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Volume 2 Research Reports Abr - Dri Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education Prague, Czech Republic July 16-21, 2006 Volume 2 Research Reports Abr - Dri 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 2 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 2-ii PME30 — 2006 TABLE OF CONTENTS VOLUME 2 Research Reports Abrahamson, Dor & Cendak, Rose M. 2-1 The odds of understanding the law of large numbers:A design for grounding intuitive probability in combinatorial analysis Adler, Jill & Davis, Zain 2-9 Imaginary-symbolic relations, pedagogic resources and the constitution of mathematics for teaching in in-service mathematics teacher education Akkoç, Hatice & Ogan-Bekiroglu, Feral 2-17 Relationship between pre-service mathematics teachers’ teaching and learning beliefs and their practices Al-Murani, Thabit 2-25 Teachers’ awareness of dimensions of variation: A mathematics intervention project Amado, Nélia & Carreira, Susana 2-33 The student teacher and the others: Multimembership on the process of introducing technology in the classroom Amato, Solange Amorim 2-41 Improving student teachers’ understanding of fractions Amit, Miriam & Jan, Irma 2-49 Autodidactic learning of probabilistic concepts through games Antonini, Samuele 2-57 Graduate students’ processes in generating examples of mathematical objects Antonini, Samuele & Mariotti, Maria Alessandra 2-65 Reasoning in an absurd world: Difficulties with proof by contradiction PME30 — 2006 2-iii Arzarello, Ferdinando & Bazzini, Luciana & Ferrara, Francesca & 2-73 Robutti, Ornella & Sabena, Cristina & Villa, Bruna Will Penelope choose another bridegroom?Looking for an answer through signs Athanasiou, Chryso & Philippou, George N. 2-81 Motivation and percetions of classroom culture in mathematics of students across grades 5 to 7 Ayalon, Michal & Even, Ruhama 2-89 Deductive reasoning: Different conceptions and approaches Babai, Reuven 2-97 The tendency to use intuitive rules among students with different Piagetian cognitive levels Ball, Lynda & Stacey, Kaye 2-105 Coming to appreciate the pedagogical uses of CAS Bardini, Caroline & Stacey, Kaye 2-113 Students’ conceptions of m and c: How to tune a linear function Bayazit, İbrahim & Gray, Eddie 2-121 A contradiction between pedagogical content knowledge and teaching indications Benítez Mojica, David & Santos Trigo, Manuel 2-129 Identifying and supporting mathematical conjectures through the use of dynamic software Benko, Palma & Maher, Carolyn A. 2-137 Students constructing representations for outcomes of experiments Berezovski, Tanya & Zazkis, Rina 2-145 Logarithms: Snapshots from two tasks Bergsten, Christer 2-153 Trying to reach the limit – The role of algebra in mathematical reasoning Bikner-Ahsbahs, Angelika 2-161 Semiotic sequence analysis – Constructing epistemic types empirically 2-iv PME30 — 2006 Bingolbali, Erhan & Monaghan, John & Roper, Tom 2-169 Service teaching: Mathematical education of students of client departments Biza, Irene & Christou, Constantinos & Zachariades, Theodossios 2-177 Students’ thinking about the tangent line Boero, Paolo 2-185 Habermas’ theory of rationality as a comprehensive frame for conjecturing and proving in school Bonotto, Cinzia 2-193 Extending students’ understanding of decimal numbers via realistic mathematical modeling and problem posing Borba, Marcelo C. & Zulatto, Rúbia B. A 2-201 Different media, different types of collective work in online continuing teacher education: Would you pass the pen, please? Bosch, Marianna & García, Fco. Javier & Gascón, Josep & Ruiz 2-209 Higueras, Luisa Reformulating “mathematical modelling” in the framework of the Anthropological Theory of Didactics Bragg, Leicha A. 2-217 Students’ impressions of the value of games for the learning of mathematics Breiteig, Trygve & Grevholm, Barbro 2-225 The transition from arithmetic to algebra: to reason, explain, argue, generalize and justify Brodie, Karin 2-233 Resisting reform pedagogy: teacher and learner contributions Brown, Jill P. 2-241 Manifestations of affordances of a technology-rich teaching and learning environment (TRTLE) Bruno, Alicia & Cabrera, Noemí 2-249 Types of representations of the number line in textbooks PME30 — 2006 2-v Campbell, Stephen R. 2-257 Educational Neuroscience: New horizons for research in mathematics education Canada, Daniel L. 2-265 Variability in a probability context: Developing pre-service teachers’ understanding Cavanagh, Michael 2-273 Implementing a reform-oriented mathematics syllabus: A survey of secondary teachers Chaachoua, Hamid & Bittar, Marilena & Nicaud, Jean-François 2-281 Student’s modelling with a lattice of conceptions in the domain of linear equations and inequations Cheng, Ying-Hao & Lin, Fou-Lai 2-289 Using reading and coloring to enhance incomplete prover’s performance in geometry proof Chick, Helen & Baker, Monica & Pham, Thuy & Cheng, Hui 2-297 Aspects of teachers’ pedagogical content knowledge for decimals Chin, Erh-Tsung & Lin, Yung-Chi & Ko, Yann-Tyng & Chien, Chi- 2-305 Tung & Tuan, Hsiao-Lin Collaborative action research on implementing inquiry-based instruction in an eighth grade mathematics class: An alternative mode for mathematics teacher professional development Chiu, Mei-Shiu 2-313 Routine and novel mathematical solutions: Central-cognitive or peripheral-affective participation in mathematics learning Cifarelli, Victor V. & Cai, Jinfa 2-321 The role of self-generated problem posing in mathematics exploration Clarke, Barbara & Clarke, Doug M. & Horne, Marj 2-329 A longitudinal study of children’s mental computation strategies 2-vi PME30 — 2006 Clarke, Doug M. & Sukenik, Michal & Roche, Anne & Mitchell, Annie 2-337 Assessing fraction understanding using task-based interviews Collet, Christina & Bruder, Regina 2-345 Evaluation of a teaching concept for the development of problem solving competences in connection with self-regulation Consogno, Valeria & Gazzolo, Teresa & Boero, Paolo 2-353 Developing probability thinking in primary school: A case study on the constructive role of natural language in classroom discussions Cooper, Tom J. & Baturo, Annette R. & Grant, Edlyn J. 2-361 Collaboration with teachers to improve mathematics learning: Pedagogy at three levels Cooper, Tom J. & Baturo, Annette R. & Warren, Elizabeth & Grant, Edlyn J. 2-369 “Aim High - Beat Yourself”: Effective mathematics teaching in a remote indigenous community Curry, Margaret & Mitchelmore, Michael & Outhred, Lynne 2-377 Development of children’s understanding of length, area, and volume measurement principles Davis, Brent & Simmt, Elaine & Sumara, Dennis 2-385 Mathematics-for-teaching: The cases of multiplication and division Davis, Gary E. & Pearn, Catherine A. 2-393 Generative concept images Davis, Pauline & Pampaka, Maria & Williams, Julian & Wo, Lawrence 2-401 Developmental assessment of data handling performance age 7-14 Deliyianni, Eleni & Michael, Eleni & Pitta-Pantazi, Demetra 2-409 The effect of different teaching tools in overcoming the impact of the intuitive rules Depaepe, Fien & De Corte, Erik & Verschaffel, Lieven 2-417 Investigating social and individual aspects in teacher’s approaches to problem solving PME30 — 2006 2-vii Di Martino, Pietro & Morselli, Francesca 2-425 Maths avoidance and the choice of university Diezmann, Carmel M 2-433 Primary students’ reasoning about diagrams: The building blocks of matrix knowledge Doig, Brian & Williams, Julian & Wo, Lawrence & Pampaka, Maria 2-441 Integrating errors into developmental assessment: ‘Time’ for ages 8-13 Douek, Nadia 2-449 Vygotsky’s everyday concepts/scientific concepts dialectics in school context: A case study Doyle, Katherine 2-457 Creating mathematical models with structure Dreyfus, Tommy & Hadas, Nurit & Hershkowitz, Rina & Schwarz, 2-465 Baruch Mechanisms for consolidating knowledge constructs Drijvers, Paul & Kieran, Carolyn & Boileau, André & Hitt, Fernando & 2-473 Tanguay, Denis & Saldanha, Luis & Guzman, José Reconciling factorizations made with CAS and with paper-and- pencil: The power of confronting two media 2-viii PME30 — 2006 THE ODDS OF UNDERSTANDING THE LAW OF LARGE NUMBERS: A DESIGN FOR GROUNDING INTUITIVE PROBABILITY IN COMBINATORIAL ANALYSIS Dor Abrahamson and Rose M. Cendak University of California, Berkeley Twenty-eight Grade 4 – 6 students participated in 1 hr. clinical interviews in a design-based study that investigated: (a) probability-related intuitions; (b) the affordances of a set of innovative mixed-media learning tools for articulating these intuitions; and (c) the utility of the learning-axes-and-bridging-tools framework supporting diagnosis, design, and data-analysis. Students intuited qualitative predictions of mean and variance, yet only through grounding computer-based simulations of probability experiments in discrete–scalar, non-uniform, multiplicative transformations on a special combinatorial space, the combinations tower, could students articulate their intuitions. We focus on a key learning axis, students’ confusing likelihoods of unique events with those of classes
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