Table Of ContentInternational Perspectives on the Teaching and
Learning of Mathematical Modelling
Gloria Ann Stillman
Werner Blum
Gabriele Kaiser Editors
Mathematical
Modelling and
Applications
Crossing and Researching Boundaries
in Mathematics Education
International Perspectives on the Teaching
and Learning of Mathematical Modelling
Editorial Board IPTL
Series editors
Gabriele Kaiser, University of Hamburg, Germany
Gloria Ann Stillman, Australian Catholic University, Australia
Editorial board
Maria Salett Biembengut, Pontifical Catholic University of Rio Grande do
Sul - PUCRS, Brazil
Werner Blum, University of Kassel, Germany
Helen Doerr, Syracuse University, USA
Peter Galbraith, University of Queensland, Australia
Toshikazu Ikeda, Yokohoma National University, Japan
Mogens Niss, Roskilde University, Denmark
Jinxing Xie, Tsinghua University, China
This book series will publish various books from different theoretical perspectives
around the world focusing on Teaching and Learning of Mathematical Modelling at
Secondary and Tertiary level. Fully refereed books arising from the biennial
conference called ICTMA, organised by the ICMI affiliated Study Group ICTMA
(International Community of Teachers of Mathematical Modelling and Applications)
will also be published in this series. These books display the worldwide state-of-
the-art in this field and will be of interest for a wider audience than the conference
participants.
ICTMA is a worldwide unique group, in which not only mathematics educators
aiming for education at school level are included but also applied mathematicians
interested in teaching and learning modelling at tertiary level are represented.
ICTMA discusses all aspects related to Teaching and Learning of Mathematical
Modelling at Secondary and Tertiary Level, e.g. usage of technology in modelling,
psychological aspects of modelling and its teaching, modelling competencies,
modelling examples and courses, teacher education and teacher education courses.
More information about this series at http://www.springer.com/series/10093
Gloria Ann Stillman • Werner Blum
Gabriele Kaiser
Editors
Mathematical Modelling
and Applications
Crossing and Researching Boundaries
in Mathematics Education
Editors
Gloria Ann Stillman Werner Blum
Australian Catholic University University of Kassel
Ballarat, VIC, Australia Kassel, Germany
Gabriele Kaiser
University of Hamburg
Hamburg, Germany
ISSN 2211-4920 ISSN 2211-4939 (electronic)
International Perspectives on the Teaching and Learning of Mathematical Modelling
ISBN 978-3-319-62967-4 ISBN 978-3-319-62968-1 (eBook)
https://doi.org/10.1007/978-3-319-62968-1
Library of Congress Control Number: 2017954330
© Springer International Publishing AG 2017
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of
the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microfilms or in any other physical way, and transmission or information
storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology
now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this book
are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the
editors give a warranty, express or implied, with respect to the material contained herein or for any errors
or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims
in published maps and institutional affiliations.
Printed on acid-free paper
This Springer imprint is published by Springer Nature
The registered company is Springer International Publishing AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Series Preface
Applications and modelling and their learning and teaching in schools and universi-
ties have become a prominent topic in the last decades in view of the growing
worldwide relevance of the usage of mathematics in science, technology and every-
day life. There is consensus that modelling should play an important role in math-
ematics education, and the situation in schools and universities is slowly changing
to include real-world aspects, frequently with modelling as real-world problem-
solving, in several educational jurisdictions. Given the worldwide impending short-
age of students who are interested in mathematics and science, it is essential to
discuss accelerating possible changes of mathematics education in school and ter-
tiary education towards the inclusion of real-world examples and the competencies
to use mathematics to solve real-world problems.
This innovative book series established by Springer, International Perspectives
on the Teaching and Learning of Mathematical Modelling, aims at promoting aca-
demic discussion on the teaching and learning of mathematical modelling at various
educational levels all over the world. The series will publish books from different
theoretical perspectives from around the world, dealing with teaching and learning
of mathematical modelling in schooling and at tertiary level. This series will also
enable the International Community of Teachers of Mathematical Modelling and
Applications (ICTMA), an International Commission on Mathematical Instruction-
affiliated study group, to publish books arising from its biennial conference series.
ICTMA is a unique worldwide group where not only mathematics educators deal-
ing with education at school level are included but also applied mathematicians
interested in teaching and learning modelling at tertiary level are represented as
well. Four of these books published by Springer have already appeared.
The planned books will display the worldwide state of the art in this field, most
recent educational research results and new theoretical developments and will be of
interest for a wide audience. Themes dealt with in the books will be teaching and
learning of mathematical modelling in schooling and at tertiary level including the
usage of technology in modelling, psychological, social and cultural aspects of
modelling and its teaching, modelling competencies, curricular aspects, modelling
examples and courses, teacher education and teacher education courses. The book
v
vi Series Preface
series aims to support the discussion on mathematical modelling and its teaching
internationally and will promote the teaching and learning of mathematical model-
ling and research of this field all over the world in schools and universities.
The series is supported by an editorial board of internationally well-known
scholars, who bring in their long experience in the field as well as their expertise to
this series. The members of the editorial board are Maria Salett Biembengut (Brazil),
Werner Blum (Germany), Helen Doerr (USA), Peter Galbraith (Australia),
Toshikazu Ikeda (Japan), Mogens Niss (Denmark) and Jinxing Xie (China).
We hope this book series will inspire readers in the present and the future to
promote the teaching and learning of mathematical modelling all over the world.
Gloria Ann Stillman Gabriele Kaiser
Ballarat, Australia Hamburg, Germany
Contents
1 Crossing Boundaries in Mathematical Modelling and Applications
Educational Research and Practice ....................................................... 1
Gloria Ann Stillman, Werner Blum, and Gabriele Kaiser
Part I N ew Approaches in Research, Teaching and Practice
from Crossing Boundaries
2 Mathematical Modelling with Increasing Learning Aids:
A Video Study .......................................................................................... 25
Deike S. Alfke
3 Modelling with Statistical Data: Characterisation
of Student Models ................................................................................... 37
Àngels Aymerich, Núria Gorgorió, and Lluís Albarracín
4 How Teachers Can Promote Mathematising by Means
of Mathematical City Walks ................................................................... 49
Nils Buchholtz
5 L’Hospital’s Weight Problem: Testing the Boundaries
Between Mathematics and Physics and Between
Application and Modelling ..................................................................... 59
France Caron and Kathleen Pineau
6 Representations of Modelling in Mathematics Education .................. 71
Helen M. Doerr, Jonas B. Ärlebäck, and Morten Misfeldt
7 The Primacy of ‘Noticing’: A Key to Successful Modelling ................ 83
Peter Galbraith, Gloria Ann Stillman, and Jill P. Brown
8 Combining Models Related to Data Distribution
Through Productive Experimentation .................................................. 95
Takashi Kawakami
vii
viii Contents
9 Reconciling Intuitions and Conventional Knowledge:
The Challenge of Teaching and Learning Mathematical
Modelling ................................................................................................. 107
Azita Manouchehri and Stephen T. Lewis
10 A Modelling Perspective in Designing Teacher
Professional Learning Communities ..................................................... 117
Nicholas Mousoulides, Marilena Nicolaidou, and Maria Evagorou
11 Mathematical Modelling and Proof by Recurrence:
An Analysis from a Wittgensteinian Perspective ................................. 129
Bárbara Nilvada Palharini, Emerson Tortola,
and Lourdes Maria Werle de Almeida
12 Quality Criteria for Mathematical Models
in Relation to Models’ Purposes: Their Usefulness
in Engineering Education ....................................................................... 141
Jacob Perrenet, Bert Zwaneveld, Kees van Overveld,
and Tijn Borghuis
13 Ethnomodelling as the Mathematization of Cultural
Practices ................................................................................................... 153
Milton Rosa and Daniel Clark Orey
14 Enabling Anticipation Through Visualisation
in Mathematising Real-W orld Problems in a Flipped
Classroom ................................................................................................ 163
Gloria Ann Stillman
15 Measuring Metacognitive Modelling Competencies ............................ 175
Katrin Vorhölter
Part II R esearching Boundaries in Mathematical
Modelling Education
16 The Mathematical Modelling Competencies Required
for Solving Engineering Statics Assignments ....................................... 189
Burkhard Alpers
17 Pre-service Teachers’ Levels of Reflectivity After Mathematical
Modelling Activities with High School Students .................................. 201
Rita Borromeo Ferri
18 Context and Understanding: The Case of Linear Models ................... 211
Jill P. Brown
19 Difficulties in Teaching Modelling: A French- Spanish
Exploration .............................................................................................. 223
Richard Cabassut and Irene Ferrando
Contents ix
20 How Students Connect Mathematical Models to Descriptions
of Real- World Situations ........................................................................ 233
Dirk De Bock, Nele Veracx, and Wim Van Dooren
21 Mathematical Modelling Strategies and Attitudes
of Third Year Pre- service Teachers ....................................................... 243
Rina Durandt and Gerrie J. Jacobs
22 Exploring the Notion of Mathematical Literacy
in Curricula Documents ......................................................................... 255
Peter Frejd and Vincent Geiger
23 Design and Implementation of a Tool for Analysing
Student Products When They Solve Fermi Problems .......................... 265
César Gallart, Irene Ferrando, Lluís M. García-Raffi,
Lluís Albarracín, and Núria Gorgorió
24 Implementing Modelling into Classrooms: Results
of an Empirical Research Study ............................................................ 277
Jana Kreckler
25 A Commognitive Perspective on Pre-service Secondary
Teachers’ Content Knowledge in Mathematical Modelling ................ 289
Joo Young Park
26 Mathematics Teachers’ Learning at the Boundaries
of Teaching and Workplace .................................................................... 301
Giorgos Psycharis and Despina Potari
27 Case Study of Pre-service Teacher Education
for Mathematical Modelling and Applications
Connecting Paintings with Mathematics .............................................. 313
Akihiko Saeki, Masafumi Kaneko, and Daisuke Saito
28 Inquiry and Modelling in a Real Archaeological Context ................... 325
Gemma Sala Sebastià, Vicenç Font Moll, Joaquim
Giménez Rodríguez, and Berta Barquero Farràs
29 Students’ Overreliance on Linearity in Economic
Applications: A State of the Art ............................................................. 337
Daam Van Reeth and Dirk De Bock
Part III P edagogical Issues for Teachers and Teacher
Educators Using Mathematical Modelling
and Applications
30 Teaching Modelling and Systemic Change ........................................... 351
Hugh Burkhardt and Malcolm Swan
