Mathematical modelling as bridge between school and university

The paper reports on university seminars of mathematical modelling at school, that were held together by the departments of mathematics and mathematics education and some schools in Hamburg. Prospective teachers together with students in upper secondary level carried out modelling examples either in ordinary lessons or special afternoon groups. They tackled authentic problems proposed by applied mathematicians working in industry. In this paper we consider three of these examples in more detail and describe students’ attempts at solving the problem. After a description of the theoretical framework of modelling in schools and the framework and the structure of the course diverse modelling attempts by students to solve two modelling examples are presented. Finally an evaluation of the series of seminars is presented. ZDM-Classification: C70, M10 In the following sections, we describe modelling examples and experiences from a series of university seminars on modelling in schools conducted at the University of Hamburg (in cooperation with Claus Peter Ortlieb and Jens Struckmeier from the Department of Mathematics). 1 Theoretical framework for modelling in mathematics education The relevance of promoting applications and mathematical modelling in schools is widely accepted. For example the PISA study emphasises as goal of mathematics education to develop within students the capacity to use mathematics in their present and future lives. It means that students should understand the relevance of mathematics in everyday life, in our environment and for sciences. This perception of the objectives of mathematics teaching has impact on the structuring of mathematics lessons. It is insufficient to simply impart competencies for applying mathematics only within the framework of school curriculum. Instead more mathematics teaching should deal with examples from which • students understand the relevance of mathematics in everyday life, in our environment and for the sciences, • students acquire competencies that enable them to solve real mathematics problems including problems in everyday life, in our environment and in the sciences. This demand for new ways of structuring mathematics teaching meets the goals for more reality-oriented mathematics education as postulated in many didactical positions since the middle or the end of the 20 century. It has been agreed that mathematics teaching should not be reduced to just reality based examples but that these should play a central role in education (for an overview see Kaiser-Meßmer 1986, Blum 1996, Maaß 2004, Kaiser & Maaß 2006). As consequence it became clear that besides the application of standard mathematical procedures (such as applying well-known algorithms) in real world context and real world contexts serving as illustrations of mathematical concepts (e.g. usage of debts for the introduction of negative numbers), modelling problems as reality based contextual examples are increasingly important. In addition, it has been discussed that students should acquire competencies which enable them to solve real mathematical problems from different extramathematical areas. Within the actual discussion it is stressed that it is not sufficient at all to deal with modelling examples in lessons but that the stimulation of modelling competencies through self-initiative is of central importance (see Maaß 2004). Since several years, within the mathematics didactical discussion, the following ideal-typical description of modelling processes has become widely accepted. A real world situation is the process’ starting point. Then the situation is idealised, i.e. simplified or structured in order to get a real world model. Then this real world model is mathematized, i.e. translated into mathematics so that it leads to a mathematical model of the original situation. Mathematical considerations during the mathematical model produce mathematical results, which must be reinterpreted into the real situation. The adequacy of the results must be checked, i.e. validated. In ZDM 2006 Vol. 38 (2) Analyses 197 the case of an unsatisfactory problem solution, which happens quite frequently in practice, this process must be iterated.