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Volume 3 Research Reports Ead - Kou Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education Prague, Czech Republic July 16-21, 2006 Volume 3 Research Reports Ead - Kou Editors: Jarmila Novotná Hana Moraová, Magdalena Krátká, Naďa Stehlíková Charles University in Prague Faculty of Education Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education Volume 3 Editors Jarmila Novotná Hana Moraová Magdalena Krátká Naďa Stehlíková Charles University in Prague Faculty of Education Czech Republic The Proceedings are also available on CD-ROM Copyright © 2006 left to the authors All rights reserved ISSN 0771-100X Logo: Ivan Špirk Cover Design: Jarmila Novotná & Ivan Špirk Printing: Atelier Guimaec, s.r.o 3-ii PME30 — 2006 TABLE OF CONTENTS VOLUME 3 Research Reports Eade, Frank & Dickinson, Paul 3-1 Exploring realistic mathematics education in English schools Ebersbach, Mirjam & Resing, Wilma C. M. 3-9 Reasoning about non-linearity in 6- to 9-year-olds: the role of task presentation Eichler, Andreas 3-17 Teachers, classroom, students – A holistic view on beliefs concerning the educational practice Elia, Iliada & Gagatsis, Athanasios 3-25 The effects of different modes of representation on problem solving: Two experimental programs Ell, Fiona 3-33 Can moderate hermeneutics help us to understand learning and teaching in the mathematics classroom? Evans, Jeff & Zan, Rosetta 3-41 Sociocultural approaches to emotions in mathematics education: Initial comparisons Falkenberg, Thomas 3-49 Moral education in the teaching of mathematics Ferrando, Elisabetta 3-57 The abductive system Ferrara, Francesca 3-65 Remembering and imagining: moving back and forth between motion and its representation Ferrari, Pier Luigi 3-73 From verbal texts to symbolic expressions: A semiotic approach to early algebra PME30 — 2006 3-iii Finnane, Maureen 3-81 Verbal memory span limitations as a factor in early mathematical learning difficulties Fox, Jillian 3-89 Connecting algebraic development to mathematical patterning in early childhood Frade, Cristina & Winbourne, Peter & Braga, Selma Moura 3-97 Aline’s and Julia’s stories: Reconceptualizing transfer from a situated point of view Francisco, John & Hähkiöniemi, Markus 3-105 Insights into students’ algebraic reasoning Fried, Michael N. & Amit, Miriam 3-113 Proof, authority, and agency: Intimations from an 8th grade classroom Fuglestad, Anne Berit 3-121 Students’ thoughts about ICT in school mathematics Fujita, Taro & Jones, Keith 3-129 Primary trainee teachers’ understanding of basic geometrical figures in Scotland Gagatsis, Athanasios & Elia, Iliada & Panaoura, Areti & Gravvani, 3-137 Katerina & Spyrou, Panayotis An empirical four-dimensional model for the undestanding of function Gal, Hagar & Lin, Fou-Lai & Ying, Jia-Ming 3-145 The hidden side in Taiwanese classrooms - Through the lens of PLS in geometry Gallardo, Aurora & Hernández, Abraham 3-153 The zero and negativity among secondary school students García Cruz, Juan Antonio & Garrett, Alexandre Joaquim 3-161 Students’ actions in open and multiple-choice questions regarding undestanding of averages George, Patricia 3-169 Paradoxes:The interplay of gender, social class and mathematics in the Caribbean 3-iv PME30 — 2006 Gervasoni, Ann 3-177 Insights about the addition strategies used by grade 1 and grade 2 children who are vulnerable in number learning Giraldo, Victor & Carvalho, Luiz Mariano 3-185 A generic organizer for the enrichment of the concept image of derivative Gkolia, Chrysanthi & Jervis, Alan 3-193 Pupils’ attitudes to integrated learning systems in mathematics Gomes, Geraldo & Meira, Luciano 3-201 The discourse of logical necessity: Rules for action in pre-school mathematics Goos, Merrilyn & Bennison, Anne 3-209 An online community of practice for pre-service and beginning teachers of secondary mathematics Grzegorczyk, Ivona & Stylianou, Despina A. 3-217 Development of abstract mathematical thinking through artistic patterns Halverscheid, Stefan & Henseleit, Melanie & Lies, Klaus 3-225 Rational numbers after elementary school: Realizing models for fractions on the real line Halverscheid, Stefan & Rolka, Katrin 3-233 Student beliefs about mathematics encoded in pictures and words Harel, Guershon & Koichu, Boris & Manaster, Alfred 3-241 Algebra teachers’ ways of thinking characterizing the mental act of problem posing Hazin, Izabel & da Rocha Falcão, Jorge T. & Leitão, Selma 3-249 Mathematical impairment among epileptic children Hegedus, Stephen & Dalton, Sara & Cambridge, Laura & Davis, Gary 3-257 Patterns of participation in networked classrooms Heggem, Tore & Iversen, Kjærand 3-265 Teacher training students’ probabilistic reasoning about compound stochastic events in an ICT environment PME30 — 2006 3-v Heinze, Aiso & Reiss, Kristina & Groß, Christian 3-273 Learning to prove with heuristic worked-out examples Heirdsfield, Ann & Lamb, Janeen 3-281 Teacher actions: Enhancing the learning of mental computation in Year 2 Hejný, Milan & Jirotková, Darina & Kratochvílová, Jana 3-289 Early conceptual thinking Hershkowitz, Rina & Hadas, Nurit & Dreyfus, Tommy 3-297 Diversity in the construction of a group's shared knowledge Hoch, Maureen & Dreyfus, Tommy 3-305 Structure sense versus manipulation skills: An unexpected result Horne, Marj & Livy, Sharyn 3-313 Young children developing place value understandings Iannece, Donatella & Mellone, Maria & Tortora, Roberto 3-321 New insights into learning processes from some neuroscience issues Iannone, Paola & Cockburn, Anne D. 3-329 Fostering conceptual mathematical thinking in the early years: A case study Inglis, Matthew & Simpson, Adrian 3-337 The role of mathematical context in evaluating conditional statements Izsák, Andrew 3-345 Knowledge for teaching fraction arithmetic: Partitioning drawn representations Jaworski, Barbara & Goodchild, Simon 3-353 Inquiry community in an activity theory frame Jay, Tim 3-361 Gender differences in patterns of strategy use amongst secondary school mathematics students Johnsen Høines, Marit & Lode, Beate 3-369 Positioning of a subject based and investigative dialogue in practice teaching 3-vi PME30 — 2006 Jungwirth, Helga 3-377 Everyday computer-based maths teaching: The predominance of practical activities Kaasila, Raimo & Hannula, Markku S. & Laine, Anu & Pehkonen, Erkki 3-385 Faciliators for change of elementary teacher student’s view of mathematics Kaiser, Gabriele 3-393 The mathematical beliefs of teachers about applications and modelling – Results of an empirical study Karp, Alexander 3-401 An analysis of solving groups of problems (toward the study of problem solving instruction) Karsenty, Ronnie 3-409 Advancing low achievers in secondary schools by using non- professional mathematics tutoring Kazima, Mercy & Adler, Jill 3-417 Mathematical knowledge for teaching: Adding to the description through a study of probability in practice Keisoglou, Stefanos & Kynigos, Chronis 3-425 Measurements with a physical and a virtual quadrant: Students’ understanding of trigonometric tangent Khisty, Lena Licón 3-433 Language and mathematics: Towards social justice for linguistically diverse students Kidron, Ivy & Picard, Thierry Dana 3-441 Concept definition, concept image and the discrete – continuous interplay Knapp, Jessica & Weber, Keith 3-449 Reconceiving strategic knowledge in proving from the student’s perspective Koichu, Boris & Berman, Abraham & Moore, Michael 3-457 Patterns of middle school students’ heuristic behaviors in solving seemingly familiar problems PME30 — 2006 3-vii Kontoyianni, Katerina & Modestou, Modestina & Erodotou, Maria & 3-465 Ioannou, Polina & Constantinides, Athinos & Parisinos, Marinos & Gagatsis, Athanasios Improper proportional reasoning: A comparative study in high school Koukkoufis, Andreas & Williams, Julian 3-473 Integer instruction: A semiotic analysis of the “compensation strategy” 3-viii PME30 — 2006 EXPLORING REALISTIC MATHEMATICS EDUCATION IN ENGLISH SCHOOLS Frank Eade, Paul Dickinson Manchester Metropolitan University Institute of Education This paper provides an account of the initial phases of a project involving the trialing of a new approach to teaching in lower secondary schools in England. The method being trialed is based on Realistic Mathematics Education (RME), originally developed in the Netherlands. The paper describes background concerns and research methods, initial findings and explores emerging issues for the project. INTRODUCTION The TIMSS (1995) study stimulated a great deal of research into international comparisons of approaches to teaching mathematics in schools. Indeed the National Numeracy Strategy (NNS), developed as a direct response to the perceived underachievement of pupils in England, is now part of the framework for teaching mathematics in all primary and secondary schools in England. In exploring the use of an approach to teaching from one country in another we need to be aware of a range of issues including the desired outcomes of teaching mathematics, social and cultural norms, curriculum compatibility and pedagogical concerns. This paper discusses teaching in a variety of countries and outlines the initiation of the use of Realistic Mathematics Education, originally developed in the Netherlands, in lower secondary schools in England. BACKGROUND Mathematics in England In response to TIMSS (1995) The National Numeracy Strategy was developed to raise the standard of mathematics in primary schools, borrowing techniques and strategies from a range of relatively
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