The Learning and Teaching of Mathematical Modelling

209 Pages·2020·4.584 MB·English

by  Mogens Niss, Werner Blum

THE LEARNING AND TEACHING
OF MATHEMATICAL MODELLING
This book takes stock of the state of affairs of the teaching and learning of
mathematical modelling with regard to research, development and practice. It
provides a conceptual framework for mathematical modelling in mathematics
education at all education levels, as well as the background and resources for teachers
to acquire the knowledge and competencies that will allow them to successfully
include modelling in their teaching, with an emphasis on the secondary school
level. Mathematics teachers, mathematics education researchers and developers will
benefit from this book.
Expertly written and researched, this book includes a comprehensive overview
of research results in the field, an exposition of the educational goals associated with
modelling, the essential components of modelling competency and an extensive
discussion of didacticopedagogical challenges in modelling. Moreover, it offers a
wide variety of illuminating cases and best-practice examples in addition to insights
into the focal points for future research and practice.
The Learning and Teaching of Mathematical Modelling is an invaluable resource for
teachers, researchers, textbook authors, secondary school mathematics teachers,
undergraduate and graduate students of mathematics as well as student teachers.
Mogens Niss is Professor of Mathematics and Mathematics Education at Roskilde
University (RUC), Denmark.
Werner Blum is Professor of Mathematics Education at the University of Kassel,
Germany.
IMPACT: Interweaving Mathematics Pedagogy
and Content for Teaching
The Learning and Teaching of Mathematical Modelling
Mogens Niss and Werner Blum
The Learning and Teaching of Geometry in Secondary Schools
A Modeling Perspective
Pat Herbst, Taro Fujita, Stefan Halverscheid and Michael Weiss
The Learning and Teaching of Algebra
Ideas, Insights and Activities
Abraham Acravi, Paul Drijvers and Kaye Stacey
THE LEARNING AND
TEACHING OF
MATHEMATICAL
MODELLING
Mogens Niss and Werner Blum
First published 2020
by Routledge
2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN
and by Routledge
52 Vanderbilt Avenue, New York, NY 10017
Routledge is an imprint of the Taylor & Francis Group, an informa business
© 2020 Mogens Niss and Werner Blum
The right of Mogens Niss and Werner Blum to be identified as authors of
this work has been asserted by them in accordance with sections 77 and 78
of the Copyright, Designs and Patents Act 1988.
All rights reserved. No part of this book may be reprinted or reproduced or
utilised in any form or by any electronic, mechanical, or other means, now
known or hereafter invented, including photocopying and recording, or in
any information storage or retrieval system, without permission in writing
from the publishers.
Trademark notice: Product or corporate names may be trademarks or
registered trademarks, and are used only for identification and explanation
without intent to infringe.
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging-in-Publication Data
A catalog record for this book has been requested
ISBN: 978-1-138-73067-0 (hbk)
ISBN: 978-1-138-73070-0 (pbk)
ISBN: 978-1-315-18931-4 (ebk)
Typeset in Bembo and Stone Sans
by Apex CoVantage, LLC
CONTENTS
Series foreword vi
Acknowledgements viii
1 Introduction 1
2 Conceptual and theoretical framework – models and
modelling: what and why? 6
3 Modelling examples 35
4 Modelling competency and modelling competencies 76
5 Challenges for the implementation of mathematical modelling 90
6 What we know from empirical research – selected findings
on the teaching and learning of mathematical modelling 111
7 Cases of mathematical modelling from educational practices 145
8 Focal points for the future 189
Index 195
SERIES FOREWORD
IMPACT, an acronym for lnterweaving Mathematics Pedagogy and Content for Teaching,
is a series of textbooks dedicated to mathematics education and suitable for teacher
education. The leading principle of the series is the integration of mathematics
content with topics from research on mathematics learning and teaching. Elements
from the history and the philosophy of mathematics, as well as curricular issues, are
integrated as appropriate.
In mathematics, there are many textbook series representing internationally
accepted canonical curricula, but such a series has so far been lacking in mathemat-
ics education. It is the intention of IMPACT to fill this gap.
The books in the series will focus on fundamental conceptual understanding
of the central ideas and relationships, while often compromising on the breadth
of coverage. These central ideas and relationships will serve as organizers for the
structure of each book. Beyond being an integrated presentation of the central ideas
of mathematics and its learning and teaching, the volumes will serve as guides to
further resources.
We are proud to present a book on the topic of modelling by two renowned
authors. What prompted us to address this area? Only a few years ago, Felix Klein’s
third volume in the Elementarmathematik series – having been published in 1902 for
the first time – appeared in English translation. This volume is – roughly speaking –
devoted to “applied mathematics”. Thus, applied mathematics should be an integral
part of school mathematics, as Klein claimed one hundred years ago.
Working in mathematics is inextricably linked with translating from real-world
contexts into mathematics and back. This is what “Mathematical Modelling”
means, and thus modelling is significant for modern mathematics. Modelling
Series foreword vii
enhances students’ mathematical understanding of reality and supports their learn-
ing of mathematical process competencies.
Series editors
Tommy Dreyfus (Israel), Nathalie M. Sinclair (Canada)
and Günter Törner (Germany)
Series Advisory Board
Abraham Arcavi (Israel), Michèle Artigue (France), Jo Boaler
(USA), Hugh Burkhardt (Great Britain), Willi Dörfler (Austria),
Koeno Gravemeijer (The Netherlands), Angel Gutiérrez (Spain),
Gabriele Kaiser (Germany), Carolyn Kieran (Canada), Jeremy
Kilpatrick (USA), Jürg Kramer (Germany), Frank K. Lester (USA),
Fou-Lai Lin (Republic of China Taiwan), John Monaghan (Great
Britain/ Norway), Mogens Niss (Denmark), Alan H. Schoenfeld
(USA), Peter Sullivan (Australia), Michael 0. Thomas (New
Zealand) and Patrick W. Thompson (USA)
ACKNOWLEDGEMENTS
The authors would like to thank the editors of the IMPACT series, Günter Törner,
Tommy Dreyfus and Nathalie Sinclair, for their continuing support and their con-
structive patience as we worked on this lengthy project. We also want to thank
Hugh Burkhardt and Caroline Yoon for their very helpful, encouraging and con-
structive comments and suggestions concerning the draft manuscript, and Martin
Niss for his careful reading of the revised manuscript and his ensuing suggestions.
Finally, we want to thank numerous colleagues who have provided significant
advice and information to support our writing of the book; no one named, no one
forgotten. Last, but certainly not least, we want to thank our wives, Elke Blum and
Kirsten Niss, for their infinite patience with us during the extended and sometimes
dense periods of hard work over the last several years.
1
INTRODUCTION
1.1 Prologue
Mathematics has been around for at least five thousand years. Throughout its exis-
tence, mathematics has been applied to deal with a host of issues, situations and
phenomena outside of mathematics itself. This fact is reflected in the five-fold
nature of mathematics (Niss, 1994): Mathematics is a fundamental science that deals
with its own internally generated issues; it is an applied science that addresses prob-
lems and questions in scientific disciplines other than mathematics; it is a system
of instruments for practice in culture and society; it is a field of aesthetic expression and
experience; and it is an educational subject with a multitude of different manifestations
that, in various ways, reflect the other four facets of mathematics.
This means that mathematics as a discipline never lived in “splendid isolation”
from the surrounding world. On the contrary, there have always been intimate
connections between mathematics and other disciplines and fields of practice –
oftentimes collectively called “extra-mathematical domains”. When mathematics
and one or more of these domains meet, the encounter must involve both math-
ematics and the domain(s); neither side can be discarded. Sometimes the encounter
is easy and straightforward, e.g., when only counting and elementary arithmetic
are involved. Sometimes it is highly complex and difficult, as when a sophisticated
mathematical theory is brought to bear on a new domain for the first time like for
example in the theory of general relativity.
The purpose of involving mathematics in dealing with situations belonging to
extra-mathematical domains is to help answer questions that arise in such situations.
Perhaps the questions simply cannot be answered without the use of mathemat-
ics. Perhaps they can be answered in a better, faster or easier manner by way of
mathematics. Invoking and activating mathematics to deal with a situation in an
extra-mathematical domain (which for brevity will simply be called a “context”