Challenges in finding suitable materials
One explanation of the absence of modelling in everyday classrooms often put forward is the lack of appropriate teaching materials. This explanation no longer seems valid considering the multitude of existing materials. In the last three decades, a plethora of valuable and feasible teaching materials (books, collections of examples, reports on experiences) have been published in many countries, including Australia, Denmark, Germany, Japan, the UK, and the USA. In the following section, we will refer to some of these materials.
Many examples of modelling tasks and learning environments for modelling can be found in the series of Proceedings of the ICTMA conferences. Under www. ictma.net/literature.html, all ICTMA books are listed. The organisation COMAP (the Consortium for Mathematics and its Applications, also see section 7.4) has produced, in the last four decades, materials containing modelling examples; see www.comap.com. We mention here the book For All Practical Purposes (the last edition of which is COMAP, 2013) and the GAIMME Report, produced in collaboration with SIAM (COMAP, 2016). The GAIMME Report contains a wide range of modelling examples from elementary to upper secondary level and from word problems to modelling projects; several examples are accompanied by detailed hints concerning how these examples can be used in the classroom. Since 1999, COMAP has organised two annual international modelling competitions for teams of college students, the Mathematical Contest in Modeling (MCM) and the Interdisciplinary Contest in Modeling (ICM). The modelling problems as well as possible solutions are available at www.comap.com/undergraduate/contest. Since 2015, COMAP, in cooperation with NeoUnion, has also organised an international modelling contest for teams of upper secondary school students, the International Mathematical Modeling Challenge (IM2C). The modelling problems and possible solutions are available at www.immchallenge.org. The Australian branch of this contest has established its own website (www.immchallenge.org.au), which also contains the book by Galbraith and Holton (2018); also see section 7.3.
The Shell Centre for Mathematical Education at Nottingham University (also see section 7.4) has developed, over the last five decades, a multitude of modelling materials for the lower secondary level (for an overview, see Burkhardt, 2018). These materials comprise modules for modelling projects where students are expected to work in groups for several hours, developed by Malcolm Swan and others in the Numeracy Through Problem Solving project: “Design a board game”, “Produce a quiz show”, “Plan a trip”, “Be a paper engineer”, and “Be a shrewd chooser”. In the Bowland Mathematics project (see www.bowlandmaths.org.uk), the Shell team developed modelling units expected to take four to five hours, such as “Reducing road accidents” and “How risky is life”. In the Mathematics Assessment project, a collaboration between the Shell Centre and the University of California at Berkeley, the modelling lesson materials supported formative assessment for learning (see www.map.mathshell.org/lessons.php and also cf. section 5.5).
Another rich source of modelling examples is the collection of modelling activities developed by Lesh and his group within the framework of the “models and modelling perspective on teaching, learning, and problem solving” (see Lesh & Doerr, 2003, and section 6.9). Although primarily developed as a research tool, the model development sequences of this project can also be used for classrooms ranging from primary to upper secondary school levels, as well as for teacher education. A typical sequence consists of three parts: a “Model Eliciting Activity” (MEA, see section 2.7), followed by a “Model Exploration Activity” (MXA), and finally a “Model Application Activity” (MAA). Teaching methods for MEAs are described in detail in Moore et al. (2018); for more information on the design principles for these activities, see Brady et al. (2018).
Especially for the German speaking world, the series of books Materialien fur einen realitdtsbezogenen Mathematikunterricht (Materials for Reality-Oriented Mathematics Teaching), edited by the so-called German Istron group, presents a wealth of tried and tested modelling examples for all school grades, with an emphasis on the lower secondary level. Since 1993, 24 volumes have been published (see www.istron.mathematik.uni-wuerzburg.de/istron/index.html@p=1033.html for an overview). Particularly interesting is a recent volume (Siller et al., 2018), which is a “best of” collection from all previous Istron books (for details of the group and especially the “best of” volume, see section 7.4).
However, one issue is the availability of resource materials that teachers can find if they search for it. Another issue is the presence or absence of sections and tasks on modelling in prevalent textbooks. In this respect, the situation is very diverse within and across countries. Since it is well known from research that textbooks constitute the predominant teaching materials upon which teachers base their teaching, it is evident that textbooks without suitable modelling sections and tasks generate marked challenges to the implementation of modelling work in everyday teaching.
Another challenge is rooted in the fact that the modelling examples, tasks, units and materials found in different types of literature display a wide range of content and quality. If a teacher looks for modelling material suitable for a certain lesson or teaching unit, it may be difficult to find something that is appropriate with respect to the mathematical topics and the extra-mathematical contexts involved (if it is desirable at all to match curricular topics and modelling topics closely, see section 5.2). The contexts and the problems considered therein may range from purely dressed-up to authentic. Burkhardt (1981) classifies modelling problems as Action, Believable, Curious, Dubious and Educational. The notion of an “authentic problem” may have several different meanings (see Palm, 2002). The strictest meaning is that the context and the problem to be dealt with come directly from a genuine field of practice in industry, business, science, society or everyday life. There are certainly examples for all levels that satisfy this demand (see Kaiser et al., 2013). However, this demand is often too strict for educational purposes since authentic problems tend to go beyond the reach of school mathematics either in terms of the mathematics involved (be it differential or difference equations, functions of several variables or advanced probability distributions or stochastic processes, or discrete mathematics) or in terms of the knowledge needed from other fields (such as physics, biology, engineering or economics). A less strict notion of authenticity (see Niss, 1992; Palm, 2002, 2007; Vos, 2011,2015) requires the contexts and problems to be constructed in such a way that they might occur in real practice and people from the practice area find them credible, albeit simplified. Crucial notions here are honesty and credibility (Carreira & Baioa, 2018), that is, the teacher ought to make it clear to the students in what respects a context or problem is not authentic in the strict sense but that it deserves to be taken seriously nevertheless. According to Vos (2018), criteria for authenticity in an educational context are: an out-of-school origin and a certification of originality (by suitable artefacts or by expert testimony). However, inauthentic contexts may in fact be suitable for educational purposes, depending on the teaching and learning goals (see section 2.7). For instance, when motivating the study or practise of certain mathematical topics is the primary aim, dressed-up problems might very well be appropriate, and the same holds when specific sub-competencies of modelling are to be practised or drilled. The question of how close a task context is to reality also has a subjective aspect. Students might hold a rather narrow view of authenticity, oriented towards their own personal life, and at the same time be more generous about the demand for authenticity since they know that instruction has multiple purposes and school is different from real life.
The more general question behind these considerations about the authenticity of tasks is what a “good” modelling task should look like. As has been said, this depends strongly on the goals the teacher wants to pursue by a given task. However, there are certain criteria that can be applied to judge whether a given task is suitable for the intended purposes. These criteria comprise, among others, the following (see MaaB, 2010; Borromeo Ferri, 2017):
- 1 Degree of precision or openness of the task question: Does the formulation of the question itself suggest a solution approach, or does a more precise question have to be developed during the solution process?
- 2 The kind of information given: Does the task contain more or less information than is necessary for dealing with it?
- 3 Complexity of the question: Is an approach immediately recognisable, and will one loop in the modelling cycle be sufficient to arrive at a satisfactory solution, or are different approaches, or several loops, likely to be necessary?
- 4 The kind of real-world context: Is the context accessible and understandable by the students dealing with the task? Is the context credible, and is it relevant for students’ lives?
- 5 Extent and level of the mathematical content: What kind of mathematics is suitable or necessary for solving the task? Is this mathematics accessible to the students?
