The ubiquity of mobile devices has made mobile touchscreen interaction a promising avenue for learning. Although children start using educational apps from early age, only a few apps adhere to interaction design recommendations and undergo empirical evaluation of their educational potential. In this paper, we describe the concept and design of an app developed for promoting early math learning through finger-based multi-touch interactions in kindergarten children. Drawing from research on embodied cognition, our app provides games for fostering finger counting as well as finger-based representation of cardinal magnitude and part-whole relations. The app's efficiency was empirically evaluated through a pre-post-intervention study design with two control conditions. Results revealed that a short-term intervention with the math app did not significantly improve children's math skills when compared to learning gains of both a content-matched, unplugged math training program and a passive, waiting list control group. We discuss possible methodological reasons underlying these results, considering key curricular differences for the effectiveness of app-based interventions in early childhood education. Finally, we reflect on the appropriateness of complex multitouch interaction for young kindergarten children and suggest future directions for research in child-centered interaction design.