Differential equations models in upper secondary school by the use of CAS.
Danish University of Education
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.
Blomhøj, M. & Jensen, T. H. (2003). Developing mathematical modeling
competence: Conceptual clarification and educational planning. In
Teaching Mathematics and its Appl. Vol 22 (3) pp 123-139
Carstensen, J. & Frandsen, J (1999). Mat 3H. Systime Denmark. Freudenthal,
H. (1991). Revisiting mathematics education Kluwer
Cobb, P., Stephan, M., McClain, K. & Gravemeijer, K. (2001). Participating
in Classroom Mathematical Practices. In The Journal of Learning
Sciences 10(1&2), 113-163 Lawrence Erlbaum Associates, Inc.
Cobb, P., Yackel, E. & McClain, K. (eds) (2000). Symbolising and
communicating in mathematics classrooms. Perspectives on discourse,
tools, and instructional design. Lawrence Erlbaum Ass. Mahwah, New
Gravemeijer, K., Lehrer, R. et.al. (2002). Symbolizing, Modeling
and Tool use in Mathematics Education Kluwer
Hjersing, N., Hammershøj, P. & Jørgensen, B. (2004). Modeller
i Derive. Differentialligninger og modelbygning. Højt niveau i
matematik. (Models in Derive. Differential equations and modelling)
Copenhagen Kelly, A & Lesh, R (2000). Handbook of research design
in mathematics and science education.
Erlbaum. Noss, R. & Hoyles, C. (1996). Windows on mathematical
meanings. Learning cultures and computers. Kluwer