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INSTITUT NATIONAL DE RECHERCHE PDAGOGIQUE

Proceedings of the Sixth Congress of the European Society for Research

in Mathematics Education

January 28th-February 1st 2009Lyon (France)

Viviane Durand-Guerrier, Sophie Soury-Lavergne & Ferdinando Arzarello (eds.)

INSTITUT NATIONAL DE RECHERCHE PDAGOGIQUE, 2010

ISBN 978-2-7342-1190-7 Ref.: BR066

Editorial boardMaha Abboud-BlanchardJanet AinleyPaul AndrewsFerdinando ArzarelloGiorgio T. BagniPatti BarberChrister BergstenMorten BlomhjMarianna BoschRosa Maria BottinoLeanor CamargoSusana CarreiraJos CarrilloClaire CazesMargarida Alexandra CesarGiampaolo ChiappiniSarah CrafterGuida de AbreuJean-Luc DorierPaul DrijversAndreas EichlerMarie-Thrse Farrugi

Marei FetzerFulvia FuringhettiPatrick GibelJuan GodinoNria GorgoriGhislaine GueudetMarkku S. HannulaMatthias HattermannLisa HefendehlHebekerStephen HegedusAlena HospesovaElia IliadaEva JablonkaUffe JankvistIvy KidronAlain KuzniakJeanBaptiste LagrangeRoza LeikinFlorence LigozatKatja MaassJoanna Mamona-DawnsMaria Alessandra Mariotti

Alain MercierJohn MonaghanCandia MorganMaria Gabriella OttavianiMarilena PanziaraBirgit PepinDave PrattSusanne PredigerKristina ReissTim RowlandFilip RoubicekLeonor SantosWolfgang SchloeglmannGrard SensevyNada StehlikovaConstantinos TzanakisPaul VanderlindJan van MaanenKjersti WgeGeoff WakeHans-Georg WeigandFloriane Wozniak

Publishing assistanceService des publications, INRP

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CERME 6 TABLE OF CONTENTS

GENERAL INTRODUCTION .........................................................................................................XVIII PLENARY 1 Signs, gestures, meanings: Algebraic thinking from a cultural semiotic perspective ................ XXXIII Luis Radford PLENARY 2 Mathematics education as a network of social practices ................................................................ LIV Paola Valero SPECIAL PLENARY SESSION Ways of working with different theoretical approaches in mathematics education research ............. 1 Introduction.......................................................................................................................................... 2 Tommy Dreyfus Networking of theories: why and how? ............................................................................................... 6 Angelika Bikner-Ahsbahs People and theories .............................................................................................................................16 John Monaghan Discussion ...........................................................................................................................................24 WORKING GROUP 1 Multimethod approaches to the multidimensional affect in mathematics education ...................................................................................................................26 Introduction.........................................................................................................................................28 Markku S. Hannula, Marilena Pantziara, Kjersti Wge, Wolfgang Schlglmann The effect of achievement, gender and classroom context on upper secondary students' mathematical beliefs ............................................................................34 Markku S. Hannula Changing beliefs as changing perspective ..........................................................................................44 Peter Liljedahl Maths and me: software analysis of narrative data about attitude towards math ........................................................54 Pietro Di Martino Students beliefs about the use of representations in the learning of fractions...................................64 Athanasios Gagatsis, Areti Panaoura, Eleni Deliyianni, Iliada Elia Efficacy beliefs and ability to solve volume measurement tasks in different representations ...........74 Paraskevi Sophocleous, Athanasios Gagatsis

TABLE OF CONTENTS

Proceedings of CERME 6, January 28th-February 1st 2009, Lyon France INRP 2010

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Students motivation for learning mathematics in terms of needs and goals......................................84 Kjersti Wge Mathematical modeling, self-representation and self-regulation........................................................94 Areti Panaoura, Andreas Demetriou, Athanasios Gagatsis Endorsing motivation: identification of instructional practices........................................................ 104 Marilena Pantziara, George Philippou The effects of changes in the perceived classroom social culture on motivation in mathematics across transitions .............................................................................. 114 Chryso Athanasiou, George N. Philippou After I do more exercise, I won't feel scared anymore Examples of personal meaning from Hong-Kong ............................................................................ 124 Maike Vollstedt Emotional knowledge of mathematics teachers retrospective perspectives of two case studies ............................................................................................................................ 134 Ilana Lavy, Atara Shriki Humour as a means to make mathematics enjoyable ....................................................................... 144 Pavel Shmakov, Markku S. Hannula Beliefs: a theoretically unnecessary construct? ................................................................................ 154 Magnus sterholm Categories of affect some remarks................................................................................................. 164 Wolfgang Schlglmann WORKING GROUP 2 Argumentation and proof .................................................................................................................. 174 Introduction....................................................................................................................................... 176 Maria Alessandra Mariotti, Leanor Camargo, Patrick Gibel, Kristina Reiss Understanding, visualizability and mathematical explanation ......................................................... 181 Daniele Molinini Argumentation and proof: a discussion about Toulmin's and Duval's models ................................. 191 Thomas Barrier, Anne-Ccile Math, Viviane Durand-Guerrier Why do we need proof ...................................................................................................................... 201 Kirsti Hemmi, Clas Lfwall Proving as a rational behaviour: Habermas' construct of rationality as a comprehensive frame for research on the teaching and learning of proof................................. 211 Francesca Morselli, Paolo Boero Experimental mathematics and the teaching and learning of proof.................................................. 221 Maria G. Bartolini Bussi Conjecturing and proving in dynamic geometry: the elaboration of some research hypotheses...... 231 Anna Baccaglini-Frank, Maria alessandra Mariotti The algebraic manipulator of alnuset: a tool to prove ...................................................................... 241 Bettina Pedemonte Visual proofs: an experiment ............................................................................................................ 251 Cristina Bardelle

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Teachers views on the role of visualisation and didactical intentions regarding proof................... 261 Irene Biza, Elena Nardi, Theodossios Zachariades Modes of argument representation for proving the case of general proof ..................................... 271 Ruthi Barkai, Michal Tabach, Dina Tirosh, Pessia Tsamir, Tommy Dreyfus Mathematics teachers reasoning for refuting students invalid claims............................................ 281 Despina Potari, Theodossios Zachariades, Orit Zaslavsky Student justifications in high school mathematics............................................................................ 291 Ralph-Johan Back, Linda Mannila, Solveig Wallin Is that a proof?: an emerging explanation for why students dont know they (just about) have one .................................................................. 301 Manya Raman, Jim Sandefur, Geoffrey Birky, Connie Campbell, Kay Somers Can a proof and a counterexample coexist? A study of students conceptions about proof ........ 311 Andreas J. Stylianides, Thabit Al-Murani Abduction and the explanation of anomalies: the case of proof by contradiction............................ 322 Samuele Antonini, Maria Alessandra Mariotti Approaching proof in school: from guided conjecturing and proving to a story of proof construction......................................................................................................... 332 Nadia Douek WORKING GROUP 3 On stochastic thinking .................................................................................................................... 343 Introduction....................................................................................................................................... 344 Andreas Eichler, Maria Gabriella Ottaviani, Floriane Wozniak, Dave Pratt Chance models: building blocks for sound statistical reasoning ...................................................... 348 Herman Callaert Recommended knowledge of probability for secondary mathematics teachers ............................... 358 Irini Papaieronymou Statistical graphs produced by prospective teachers in comparing two distributions....................... 368 Carmen Batanero, Pedro Arteaga, Blanca Ruiz The role of context in stochastics instruction.................................................................................... 378 Andreas Eichler Does the nature and amount of posterior information affect preschoolers inferences .................... 388 Zoi Nikiforidou, Jenny Pange Students Causal explanations for distribution ................................................................................. 394 Theodosia Prodromou, Dave Pratt Greek students ability in probability problem solving .................................................................... 404 Sofia Anastasiadou WORKING GROUP 4 Algebraic thinking and mathematics education................................................................................ 413 Introduction....................................................................................................................................... 415 Janet Ainley, Giorgio T. Bagni, Lisa Hefendehl-Hebeker, Jean-Baptiste Lagrange

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The effects of multiple representations-based instruction on seventh grade students algebra performance .............................................................................. 420 Oylum Akkus, Erdinc Cakiroglu Offering proof ideas in an algebra lesson in different classes and by different teachers.................. 430 Michal Ayalon, Ruhama Even Rafael Bombellis Algebra (1572) and a new mathematical object: a semiotic analysis ............. 440 Giorgio T. Bagni Cognitive configurations of pre-service teachers when solving an arithmetic-algebraic problem................................................................................. 449 Walter F. Castro, Juan D. Godino Transformation rules: a cross-domain difficulty............................................................................... 459 Marie-Caroline Croset Interrelation between anticipating thought and interpretative aspects in the use of algebraic language for the construction of proofs in elementary number theory......... 469 Annalisa Cusi Epistemography and algebra............................................................................................................. 479 Jean-Philippe Drouhard Smi culture and algebra in the curriculum ...................................................................................... 489 Anne Birgitte Fyhn Problem solving without numbers An early approach to algebra.................................................. 499 Sandra Gerhard The ambiguity of the sign .......................................................................................................... 509 Bernardo Gmez, Carmen Buhlea Behind students spreadsheet competencies: their achievement in algebra? A study in a French vocational school .............................................................................................. 519 Mariam Haspekian, Eric Bruillard Developing Katys algebraic structure sense .................................................................................... 529 Maureen Hoch, Tommy Dreyfus Childrens understandings of algebra 30 years on: what has changed?............................................ 539 Jeremy Hodgen, Dietmar Kuchemann, Margaret Brown, Robert Coe Presenting equality statements as diagrams ...................................................................................... 549 Ian Jones Approaching functions via multiple representations: a teaching experiment with Casyopee .......... 559 Jean-Baptiste Lagrange, Tran Kiem Minh Equality relation and structural properties ........................................................................................ 569 Carlo Marchini, Anne Cockburn, Paul Parslow-Williams, Paola Vighi Structure of algebraic competencies ................................................................................................. 579 Reinhard Oldenburg Generalization and control in algebra ............................................................................................... 589 Mabel Panizza From area to number theory: a case study ........................................................................................ 599 Maria Iatridou, Ioannis Papadopoulos Allegories in the teaching and learning of mathematics ................................................................... 609 Reinert A. Rinvold, Andreas Lorange

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Role of an artefact of dynamic algebra in the conceptualisation of the algebraic equality .............. 619 Giampaolo Chiappini, Elisabetta Robotti, Jana Trgalova Communicating a sense of elementary algebra to preservice primary teachers ............................... 629 Franziska Siebel, Astrid Fischer Conception of variance and invariance as a possible passage from early school mathematics to algebra...................................................... 639 Ilya Sinitsky, Bat-Sheva Ilany Growing patterns as examples for developing a new view onto algebra and arithmetic.................. 649 Claudia Bttinger, Elke Sbbeke Steps towards a structural conception of the notion of variable ....................................................... 659 Annika M. Wille WORKING GROUP 5 Geometrical thinking ......................................................................................................................... 669 Introduction....................................................................................................................................... 671 Alain Kuzniak, Iliada Elia, Matthias Hattermann, Filip Roubicek The necessity of two different types of applications in elementary geometry.................................. 676 Boris Girnat A French look on the Greek geometrical working space at secondary school level ........................ 686 Alain Kuzniak, Laurent Vivier A theoretical model of students geometrical figure understanding ................................................. 696 Eleni Deliyianni, Iliada Elia, Athanasios Gagatsis, Annita Monoyiou, Areti Panaoura Gestalt configurations in geometry learning..................................................................................... 706 Claudia Acua Investigating comparison between surfaces...................................................................................... 716 Paola Vighi The effects of the concept of symmetry on learning geometry at French secondary school ............ 726 Caroline Bulf The role of teaching in the development of basic concepts in geometry: how the concept of similarity and intuitive knowledge affect students perception of similar shapes........................ 736 Mattheou Kallia, Spyrou Panagiotis The geometrical reasoning of primary and secondary school students ............................................ 746 Georgia Panaoura, Athanasios Gagatsis Strengthening students understanding of proof in geometry in lower secondary school ............. 756 Susumu Kunimune, Taro Fujita, Keith Jones Written report in learning geometry: explanation and argumentation.............................................. 766 Slvia Semana, Leonor Santos Multiple solutions for a problem: a tool for evaluation of mathematical thinking in geometry....... 776 Anat Levav-Waynberg, Roza Leikin The drag-mode in three dimensional dynamic geometry environments Two studies ................... 786 Mathias Hattermann 3D geometry and learning of mathematical reasoning ..................................................................... 796 Joris Mithalal

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In search of elements for a competence model in solid geometry teaching Establishment of relationships .......................................................................................................... 806 Edna Gonzlez, Gregoria Guilln Students 3D geometry thinking profiles .......................................................................................... 816 Marios Pittalis, Nicholas Mousoulides, Constantinos Christou WORKING GROUP 6 Language and mathematics ............................................................................................................. 826 Introduction....................................................................................................................................... 828 Candia Morgan Imparting the language of critical thinking while teaching probability............................................ 833 Einav Aizikovitsh, Miriam Amit Toward an inferential approach analyzing concept formation and language processes ................... 842 Stephan Humann, Florian Schacht Iconicity, objectification, and the math behind the measuring tape: An example from pipe-trades training .............................................................................................. 852 Lionel LaCroix Mathematical reflection in primary school education: theoretical foundation and empirical analysis of a case study .......................................................... 862 Cordula Schlke, Heinz Steinbring Surface signs of reasoning ................................................................................................................ 873 Nathalie Sinclair, David Pimm A teachers use of gesture and discourse as communicative strategies in concluding a mathematical task .................................................................................................... 884 Raymond Bjuland, Maria Luiza Cestari, Hans Erik Borgersen A teachers role in whole class mathematical discussion: facilitator of performance etiquette? ..... 894 Thrse Dooley Use of words Language-games in mathematics education ............................................................ 904 Michael Meyer Speaking of mathematics Mathematics, every-day life and educational mathematics discourse ............................................................................................ 914 Eva Riesbeck Communicative positionings as identifications in mathematics teacher education.......................... 924 Hans Jrgen Braathe Teachers collegial reflections of their own mathematics teaching processes Part 1: An analytical tool for interpreting teachers` reflections........................................................ 934 Kerstin Bruning, Marcus Nhrenbrger Teachers reflections of their own mathematics teaching processes Part 2: Examples of an active moderated collegial reflection........................................................... 944 Kerstin Bruning, Marcus Nhrenbrger Internet-based dialogue: a basis for reflection in an in-service mathematics teacher education program ................................................................. 954 Mario Snchez

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The use of algebraic language in mathematical modelling and proving in the perspective of Habermas theory of rationality....................................................................... 964 Paolo Boero, Francesca Morselli Objects as participants in classroom interaction............................................................................... 974 Marei Fetzer The existence of mathematical objects in the classroom discourse .................................................. 984 Vicen Font, Juan D. Godino, Nria Planas, Jorge I. Acevedo Mathematical activity in a multi-semiotic environment ................................................................... 993 Candia Morgan, Jehad Alshwaikh Engaging everyday language to enhance comprehension of fraction multiplication ..................... 1003 Andreas O. Kyriakides Tensions between an everyday solution and a school solution to a measuring problem................ 1013 Frode Rnning Linguistic accomplishment of the learning-teaching processes in primary mathematics instruction................................................................................................. 1023 Marcus Schtte Mathematical cognitive processes between the poles of mathematical technical terminology and the verbal expressions of pupils ............................................................................................... 1033 Rose Vogel, Melanie Huth WORKING GROUP 7 Technologies and resources in mathematical education ............................................................... 1043 Introduction..................................................................................................................................... 1046 Ghislaine Gueudet, Rosa Maria Bottino, Giampaolo Chiappini, Stephen Hegedus, Hans-Georg Weigand Realisation of mers (multiple extern representations) and melrs (multiple equivalent linked representations) in elementary mathematics software .................................................................... 1050 Silke Ladel, Ulrich Kortenkamp The impact of technological tools in the teaching and learning of integral calculus...................... 1060 Alejandro Lois, Liliana Milevicich Using technology in the teaching and learning of box plots........................................................... 1070 Ulrich Kortenkamp, Katrin Rolka Dynamical exploration of two-variable functions using virtual reality .......................................... 1081 Thierry Dana-Picard, Yehuda Badihi, David Zeitoun, Oren David Israeli Designing a simulator in building trades and using it in vocational education .............................. 1091 Annie Bessot, Colette Laborde Collaborative design of mathematical activities for learning in an outdoor setting ....................... 1101 Per Nilsson, Hkan Sollervall, Marcelo Milrad Student development process of designing and implementing exploratory and learning objects ........................................................................................................................ 1111 Chantal Buteau, Eric Muller How can digital artefacts enhance mathematical analysis teaching and learning........................... 1121 Dionysis I. Diakoumopoulos

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A learning environment to support mathematical generalisation in the classroom ........................ 1131 Eirini Geraniou, Manolis Mavrikis, Celia Hoyles, Richard Noss Establishing a longitudinal efficacy study using SimCalc MathWorlds ..................................... 1141 Stephen Hegedus, Luis Moreno, Sara Dalton, Arden Brookstein Interoperable Interactive Geometry for Europe First technological and educational results and future challenges of the Intergeo project.................................................................................. 1150 Ulrich Kortenkamp, Axel M. Blessing, Christian Dohrmann, Yves Kreis, Paul Libbrecht, Christian Mercat Quality process for dynamic geometry resources: the Intergeo project.......................................... 1161 Jana Trgalova, Ana Paula Jahn, Sophie Soury-Lavergne New didactical phenomena prompted by TI-Nspire specificities The mathematical component of the instrumentation process........................................................ 1171 Michle Artigue, Caroline Bardini Issues in integrating cas in post-secondary education: a literature review ..................................... 1181 Chantal Buteau, Zsolt Lavicza, Daniel Jarvis, Neil Marshall The long-term project Integration of symbolic calculator in mathematics lessons The case of calculus ........................................................................................................................ 1191 Hans-Georg Weigand, Ewald Bichler Enhancing functional thinking using the computer for representational transfer........................... 1201 Andrea Hoffkamp The Robot Race: understanding proportionality as a function with robots in mathematics class .............................. 1211 Elsa Fernandes, Eduardo Ferm, Rui Oliveira Internet and mathematical activity within the frame of Sub14 ................................................... 1221 Hlia Jacinto, Nlia Amado, Susana Carreira A resource to spread math research problems in the classroom ..................................................... 1231 Gilles Aldon, Viviane Durand-Guerrier The synergy of students use of paper-and-pencil techniques and dynamic geometry software: a case study ..................................................................................................................................... 1241 Nria Iranzo, Josep Maria Fortuny Students utilization schemes of pantographs for geometrical transformations: a first classification ......................................................................................................................... 1250 Francesca Martignone, Samuele Antonini The utilization of mathematics textbooks as instruments for learning ........................................... 1260 Sebastian Rezat Teachers beliefs about the adoption of new technologies in the mathematics curriculum ........... 1270 Marilena Chrysostomou, Nicholas Mousoulides Systemic innovations of mathematics education with dynamic worksheets as catalysts ............... 1280 Volker Ulm A didactic engineering for teachers education courses in mathematics using ICT ........................ 1290 Fabien Emprin Geometers sketchpad software for non-thesis graduate students: a case study in Turkey ............ 1300 Berna Cantrk-Gnhan, Deniz zen

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Leading teachers to perceive and use technologies as resources for the construction of mathematical meanings .............................................................................. 1310 Eleonora Faggiano The teachers use of ICT tools in the classroom after a semiotic mediation approach................... 1320 Mirko Maracci, Maria Alessandra Mariotti Establishing didactical praxeologies: teachers using digital tools in upper secondary mathematics classrooms ...................................... 1330 Mary Billington Dynamic geometry software: the teachers role in facilitating instrumental genesis ..................... 1340 Nicola Bretscher Instrumental orchestration: theory and practice.............................................................................. 1349 Paul Drijvers, Michiel Doorman, Peter Boon, Sjef van Gisbergen Teaching Resources and teachers professional development: towards a documentational apProach of didactics .......................................................................... 1359 Ghislaine Gueudet, Luc Trouche An investigative lesson with dynamic geometry: a case study of key structuring features of technology integration in classroom practice.............. 1369 Kenneth Ruthven Methods and tools to face research fragmentation in technology enhanched mathematics education........................................................................... 1379 Rosa Maria Bottino, Michele Cerulli The design of new digital artefacts as key factor to innovate the teaching and learning of algebra: the case of Alnuset .......................................................................................................................... 1389 Giampaolo Chiappini, Bettina Pedemonte Casyope in the classroom: two different theory-driven pedagogical approaches......................... 1399 Mirko Maracci, Claire Cazes, Fabrice Vandebrouck, Maria Alessandra Mariotti Navigation in geographical space ................................................................................................... 1409 Christos Markopoulos, Chronis Kynigos, Efi Alexopoulou, Alexandra Koukiou Making sense of structural aspects of equations by using algebraic-like formalism...................... 1419 Foteini Moustaki, Giorgos Psycharis, Chronis Kynigos Relationship between design and usage of educational software: the case of Aplusix .................. 1429 Jana Trgalova, Hamid Chaachoua WORKING GROUP 8 Questions and thoughts for researching cultural diversity and mathematics education ............... 1439 Introduction..................................................................................................................................... 1440 Guida de Abreu, Sarah Crafter, Nria Gorgori A survey of research on the mathematics teaching and learning of immigrant students................ 1443 Marta Civil Parental resources for understanding mathematical achievement in multiethnic settings.............. 1453 Sarah Crafter Discussing a case study of family training in terms of communities of practices and adult education.............................................................. 1462 Javier Dez-Palomar, Montserrat Prat Moratonas

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Understanding Ethnomathematics from its criticisms and contradictions...................................... 1473 Maria do Carmo Domite, Alexandre Santos Pais Using mathematics as a tool in Rwandan workplace settings: the case of taxi drivers .................. 1484 Marcel Gahamanyi, Ingrid Andersson, Christer Bergsten Parents experiences as mediators of their childrens learning: the impact of being a parent-teacher ............................................................................................... 1494 Rachael McMullen, Guida de Abreu Batiks: another way of learning mathematics ................................................................................. 1506 Luclia Teles, Margarida Csar The role of Ethnomathematics within mathematics education ....................................................... 1517 Karen Franois WORKING GROUP 9 Different theoretical perspectives and approaches in mathematics education research .............. 1527 Introduction..................................................................................................................................... 1529 Susanne Prediger, Marianna Bosch, Ivy Kidron, John Monaghan, Grard Sensevy Research problems emerging from a teaching episode: a dialogue between TDS and ATD ......... 1535 Michle Artigue, Marianna Bosch, Joseph Gascn, Agns Lenfant Complementary networking: enriching understanding................................................................... 1545 Ferdinando Arzarello, Angelika Bikner-Ahsbahs, Cristina Sabena Interpreting students reasoning through the lens of two different languages of description: integration or juxtaposition? ........................................................................................................... 1555 Christer Bergsten, Eva Jablonka Coordinating multimodal social semiotics and institutional perspective in studying assessment actions in mathematics classrooms............................................................ 1565 Lisa Bjrklund-Boistrup, Staffan Selander Integrating different perspectives to see the front and the back: The case of explicitness ............. 1575 Uwe Gellert The practice of (university) mathematics teaching: mediational inquiry in a community of practice or an activity system........................................... 1585 Barbara Jaworski An interplay of theories in the context of computer-based mathematics teaching: how it works and why ..................................................................................................................... 1595 Helga Jungwirth On the adoption of a model to interpret teachers use of technology in mathematics lessons ....... 1605 Jean-Baptiste Lagrange, John Monaghan The joint action theory in didactics: why do we need it in the case of teaching and learning mathematics?........................................... 1615 Florence Ligozat, Maria-Luisa Schubauer-Leoni Teachers didactical variability and its role in mathematics education .......................................... 1625 Jarmila Novotn, Bernard Sarrazy The potential to act for low achieving students as an example of combining use of different theories .................................................................... 1635 Ingolf Schfer

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Outline of a joint action theory in didactics.................................................................................... 1645 Grard Sensevy The transition between mathematics studies at secondary and tertiary levels; individual and social perspectives................................................................................................... 1655 Erika Stadler Combining and Coordinating theoretical perspectives in mathematics education research........... 1665 Tine Wedege Comparing theoretical frameworks in didactics of mathematics: the GOA-model........................ 1675 Carl Winslow WORKING GROUP 10 Mathematical curriculum and practice ............................................................................................ 1685 Introduction..................................................................................................................................... 1688 Leonor Santos, Jos Carrillo, Alena Hospesova, Maha Abboud-Blanchard Effective blended professional development for teachers of mathematics: Design and evaluation of the UPOLA Program.......................................................................... 1694 Lutz Hellmig Teachers efficacy beliefs and perceptions regarding the implementation of new primary mathematics curriculum.................................................................................................................. 1704 Isil Isler, Erdine Cakiroglu Curriculum management in the context of a mathematics subject group ....................................... 1714 Cludia Canha Nunes, Joo Pedro da Ponte Gestures and styles of communication: are they intertwined?........................................................ 1724 Chiara Andr Teachers subject knowledge: the number line representation ....................................................... 1734 Maria Doritou, Eddie Gray Communication as social interaction. Primary School Teacher Practices...................................... 1744 Antonio Guerreiro, Lurdes Serrazina Experimental devices in mathematics and physics standards in lower and upper secondary school, and their consequences on teachers practices ................... 1751 Fabrice Vandebrouck, Cecile de Hosson, Aline Robert Professional development for teachers of mathematics: opportunities and change........................ 1761 Marie Joubert, Jenni Back, Els De Geest, Christine Hirst, Rosamund Sutherland Teachers perception about infinity: a process or an object?.......................................................... 1771 Maria Kattou, Michael Thanasia, Katerina Kontoyianni, Constantinos Christou, George Philippou Perceptions on teaching the mathematically gifted......................................................................... 1781 Katerina Kontoyianni, Maria Kattou, Polina Ioannou, Maria Erodotou, Constantinos Christou, Marios Pittalis

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The nature on the numbers in grade 10: A professional problem................................................... 1791 Mirne Larguier, Alain Bronner A European project for professional development of teachers through a research based methodology: The questions arisen at the international level, the Italian contribution, the knot of the teacher-researcher identity...................................................................................... 1801 Nicolina A. Malara, Roberto Tortora Why is there not enough fuss about affect and meta-affect among mathematics teachers?........... 1811 Manuela Moscucci The role of subject knowledge in Primary Student teachers approaches to teaching the topic of area ............................................................................................................ 1821 Carol Murphy Developing of mathematics teachers community: five groups, five different ways ..................... 1831 Regina Reinup Foundation knowledge for teaching: contrasting elementary and secondary mathematics............ 1841 Tim Rowland Results of a comparative study of future teachers from Australia, Germany and Hong Kong with regard to competences in argumentation and proof................................................................ 1851 Bjrn Schwarz, Gabriele Kaiser Kates conceptions of mathematics teaching: Influences in the first three years ........................... 1861 Fay Turner Pre-service teacher-generated analogies for function concepts ...................................................... 1871 Behiye Ubuz, Ayegl Erylmaz, Utkun Aydn, Ibrahim Bayazit Technology and mathematics teaching practices: about in-service and pre-service teachers ........ 1880 Maha Abboud-Blanchard Teachers and triangles..................................................................................................................... 1890 Sylvia Alatorre, Mariana Saz Mathematics teacher education research and practice: researching inside the MICA program ..... 1901 Joyce Mgombelo, Chantal Buteau Cognitive transformation in professional development: some case studies ................................... 1911 Jorge Soto-Andrade What do student teachers attend to?................................................................................................ 1921 Nada Stehlkov The mathematical preparation of teachers: A focus on tasks.......................................................... 1931 Gabriel J. Stylianides, Andreas J. Stylianides Problem posing and development of pedagogical content knowledge in pre-service teacher training......................................................................................................... 1941 Marie Tich, Alena Hospesov Sustainability of professional development .................................................................................... 1951 Stefan Zehetmeier A collaborative project as a learning opportunity for mathematics teachers.................................. 1961 Maria Helena Martinho, Joo Pedro da Ponte Reflection on Practice: content and depth....................................................................................... 1971 Christina Martins, Leonor Santos Developing mathematics teachers education through personal reflection and collaborative inquiry: which kinds of tasks?............................................................................ 1981 Angela Pesci

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The learning of mathematics teachers working in a peer group ..................................................... 1991 Martha Witterholt, Martin Goedhart Use of focus groups interviews in mathematics educational research............................................ 2000 Bodil Kleve Analyses of interactions in a collaborative context of professional development.......................... 2010 Maria Cinta Muoz-Cataln, Jos Carrillo, Nuria Climent Adapting the knowledge quarter in the Cypriot mathematics classroom ....................................... 2020 Marilena Petrou Professional knowledge in an improvisation episode: the importance of a cognitive model ......... 2030 C. Miguel Ribeiro, Rute Monteiro, Jos Carrillo WORKING GROUP 11 Applications and modelling ............................................................................................................. 2040 Introduction..................................................................................................................................... 2042 Morten Blomhj Mathematical modelling in teacher education Experiences from a modelling seminar .............. 2046 Rita Borromeo Ferri, Werner Blum Designing a teacher questionnaire to evaluate professional development in modelling ................ 2056 Katja Maa, Johannes Gurlitt Modeling in the classroom Motives and obstacles from the teachers perspective ..................... 2066 Barbara Schmidt Modelling in mathematics teachers professional development ..................................................... 2076 Jeroen Spandaw, Bert Zwaneveld Modelling and formative assessment pedagogies mediating change in actions of teachers and learners in mathematics classrooms ......................................................................................... 2086 Geoff Wake Towards understanding teachers beliefs and affects about mathematical modelling.................... 2096 Jonas Bergman rlebck The use of motion sensor can lead the students to understanding the cartesian graph ................... 2106 Maria Lucia Lo Cicero, Filippo Spagnolo Interacing populations in a restricted habitat Modelling, simulation and mathematical analysis in class.................................................................................................. 2116 Christina Roeckerath Aspects of visualization during the exploration of quadratic world via the ICT Problem fireworks ....................................................................................................................... 2126 Mria Lalinsk, Janka Majherov Mathematical modeling in class with and without technology....................................................... 2136 Hans-Stefan Siller, Gilbert Greefrath The ecology of mathematical modelling: constraints to its teaching at university level ............. 2146 Berta Barquero, Marianna Bosch, Josep Gascn The double transposition in mathematisation at primary school .................................................... 2156 Richard Cabassut Exploring the use of theoretical frameworks for modelling-oriented instructional design ............ 2166 F.J. Garca, L. Ruiz-Higueras

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Study of a practical activity: engineering projects and their training context ................................ 2176 Avenilde Romo Vzquez Fitting models to data: the mathematising step in the modelling process ...................................... 2186 Ldia Serrano, Marianna Bosch, Josep Gascn What roles can modelling play in multidisciplinary teaching......................................................... 2196 Mette Andresen Modelling in environments without numbers A case study......................................................... 2206 Roxana Grigoras Modelling activities while doing experiments to discover the concept of variable........................ 2216 Simon Zell, Astrid Beckmann Modeling with technology in elementary classrooms..................................................................... 2226 N. Mousoulides, M. Chrysostomou, M. Pittalis, C. Christou WORKING GROUP 12 Advanced mathematical thinking .................................................................................................... 2236 Introduction..................................................................................................................................... 2238 Roza Leikin, Claire Cazes, Joanna Mamona-Dawns, Paul Vanderlind A theoretical model for visual-spatial thinking............................................................................... 2246 Conceio Costa, Jos Manuel Matos, Jaime Carvalho e Silva Secondary-tertiary transition and students difficulties: the example of duality ............................ 2256 Martine De Vleeschouwer Learning advanced mathematical concepts: the concept of limit ................................................... 2266 Antnio Domingos Conceptual change and connections in analysis ............................................................................. 2276 Kristina Juter Using the onto-semiotic approach to identify and analyze mathematical meaning in a multivariate context.................................................................................................................. 2286 Mariana Montiel, Miguel R. Wilhelmi, Draga Vidakovic, Iwan Elstak Derivatives and applications; development of one students understanding .................................. 2296 Gerrit Roorda, Pauline Vos, Martin Goedhart Finding the shortest path on a spherical surface: academics and reactors in a mathematics dialogue.................................................................. 2306 Maria Kaisari, Tasos Patronis Number theory in the national compulsory examination at the end of the French secondary level: between organising and operative dimensions.............. 2316 Vronique Battie Defining, proving and modelling: a background for the advanced mathematical thinking............ 2326 Mercedes Garca, Victoria Snchez, Isabel Escudero Necessary realignments from mental argumentation to proof presentation ................................... 2336 Joanna Mamona-Downs, Martin Downs An introduction to defining processes ............................................................................................ 2346 Ccile Ouvrier-Buffet Problem posing by novice and experts: comparison between students and teachers ..................... 2356 Cristian Voica, Ildik Pelczer

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Advanced mathematical knowledge: how is it used in teaching?................................................... 2366 Rina Zazkis, Roza Leikin Urging calculus students to be active learners: what works and what doesn't................................ 2376 Buma Abramovitz, Miryam Berezina, Boris Koichu, Ludmila Shvartsman From numbers to limits: situations as a way to a process of abstraction ........................................ 2386 Isabelle Bloch, Imne Ghedamsi From historical analysis to classroom work: function variation and long-term development of functional thinking ........................................... 2396 Renaud Chorlay Experimental and mathematical control in mathematics ................................................................ 2406 Nicolas Giroud Introduction of the notions of limit and derivative of a function at a point.................................... 2416 Jn Gunaga Factors influencing teachers design of assessment material at tertiary level ................................ 2426 Marie-Pierre Lebaud Design of a system of teaching elements of group theory .............................................................. 2436 Ildar Safuanov WORKING GROUP 13 Comparative studies in mathematics education............................................................................. 2446 Introduction..................................................................................................................................... 2447 Eva Jablonka, Paul Andrews, Birgit Pepin Comparing Hungarian and English mathematics teachers professional motivations.................... 2452 Paul Andrews Spoken mathematics as a distinguishing characteristic of mathematics classrooms in different countries ....................................................................................................................... 2463 David Clarke, Xu Li Hua Mathematical behaviors of successful students from a challenged ethnic minority....................... 2473 Tiruwork Mulat, Abraham Arcavi A problem posed by J. Mason as a starting point for a Hungarian-Italian teaching experiment within a European project........................................ 2483 Giancarlo Navarra, Nicolina A. Malara, Andrs Ambrus A comparison of teachers beliefs and practices in mathematics teaching at lower secondary and upper secondary school ..................................... 2494 Hans Kristian Nilsen Mathematical tasks and learner dispositions: A comparative perspective...................................... 2504 Birgit Pepin Elite mathematics students in Finland and Washington: access, collaboration, and hierarchy ............................................................................................... 2513 Jennifer von Reis Saari International comparative research on mathematical problem solving: Suggestions for new research directions......................................................................................... 2523 Constantinos Xenofontos

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WORKING GROUP 14 Early years mathematics ................................................................................................................ 2533 Introduction..................................................................................................................................... 2535 Patti Barber Girls and boys in the land of mathematics 6 to 8 years old childrens relationship to mathematics interpreted from their drawings............... 2537 Pivi Perkkil, Eila Aarnos Numbers are actually not bad Attitudes of people working in German kindergarten about mathematics in kindergarten ............ 2547 Christiane Benz Learning mathematics within family discourses............................................................................. 2557 Birgit Brandt, Kerstin Tiedemann Orchestration of mathematical activities in the kindergarten: the role of questions....................... 2567 Martin Carlsen, Ingvald Erfjord, Per Sigurd Hundeland Didactical analysis of a dice game.................................................................................................. 2577 Jean-Luc Dorier, Cline Marchal Tell them that we like to decide for ourselves Childrens agency in mathematics education.................................................................................. 2587 Troels Lange Exploring the relationship between justification and monitoring among kindergarten children .......................................................................................................... 2597 Pessia Tsamir, Dina Tirosh, Esther Levenson Early years mathematics The case of fractions............................................................................ 2607 Ema Mamede Only two more sleeps until the school holidays: referring to quantities of things at home............ 2617 Tamsin Meaney Supporting children potentially at risk in learning mathematics Findings of an early intervention study........................................................................................... 2627 Andrea Peter-Koop The structure of prospective kindergarten teachers proportional reasoning.................................. 2637 Demetra Pitta-Pantazi, Constantinos Christou How can games contribute to early mathematics education? A video-based study .................... 2647 Stephanie Schuler, Gerald Wittmann Natural differentiation in a pattern environment (4 year old children make patterns) ................... 2657 Ewa Swoboda Can you do it in a different way?.................................................................................................... 2667 Dina Tirosh, Pessia Tsamir, Michal Tabach WORKING GROUP 15 Theory and research on the role of history in mathematics education .......................................... 2677 Introduction..................................................................................................................................... 2679 Fulvia Furinghetti, Jean-Luc Dorier, Uffe Jankvist, Jan van Maanen, Constantinos Tzanakis The teaching of vectors in mathematics and physics in France during the 20th century ............... 2682 Ciss Ba, Jean-Luc Dorier

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Geometry teaching in Iceland in the late 1800s and the van Hiele theory...................................... 2692 Kristn Bjarnadttir Introducing the normal distribution by following a teaching approach inspired by history: an example for classroom implementation in engineering education............................................. 2702 Mnica Blanco, Marta Ginovart Arithmetic in primary school in Brazil: end of the nineteenth century .......................................... 2712 David Antonio Da Costa Historical pictures for acting on the view of mathematics.............................................................. 2722 Adriano Dematt, Fulvia Furinghetti Students beliefs about the evolution and development of mathematics ........................................ 2732 Uffe Thomas Jankvist Using history as a means for the learning of mathematics without losing sight of history: the case of differential equations .................................................................................................... 2742 Tinne Hoff Kjeldsen What works in the classroom Project on the history of mathematics and the collaborative teaching practice .............................. 2752 Snezana Lawrence Intuitive geometry in early 1900s Italian middle school................................................................. 2762 Marta Menghini The appropriation of the New Math on the Technical Federal School of Parana in 1960 and 1970 decades ............................................................................................................... 2771 Brbara Winiarski Diesel Novaes, Neuza Bertoni Pinto History, heritage, and the UK mathematics classroom................................................................... 2781 Leo Rogers Introduction of an historical and anthropological perspective in mathematics: an example in secondary school in France...................................................................................... 2791 Claire Tardy, Vivianne Durand-Guerrier The implementation of the history of mathematics in the new curriculum and textbooks in Greek secondary education ......................................................................................................... 2801 Yannis Thomaidis, Constantinos Tzanakis

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CERME 6 GENERAL INTRODUCTION

GENERAL INTRODUCTION


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INTRODUCTION TO CERME 6 BY BARBARA JAWORSKI PRESIDENT OF ERME

EUROPEAN SOCIETY FOR RESEARCH IN MATHEMATICS EDUCATION

CERME is the two-yearly congress of ERME, the European Society for Research in Mathematics Education. CERME 6 marks more than a decade of ERME and it is important to recognise the achievements of the society over this time. In May 1997, a group of 16 scholars from different European countries met in Osnabrck, Germany, for three days to discuss the formation of a European society in mathematics education. In true European spirit, we decided that we wanted a society which would bring together researchers from across Europe, particularly including colleagues from Eastern Europe, fostering communication, cooperation and collaboration. We wanted a conference that would explicitly provide such opportunity. We wanted especially to encourage and contribute to the education of young researchers. Thus ERME was born and began to take shape. We decided on a two-yearly conference, or congress as it later became known, and the name CERME emerged Congress of the European Society for Research in Mathematics Education. CERME should have a group structure in which researchers would have sufficient time to really get to know each other, share and discuss their research and engage in deep scholarly debate. The first CERME was planned for February 1999, at Osnabrck. The Program Committee took very seriously the aims for the conference expressed at the 1997 meeting. Seven working groups were planned and 12 hours were provided for work in a group. To avoid most of the conference time being taken up by paper presentation, it was decided there would be no oral presentations at the conference. Papers would be presented in written form before the conference with sufficient time for group participants to read the papers. The 12 hours would be spent discussing the papers and working on themes and issues suggested by the papers and the group leaders. Over the succeeding years, a group led by Konrad Krainer (Austria) and Paolo Boero (Italy) developed a plan and style for a YERME summer school (YESS). The first summer school was held in Klagenfurt, Austria in August 2002. Like CERME, the summer school was based around groups, working on papers submitted by the young researchers, each with an international expert as leader. The pattern of CERME and YERME has developed so they take place in alternative years, the group structure being developed and carried forwards from one to the next. We had CERME 2 in Marianske Lazne, Czech Republic in 2001; CERME 3 in Bellaria, Italy in 2003; and YESS 2 in Podebrady, Czech Republic in 2004. CERME 4 took place in Saint Feliu, Spain in February, 2005 and YESS 3 in Jyvskyl, Finland in August 2006. CERME 5 was held in Cyprus in February 2007, and YESS



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4 in Trabzon, Turkey in August, 2008. CERME 6 will take place in Lyon, France in 2009 and YESS 5 in Palermo, Italy in 2010. People came from these events speaking of inspirational experiences. It seemed clear that the events generated something that we came to call the CERME Spirit. Based fundamentally on the three Cs, communication, cooperation and collaboration, the CERME Spirit was about the inspiration that derives from serious scholarly tackling of ideas and concepts in key areas and of mathematics education research with colleagues from multiple nations, facilitated by the group design of the events. However, the group design was not without its critics. Some critics felt constrained by the requirement to spend a conference, largely, in just one group. Some felt that a conference ought to offer a greater variety of opportunity to participants. Participants should be free to choose where to be at any time. However, the group work at CERME or YESS would be seriously disrupted if participants were to hop from group to group, not engaging seriously with the work in any one. Some suggested that perhaps planning could allow participants to take part in two groups, so that engagement in both could be serious. Such ideas have been considered by the ERME Board and Programme Committees but so far we have remained faithful to the initial conception. Many participants have said in evaluation of the events that the opportunity to spend serious time in one group allowed them to really get to know researchers from other countries, and that this contributed significantly to the depth of thinking that was possible. We want to encourage wider participation to ongoing activity in our Society, with more nations contributing to hosting events and a secure financial platform for continuing our inclusive communication, cooperation and collaboration within Europe. Further details of ERME can be found at the following site: http://ermeweb.free.fr/

Barbara Jaworski President of ERME



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PRESENTATION OF CERME 6 BY FERDINANDO ARZARELLO, CHAIR OF THE SCIENTIFIC INTERNATIONAL COMMITTEE

As pointed out in the document written by our President, CERME is a Congress designed to foster a communicative spirit in European mathematics education according to the three Cs of ERME: communication, cooperation and collaboration. It deliberately moves away from research presentations by individuals towards collaborative group work. Its main feature is a number (15) of thematic groups whose members have worked together in a common research area. In addition to the working group sessions, there was:

Two plenary lectures and a panel; Two parallel 1 hour sessions where the participants had the opportunity of

debating with the plenarists;

A poster session; Final parallel sessions (repeated twice), where each group has presented its work

to the interested participants;

Policy and purpose sessions to negotiate the work and directions of ERME. The philosophy of our Congress is based on the following two issues:

i. We need to know more about the research which has been done and is ongoing, and the research groups and research interests in different European countries;

ii. We need to provide opportunities for cooperation in research areas and for inter-European collaboration between researchers in joint research projects.

In organising this Conference we considered both the ERME spirit and the observations from the questionnaires filled by the participants, which mainly concerned the plenary events. Consequently, the following structure was planned:

Two plenary lectures of 75 minutes; each plenarist had a reactor: they had 60 minutes for their two presentations, and then there was 15 minutes for questions from the floor. Moreover the interested people had the opportunity to meet the plenarists in an informal meeting in another day.

An other event is the special 2 hour plenary of the last day, which had three participants: the aim was to discuss a topic emerging from previous CERMEs,



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analysing it from different standpoints and to give people the possibility of a wide debate.

The structure of the Working Groups was essentially the same: each group had more of 12 hours for discussing its topic. In the final Sunday session each group have presented the results of its work in two consecutive one hour slots, according to the model experienced in CERME 5, which had received the approval of the participants. I think that all of us had a very exciting week, plenty of interesting scientific and social opportunities. In particular I underline the lecture of Prof. E. Ghys http://www.dimensions-math.org and the discussion on a Project of a European Journal of Mathematical Education. I wish to thank the local organisers, and particularly Viviane Durand-Guerrier, for the enormous work they have done to make possible the realisation of this Conference.

Ferdinando Arzarello Chair of the scientific international committee



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QUALITY AND INCLUSION IN CERME 6: A PROPOSED REVIEW

The European Society for Research in Mathematics Education (ERME), and its principal activities CERME (2-yearly Congress of ERME) and YERME (meetings of Young researchers in ERME) are committed to the three Cs: Communication, Cooperation and Collaboration in research in mathematics education. Over the years in which ERME has existed, the community has developed what has become known as The CERME Spirit. These words capture a practical manifestation of the objectives expressed in the three Cs. The phrase refers to an inclusivity of working in which people genuinely work together, in which all are welcome, and in which members work hard to ensure that all can take a full part in activity. A major factor and issue that of the language of our work has been addressed seriously; different groups devising their own approaches to their working language. However, these things are not straightforward and issues arise as soon as we construct practical situations. The main example of this concerns the scientific quality of our work in mathematics education research. Of course we aspire to a high quality of scientific work, just as we aspire to operate in fully inclusive ways. Ideally we should like there to be compatibility between the two. But what does or can this look like in practice? These issues face group leaders as soon as they set out to construct a programme of work for their group, starting with a call for papers. Responding to this call, we see that many papers are now received for all groups. This suggests that researchers in our field want to be part of CERME and offer their work to colleagues in CERME. From an inclusive point of view, all papers should be welcome and all those wishing to participate should have a place. However, from a scientific point of view, papers should be reviewed according to scientific criteria, those that are of a suitable scientific quality (according to the group leaders) should be accepted and others rejected. In practice this means that authors of rejected papers may not be able to attend the congress since funding depends on an accepted paper. The practice seems to go against principles of inclusion. The ERME Board, and Programme Committees of CERME conferences have been aware of these issues and have addressed them by creating a two stage review process. For presentation of papers at the congress, a much more open attitude should be taken to the criteria, aiming to include as many participants as possible. At this stage, feedback to prospective participants should detail what is required for a paper to be acceptable for the scientific proceedings following the congress. Papers not meeting these requirements would not be accepted for the proceedings. Of course, it is then up to the group leaders to determine how to make the necessary decisions: what is acceptable for presentation, and what are the more strict criteria for publication? They also have to decide how to conduct the work of the group in an



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inclusive way. Similarly those organising YERME events have to decide how to ensure both quality and inclusion in practice. Our sixth CERME achieved, it therefore feels like a time to review these issues and procedures. For this purpose, a small group of interested members of ERME has agreed to survey participants in CERME 6 and seek views on the processes and issues that are involved. We have included an opportunity to comment in the evaluation questionnaire for CERME 6 and possibility to send us a personal communication (written) to express your views in more detail. We have also asked group leaders, present and past, to tell us how they have made decisions and what difficulties if any there have been. As a result of analysing the information received we hope to write a paper for a scientific edited book on the topic of inclusion and quality. Such a paper could also act as a basis for future policy in ERME, CERME and YERME.

Barbara Jaworski, Ferdinando Arzarello

M. Alessandra Marriotti Constantinos Christou

Joao Pedro da Ponte



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SCIENTIFIC PROGRAM

CERME 6 PLENARY CONFERENCES Jan 28, 15:30 - 16:45 Luis Radford, Universit Laurentienne, Ontario, Canada. SIGNS, GESTURES, MEANINGS: ALGEBRAIC THINKING FROM A CULTURAL SEMIOTIC PERSPECTIVE. Reactor: Heinz Steinbring (Duisburg-Essen University) Summary. In this presentation I will deal with the ontogenesis of algebraic thinking. Drawing on a cultural semiotic perspective, informed by current anthropological and embodied theories of knowing and learning, in the first part of my talk I will comment on the shortcomings of traditional mental approaches to cognition. In tune with contemporary research in neuroscience, cultural psychology, and semiotics, I will contend that we are better off conceiving of thinking as a sensuous and sign-mediated activity embodied in the corporeality of actions, gestures, and artifacts. In the second part of my talk, I will argue that algebraic thinking can be characterized in accordance with the semiotic means to which the students resort in order to express and deal with algebraic generality. I will draw upon results obtained in the course of a 10-year longitudinal classroom research project to offer examples of students forms of algebraic thinking. Two of the most elementary forms of algebraic thinking identified in our research are characterized by their contextual and embodied nature; they rely extensively upon rhythm and perceptual and deictic (linguistic and gestural) mechanisms of meaning production. Furthermore, keeping in line with the situated nature of the students mathematical experience, signs here usually designate their objects in an indexical manner. These elementary forms of algebraic thinking differ from the traditional onebased on the standard alphanumeric symbolismin that the latter relies on sign distinctions of a morphological kind. Here signs cease to designate objects in the usual indexical sense to give rise to symbolic processes of recognition and manipulation governed by sign shape. The aforementioned conception of thinking in general and the ensuing distinction of forms of algebraic thinking shed some light on the kind of abstraction that is entailed by the use of standard algebraic symbolism. They intimate some of the conceptual shifts that the students have to make in order to gain fluency in a cultural sophisticated form of mathematical thinking. Voice, gesture, and rhythm fade away. Embodied and contextual ways of signifying are then replaced with a perceptual activity where differences and similarities are a matter of morphology, and where meaning becomes relational. Jan 29, 9:15 - 10:30



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Paola Valero, Aalborg University, Denmark. ATTENDING TO SOCIAL CHANGES IN EUROPE: CHALLENGES FOR MATHEMATICS EDUCATION RESEARCH IN THE 21ST CENTURY Reactor: Margarida Alexandra da Piedade Silva Cesar (Lisbon University) Summary. Based on an analysis of mathematics education research as an academic field and on current social, political and economic transformations in many European countries, I would argue for the need to rethink and enlarge definitions and views of mathematics education as a scientific field of study in order to provide better understandings and alternatives for practice in the teaching and learning of mathematics today. I will explore the notion of the network of mathematics education practices as a complex, multi-layered space of social practice where the meanings of the teaching and learning of mathematics are constituted. I will illustrate the potentiality of this notion to envision possible research paths in the field. I will illustrate these with the research that my colleagues and I have been carrying on multicultural classrooms in Denmark; as well as will offer examples of other research studies in Europe and other parts of the world where I see that the discipline is gaining newer insights that could allow attending to the social changes and challenges of the 21st century. Feb 1st, 11:00 13:00 SPECIAL PLENARY: WAYS OF WORKING WITH DIFFERENT THEORETICAL APPROACHES IN MATHEMATICS EDUCATION RESEARCH Speakers: Angelika Bikner-Ahsbahs, Bremen University, Germany John Monaghan, University of Leeds, United Kingdom Chair: Tommy Dreyfus, Tel Aviv University, Israel Structure : This plenary activity is planned to last 2 hours and will comprise five parts Introduction (T. Dreyfus, 5 min) Networking of theories why and how? (A. Bikner-Ahsbahs, 25 min + 5 min for clarifications) Taking the appropriate parts from a variety of theories (J. Monaghan, 25 min + 5 min for clarifications) Questions to the floor (T. Dreyfus, 10 min) Questions and contributions from the audience with reactions from the speakers (45 min)



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Background. The development and elaboration of theoretical constructs that allow research in mathematics education to progress has long been a focus of mathematics education researchers in Europe. This focus has found its expression in many CERME working groups: some are focused around a specific theoretical approach and others allow researchers from different theoretical traditions and backgrounds to meet and discuss. More specifically, relationships between theories have been made the explicit focus of attention of the theory working group that started at CERME 4 in 2005. The present plenary activity inserts itself in this line of work of CERME, and aims to broaden the discussion about relationships between theories to include members of all CERME working groups. Abstract by Angelika Bikner-Ahsbahs: Networking of theories why and how? Research in mathematics education addresses teaching and learning of mathematics in a wide sense. For example, theories about learning fractions may tell a lot of different things about learning fractions. Some of them are about mistakes and why some mistakes are stable. Others may tell us about how students can be motivated to learn fractions. There are theories about how fraction concepts can be built best, which students imaginations accompany learning fractions and what abstraction processes can be observed. In addition, we have to distinguish between theories for gifted students and theories for students with special needs, etc. These considerations already show that research objects within mathematics education are complex. This complexity has led to a large variety of theoretical approaches. Every successful new theoretical view broadens or deepens insight in a phenomenon, hence, enriches our knowledge about the phenomenon. Therefore, it seems necessary to regard the large diversity of theories as richness. However, the rich diversity of theoretical approaches engenders problems of understanding and communicating. Sometimes we find the same terms meaning different things, for example the different concepts of abstraction, mathematising and constructing. However, we also find different words for the same or similar meanings, for example reification and constructing can both mean building a new knowledge entity. Hence, a large diversity of theories can be regarded as richness but it also causes difficulties for researchers to understand each other and for teachers and teacher trainers to make use of research results in an adequate way. These problems raise the following questions: How could researchers gainfully frame the use of the diversity of theoretical approaches? What kind of benefit can be gained through such frames? How can theories be made more useful for practitioners? In the plenary talk, networking of theories is proposed to be a fruitful approach to frame the diversity of theories or theoretical approaches. It has been practiced and reflected on since 2005 (CERME4) within a group of researchers networking their theories. This work has already shown that networking of theories means more than creating a consistent frame to investigate a research question it is a systematic way of theory development. In the plenary talk, an example is used to clarify the meaning



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and to describe some benefit of it for the research and the practice of teaching and learning mathematics. Abstract by John Monaghan: Taking the appropriate parts from a variety of theories. I will argue the case for taking the appropriate parts from a variety of theories according to needs of the research rather than trying to merge theories. One part of my argument is who I and, if I may extend this, who most CERME participants are working mathematics education researchers. Mathematics education research is demanding and does not (except for a few gifted individuals) allow researchers to become specialist philosophers, psychologists and /or sociologists; but we may find it useful to use the ideas of philosophers, psychologists and /or sociologists. Another part of my argument will concern theoretical frameworks within mathematics education and I will argue for caution with regard to attempts to merge such theories. These theories have, in general, distinct historical roots, developed in academic communities which have appropriated constructs in specific ways and the grain sizes of their analyses often differ. Attempting to merge whole theories, as opposed to appropriating constructs, comes with a real danger of creating an ill-formed hybrid. So will I argue that mathematics education researchers should pick a little bit from this theory and a little bit from that theory? Well, yes, I will but with caution! I will argue that the bits we pick depend on the situation, the specific focus of the research in which we are engaged, and the consistency of bits we pick. I have avoided referring to specific theories in this abstract but I will detail theories in my talk and I will also use research studies as cases to exemplify my arguments.

WORKING GROUPS 15 working groups: 7 sessions, 1 or 2 per day, duration 1h30 or 2h Final group reports: Sunday Feb 1st, 8:30 - 10:30 Poster Session: Thursday Jan. 29 17:15 - 18:30

N.B. The posters remain during the all congress in the hall of the THEMIS. During the poster session, the authors were present.

Group 1: Affect and mathematical thinking - This includes the role of beliefs, emotions, and other affective factors Markku Hannula, Finland (Chair); Tine Wedege, Norway; Marilena Pantziara, Cyprus.



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Group 2: Argumentation and proof - This includes epistemological and historical studies, learning issues and classroom situations Maria Alessandra Mariotti, Italy (Chair); Patrick Gibel, France; Leonor Camargo, Colombia; Kristina Reiss, Germany. Group 3: Stochastic thinking - This includes epistemological and educational issues, pupils cognitive processes and difficulties, and curriculum issues Andreas Eichler, Germany (Chair); Maria Gabriella Ottaviani, Italy; Dave Pratt, United kingdom; Floriane Wozniak, France. Group 4: Algebraic thinking - This includes epistemological and educational issues, pupils cognitive processes and difficulties, and curriculum issues Chair: Giorgio Bagni, Italy (Chair); Janet Ainley, United Kingdom; Lisa Hefendehl-Hebeker, Germany; JeanBaptiste Lagrange, France. Group 5: Geometrical thinking - This includes epistemological and educational issues, pupils cognitive processes and difficulties, and curriculum issues Alain Kuzniak, France (Chair); Iliada Elia, Cyprus; Mathias Hattermann Germany; Filip Roubicek, Czech Republic. Group 6: Mathematics and language - This includes semiotics and communication in classrooms, social processes in learning and teaching mathematics Candia Morgan, United Kingdom (Chair); Marie-Thrse Farrugia (Malta); Marei Fetzer (Germany); Alain Mercier, France. Group 7: Technologies and resources in mathematical education - This includes teaching and learning environments Ghislaine Gueudet, France (Chair); Rosa Maria Bottino, Italy; Stephen Hegedus, United States of America; Hans-Georg Weigand, Germany.



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Group 8: Cultural diversity and mathematics education - This includes students' diverse backgrounds and identities, social and cultural processes involved, political issues in the educational and school policies. Chair: Guida de Abreu, United Kingdom (Chair); Nuria Gorgorio, Spain; Sarah Crafter, United Kingdom. Group 9: Different theoretical perspectives / approaches in research in mathematics education - This includes ways of linking theory and practice and paradigms of research in ME. Susanne Prediger, Germany (Chair); Marianna Bosch, Spain; Ivy Kidron, Israel; John Monaghan, United kingdom; Grard Sensevy, France. Group 10: From a study of teaching practices to issues in teacher education - This includes teachers beliefs and the role of the teacher in the classroom, as well as strategies for teacher education and links between: theory and practice, research and teaching and teacher education, collaborative research. Chair: Leonor Santos (Portugal) Jos Carrillo, Spain; Alena Hospesova, Czech Republic; Maha Abboud-Blanchard, France. Group 11: Applications and modelling - This includes theoretical and empirical-based reflections on: the modelling process and necessary competencies, adequate applications and modelling examples, epistemological and curricular aspects, beliefs and attitudes, assessment and the role of technology. Morten Blomhoej, Denmark (Chair); Susana Carreira, Portugal; Katja Maass, Germany; Geoff Wake, United Kingdom. Group 12: Advanced mathematical thinking - This includes conceptual attainment, proof techniques, problem-solving, processes of abstraction, at the upper secondary and tertiary educational level. Roza Leikin, Israel (Chair); Claire Cazes, France; Joanna Mamona-Downs, Greece; Paul Vanderlind, Sweden.



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Group 13: Comparative Studies in Mathematics Education - It includes questions surrounding mathematics teaching and learning in the classroom, learners and teachers experiences and identities, and policy issues in different cultures and/or countries. Eva Jablonka, Sweden (Chair); Paul Andrews, United Kingdom; Birgit Pepin, United kingdom; Pasi Reinikainen, Finland. Group 14: Early Years Mathematics . This Working Group deals with the research domain of mathematics learning and mathematics education in the early years, age 3 to 7- In the last decades the interest in this topic has increased immensely. Gtz Krummheuer, Germany (Chair); Patti Barber, United Kingdom; Demetra Pitta-Pantazi, Cyprus; Ewa Swoboda, Poland. Group 15: Theory and research on the role of history in Mathematics Education - The integration of history of mathematics in mathematics education is a subject which has received increasing attention during the last decades. Chair: Fulvia Furinghetti , Italy (Chair); Jean-Luc Dorier, France; Uffe Thomas Jankvist, Denmark; Costantinos Tzanakis, Greece.



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YERME - YOUNG ERME YERME is an organization aiming at creating collaboration and mutual support among young researchers of different countries in the field of mathematics education. The two main activities of YERME are: 1. YESS YERME Summer Schools The aims of the Summer Schools are:

To let people from different countries meet and establish a friendly and cooperative style of work in mathematics education research;

To let people compare and integrate their preparation in mathematics education research in a peer discussion climate with the help of highly qualified and differently oriented experts;

To let people present their research ideas, theoretical difficulties, methodological problems, and preliminary research results, in order to get suggestions (from other participants and experts) about possible developments, different perspectives, etc. and open the way to possible connections with nearby research projects and co-operation with researchers in other countries.

YESS1 took place in Klagenfurt, Austria, 2002; YESS2 at Podebrady, Czech Republic, 2004; YESS3 at Jyvskyl, Finland, 2006 and YESS4 at Trabzon, Turkey, 2008. YESS5 will take place in Italy (August 2010). Ph.D., Master and post-graduate students and other people entering Mathematics Education research are invited to take part in YESS summer schools. 2. YERME day The YERME-day takes place the day before CERME. The spirit is the same as YESS. Young European researchers take part in Discussion Groups and Working Groups. The topics of these groups are close to young researchers' interests. This kind of organization allows European students to meet and start to build links between different countries. They also have the opportunity to work with experts in the research education field. The program of the YERME-Day 2009 (January, 27th and 28th) is available on the YERME Website http://yerme.eu .



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CERME 6 PLENARY 1 Signs, gestures, meanings:

Algebraic thinking from a cultural semiotic perspective Luis Radford, Universit Laurentienne, Ontario, Canada Reactor: Heinz Steinbring (Duisburg-Essen University) Summary. In this presentation I will deal with the ontogenesis of algebraic thinking. Drawing on a cultural semiotic perspective, informed by current anthropological and embodied theories of knowing and learning, in the first part of my talk I will comment on the shortcomings of traditional mental approaches to cognition. In tune with contemporary research in neuroscience, cultural psychology, and semiotics, I will contend that we are better off conceiving of thinking as a sensuous and sign-mediated activity embodied in the corporeality of actions, gestures, and artifacts. In the second part of my talk, I will argue that algebraic thinking can be characterized in accordance with the semiotic means to which the students resort in order to express and deal with algebraic generality. I will draw upon results obtained in the course of a 10-year longitudinal classroom research project to offer examples of students forms of algebraic thinking. Two of the most elementary forms of algebraic thinking identified in our research are characterized by their contextual and embodied nature; they rely extensively upon rhythm and perceptual and deictic (linguistic and gestural) mechanisms of meaning production. Furthermore, keeping in line with the situated nature of the students mathematical experience, signs here usually designate their objects in an indexical manner. These elementary forms of algebraic thinking differ from the traditional onebased on the standard alphanumeric symbolismin that the latter relies on sign distinctions of a morphological kind. Here signs cease to designate objects in the usual indexical sense to give rise to symbolic processes of recognition and manipulation governed by sign shape. The aforementioned conception of thinking in general and the ensuing distinction of forms of algebraic thinking shed some light on the kind of abstraction that is entailed by the use of standard algebraic symbolism. They intimate some of the conceptual shifts that the students have to make in order to gain fluency in a cultural sophisticated form of mathematical thinking. Voice, gesture, and rhythm fade away. Embodied and contextual ways of signifying are then replaced with a perceptual activity where differences and similarities are a matter of morphology, and where meaning becomes relational

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