Desempleo urbano

Modelling in mathematics teaching in the Nordic countries has by and large been developing parallel to each other – thanks to a common educational culture and cross- national co-operation among didactical institutions. In the following I shall give examples of how modelling is included at different educational levels.

The Nordic educational system can roughly speaking be divided into 9-10 years of general primary school (age 6-16) including what in many countries is called lower secondary level, 2-4 years of upper secondary school (age 16-19), different kinds of tertiary level education including university studies, and on top a PhD- programme. Modelling is part of the programme at all levels:

Primary school

Mathematics teaching in primary school has always emphasised practical use of mathematics in everyday life. The modelling discourse, however, served as a legitimization of a teaching practice which had already been typical for years. In the Danish curriculum from 1993 it is said that instruction should enable students to recognize the possibilities and limitations in using mathematical models. It is also said that students should be aware of the fact that a model always represents a simplification of an extra-mathematical situation, and that the construction of a model always is relative to the constructor; different constructors will not necessarily construct the same model given the same situation. The Swedish curriculum emphasises students’ capability of analysing models critically. The Norwegian curriculum does not use the term modelling explicitly but it says that students should work with problems in a realistic context, for example in projects7.

Upper secondary school

7 _{For details on primary school curricula, check}

http://www3.skolverket.se/ki/SV/0102/sf/11/ol/S_2087.HTML#hit (Sweden),

http://www.ls.no/L97/L97/ (Norway) and http://www.faellesmaal.uvm.dk/fag/Matematik/formaal.html

Modelling was incorporated in curricula in the late 80s. In the general upper secondary education in Denmark (the Gymnasium) it is called ‘the model aspect’ indicating that modelling is not a specific mathematical topic like functions (Hirsberg & Hermann 1991). Several attempts have been made in order to deal with real or realistic problems but most of the modelling tasks are still highly structured and fairly closed tasks which do not leave much room for the student’s own choice of model, method, and technique and which do not give rise to reflection and evaluation. Some of the problems are discussed in Blomhøj (1991), and a few successful examples are documented in von Essen (1991) and Ebbensgaard (1995). In the Swedish and the Norwegian curriculum for upper secondary level modelling is explicitly connected to information technology; it says that students should be able to use graphic calculators and computers appropriate in the modelling process8.

University-studies

Modelling courses are normally not included in the programme for mathematics at university level. Nevertheless this is the case at IMFUFA (Institute for the Study of Mathematics and Physics and their Functions in Teaching, Research and Applications) at Roskilde University in Denmark, which has been a leading institute for putting modelling and project work on the agenda in all the Nordic countries (and international as well). In the mid 90s IMFUFA introduced an introductory course in modelling for students enrolling for mathematics and natural science. It is called BASE (Basic analysis, modelling and simulation) (Blomhøj & Jensen 2002). The course is meant to give students different experiences with mathematizing and analysing already constructed models. Half of the course is rather traditional, but the other half is used for students’ own project work based on modelling problems.

In addition to this introductory course, the master degree in mathematics at IMFUFA comprises three projects one of which must be a modelling project. Some of the final ‘main tasks’ are also modelling projects – some are technological (building new or analysing existing models for authentic use), and some are didactical (discussing the arguments for modelling, analysing the modelling competence etc.). It should be added that modelling has also been the issue of many reports and main tasks from other universities and from other Nordic countries.

PhD-studies

Modelling has also been the object of many PhD-studies. Morten Blomhøj’s doctoral thesis (1992) is concerned with modelling and use of spreadsheets at the end of primary school (or lower secondary level) in Denmark. He concludes that modelling has a potential with regards to understanding mathematical concepts and linking students’ own experiences to abstract mathematics. Iben Maj Christiansen’s thesis (1996) focuses on the critical potentials in working with real modelling problems in upper secondary school in Denmark. She labels the modelling context in school practice virtual reality, and she notices that this virtuality could bring the student to a

8 _{For details on upper secondary school curricula, check}

http://www3.skolverket.se/ki/SV/0102/sf/21/ol/ABC_AM.HTML#hit (Sweden), http://www.ls.no/eway/?pid=207 (Norway) and

proper distance to the real situation in order to be able to see the model in a critical perspective. Hugo Wikström’s thesis (1997) is dealing with modelling of dynamic systems in upper secondary school in Sweden and focuses on students’ understanding of concepts in calculus. Claus Michelsen’s thesis (2001) analyses the potentials of a cross curricula modelling project in mathematics and physics on upper secondary level in Denmark. Torulf Palm’s thesis (2002) analyses the realism in tasks used in the final examination in Sweden and Finland. Modelling and use of technology in teacher education in Sweden is the object of Thomas Lingefjärd’s thesis (2000). He concludes that prospective teachers in mathematics have a positive attitude towards modelling and use of technology, but that students sometimes ascribe an authority to technology, which prevents them from being reflective and critical. My own thesis (Antonius 2003) focuses on how to assess modelling in a final examination. I have analysed the potentials and the difficulties in substituting the traditional written examination with a project examination and I conclude that a project examination in many ways seems to be an appropriate assessment in the sense that it reflects goals and intentions, teaching practice and students’ learning. However, there seem to be problems with reliability (due to lack of assessment criteria) and authentication.