In educational reform often the level of subject specific teaching and learning is not adequately considered (Altrichter & Maag Merki, 2010). The interdiscipinary research project FaBiT (Doff et al., 2014) aims at filling this gap by investigating empirically how change in teaching and learning happens and how this can be arranged. The mathematics project of FaBiT addresses the change of understanding functions in the transition to high school level. Through a design-based research approach it investigates how the transformation of the concept of functions can be initiated and how this leads to change in teaching and learning. Since students at the initial stage of higher secondary come from a wide range of feeder schools they are not equally prepared for learning advanced mathematics. In addition, they have not developed one but many concepts of functions such as linear, or quadratic functions (if any) and their function concepts are imprinted by the way functions are taught at lower secondary (Bikner-Ahsbahs et al., 2015). We talk about a fragmented concept of functions. This fragmentation shall be overcome by tasks aiming to use functions more flexibly. The approach for task design addresses formulas as expressions of functions (Malle, 1993), e.g. for the cone volume (Fig.1). If students at this stage take the cone as a geometric object, they are surprised that the volume is dublicated by doubling the height; but when looking at the algebraic expression, the proportional relationship between h and V easily clarifies this. Fig. 1: The cone volume We expect that such tasks will both, deepen the understanding of the types of functions learned before and build a more comprehensive and flexible function concept. Results from a pilot study on three clinical interviews with pairs of students indicate that blending formulas with functions is a challenge which must be designed carefully.

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