliefs questionnaire to nationality, gender and age Paul Andrews, Jose Diego –Mantecón, Peter Op ‘t Eynde, Judy Sayers Students’ motivation in mathematics and gender differences in grades 6 and 7 Chryso Athanasiou, George Philippou Refining the mathematics-related beliefs questionnaire (MRBQ) Jose Diego-Mantecón, Paul Andrews, Peter Op ‘t Eynde Mathematics teachers’ desire to develop Markku S. Hannula, Madis Lepik, Tiiu Kaljas Mathematics is - favourite subject, boring or compulsory Kirsti Hoskonen Students’ beliefs and attitudes concerning mathematics and their effect on mathematical ability Eleftherios Kapetanas, Theodosios Zachariades The notion of children’s perspectives Troels Lange Belief change as conceptual change Peter Liljedahl, Katrin, Rolka, Betinna Rösken Changes in students’ motivational beliefs and performance in a self-regulated mathematical problem-solving environment Andri Marcou, Stephen Lerman About mathematical belief systems awareness Manuela Moscucci Efficacy beliefs, problem posing, and mathematics achievement Aristoklis A. Nicolaou, George N. Philippou Students’ self regulation of emotions in mathematics learning Peter Op ‘t Eynde, Erik De Corte, Inge Mercken The impact of recent metacognitive experiences on preservice teachers’ selfrepresentation in mathematics and its teaching Areti Panaoura Is motivation analogous to cognition? Marilena Pantziara, Demetra Pitta-Pantazi, GeorgePhilippou Identifying dimensions of students’ view of mathematics Bettina Rösken, Markku Hannula, Erkki Pehkonen, Raimo Kaasila, Anu Laine Student errors in task-solving processes Wolfgang Schlöglmann

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Contents Influence of didactical games on pupils’ attitudes towards mathematics and process of its teaching Peter Vankúš Intrinsic and extrinsic motivation versus social and instrumental rationale for learning mathematics Kjersti Wæge Potential for change of views in the mathematics classroom? Tine Wedege, Jeppe Skott

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WORKING GROUP 3. Building structures in mathematical knowledge

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Building structures in mathematical knowledge Milan Hejný, Graham Littler Constructing Multiplication: Different strategies used by pupils Joana Brocardo, Lurdes Serrazina, Isabel Rocha Students’ thinking about fundamental real numbers properties Eustathios Giannakoulias, Alkeos Souyoul, Theodossios Zachariades Schema A ± B = C as the basis of arithmetic structure Milan Hejný, Jana Slezáková Recognising an algebraic structure Maureen Hoch, Tommy Dreyfus Creating a mental image of dice blackjack game Antonín Jančařík Classification, manipulation and communication: Work with pupils and student teachers Darina Jirotková, Graham H. Littler Investigating the processing structures of students’ inductive reasoning in mathematics Eleni Papageorgiou, Constantinos Christou Empirical Hierachy of pupils’ attainment of measurement in early primary school years Alexandra Petridou, Maria Pampaka, Constantia Hadjidemetriou How do students from primary school discovery the regularity Marta Pytlak Reflection on activity – effect relationships in solving word problems Ana Isabel Roig, Salvador Llinares Children’s perceptions on infinity: Could they be structured? Mihaela Singer, Cristian Voica The role of spatial configurations in early numeracy problems Fenna Van Nes, Jan de Lange Students’ ability in solving line symmetry tasks Xenia Xistouri

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WORKING GROUP 4. Argumentation and proof

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Argumentation and proof Maria Alessandra Mariotti, Kirsti Hemmi, Viviane Durrand Guerrier Indirect proof: An interpreting model Samuele Antonini, Maria Alessandra Mariotti Mathematics learning and the development of general deductive reasoning Michal Ayalon, Ruhama Even

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Contents How to decide? Students’ ways of determining the validity of mathematical statements Orly Buchbinder, Orit Zaslavsky Cabri’s role in the task of proving within the activity of building part of an axiomatic system Leonor Camargo, Carmen Samper, Patricia Perry Some remarks on the theorem about the infinity of prime numbers Ercole Castagnola, Roberto Tortora Proofs problems in elementary number theory: Analysis of trainee teachers’ productions Annalisa Cusi, Nicolina A. Malara Relationship between beginner teachers in mathematics and the mathematical concept of implication Virginie Deloustal-Jorrand Using the Van Hiele theory to analyse the teaching of geometrical proof at grade 8 in Shanghai Liping Ding, Keith Jones Analysis of conjectures and proofs produced when learning trigonometry Jorge Fiallo, Angel Gutiérrez Analysis of the teacher’s arguments used in the didactical management of a problem solving situation Patrick Gibel Structural relationships between argumentation and proof in solving open problems in algebra Bettina Pedemonte Mathematical proof: Teachers’ beliefs and practices Antonis Sergis The mental models theory of deductive reasoning: Implications for proof instruction Andreas J. Stylianides, Gabriel J. Stylianides Reviewing textbook proofs in class: A struggle between proof structure, components and details Stine Timmermann

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WORKING GROUP 5. Stochastic Thinking

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Developing stochastic thinking Rolf Biehler, Maria Meletiou, Maria Gabriella Ottaviani, Dave Pratt Understanding confidence intervals Herman Callaert Conditional probability problems and contexts. The diagnostic test context Marta Carles, Μ. Pedro Huerta A microworld to implant a germ of probability Michele Cerulli, Augusto Chioccariello, Enrica Lemut The impact of a typical classroom practice on students’ statistical knowledge Andreas Eichler The “same” problem in three presentation formats: Different percentages of success and thinking processes Μ. Pedro Huerta, Μα Ángeles Lonjedo The relationship between local and global perspectives on randomness Peter Johnston-Wilder, Dave Pratt

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Contents Transparent urns and colored tinker – cubes for natural stochastics in primary school Laura Martignon, Kathryn Laskey, Elke Kurz-Milcke Constructing stochastic simulations with a computer tool – Students’ competencies and difficulties Carmen Maxara, Rolf Biehler A cross – national comparison of introductory statistics students’ prior knowledge of graphs Maria Meletiou-Mavrotheris, Carl Lee Sample space and the structure of probability combinations in preschoolers Zoi Nikiforidou, Jenny Pange Looking for randomness in tasks of prospective teachers Efi Paparistodemou, Despina Potari, Demetra Pitta Making connections between the two perspectives on distribution Theodosia Prodromou

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WORKING GROUP 6. Algebraic Thinking

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Working on algebraic thinking Luis Puig, Janet Ainley, Abraham Arcavi, Giorgio Bagni Research impacting on student learning: How construction tasks influenced learners’ thinking Shafia Abdul-Rahman Elementary school students’ understanding and use of the equal sign Vassiliki Alexandrou-Leonidou, George Philippou A contribution of ancient Chinese algebra: Simultaneous equations and counting rods Giorgio T. Bagni Patterns and generalization: the influence of visual strategies Ana Barbosa, Pedro Palhares, Isabel Vale Matrices as Peircean diagrams: A hypothetical learning trajectory Willibald Dörfler Do you see what I see? Issues arising from a lesson on expressing generality Helen Drury Signs used as algebraic tools – A case study Astrid Fischer Problems of a linear kind: From Vallejo to Peacock Bernardo Gómez Developmental assessment of algebraic performance Constantia Hadjidemetriou, Maria Pampaka, Alexandra Petridou, Julian Williams, Lawrence Wo Integrating the learning of algebra with technology at the European level: Two examples in the ReMath project Jean-Baptiste Lagrange, Giampaolo Chiappini Research and practice in algebra: Interwoven influences John Mason Distinguishing approaches to solving true/false number sentences Marta Molina, Encarnación Castro, John Mason Student difficulties in understanding the difference between algebraic expressions and the concept of linear equation Irini Papaieronymou

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Contents Teachers’ practices with spreadsheets and the development of algebraic activity Kirsty Wilson, Janet Ainley

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WORKING GROUP 7. Geometrical Thinking

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From geometrical thinking to geometrical work Alain Kuzniak, Athanasios Gagatsis, Matthias Ludwig, Carlo Marchini The use of everyday objects and situations in mathematics teaching: The symmetry case in French geometry teaching Caroline Bulf Geometrical working space, a tool for comparison Catherine Houdement Comparison of observation of new space and its objects by sighted and non-sighted pupils Iveta Kohanová Assessing the attainment of analytic – descriptive geometrical thinking with new tools George Kospentaris, Panagiotis Spyrou Horizon as epistemological obstacle to understanding infinity Magdalena Krátká Geometrical ridigity and the use of dragging in a dynamic geometry environment Victor Larios-Osorio The utilisation of video enriched microworlds based on dynamic geometry environments Markus Mann, Matthias Ludwig Geometrical tiles as a tools for revealing structures Carlo Marchini, Paola Vighi The process of composition and decomposition of geometric figures within the frame of dynamic transformations Christos Markopoulos, Despina Potari, Eftychia Schini Problem solving in geometry: The case of the illusion of proportionality Modestina Modestou, Iliada Elia, Athanasios Gagatsis, Giorgos Spanoudes Spatial abilities in relation to performance in geometry tasks Georgia Panaoura, Athanasios Gagatsis, Charalambos Lemonides Spatial ability as a predictor of students’ performance in geometry Marios Pittalis, Nicholas Mousoulides, Constantinos Christou Computer geometry as mediator of mathematical concepts Paola Vighi

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WORKING GROUP 8. Language and Mathematics

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Multiple perspectives on language and mathematics: Introduction and post – script Candia Morgan, Konstantinos Tatsis, Hana Moraová, Jarmila Novotná, Margarida César, Birgit Brandt, Elmar Cohors-Fresenborg, Christa Kaune Driving spontaneous processes in mathematical tasks Rossella Ascione, Maria Mellone Communities of practice in online mathematics discussion boards: Unpicking threads Jenni Back, Nick Pratt Pupils’ mathematical reasoning expressed through gesture and discourse: A case study from a sixth-grade lesson Raymond Bjuland, Maria Luiza Cestari, Hans Erik Borgersen

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Contents How mathematical signs work in a class of students with special needs: Can the interpretation process become operative? Isabelle Bloch Analyzing the constructive function of natural language in classroom discussions Paolo Boero, Valeria Consogno Assessment in the mathematics classroom. Studies of interaction between teacher and pupil using a multimodal approach Lisa Björklund Boistrup Certainty and uncertainty as attitudes for students’ participation in mathematical classroom interaction Birgit Brandt Modelling classroom discussions and categorizing discursive and metacognitive activities Elmar Cohors-Fresenborg, Christa Kaune The language of friendship: Developing sociomathematical norms in the secondary school classroom Julie-Ann Edwards The use of a semiotic model to interpret meanings for multiplication and division Marie Therese Farrugia “Why should I implement writing in my classes?” An empirical study on mathematical writing Marei Fetzer Issues in analysis of individual discourse concurrent with solving a mathematical problem Boris Koichu Authority relations in the acquisition of the mathematical register at home and at school Tamsin Meaney Idea generation during mathematical writing: Hard work or a process of discovery? Morten Misfeldt Students’ mathematical interactions and teachers’ reflections on their own interventions. Part 1: Epistemological analysis of students’ mathematical communication Marcus Nührenbörger, Heinz Steinbring Students’ mathematical interactions and teachers’ reflections on their own interventions. Part 2: Analysis of the collegial reflection on students’ mathematical communication Marcus Nührenbörger, Heinz Steinbring Children’s talk about mathematics and mathematical talk Päivi Perkkilä, Eila Aarnos The influence of learners’ limited language proficiency on communication obstacles in bilingual teaching/learning of mathematics Jana Petrová, Jarmila Novotná A question of audience, a matter of address David Pimm, Ruth Beatty, Joan Moss Selected problems in communication between the teacher and the pupil explored from the semiotic viewpoint Filip Roubíček Obstacles in mathematical discourse during researcher – student interaction Jana Slezáková, Ewa Swoboda Writing mathematics through dominant discourses: The case of a Greek school mathematics magazine Anastasia G. Stamou, Anna Chronaki

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Contents Investigating the influence of social and sociomathematical norms in collaborative problem solving 1321 Konstantinos Tatsis

WORKING GROUP 9. Tools and technologies in mathematical didactics

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Tools and technologies in mathematical didactics 1332 Chronis Kynigos, Caroline Bardini, Bärbel Barzel, Michela Maschietto Teacher´s practices and degree of ICT integration 1339 Teresa Assude Integration of computer algebra in an open learning environment 1349 Bärbel Barzel Teaching analysis in dynamic geometry environments 1359 Irene Biza, Dionissis Diakoumopoulos, Alkeos Souyoul Online resources in mathematics: Teachers’ genesis of use 1369 Laetitia Bueno-Ravel, Ghislaine Gueudet Strange 3D plots 1379 Thierry Dana-Picard, Ivy Kidron, David Zeitoun Tool use in a technology – Rich learning arrangement for the concept of function 1389 Paul Drijvers, Michiel Doorman, Peter Boon, Sjef van Gisbergen, Koeno Gravemeijer Analysis of teacher education in mathematics and ICT 1399 Fabien Emprin Developing tasks and teaching with ICT in mathematics in an inquiry community 1409 Anne Berit Fuglestad Technology that mediates and participates in mathematical cognition 1419 Stephen Hegedus, Sara Dalton, Luis Moreno-Armella Three dimensional constructions using an absolute frame of reference in a computer simulated 3D space 1429 Chronis Kynigos, Efi Alexopoulou, Maria Latsi Change in perception of prospective teachers regarding the image of the teacher as a result of engagement in a computerized environment 1439 Ilana Lavy, Atara Shriki Mathematics learning with the use of the balance, a computer programme from enciclomedia 1449 María-Dolores Lozano, María Trigueros Memorizing algebraic formulas: The support of a microworld 1460 Laura Maffei, Maria Alessandra Mariotti Flexibility and cooperation: Virtual learning environments in online undergraduate mathematics 1470 Morten Misfeldt, Anders Sanne Algebraic modelling using a dynamically linked geometry and computer algebra system 1480 Reinhard Oldenburg Cognitive styles, dynamic geometry and performance in area tasks 1489 Demetra Pitta-Pantazi, Constantinos Christou Meanings for fraction as number-measure by exploring the number line 1499 Giorgos Psycharis, Maria Latsi, Chronis Kynigos To sense and to visualize functions: The case of graph stretching 1509 Geula Sever, Michal Yerushalmy CERME 5 (2007)

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Contents Tools that force reflection Yianna Sirivianou, John Threlfall Educational software based on the theory of constructivism Mária Slavíčková Teaching with a symbolic calculator – The first years of a long-term project Hans-Georg Weigand

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WORKING GROUP 10. Mathematics education in multicultural settings

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Researching diversity in mathematics education Margarida César, Guida de Abreu, Núria Gorgorió Social representations and multicultural mathematics teaching and learning Guida de Abreu, Núria Gorgorió Landscapes of learning in a multicultural mathematics classroom Helle Alrø, Ole Skovsmose, Paola Valero Young children’s number sense in China, England and Finland Pirjo Aunio, Carol Aubrey, Ray Godfrey, Yan Liu, Pan Yuejuan Filling the gap between global and local mathematics Darlinda Moreira Multiple identities in the mathematics classroom: A theoretical perspective Diana Stentoft Storying mathematical identities with cultural models Julian Williams, Laura Black, Paul Hernandez-Martinez, Paulin Davis, Graeme Hutcheson, Su Nicholson, Maria Pampaka, Geoff Wake

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WORKING GROUP 11. Different theoretical perspectives and approaches in research in mathematics education Different theoretical perspectives in research from teaching problems to research problems Ferdinando Arzarello, Marianna Bosch, Agnès Lenfant, Susanne Prediger Ostensives through the lenses of two theoretical frameworks Ferdinando Arzarello, Ornella Robutti, Cristina Sabena How do theories influence the research on teaching and learning limits of functions? Christer Bergsten Integrating research teams: The TELMA approach Michele Cerulli, Jean Philippe Georget, Mirko Maracci, Giorgos Psycharis, Jana Trgalova Construction of knowledge by primary pupils: The role of whole-class interaction Thérèse Dooley Emergence or structure: A comparison of two sociological perspectives on mathematics classroom practice Uwe Gellert An activity theory perspective of didacticians’ learning within a mathematics teaching development research project Simon Goodchild

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Contents Theory in developmental research in mathematics teaching and learning: Social practice theory and community of inquiry as analytical tools Barbara Jaworski On the nature of the mathematical knowledge under construction in the classroom Maria Kaldrimidou, Haralambos Sakonidis, Marianna Tzekaki Social interaction in learning processes as seen by three theoretical frameworks Ivy Kidron, Agnès Lenfant, Angelika Bikner-Ahsbahs, Michèle Artigue, Tommy Dreyfus Students’ difficulties in vector spaces theory from two different theoretical perspectives Mirko Maracci Using mixed-methods methodology to investigate Cypriot preservice teachers’ mathematics content knowledge Marilena Petrou From teaching problems to research problems. Proposing a way of comparing theoretical approaches Susanne Prediger, Kenneth Ruthven Towards a cultural theory of learning Luis Radford An anthropological approach to metacognition: The “study and research courses” Esther Rodríguez, Marianna Bosch, Josep Gascón Conditions and constraints in the teaching of statistics: The scale of levels of determination Floriane Wozniak

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 José Carrillo, Leonor Santos, Liz Bills, Alain Marchive Teachers’ activity in exercises-based lessons. Some case studies Maha Abboud-Blanchard, Claire Cazes, Fabrice Vandebrouck Expressing generality: Focus on teachers’ use of algebraic notation Claire Vaugelade Berg Striving to “know what is to be done”: The role of the teacher Laurinda Brown, Alf Coles “You don’t need a tables book when you have butter beans!” Is there a need for mathematics pedagogy here? Dolores Corcoran Facilitate research activities at the primary level: Intentional communities of practice, teaching practices, exchanges about these practices Jean-Philippe Georget Adapting the hypothetical learning trajectory notion to secondary preservice teacher training Pedro Gómez, Maria José González, Jose Luis Lupiáñez Formative assessment: Tools for transforming school mathematics towards a dialogic practice? Jeremy Hodgen Developing strategies and materials impact on teacher education Marie Hofmannová, Jarmila Novotná CERME 5 (2007)

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Contents Differences and similarities in (qualified) pedagogical reflection Alena Hošpesová, Marie Tichá, Jana Macháčková Pre-service teachers’ representations of division of fractions Mine Isiksal, Erdinc Cakiroglu A task aimed at leading teachers to promoting a constructive early algebra approach Nicolina A. Malara, Giancarlo Navarra The professional development of a novice teacher in a collaborative context: An analysis of classroom practice Maria Cinta Muñoz-Catalán, José Carrillo, Nuria Climent From studies of cooperative learning practices towards a model of intervention on mathematics teachers Angela Pesci Teachers’ mathematical knowledge and pedagogical practices in the teaching of derivative Despina Potari, Theodosis Zachariades, Constantinos Christou, George Kyriazis, Demetra Pitta-Pantazi Prospective primary teachers’ use of mathematics teaching handbooks Tim Rowland The project work and the collaboration on the initial teacher training Leonor Santos, Alexandra Bento Primary teachers’ attitudes towards and beliefs about mathematics teaching: the collective culture of one English primary school Judy Sayers The teaching modes: A conceptual framework for teacher education Rosa Antónia Tomás Ferreira The mathematics content knowledge of beginning teachers: The case of Amy Fay Turner Training mathematics teachers in a community of learners (COL) Nellie C. Verhoef, Cees Terlouw Exemplification in the mathematics classroom: What is it like and what does it imply? Iris Zodik, Orit Zaslavsky

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WORKING GROUP 13. Modelling and applications

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Modelling and applications – Differentiating perspectives and delineating commonalties Gabriele Kaiser, Bharath Sriraman, Morten Blomhøj, Fco. Javier Garcia Understanding of “modelling” Mette Andresen Using research and study courses for teaching mathematical modelling at university level Berta Barquero, Marianna Bosch, Josep Gascón The teaching calculus with applications experiment succeeded – Why and what else? Abraham Berman, Igor M. Verner, Shuki Aroshas Learning the integral concept through mathematical modelling Morten Blomhøj, Tinne Hoff Kjeldsen Personal experiences and extra-mathematical knowledge as an influence factor on modelling routes of pupils Rita Borromeo Ferri Making mathematical literacy a reality in classrooms Hugh Burkhardt

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Contents Modelling bungy jumping: Why is it so difficult? Ana Paula Canavarro Mathematical modelling and parallel discussions Jonei Cerqueira Barbosa Comparison of mathematization in microcomputer based laboratory (MBL) and verification – Type laboratory (VTL) in physics Murad Jurdak, Saouma BouJaoude, Norma Ghumrawi Modelling tasks for low achieving students – First results of an empirical study Katja Maaß Tracing students’ modelling processes in elementary and secondary school Nicholas Mousoulides, Bharath Sriraman, Marios Pittalis, Constantinos Christou A meta-perspective on the nature of modelling and the role of mathematics Irit Peled The role of mathematical knowledge in a practical activity: Engineering projects at university level Avenilde Romo Vázquez Derivatives in applications: How to describe students’ understanding Gerrit Roorda, Pauline Vos, Martin Goedhart The functional algebraic modelling at Secondary level Noemí Ruiz, Marianna Bosch, Josep Gascón Mathematical modelling in school – Experiences from a project integrating school and university Björn Schwarz, Gabriele Kaiser Personal meaning in relation to modelling problems Katrin Vorhölter Interpreting velocity and stopping distance; complementarity, context and mathematics Pauline Vos, Gerrit Roorda Measuring perceived self-efficacy in applying mathematics Geoff D. Wake, Maria Pampaka

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WORKING GROUP 14. Advanced mathematical thinking

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Advanced mathematical thinking Joanna Mamona-Downs Lagrange’s theorem: What does the theorem mean? Buma Abramovitz, Miryam Berezina, Abraham Berman, Ludmila Shvartsman University students generating examples in real analysis: Where is the definition? Samuele Antonini, Fulvia Furinghetti, Francesca Morselli, Elena Tosetto Is there equality in equation? Iiris Attorps, Timo Tossavainen Analysis of the autonomy required from mathematics students in the French lycee Corine Castela Local and global perspectives in problem solving Martin Downs, Joanna Mamona-Downs The application of the abductive system to different kinds of problems Elisabetta Ferrando University students’ difficulties with formal proving and attempts to overcome them Justyna Hawro

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Contents The interplay between syntactic and semantic knowledge in proof production: Mathematicians’ perspectives Paola Iannone, Elena Nardi Belief bias and the study of mathematics Matthew Inglis, Adrian Simpson Students’ concept development of limits Kristina Juter Habits of mind associated with advanced mathematical thinking and solution spaces of mathematical tasks Roza Leikin Mathematical background and problem solving: How does knowledge influence mental dynamics in game theory problems? Francesca Martignone Student generated examples and the transition to advanced mathematical thinking Maria Meehan Understanding of systems of equations in linear algebra Maria Trigueros Students’ choices between informal and formal reasoning in a task concerning differentiability Antti Viholainen Interplay between research and teaching from the perspective of mathematicians Carl Winsløw, Lene Møller Madsen Advancing mathematical thinking: Looking back at one problem Rina Zazkis, Mark Applebaum

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WORKING GROUP 15. Comparative studies in mathematics education

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Comparative Studies in mathematics education – comparing the incomparable? Birgit Pepin, Eva Jablonka, Richard Cabassut Development of the mathematics education system in Iceland in the 1960s in comparison to three neighbouring countries Kristín Bjarnadóttir Policy change, graphing calculators and ‘High stakes examinations’: A view across three examination systems Roger G. Brown Examples of comparative methods in the teaching of mathematics in France and in Germany Richard Cabassut Types of algebraic activities in two classes taught by the same teacher Tammy Eisenmann, Ruhama Even Proportion in school mathematics textbooks: A comparative study João Pedro da Ponte, Sandra Marques Factors related to students’ mathematical literacy in Finland and Sweden Jukka Törnroos A comparative study of assessment activity involving 8 pre-service teachers: What referent for the assessor? Marc Vantourout

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Contents The construction of personal meaning – A comparative case study in Hong Kong and Germany 2473 Maike Vollstedt

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