1
INTRODUCTION
1.1 Prologue
Mathematics has been around for at least five thousand years. Throughout its existence, 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 problems 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 âsplendid isolationâ from the surrounding world. On the contrary, there have always been intimate connections between mathematics and other disciplines and fields of practice â oftentimes collectively called âextra-mathematical domainsâ. When mathematics and one or more of these domains meet, the encounter must involve both mathematics 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 mathematics. 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 âcontextâ when appropriate) necessarily happens via an explicit or implicit construction of a mathematical model. Constructing such a model â or, differently put, undertaking mathematical modelling â consists of representing the main elements of a context with mathematical entities and the questions pertaining to the context with mathematical questions. The whole enterprise then consists of seeking answers to mathematical questions and interpreting these answers in terms of the context.
If the application of mathematics to other areas or disciplines is brought about by mathematical modelling, the people using mathematics in such contexts must be able to undertake mathematical modelling. Since an increasing proportion of the population will use mathematics in such contexts, the education system must equip learners with this ability; contributing to making this happen is the driving force behind this book.
1.2 What is this book about?
This book, which forms part of the IMPACT series, is about the learning and teaching of mathematical modelling. It has been written based on the following four observations, most of which will be further explained and elaborated on in various chapters of the book.
The first observation is that all students must learn to put mathematics to use in a wide variety of contexts. It is essential for societies and their citizens that these contexts include everyday practical and leisurely life with family and friends, occupational and professional life in the work force, the civic and societal life of citizens being concerned with culture and society, as well as life in specialised professional fields and academic disciplines that use mathematics to a significant extent. This is the main reason why mathematics is, by far, the worldâs most taught subject, measured by the number of students who study it in primary, secondary or tertiary education or by the number of lesson hours mathematics is taught. Since mathematics is used outside of the discipline itself by way of mathematical models and mathematical modelling, the education system must enable students to work with mathematical models and to undertake mathematical modelling.
The second observation is that mathematical modelling is, cognitively speaking, a difficult and demanding enterprise. Knowledge of and skills in pure mathematics, even if very solid and well founded, are not sufficient prerequisites for students to be(come) able to engage effectively and successfully in modelling activities. Much more is needed. In this book, this observation is further investigated and explained by a broad array of theoretical and empirical considerations.
The third observation is that mathematical modelling can, in fact, be successfully taught to and learnt by students. This requires teaching and learning to take place in environments that are diverse, multi-faceted and activity rich. For that to happen, we need teachers who are mathematically, didactically and pedagogically competent and committed. In particular, teachers need to be well versed in the teaching and learning of mathematical modelling themselves. This observation, too, will be further discussed in theoretical and empirical terms in this book, with particular regard to how the conditions for successful teaching and learning of mathematical modelling can be met.
The fourth and final observation is that, in spite of the previous observation, there is still a variety of strong barriers and challenges to be overcome if we want to ensure models and modelling an appropriate place and role in the teaching and learning of mathematics in ordinary classrooms throughout the world. We can observe, though, that more recent mathematics curricula and standards in several countries (e.g., the USA, Germany, Singapore, China, and Chile, to name only a few) include modelling as a compulsory component. Nevertheless, these barriers and challenges call for further investment of mental and material resources in research and development on teaching and learning mathematical modelling at all education levels.
Against this background, the primary focus of this book will be the teaching and learning of modelling, whereas the teaching and learning of given models and applications of mathematics will be of derived or secondary importance. Our overarching ambition with this book is to provide an up-to-date outline of the state of modelling in mathematics education and to indicate how modelling can be used in educational settings, with an emphasis on the secondary school level. We will take stock of the progress made in research and development, with a glance at selected educational practices in the field. One may describe this book as a research-based introduction to the didactics of mathematical modelling for interested parties who (are to) teach mathematical modelling at schools or universities, including teacher training institutions, or want to engage in professional development activities. It aims to provide important insights into the learning and teaching of mathematical modelling.
The structure of the book is the following. Chapter 2 is devoted to setting the stage of mathematical models and modelling by offering a detailed general conceptual and theoretical framework of the fundamental notions involved in this area of study. The chapter also deals with the cognitive aspects of mathematical modelling and with the role of models and modelling in the education system. In order to to provide flesh and blood to the largely theoretical exposition in Chapter 2, in Chapter 3, we present several modelling examples as a varied source of instantiations of our considerations and expositions in the chapters to come. Even though the examples are presented without specific regard to their actual or potential role in the teaching and learning of mathematics, each of them is accessible to lower or upper secondary students and, of course, to their teachers. Moreover, all of them have, in some form or another, been used in real classrooms. The concepts of modelling competency and (sub-)competencies are introduced and discussed in Chapter 4. Chapter 5 focuses on the challenges and barriers to the inclusion of mathematical modelling in mathematics education that have been encountered in different places. These must be overcome if mathematical modelling is to become a substantive component of the teaching and learning of mathematics. While multiple references to theoretical and empirical research are found in all chapters of the book, Chapter 6 presents a comprehensive survey of empirical research on a variety of key aspects of mathematical modelling in mathematics education. Research and development in the didactics of mathematical modelling are the main focus of this book. However, educational practices of mathematical modelling at different levels must not be excluded from this book, especially since it is still an unusual â if not outright esoteric â topic in many mathematics curricula around the world. Chapter 7 is therefore devoted to presenting a few selected cases of different states of implementation of mathematical modelling in the teaching and learning of mathematics. Finally, Chapter 8 attempts, in a more global way, to take stock of what we know and have accomplished in the field of didactics of mathematical modelling, providing a point of departure for looking into future needs and challenges.
By its very nature, this book is, in large part, an exposition of what already exists and what has been done in the field of mathematical modelling, by others as well as by ourselves. However, in several places, we offer conceptualisations, approaches and perspectives that are new to the field. In addition, we include some novel views on the field. We realise that these are our views and that others working in this field may prefer other takes.
A final remark on the nature of this book in the fauna of other books about mathematics education is warranted. The didactics of mathematical modelling differ from many other sub-fields of mathematics education in that it is not entirely subsumed under the umbrella of mathematics. Extra-mathematical needs, demands, facts, aspects and elements are necessarily present in crucial ways in any kind of mathematical modelling, even in its highly idealised and stylised manifestations. One consequence of this is that the teaching and learning of mathematical modelling unavoidably must transgress the borders of mathematics to move into domains ruled by other sorts of preoccupations, forms of knowledge and methodologies than the ones characteristic of mathematics. Therefore, mathematical modelling cannot be addressed solely by mathematical means, so students and teachers engaging in mathematical modelling must locate, adopt and activate knowledge outside of mathematics as a prerequisite to or a part of their modelling work. For many a student or teacher, this presents major challenges.
1.3 What this book is not about
In mathematics education research and development, it is not unusual to invoke the term âmodellingâ in several ways, many of which have nothing to do with mathematical modelling as the term is understood and used in this book.
The most widespread alternative use of the term âmodellingâ is encountered in attempts to model studentsâ mathematical thinking, mathematical problem solving, mathematical mistakes, mathematical behaviour, etc. In such attempts, âmodellingâ means establishing a conceptual and theoretical framework designed to make sense of studentsâ engagement with mathematics by interpreting their actions, behaviour and statements. Only in the special case of âmodelling studentsâ mathematical modellingâ (see Lesh et al., 2010) is this notion of modelling relevant to this book. In the theory of cognitive apprenticeship (see, e.g., Brown et al., 1989), the first of several phases is also called âmodellingâ. In this phase, the teacher demonstrates, as an expert (a role model), how to tackle a typical task in the topic area that the students are to address. If this task is a modelling task, this means the teacher serves as a model of how to model a situation. Again, this use of âmodelâ and âmodellingâ according to cognitive apprenticeship is not what we are dealing with in this book.
Quite a different notion of âmodellingâ is found outside of mathematics education â in the logical foundations of mathematics and in mathematical logic as a separate topic â in which a concrete mathematical theory is perceived as a realisation â a model â of some abstract axiom system. From this perspective, mathematical modelling is the activity of identifying mathematical entities which display the properties of a given axiom system. The present book does not adopt this perspective.
Similarly, it often happens in work within the discipline of mathematics that one mathematical theory, say linear algebra (or general topology), is introduced to model another mathematical theory, say Euclidean geometry (or real analysis). While such (intra-mathematical) modelling is indeed both very important and highly illuminating, this notion of modelling does not form part of this book. A somewhat analogous situation is found with regard to the notions of horizontal and vertical mathematisation, initially proposed by Treffers in 1978 (Treffers, 1993) and later supported by Freudenthal (1991). Horizontal mathematisation is the process of building a mathematical model of some situation outside of mathematics, which is what we call mathematical modelling in this book. Vertical mathematisation, in contrast, is the process of subjecting a problem formulated within mathematics to internal mathematical treatment in order to solve the problem. In the context of mathematics education, vertical mathematisation is typically introduced once a horizontal mathematisation has been undertaken. In this book, we have not adopted the notion of vertical mathematisation as this term is at odds with key notions in our exposition of mathematical modelling.
References
Brown, J.S., Collins, A. & Duguid, P. (1989). Situated cognition and the culture of learning. In: Educational Researcher 18, 32â42.
Freudenthal, H. (1991). Revisiting Mathematics Education: China Lectures. Dordrecht: Kluwer Academic Publishers.
Lesh, R., Galbraith, P.L., Haines, C.R. & Hurford, A. (Eds.) (2010). Modeling Studentsâ Mathematical Modeling Competencies: ICTMA 13. New York, NY: Springer.
Niss, M. (1994). Mathematics in society. In: R. Biehler, R.W. Scholz, R. Strässer & B. Winkelmann (Eds.), Didactics of Mathematics as a Scientific Discipline (pp. 367â378). Dordrecht: Kluwer Academic Publishers.
Treffers, A. (1993). Wiskobas and Freudenthal: Realistic mathematics education. In: Educational Studies in Mathematics 25, 85â108.
