Modelling motion: from trace graphs to instantaneous change

Utrecht University, the Netherlands

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Students in secondary education experience mathematics and physics as strictly separate disciplines. They do not realise for instance that the mathematics used to describe change (calculus) is used in the topic kinematics in physics. The goal of this research was to examine whether it is possible to develop understanding of both subjects and of their mutual relationship. Furthermore, it has been examined what role computer tools could play in learning mathematics and physics.

Doorman has opted for the subjects of calculus and kinematics in upper secondary education (pre-university). Teaching materials have been developed, inspired by the two topics’ mutual history; a history in which Galileo played an important role. In addition, the materials are influenced by semiotic theories on symbolising and focuses on a dialectic development of construction of meaning and of external representations. The research presented in this thesis aims at contributing to an empirically grounded instruction theory for calculus and kinematics in which computer tools are used.

It turned out that students can develop knowledge of differential calculus and kinematics using this approach. This development is supported by a series of graphs from discrete to continuous. The graphs emerged from modelling activities of students, and supported the use of mathematical and physical reasoning. A problem in that context was the transition from modelling measuring data to using mathematical formulas. There should be more attention for such transitions in mathematics education.

The dynamics in computer software and the possibility to explore a lot of situations offered students an opportunity to express their own ideas. One risk of computer use, though, was that students went through the activities too fast and too superficially. A good preparation in the teaching materials and class discussions led by the teacher should allow for a compatibility between the possibilities of the software and the students’ current reasoning. Subsequently, activities are needed for both reflection and creating a classroom consensus on what has been learned.

Finally, the students did not automatically use the possibilities to construct and trace meanings. It is therefore necessary that regular attention is given in mathematics teaching to the meaning of mathematical concepts, so that learned algorithms do not become blind automatisms that only work within mathematical questions.

Key words – science education, mathematics education, calculus, kinematics, design research, computer tools, symbolising

Full reference

Doorman, L.M. (2005), Modelling motion: from trace graphs to instantaneous change. Utrecht: CdB Press, Centre for Science and Mathematics Education, Utrecht University (CdB Scientific Library, 51) – 243 p. (including a 10 p. summary in Dutch) – ISBN 90-73346-59-2.URL: 


Dr. Michiel Doorman

Freudenthal Institute, Utrecht University

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