Ecological analysis of derivatives within the concept of mathematical site and didactic restructuration of their conceptual relationships

The mathematical site was developed as an ecological analysis model where the relationships between the mathematical subjects and concepts determined as the study area are described by utilizing historical, epistemological and didactic data. According to this model, it is possible to mention two different mathematical sites as global and local site. The global site can be expressed as a site that enables to evaluate the conceptual relationships for the study area as a whole. Local sites can be defined as global site sections where conceptual relationships are involved depending on the performance of typical tasks for the study area. The purpose of this study is to analyse derivatives around the mathematical site concept and to restructure their didactic relationships. In the study, the global site of the calculus was constructed using didactic, historical and epistemological data while taking the approach of the curriculum into account. Using praxeological analyses, local sites were also established for limit, continuity and derivative tasks which were identified through the acquisitions of the curriculum. On the basis of the local sites, sample questions from the textbooks were analysed and ecological problems were identified. The construction of a global site for calculus has demonstrated that the concept of variation is the main subject of calculus, that the concepts of limit, continuity, derivative and integral are utilized as a tool in the examination of variations in real numbers and functions, and that a hierarchical relationship can be established between the concepts of calculus. On the other hand, the results obtained from the local sites have demonstrated that the ecological relationships of the critical concepts such as neighbourhood, undefined and indeterminate forms are of great importance in the accomplishment of the technologies of the types of tasks for the limits, continuity and derivative, and that the use of different techniques together in the related tasks might contribute to strengthen the ecological connections between the objects forming the local sites.