THE TEACHING AND LEARNING MATHEMATICAL MODELLING

We have a large and growing collection of didactical research on mathematical modelling. Moreover this research even seems to have had a serious impact on the practices of mathematics teaching at least on curricula level. During the last couple of decades the introduction of mathematical modelling and applications

is probably - together with the introduction of information technology - the most prominent common features in mathematics curricula reforms around the world (Kaiser, Blomhøj and Sriraman, 2006, p. 82). Curricula reforms in many western countries, especially at secondary level have emphasised mathematical modelling as an important element in an up-to date mathematics secondary curricula preparing generally for further education. Didactical research has undoubtedly played an important role in this development. The fundamental goals in the teaching of mathematical modelling and the reasons for pursuing these goals developed and analysed in research can be pinpointed in the guidelines for mathematics teaching in many countries. Also, the general understanding of the model concept and of a modelling process expressed in many mathematics curricula is clearly influenced by didactical research. (Blum et al, 2007) and (Haines et al, 2006).

However, the way mathematical modelling and applications is organised in curricula and, especially, how these parts of the curricula are assessed reveal only a very limited influence from research. And when it comes to the level of teaching practice in the classroom it is still a pending question to which degree the many developmental modelling projects carried out and analysed in research have actually influenced the practices of teaching mathematical modelling. Influencing practices of mathematics teaching are not the only criteria for

progress in the didactical research on mathematical modelling. It is also relevant to try to evaluate the coherency of the theories developed. In the editorial Towards a didactical theory for mathematical modelling of ZDM (2, vol. 38), we argued that at a general level it is possible to identify in the field of research

 ... a global theory for teaching and learning mathematical modelling, in the sense of a system of connected viewpoints covering all didactical levels: learning goals, fundamental reasons for pursuing these goals at different levels of the educational systems, tested ideas about how to support teachers in implementing learning goals and recognised didactical challenges and dilemmas related to different ways of organising the teaching, theoretically and empirically based analyses of learning difficulties connected to modelling, and ideas about different ways of assessing students’ learning in modelling activities and related pitfalls. (Kaiser, Blomhøj and Sriraman, 2006, p. 82)

However, this “global theory” is not based on a single strong research paradigm. On the contrary, in fact, it is possible to identify a number of different approaches and perspectives in mathematics education research on the teaching and learning of modelling. This is, precisely, the reason for choosing Conceptualizations of mathematical modelling in different theoretical frameworks and for different purposes as one of the themes for the Mathematical applications and modelling in the teaching and learning of mathematics . We intended to provide a background for in-depth discussions of the theoretical basis of the different approaches within the field. Kaiser & Sriraman (2006) report about the historical development of different research perspectives and identify seven main perspectives describing the current trends in the research field.

These perspectives may have overlaps and also they do not necessarily cover the entire research area. Nevertheless, they all represent distinctive perspectives of research on the teaching and learning of mathematical modelling, and they have been developed in particular research milieus over a long period of time and all of them have produced a considerable number of research publications. The main rationale for developing a categorisation of research perspectives is of course to deepen our mutual understanding of the individual perspective and to recognise similarities and differences amongst these. The idea is not to try to judge about their relevance or their relative importance.

The main idea of the educational perspective is to integrate models and modelling in the teaching of mathematics both as means for learning mathematics and as an important competency in its own right. Accordingly classical didactical questions about educational goals and related justifications for teaching mathematical modelling at various levels and branches of the educational system, ways to organize mathematical modelling activities in different types of mathematics curricula, problems related to the implementation of modelling in school culture and teaching practices, and problems related to assessing the

students’ modelling activities are all been addressed under this research perspective.

Niss (1987, 1989) and Blum & Niss (1991) are classical references to this research perspective, which has been quite dominant in Western Europe in the last three decades. Defining and discussing the basic notions in the field – such as: model, modelling, the modelling cycle or modelling cycles, modelling applications and competency– and the meaning of these notions in relation to mathematics teaching at different educational levels is an important element in the research under the educational perspective. The introduction to the study volume gives an overview of the concept clarifications and the history of the field. (Niss, Blum & Galbraith, 2007) In my interpretation (Blomhøj, 2004), the three main arguments for teaching mathematical modelling as an integrated element in mathematics in general education especially at secondary level, which be identified in research under the educational perspective are the following:

(1)Mathematical modelling bridges the gab between students’ real life experiences and mathematics. It motivates the students’ learning of mathematics, gives direct cognitive support for the students’ conceptions, and it places mathematics in the culture as a means for describing and understanding real life situations.

(2) In the development of highly technological societies, competences for setting up, analysing, and criticising mathematical models are of crucial importance. This is the case both from an individual perspective in relation to opportunities and challenges in education and work-life, and from a societal perspective in relation to the need for an adequately educated workforce.

(3)Mathematical models of different kinds and complexity are playing important roles in the functioning and forming of societies based on high technologies. Therefore, the development of an expert as well as a layman

competence to critique mathematical models and the way models and model results are used in decision making, are becoming imperatives for the maintaining and further development of democracy.

The third argument is also part of the basis of the socio-critical perspective dealt with below, where it is further developed. However, it is important to recognize that a critical perspective on mathematical modelling and the use of mathematical models in society are also included in the educational perspective.

In the paper by Lambardo & Jacobini  the authors are reporting from their developmental project in Brazil with teaching Linear Programming and mathematical modelling to students employed in various businesses and industries, and who are taking a college degree. Working in pairs, the students were challenged to find problems from their own working life that could be addressed by means of mathematical modelling and Linear Programming. The experiment involved both a mathematics course and a course in data processing in which the students were introduced to software for optimisation. The clear connection to the students’ working life created a strong motivation for learning the “mathematics behind” and for learning how to use the software in order to reach an optimal solution to a LP problem but the students did not, by themselves, engage in reflections about model assumptions, the stability of their optimal solution or the general validity of the model, and the possible implementation of the model results in real life. So, authenticity and close connection to real life experiences do not ensure the occurrence of relevant and critical reflections among the students.

The paper by vom Hofe et al.  reports on an extensive German research project and places itself clearly within the educational perspective. The research project has the double focus characteristic for the educational perspective. On the one hand, mathematical modelling is seen as a means to challenge and develop the students’ mathematical understanding and especially their basic mathematical beliefs (Grundvorstellungen, GV), and, on the other hand, mathematical modelling is seen as an educational goal in its own right. The research is based on comprehensive data material from a longitudinal study, yearly assessing grade 5 to 10 students’ performance solving mathematical modelling tasks. The findings concerning the development of the students’ modelling competency from grade to grade in the three different school branches in the German system are presented and discussed. However, the data are also intended for pinpointing weak spots in the students’ mathematical understanding and beliefs (their GVs) at the different levels and in the different school branches, with the intention of forming a basis for designing teaching material that could help overcome identified learning difficulties in the future. The connection between the students’ mathematical beliefs (GVs) and their performance in modelling task . According the authors, it is in the processes of mathematization and interpretation that the students’ basic mathematical beliefs (GVs) can be unveiled.

Ludwig & Xu  report on a comparative study on the development of mathematical modelling competency in upper secondary students in Germany and China. Building on the conceptualisation of the mathematical modelling process by Blum & Leiss (2005), the authors define five levels of mathematical modelling competency, which they use to measure the students’ performance in different modelling tasks in the two countries. This research lies within the educational perspective with a clear focus on mathematical modelling

competency as an educational goal.

The paper by Meier reports on a comprehensive developmental European project supported by the European Union, where mathematics teachers, primarily of the secondary level, and mathematics education researchers from eleven countries work together in developing and testing mathematical modelling tasks. One of the main research questions in the project is “What is a good modelling task?”, and so far a template for assessing modelling tasks with respect to particular learning objectives has been developed by the project. The template is intended to functioning as a tool for teachers for selecting and reflecting on modelling tasks and, in the paper, template is explained and illustrated through the analysis of a particular task. The project clearly lies within the educational perspective and the research characterizing good modelling tasks tries to take both types of goals into account. Oliveira & Barbosa  have investigated tensions that elementary Brazilian teachers experience when teaching mathematical modelling. This research also seems to be within the educational perspective. However, it is not stated in the paper whether the teaching was focusing on modelling as a means for learning mathematics or as a goal. Teachers might experience different types and degrees of tensions in these two cases. However, this is not investigated in this paper and it may need further research to analyse such possible relations between tension in the practice of teaching and goal of teaching mathematical modelling.

The paper by Rodríguez belongs as already mentioned to the educational perspective and this research also have a clear dual focus on mathematical modelling as a goal and as a means for learning mathematics and, in this case, also physics. Likewise the paper by Kadijevich could also be considered to belong to the educational perspective. In this case, however, the focusing on creating a didactical setting in which the students’ can work with a problem that they conceive as a realistic problem. Moreover the main criterion for success of the students’ work is the solution of the business problem and not the development of mathematical modelling competency or the students’ learning of some particular mathematical concepts or methods. Therefore, I have placed this article under the realistic perspective.