In Germany the term 'functional thinking' was first mentioned within the reform of Meran ('Meraner Reform') guided by Felix Klein in the year 1905. This was meant as a reform of the whole mathematics and nature science education at school. Functional thinking was pointed out as a special task of the mathematics education at school. It was meant in a broad sense: as a cross regional habit of thought or fundamental idea which pervades the whole mathematics education, as thinking in variations and functional dependencies with emphasis on flow and change. Infinitesimal calculus, which was introduced to the school curricula in the course of the Meran reform, should not appear as additional content but as point of culmination achieved organically in mathematic classes. In this sense functional thinking can be seen as propaedeutics for calculus.

Many authors (e.g. Vollrath, Malle) describe three aspects of functions respectivly functional thinking: mapping aspect (functions as pointwise relation, static view), aspect of change (dynamic view), functions as objects (global view). Especially the last two mentioned aspects are difficult for students, but are necessary to understand the concepts of calculus.

Difficulties with the dynamic view of functional dependencies often result in the so-called graph-as-image misconception, when students interpret function graphs as photographical images of real situations, or when distance-time-graphs are interpreted as movements in the plane.

Another often described observation is that in calculus classes at school the main focus is on calculating procedures rather than on structural understanding.

In this project interactive learning activities based on the IGS Cinderella are developed in order to accentuate the dynamic aspect and object view of functional dependencies. The activities are designed with regard to an qualitative approach to the concepts of calculus (in the sense of propaedeutics of calculus).

The activities are used in qualitative studies in 10th grade highschool classes in Berlin, Germany.

**Links to the interactive activities:**

- Dreiecksfläche ('Area of a triangle‘) - Version 1, Version 2
- Die Reise ('The journey')
- Einbeschriebene Rechtecke ('Inscribed rectangles')

**Concept of the Activitites**

The learning activities have the following conceptual and theoretical ideas in common:

**Connection situation-graph**

The starting point is a figurative description of a functional dependency, which is simultaneously connected to a graphical representation. The graphical representation was chosen, because it relates to the aspect of change in a very eminent way.

**Language as mediator**

The students are asked to verbalise their observations in their own words. Janvier (1978) emphasizes the role of the language as a mediator between the representations of the functional dependency and the mental conceptions of the students.

**Two levels of variation**

The activities allow two levels of variation. First, one can vary within the given situation. This visualizes the dynamic aspect. To understand a dynamic situation one needs to construct an 'executable' mental model to achieve mental simulation. The idea is to support the mental simulation processes visually (Supplantation, Salomon 1994). Secondly, one can change the situation itself and watch the effects on the graph. We will call this meta-variation. Meta-variation allows the user to investigate the aspect of change in several scenarios. It is variation within the function that maps the situation to the graph of the underlying functional dependency and changes the functional dependency itself. This leads to a more global view of the dependency. Therefore meta-variation refers to the object view of the function. To understand the dynamic aspect one needs to find correlations between different points of the graph in order to describe changes. This requires a global view of the graph. For example the property 'strict monotony' of a graph is a global property and therefore refers to the object view of a functional dependency. But to describe it in terms of 'if x>y then f(x)>f(y)' one has to understand the covariation of different points of the graph.

**Low-overhead technology and practicability**

To work with the interactive activities there is no special knowledge of the technology necessary. The activities make use of the students' experience with Internet browsing (actions like dragging, using links, using buttons etc.). The students (and the teachers) can work directly on the problems without special knowledge of the software and the software's mathematical background. This is important especially with regard to time economy.

For more information and additional material see also http://www.math.tu-berlin.de/~hoffkamp/ (in german)