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Proceedings of the Problem@Web International Conference: Technology, creativity and affect in mathematical problem solving
Editors Susana Carreira, Nélia Amado, Keith Jones, and Hélia Jacinto
Copyrights © 2014 left to the authors All rights reserved May, 2014 Faro, Universidade do Algarve ISBN 978-989-8472-50-2 Cover Photo: Vilamoura Marina (Courtesy of the Municipality of Loulé) Cite as: Carreira, S., Amado, N., Jones, K., & Jacinto, H. (Eds.) (2014). Proceedings of the Problem@Web International Conference: Technology, creativity and affect in mathematical problem solving. Faro, Portugal: Universidade do Algarve. Proceedings available for download at: http://hdl.handle.net/10400.1/3750.
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TABLE OF CONTENTS Acknowledgements ………………………………………………………………………..
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Organizing Committee ……………………………………………………………………
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International Scientific Committee ………………………………………………………
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The Problem@Web Project: An overview ……………………………………………….
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Opening Note from the Editors ………………………...………………………………… Susana Carreira, Nélia Amado, Keith Jones, Hélia Jacinto
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PLENARY SESSIONS
LECTURES Math problem, Internet and digital mathematical performance ……………………… Marcelo Borba
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Roles of aesthetics and affect in mathematical problem solving …………………….... Norma Presmeg
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The curriculum, creativity and mathematical competitions ………………………….. Jaime Carvalho e Silva
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Multiple solution problems in computerized and non-computerized environments: what difference does it make? ………………………………………………………….... Michal Tabach
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PANEL Present and future roles of online math competitions in the education of twenty-first century young students ………………………………………………………………...…. João Pedro da Ponte, Keith Jones
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KEYNOTE ADDRESSES Strand 1 – Technology: strategies and representations used in technology-based problem solving approaches The use of digital tools in web-based mathematical competitions: degrees of sophistication in problem solving-and-expressing ………………………………….. Hélia Jacinto, Sandra Nobre, Susana Carreira, Nélia Amado
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Strand 2 – Creativity: students’ mathematical creativity in mathematical problem solving Highlighting creativity in children’s beyond-school mathematical problem solving Nuno Amaral, Susana Carreira
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Strand 3 – Affect: attitudes and emotions of students, parents and teachers regarding mathematical problem solving The affective relationship of youngsters and parents with mathematics and problem solving in inclusive mathematical competitions …………………………. Nélia Amado, Susana Carreira, Eugénia Castela, Rosa Antónia Tomás Ferreira
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PARALLEL SESSIONS
RESEARCH PAPERS Strand 1 – Technology: strategies and representations used in technology-based problem solving approaches Digital storytelling for improving mathematical literacy ………………………….. Giovannina Albano, Anna Pierri
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Using statistical software in basic education: difficulties and affordances ……….. Cláudia Fernandes, João Pedro da Ponte
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Correlation between different learning styles and the use of wiki in learning …… Nashwa Ismail, Julie-Ann Edwards, Gary Kinchin
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The role of peer and computer feedback in student’s problem solving …………... Júlio Paiva, Nélia Amado, Susana Carreira
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Simulating random experiments with computers in high school level teaching …. Bernard Parzysz
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Electronic forums: an added value when solving initial algebraic problems ……. M. Pilar Royo, Joaquin Giménez
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Preservice high school teachers’ construction and exploration of dynamic models of variation phenomena ……………………………………………………………… Manuel Santos-Trigo, Matías Camacho-Machín, Carmen Olvera-Martínez
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The role of using technology and creativity in developing positive dispositions toward mathematical problem solving ……………………………………………… Mehmet Türegün, Luis Conde
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Strand 2 – Creativity: students’ mathematical creativity in mathematical problem solving
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How to foster creativity in problem posing and problem solving activities ……… Cinzia Bonotto, Lisa Dal Santo
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Mathematical problems in basic education ………………………………………… Virginia Furlanetto, Maria Madalena Dullius, Italo Gabriel Neide
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Dimensions of mathematically creative processes in early childhood …………….. Melanie Münz
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Creative advantages of visual solutions to some non-routine mathematical problems ………………………………………………………………………………. Norma Presmeg
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The borrowers: using transportation, addresses, and paralelepípedos to prompt creativity using Ethnomodeling …………………………………………………….. Daniel Clark Orey, Milton Rosa
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Creativity in mathematics class: how can it emerge? ……………………………… Sandra Pinheiro, Isabel Vale
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Strategies used by elementary grade students in Mathematics Olympiad tests …. Márcia Jussara Hepp Rehfeldt, Marli Teresinha Quartieri, Maria Madalena Dullius
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Writing problem stories: developing creativity through the integration of mathematics and language …………………………………………………………... Fátima Sardinha, Pedro Palhares, Fernando Azevedo
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Revealing the inner connections of math using a clock puzzle ……………………. Sabine Stöcker-Segre
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Mathematical creativity through the eyes of future teachers ……………………... Isabel Vale, Ana Barbosa, Lina Fonseca
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Noticing creativity issues when future teachers analyzing problem solving dialogues ………………………………………………………………………………. Yuly Marsela Vanegas, Joaquin Giménez
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Strand 3 – Affect: attitudes and emotions of students, parents and teachers regarding mathematical problem solving Practice in the mathematical modelling environment ……………………………... Bárbara Cândido Braz, Lilian Akemi Kato
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Cognitive scaffolding for problem solving: use of the practical worksheet ………. Jaguthsing Dindyal, Eng Guan Tay, Khiok Seng Quek, Yew Hoong Leong, Tin Lam Toh, Pee Choon Toh, Foo Him Ho
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Affective issues in solving challenging mathematical problems within an inclusive competition ……………………………………………………………………………. Rosa A. Tomás Ferreira, Susana Carreira, Nélia Amado
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Identifying cognitive-affective interaction phenomena in a technological mathematical context ………………………………………………………………… Inés M. Gómez-Chacón
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An exploratory study on using the think-pair-share cooperative learning strategy for students to solve mathematics problems in a Hong Kong primary school …… Andrew P. Kwok, Alexandria Lau
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A mathematical congress: a window to affect in problem solving ………………... Teresa Pimentel, Isabel Vale
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Teachers’ involvement and learning in a lesson study …………………………….. João Pedro da Ponte, Marisa Quaresma, Mónica Baptista, Joana Mata-Pereira
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E-POSTERS Teaching and learning mathematics for creativity through challenging tasks …........... Ana Barbosa, Isabel Vale, Teresa Pimentel
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Developing positive dispositions toward mathematical problem solving ……………… Luis Conde, Mehmet Türegün
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Posing and solving problems in mathematical modeling scenarios ……………………. Cristina Esteley, María Mina, Mónica Villarreal
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DROIDE II – Robots in mathematics and informatics education – final results of the project ……………………………………………………………………………………... Elsa Fernandes
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Problem solving: carving out spaces for creativity, communication, and perseverance Deborah Moore-Russo, Anne Izydorczak, Harry T. O’Malley Proof and creativity in a geometrical problem from a regional Mathematical Olympiad ………………………………………………………………………………….. Juan Rodriguez, Nélia Amado, Susana Carreira, Sílvia Reis
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Information environment for mathematics education based on relationship development theory ……………………………………………………………………….. Karasawa Toshimitsu
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Problem solving in modeling scenarios with experimental activities: the role of information and communication technologies …………………………….…………….. Mónica Villarreal, María Mina, Cristina Esteley
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Reviewers …………………………………………………………………………………..
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List of participants ………………………………………………………………………...
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Sponsors ……………………………………………………………………………………
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ACKNOWLEDGEMENTS This Conference was kindly supported by several sponsors and organizations. We wish to express our special thanks to:
Câmara Municipal de Loulé Hotel Dom Pedro Golf Springer Dismel Texas Instruments Pelcor Raiz Editora Associação de Professores de Matemática (APM) Fundação para a Ciência e a Tecnologia (FCT) Região de Turismo do Algarve
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ORGANIZING COMMITTEE Nélia Amado [Chair], Faculty of Sciences and Technology, University of Algarve, Portugal Hélia Jacinto, Research Unit of the Institute of Education of the University of Lisbon, Portugal Isa Martins, Pinheiro e Rosa Schools Grouping, Faro, Portugal Jaime Carvalho e Silva, Faculty of Sciences and Technology, University of Coimbra, Portugal Juan Rodriguez, Faculty of Sciences and Technology, University of Algarve, Portugal Nuno Amaral, Basic School 2,3 Sophia de Mello Breyner Andresen, Lisbon, Portugal Rosa Antónia Tomás Ferreira, Faculty of Sciences, University of Porto, Portugal Sandra Nobre, Research Unit of the Institute of Education of the University of Lisbon, Portugal Sílvia Reis, Schools Grouping of Valpaços, Portugal Susana Carreira, Faculty of Sciences and Technology, University of Algarve, Portugal
INTERNATIONAL SCIENTIFIC COMMITTEE Keith Jones [Co-chair], University of Southampton, United Kingdom Susana Carreira [Co-chair], Faculty of Sciences and Technology, University of Algarve, Portugal Guida de Abreu, Oxford Brookes University, United Kingdom Inés Gómez-Chacón, Faculty of Mathematics, Complutense University of Madrid, Spain Isabel Vale, School of Education, Polytechnic Institute of Viana do Castelo, Portugal Jaime Carvalho e Silva, Faculty of Sciences and Technology, University of Coimbra, Portugal Joaquin Giménez, University of Barcelona, Spain João Filipe Matos, Institute of Education, University of Lisbon, Portugal João Pedro da Ponte, Institute of Education, University of Lisbon, Portugal Marcelo Borba, State University of São Paulo, Brazil Matías Camacho-Machín, University of la Laguna, Spain Michal Tabach, Tel Aviv University, Israel Nélia Amado, Faculty of Sciences and Technology, University of Algarve, Portugal Norma Presmeg, Illinois State University (Professor Emerita), United States of America Rosa Antónia Tomás Ferreira, Faculty of Sciences, University of Porto, Portugal
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THE PROBLEM@WEB PROJECT: AN OVERVIEW The Problem@Web International Conference is a satellite conference of a research project running in Portugal between December 2010 and June 2014, jointly developed by the Institute of Education of the University of Lisbon and by the University of Algarve. As the completion of the Problem@Web Projecti approaches, the research team and the project external consultant (Prof. Keith Jones) took upon themselves the task of promoting a conference aiming to present and disseminate the main results of the research undertaken within the project and to create the opportunity of sharing and discussing neighbouring perspectives and ideas from scholars and researchers in the international field. The project was launched to embrace the opportunity of studying mathematical problem solving beyond the mathematics classroom, by looking at the context of two web-based inclusive mathematical competitions, running in the south of Portugal – Sub12® and Sub14®. By addressing such a rich multi-faceted environment, the project intended to explore, in an integrated manner, issues that combine cognitive, affective and social aspects of the problem solving activity of young students. Therefore, the field of research was clearly based on inclusive mathematical competitions that occur mainly through the Internet, involving a clear digital communicational activity and inducing strong resonance with the homes and lives of students. The two competitions, Sub12 and Sub14, have been running since 2005, promoted by the Mathematics Department of the Faculty of Sciences and Technology of the University of Algarve. The Sub12 addresses students in 5th and 6th grades (10-12 year-olds) and the Sub14 addresses students in 7th and 8th grades (12-14 year-olds). The two competitions are web-based, located in the same website (http://fctec.ualg.pt/matematica/5estrelas/), have similar rules and operate in parallel. They involve two distinct phases: the Qualifying and the Final. The Qualifying phase develops entirely at distance through the website and consists of a set of ten problems each posted every two weeks. During the Qualifying phase competitors may participate individually or in small teams of two or three elements. They have to send their answers to the given problems by e-mail or through the digital form available at the website that allows attaching files. The answers to the problems are received in e-mail accounts specifically devoted to that purpose and the organization replies to every participant, with formative feedback, suggestions, clues or appraisal. Participants are allowed to resubmit revised solutions as many times as needed within the respective deadline. The Final is a half-day on-site contest held at the university campus with the presence of the finalists, their families and teachers. At the Final, participants are given a set of five problems to be solved in limited time with paper and pencil. Everyone is competing individually and there is no technology available. The written answers to the problems are corrected anonymously by a jury. In the meantime parents, teachers ix
and other accompanying guests have a program devoted to them (usually a seminar or a workshop about mathematical ideas, especially prepared for that occasion). The Final culminates with the awarding ceremony of the three winners who receive prizes and honour diplomas. Throughout the history of this competition a number of distinctive characteristics have been standing out: i) it proposes non-routine word problems, usually allowing several ways to be solved; ii) problems are not intended to fit any particular school curricular topic; iii) the main trend is on moderate mathematical challenges; iv) the competition explicitly requires participants to expose the process to find the solution; v) it is close to teachers and families in the sense that it encourages their support to the young participants; vi) opportunities for reformulating and resubmitting answers are offered to all participants; vii) all types of media to find and develop solutions are welcome; viii) communication and interaction is carried out through digital web-based and email infrastructures; ix) interesting and diverse proposed solutions are published on the competition website; x) the competitive component is concentrated on the Final phase of the competition. Based upon this context, the Problem@Web project has defined three research foci: (a) Ways of thinking and strategies in mathematical problem solving, forms of representation and expression of mathematical thinking, and technology-supported problem solving approaches; (b) Beliefs, attitudes and emotions related to mathematics and mathematical problem solving, both in school and beyond school, considering students, parents and teachers. (c) Creativity manifested in the expression of mathematical solutions to problems and its relation to the use of digital technologies. The empirical research involved two main approaches – extensive and intensive – and the data analysis combined quantitative and qualitative methods. The extensive approach drew on the following data: a database of participants’ digital productions in three editions of the competitions (emails and attached files); the finalists’ written solutions collected in the Finals; the answers to a mini-questionnaire about each proposed problem during the Qualifying phase (online); and an online survey to all students from the Algarve (online). The intensive approach aimed to in-depth study and the collected data included: interviews with participants (including former participants some years after leaving the competitions), interviews with parents (or family members) of participants; interviews with teachers who have students participating; participant observation and video-recording of the Finals; participant observation in regular classes where problems from the competitions were proposed; ethnographic observation of participants in their home environment. The project has implemented qualitative and quantitative data analysis methods, namely content analysis, case study research, statistical descriptive analysis, and multivariate statistical analysis.
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For the theoretical framework particularly relevant theories were those that suggest and endorse the expression of mathematical thinking as an integral component of the problem solving process. Therefore, our research has elected solving-and-expressing as a central unit of the problem solving activity. Moreover, as our aim is to study students’ problem solving with the digital technologies of their choice, either at school or at home, the project pays particular attention to the expression of students’ mathematical ideas mediated by digital tools. In particular, the interweaving of technological fluency with mathematical knowledge (and problem solving ability) has motivated the study of participants’ “techno-mathematical fluency”. In connection to the challenging and inclusive character of the competition, the project has considered some affective aspects surrounding it. The concept of inclusive competition, rather than the competitive element, and the idea of challenging mathematical problem of moderate degree are key notions that structure the theoretical view on the research strand devoted to affect. Parental involvement, help-seeking in problem solving, perceived difficulty in tackling the problems from the competition, along with attitudes, beliefs and emotions of participants, their parents and teachers, concerning school and beyond-school mathematics, the participation in the competitions, and problem solving in general are strategic ideas in the research developed. Finally, some characteristics of the way in which the competitions work, among which stands out an extended time for the submission of answers to the problems, the possibility of using all available resources and the voluntary nature of participation, are factors that promote the emergence of mathematical creativity. Our option in researching students’ mathematical creativity in problem solving is to divide attention between the mathematical representations and the strategies devised by the participants. Considering the final quality of a product as one of the ways in which creativity can be approached both in mathematics as in other areas, we have examined students’ creative solutions to problems by proposing analytical forms of consensual assessment of creativity and by putting emphasis on an inclusive view of creativity that accounts for novelty and usefulness in mathematical problem solving. The Problem@Web Project has so far attained a significant number of interventions in research conferences as well as several national and international publications. It is our intention to develop the work produced so far and to strengthen it through the contribution of a wider community, namely in the form of an international conference devoted to the three strands that are the pillars of the very existence of the project. Susana Carreira (Principal Investigator) NOTES i
Acronym for the project Mathematical Problem Solving: Views on an interactive web based competition (Sub12 & Sub14), funded by Fundação para a Ciência e Tecnologia under the grant PTDC/CPE-CED/101635/2008.
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OPENING NOTE FROM THE EDITORS The Problem@Web International Conference is an opportunity to gather together participants from the research community in mathematics education as well as from related research areas such as the psychology of education, technology education, mathematics popularization and other relevant fields that converge to the study of mathematical problem solving in different educational environments in the twenty-first century. The conference has elected three major topics presented as strands, technology, creativity, and affect, within which pedagogical and research perspectives were anticipated. The strands thus represent key issues that are crucially embedded in the activity of problem solving, whether in teaching or learning mathematics, both within the school and beyond the school. Technology, creativity, and affect in mathematical problem solving were the main focuses of a research project conducted in Portugal over the past three years, under a grant from the Portuguese funding agency, Fundação para a Ciência e Tecnologia. This research project, in devoting particular attention to web-based mathematical competitions, in line with the growing acceptance and importance attributed to this type of initiative all over the world, set itself the goal of sharing results, ideas, conclusions and developments achieved along its course, through a scientific meeting with interested colleagues from other countries. The scientific programme thus gravitates around the three thematic strands, offering four plenary talks given by prominent researchers from different parts of the world and three keynote addresses proposed by members of the Portuguese Problem@Web research team followed by invited reactors’ comments, but it simultaneously took in mind to encourage and welcome submissions of research papers and e-posters. Our sincere thanks are due to the plenary speakers and to the invited reactors who generously accepted to enrich the conference with their insightful inputs. It is our pleasure to have reached the final program with 27 peer-reviewed research papers and 8 e-posters, distributed across the three strands of the conference, attaining a good solid body of work that filled the three days of the conference. We want to compliment all the authors for their relevant perspectives, and we are in debt to all the reviewers who kindly and thoroughly collaborated in revising and improving the proposals, under a process of open peer-reviewing. All this contributed to the high quality of the final published papers in the conference proceedings. The conference proceedings are offering an overview of all the research presented in the form of plenary sessions, research papers and e-posters. This conference aims to be the seed of future developments, in particular, allowing the gathering of researchers who join and gain momentum for the preparation of a post-conference book that pushes research forward on issues discussed during the three days of work and debate. Therefore the plenary sessions, including both the lectures from invited authors and the keynote addresses in each of the conference strands, were seen as starting points for 1
the development of further publishable material and their abstracts are offered in the conference proceedings as the authors’ approaches to the conference key issues. The research papers are fully published in the proceedings and represent a substantial part of its content, organized under the three strands of the conference, according to the authors’ own intention of inclusion of their work in a particular strand. Moreover, the e-posters are summarized as short one-page articles and also connect to one or more of the conference strands. Altogether, these proceedings combine multiple perspectives and ways of addressing and relating technology, creativity and affect, three promising development directions in the current research on mathematical problem solving. One last word of recognition is due to the University of Algarve and to the Institute of Education of the University of Lisbon, which jointly promoted this event, contributing with their invaluable support to the organization. Susana Carreira, Nélia Amado, Keith Jones, and Hélia Jacinto
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