Differential equations models in upper secondary school by the use of CAS.

Abstract

Abstract This paper reports on a Ph.D. project, which was part of a larger project (see www.matnatverdensklasse.dk). The project comprised about 25 primary- and upper secondary schools in the Copenhagen region, and each year 80 – 100 teachers and about 1000 students participated. In the reported part of the project, each student had had a laptop at his disposal for at least two years. The Ph.D. project inquires the try out in four classes of teaching materials on differential equations. One of the objectives of the project was changes at two levels: Changes at curriculum level. The textbook (Hjersing, Hammershøj & Jørgensen 2004) realised a dynamical systems’ point of view on differential equations. Accordingly, the teaching was based on the use of laptops, with the CAS software Derive. This point of view is in contrast to the traditional approach to differential equations, which is structural algebraic-analytical. Differential equations are most commonly introduced in connection with calculus, linked to determination of integrals and considered as algebraic equations in a function and its derivative. (Carstensen & Frandsen 1999 pp 77-92). The teacher-authors of the textbook intended to change focus of attention towards understanding the dynamics of the differential equations, and they intended to challenge the students to offer interpretations of the drawings produced by the computer and relate them to the applications on ‘reality’ of the model. (Hjersing et al. 2004 p4). This is in the spirit of the new trend in math teachers’ discourse linked to the use of computer. Changes in the intentions of modelling and using models. The Ph.D. project aimed to introduce the perspective of concept formation by modelling into teaching, in accordance with the basic principles of Realistic Mathematics Education (Freudenthal 1991). This introduction implies a change of focus of attention into the design heuristics of emergent modelling (Gravemeijer & Stephan 2002 p 159). Using the notion of explorative work with allusion to the descriptions by Gravemeijer et al. in (Cobb, Yackel et al. 2000 pp 225-274) and expressive work like in (Blomhøj & Jensen 2003 p 126), though, the students’ work tended to be explorative rather than expressive in the project, since the subject of differential equations comprises a heavily accessible area to most of them. Expressive work in the area, moreover, is beyond the scope of upper secondary school teaching of differential equations.

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