Mathematical Challenges to Secondary School Students in a Guided Reinvention Teaching-Learning Strategy towards the Concept of Energy Conservation

Guiding sixteen-year-old students to rediscover the concept of energy conservation may be done in
three distinct learning steps. First, we have chosen for the students to reinvent what we call partial laws of energy conservation (e.g. Σm∙g∙h = k1). Secondly, the students are asked to combine these partial laws into more and more general laws of energy conservation (e.g. forming Σm∙g∙h + Σm∙c∙T = k3). Because a new term may always be added this process of combining laws can be continued for a long time. The result may still be only a partial law of energy conservation. A third learning step is needed in which students are to extrapolate the process of combining partial laws. If the student becomes convinced that it is indeed always possible to add an extra term to the equation when necessary, the student must now be convinced as well that the law is applicable to any situation and has thereby reinvented the general law of energy conservation. In the first two learning steps we have uncovered mathematically challenging steps which remain obscure in more traditional teachings of the concept of energy conservation. The first mathematical challenge lies in retrieving a physical law from measured data, specifically quadratic ones. The second challenge lies in combining the special cases of energy conservation into a more widely applicable case of energy conservation. In this paper we will exemplify these mathematical problems and give some possible solutions for teachers to guide students in passing these obstacles. At least partially these steps clarify the abstractness of the concept of energy and contribute to explain problems students are having in understanding this concept.