A global survey of international perspectives on modelling in mathematics education

In this article we survey the current debate on modelling and describe different perspectives on this debate. We relate these perspectives with earlier perspectives and show similarities and differences between these different approaches. ZDM-Classification: M10 The international discussion on modelling, which is amongst others documented in the last and the current issue of this journal, demonstrate that there does not exist a homogeneous understanding of modelling and its epistemological backgrounds within the international discussion on modelling. However, this is not a new situation at all and if we go back into the earlier debate on modelling we can find astonishing similarities. 1. The earlier debate on modelling in mathematics education Nearly twenty years ago, Kaiser-Messmer (1986, pp. 83) showed in her analyses that within the applications and modelling discussion of that time various perspectives could be distinguished, internationally and nationally in Germany or German-speaking countries as well. The following two main perspectives emerged from the discussion that time: • A pragmatic perspective, focussing on utilitarian or pragmatic goals, the ability of learners to apply mathematics to solve practical problems. Henry Pollak (see for example 1969) can be regarded as a prototype of this perspective. • A scientific-humanistic perspective which is oriented more towards mathematics as a science and humanistic ideals of education with focus on the ability of learners to create relations between mathematics and reality. The ‘early’ Hans Freudenthal (see for example 1973) might be viewed as a prototype of this approach. Freudenthal changed his position at the end of his life, as he tended to take pragmatic aims more into consideration (see for example 1981). Although these were the main streams of the discussion on modelling further differentiations become obvious, especially interesting on a national level within the German debate, but with significant international linkages. For a better understanding of the current approaches, a brief description of these approaches will be given. Related to the scientific-humanistic perspective the approach developed by Hans-Georg Steiner (1968) put epistemological goals into the foreground and emphasised the development of mathematical theory as an integrated part of the processes of mathematising. However, early attempts such as that of the French-speaking André Revuz (1971) are also important. He starts out from the triple situation-model-theory which means that models are constructed by starting from a situation which then leads to the development of a mathematical theory. Furthermore, an emancipatory perspective in the discussion can be identified, which is developing into socio-critical attempts of mathematics teaching (for current approaches see for example Gellert, Jablonka & Keitel 2001). A third stream, named integrative perspective, demands that applications and modelling should become subject to different levels of aims, that is to serve scientific, mathematical and pragmatic purposes but in a harmonious relation to each other. This perspective is not limited to specific aims and gets its strength from a wide range of aims and arguments (see for example Blum & Niss 1991). The various perspectives of the discussion as reconstructed by Kaiser-Messmer vary strongly due to their aims concerning application and modelling. The appropriate references suggest various dimensions of aims. Kaiser (1995, p. 69f) distinguishes the following goals: • Pedagogical goals: imparting abilities that enable students to understand central aspects of our world in a better way; • Psychological goals: fostering and enhancement of the motivation and attitude 2006, Vol. 38 (3) Analyses 303 of learners towards mathematics and mathematics teaching; • Subject-related goals: structuring of learning processes, introduction of new mathematical concepts and methods including their illustration; • Science-related goals: imparting a realistic image of mathematics as science, giving insight into the overlapping of mathematical and extra-mathematical considerations of the historical development of mathematics. Comparable dimensions of aims are stated by Blum (1996, p. 21f) although he identified and described the nuances differently. Meanwhile, the current discussion on modelling has developed further and become more differentiated. New perspectives can be identified which, as it became obvious from detailed analyses, emerged from the above described traditions or partly can be regarded as their continuations. 2. The current debate on modelling in mathematics education In the following, a classification system for present modelling approaches will be suggested by reverting to the previous differentiations summarized above but taking the current developments of the modelling discussion into consideration. This suggestion is based on recent analyses using literature mainly generated by ICMI and ICTMA activities and additional publications (see for example the reference list in the discussion document of the ICMI Study on applications and modelling in mathematics education (Blum et al. 2002, p. 279f)). This classification distinguishes various perspectives within the discussion according to their central aims in connection with modelling and describes in short words the backgrounds these perspectives are based on as well as their connection to the initial perspectives. This ensures both a continuity for the present discussion as well as accumulates current perspectives coherently into the existing