Students have various difficulties in working with functions. There the question arises, what is the influence of dynamic GeoGebra-Applets on the individual conceptions of students of lower secondary school. Based on this question, my PhD-thesis “the use of GeoGebra-Applets to support functional thinking in lower secondary school” has developed.

Functional thinking

But what is meant with „functional thinking“? Vollrath (1989) describes it as a way of thinking that is typical for dealing with functions, and he mentions 3 different aspects concerning this.   Malle (2000) formulates the following aspects of functional thinking in a slightly altered version:
  • Relation aspect: Each argument x is associated with exact one value f(x).
  • Co-variation aspect: If the argument x is changed, the value f(x) will change in a specific way and vice versa. 


In the literature are various difficulties of students in the context of functional dependencies described. Especially the following are prominent: Poorly developed co-variational aspect The co-variational aspect is very important for working with functions in practice. But empirical studies show that especially this aspect is underdeveloped with students (De Bock, Verschaffel & Janssens, 1998; Malle, 2000; Hoffkamp, 2011). Graph-as-picture error A poorly developed co-variational aspect can be seen among other things on the graph-as-picture error. This occurs in various forms and means that students see functional graphs as a photographic image of a real situation (Clement,1989; Schlöglhofer, 2000; Hoffkamp, 2011). Illusion of linearity Another problem of students is the so called “Illusion-of-linearity”. This means that linear or directly proportional models are preferably used for the description of relations (DeBock, Van Dooren, Janssens & Verschaffel, 2002; Hoffkamp, 2011). Slope-height confusion Difficulties arise also in the interpretation of slope and growth, for example if the point of maximum growth is confused with the largest function value. The confusion of height and slope leads also to difficulties in the interpretation of path-time graphs (Janvier, 1978; Clement, 1989; Hoffkamp, 2011).