Broadening the Scope of Research on Mathematical Problem Solving: A Focus on Technology, Creativity and Affect

573 Pages·2018·16.06 MB·English

by  Nélia Amado

Research in Mathematics Education
Series Editors: Jinfa Cai · James A. Middleton
Nélia Amado · Susana Carreira 
Keith Jones Editors
Broadening the
Scope of Research
on Mathematical
Problem Solving
A Focus on Technology, Creativity and
Aff ect
Research in Mathematics Education
Series editors
Jinfa Cai, Newark, DE, USA
James A. Middleton, Tempe, AZ, USA
This series is designed to produce thematic volumes, allowing researchers to access
numerous studies on a theme in a single, peer-reviewed source. Our intent for this
series is to publish the latest research in the field in a timely fashion. This design is
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mathematics education leaders—people who need the highest quality research,
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help them make decisions regarding the improvement of teaching, learning, policy,
and practice. With this vision, our mission of this book series is:
1. To support the sharing of critical research findings among members of the
mathematics education community;
2. To support graduate students and junior faculty and induct them into the research
community by pairing them with senior faculty in the production of the highest
quality peer-reviewed research papers; and
3. To support the usefulness and widespread adoption of research-based innovation.
More information about this series at http://www.springer.com/series/13030
Nélia Amado • Susana Carreira • Keith Jones
Editors
Broadening the Scope
of Research on Mathematical
Problem Solving
A Focus on Technology, Creativity and Affect
Editors
Nélia Amado Susana Carreira
Universidade do Algarve and UIDEF Universidade do Algarve and UIDEF
Instituto de Educação Instituto de Educação
Universidade de Lisboa Universidade de Lisboa
Lisbon, Portugal Lisbon, Portugal
Keith Jones
School of Education
University of Southampton
Southampton, UK
ISSN 2570-4729 ISSN 2570-4737 (electronic)
Research in Mathematics Education
ISBN 978-3-319-99860-2 ISBN 978-3-319-99861-9 (eBook)
https://doi.org/10.1007/978-3-319-99861-9
Library of Congress Control Number: 2018962920
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Preface
This book has its origins in the Problem@Web International Conference, subtitled
Technology, creativity, and affect in mathematical problem solving, which took
place in 2014 in Portugal1 (Carreira, Amado, Jones, & Jacinto, 2014). The confer-
ence was the culmination of the Problem@Web research project conducted in
Portugal (with the support of FCT, the Portuguese research funding agency) and
was also one of the project outcomes (Carreira, Jones, Amado, Jacinto, & Nobre,
2016). The event offered an opportunity to share the project results, as well as to
connect researchers and experts from various parts of the world involved in the
study of problem solving, taking into account the multiple aspects of this area of
study. One major aim of the conference was to assemble emerging and cutting-edge
knowledge on what has been a highly relevant and fundamental subject in
Mathematics Education research for several decades. The awareness that this field
of research is invigorating, and has today new extensions, developments, theoretical
approaches, practical implications, and international relevance, was one of the great
motivations of the Problem@Web conference.
To a certain extent, following up on the developments and challenges in research
on mathematical problem solving was an impetus for the Problem@Web project
that was launched to investigate youngsters’ (aged 10–14) mathematical problem
solving activity in a context beyond school, namely in the sphere of two mathemati-
cal competitions.2 These competitions are inclusive problem solving competitions,
involving the use of digital technologies, bringing families and teachers closer to the
mathematical activity of young people, and generating opportunities for the mani-
festation of mathematical creativity through original approaches to moderate math-
ematical challenges (Carreira et al., 2016).
At the Problem@Web conference, we sought submissions in the three strands of
technology, creativity, and affect in mathematical problem solving, by considering
the school context but also others such as mathematical competitions.
1 http://www.fctec.ualg.pt/problemweb2014/
2 http://fctec.ualg.pt/matematica/5estrelas/
v
vi Preface
At the end of the Problem@Web conference, on a beautiful sunny afternoon in
the Algarve, a range of conference participants gathered to discuss the prospect of
publishing the research that had been shared at the conference in the form of an
English-language edited book. The discussion around that enterprise continued and
evolved in the following months until a general plan was reached. The plan was
based on the idea of broadening the outlines of the research on specific facets of
mathematical problem solving seen as pertinent for advancing the field.
The 25 chapters that compose the book are the result of a hard, prolonged, and
very committed work of a group of authors to whom we, as editors, want to express
our deep thanks. The contributing authors include people that were presenters at the
Problem@Web conference, together with other researchers who were later invited
to join, and who enthusiastically and diligently brought their knowledge, and their
own research, into the whole. The set of manuscripts denotes the inclusive spirit of
this publication rooted in a highly participated conference in Portugal. The book
does this through the diversity of authors’ nations, the participation of young
researchers as coauthors of chapters, as well as through the variety of research per-
spectives and wealth of ideas put into place and discussed.
The fact that we have brought the “footprint” of the Problem@Web conference,
and the driving energy of the Problem@Web research project, into the book justifies
the decision to begin each of the first three parts by a chapter that has its source in
the keynote addresses that were presented at the conference. To complete each of
the first three parts of the book, we invited researchers who are well known for the
excellence of their research, and for their influential work, to act as discussants.
They accepted the challenging task of writing a critical commentary on the set of
chapters in each part. In view of their kind and full dedication to our proposal, and
the results achieved, which add to each of the themes of the book a careful, synthe-
sizing, and insightful reading, we owe words of praise and thanks to Arthur Powell,
Pietro Di Martino, and Roza Leikin.
Finally, we counted on the invaluable collaboration of a well-known author who
agreed to undertake the final part that aims to link, articulate, and integrate the three
topics that the book develops throughout the first three parts. To our colleague
Viktor Freiman, for his dedication, generosity, and commitment, our sincere thanks
for his valuable work.
We wish to emphasize our appreciation for the extensive and continuous collabo-
ration of all the authors and reviewers of the chapters, who patiently responded to
our requests and who strove to ensure that each manuscript reached its final version,
and encouraged us to conclude this project, which, although sometimes delayed by
different contingencies, never ceased to be an ambition of us all.
Our thanks go to the staff at Springer, and the editors of the Research in
Mathematics Education book series, Jinfa Cai and James A. Middleton, who stimu-
lated and supported us and enabled our proposal to go ahead and become this book.
Lisbon, Portugal Nélia Amado
Lisbon, Portugal Susana Carreira
Southampton, UK Keith Jones
Preface vii
References
Carreira, S., Amado, N., Jones, K., & Jacinto, H. (2014). Proceedings of the Problem@Web
International Conference: Technology, creativity and affect in mathematical problem solving.
Faro, Portugal: Universidade do Algarve. http://hdl.handle.net/10400.1/3750.
Carreira, S., Jones, K., Amado, N., Jacinto, H., & Nobre, S. (2016). Youngsters solving math-
ematical problems with technology: The results and implications of the Problem@Web project.
Cham, CH: Springer.
Contents
1 Broadening Research on Mathematical Problem-Solving:
An Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Nélia Amado, Susana Carreira, and Keith Jones
Part I T echnology in Mathematical Problem Solving
2 Different Levels of Sophistication in Solving and Expressing
Mathematical Problems with Digital Tools . . . . . . . . . . . . . . . . . . . . . . . 15
Hélia Jacinto, Sandra Nobre, Susana Carreira, and Nélia Amado
3 Solving Probabilistic Problems with Technologies in Middle
and High School: The French Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Bernard Parzysz
4 High School Teachers’ Use of a Dynamic Geometry System
to Formulate Conjectures and to Transit from Empirical
to Geometric and Algebraic Arguments in Problem-Solving
Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Manuel Santos-Trigo, Matías Camacho-Machín,
and Carmen Olvera-Martínez
5 The Interactive Whiteboard and the Development of Dialogic
Interaction in the Context of Problem-Solving . . . . . . . . . . . . . . . . . . . 101
Ana Paula Canavarro and Carla Sofia Pereira Reis
6 Learning Scenarios with Robots Leading to Problem-Solving
and Mathematics Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Elsa Fernandes, Paula Lopes, and Sónia Martins
7 Developing the Mathematical Eye Through Problem-Solving
in a Dynamic Geometry Environment . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Maria Alessandra Mariotti and Anna Baccaglini-Frank
ix
x Contents
8 Reflecting on Digital Technology in Mathematical
Problem-Solving Situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Arthur B. Powell
Part II C reativity in Mathematical Problem Solving
9 Mathematical Problem Solving Beyond School: A Tool
for Highlighting Creativity in Children’s Solutions . . . . . . . . . . . . . . . . 187
Susana Carreira and Nuno Amaral
10 Solving a Task with Infinitely Many Solutions: Convergent
and Divergent Thinking in Mathematical Creativity . . . . . . . . . . . . . . 219
Michal Tabach and Esther Levenson
11 The Power of Seeing in Problem Solving and Creativity:
An Issue Under Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Isabel Vale, Teresa Pimentel, and Ana Barbosa
12 Creativity and Problem Solving with Early Childhood Future
Teachers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Yuly Vanegas and Joaquín Giménez
13 Stimulating Mathematical Creativity through Constraints
in Problem-Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
Christian Bokhove and Keith Jones
14 Linking Mathematical Creativity to Problem Solving: Views
from the Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
Deborah Moore-Russo and Erica L. Demler
15 Problem-Solving and Mathematical Research Projects: Creative
Processes, Actions, and Mediations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
Inés M. Gómez-Chacón and Constantino de la Fuente
16 Mathematics Education and Creativity: A Point of View
from the Systems Perspective on Creativity . . . . . . . . . . . . . . . . . . . . . . 375
Cleyton Hércules Gontijo
17 Openness and Constraints Associated with Creativity-Directed
Activities in Mathematics for All Students . . . . . . . . . . . . . . . . . . . . . . . 387
Roza Leikin
Part III A ffect and Aesthetics in Mathematical Problem Solving
18 Students’ Attitudes in a Mathematical Problem-Solving
Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
Nélia Amado and Susana Carreira