Research has highlighted the potential of digital technology to support the development of children's number sense abilities. However, the main focus of such research has been on apps affording directed interactions, where only one solution strategy is available, and it has targeted mostly cardinality. Little is known, in these terms, about task design and implementation in more open environments where several different solution strategies are available. To explore this direction, we chose to study "TouchCounts," an open environment that combines multi-touch affordances with aural, visual and symbolic ones as well. Using tasks that were designed to address different number sense abilities, we experimented with 4-year-old preschoolers. In this paper we present two tasks, their expected potential with respect to strengthening number sense abilities, and analyses of data collected during the preschoolers' interactions with "TouchCounts." The analyses reveal that the children used different strategies in response to the two tasks, and that a broad range of abilities related to number sense were elicited, including both cardinality and ordinality. An important contribution of this study is also a theoretical framework capable of identifying children's learning in a multi-touch environment.