There has been much research about tasks and learning environments but not yet sufficient regarding those involving mobile technologies for mathematics education. This paper discusses the quality of the expanding amount of tablet applications for primary school mathematics and their eloquent integration in student-centred learning environments (LE). It argues that students' engagement in LE with tablet apps does not necessarily start with a particular a priori task designed within the LE and through a task-format, as by now reported in literature. An engaging design of touchpad apps may rather encourage explorative activities leading towards a single task formulation and its solution as a result of that process. The paper further aims at framing essential characteristics and potentials of environments enriched with tablet-apps in three main categories: mathematical and curricular meaning, didactical aspects and technical matters. Setting the focus on the didactical potentials of tablet-apps, the overarching categories are further specified finalizing with six categories: (1) mathematical contents and relation to curriculum; (2) communication, collaboration and cooperation; (3) differentiation; (4) feedback and assessment; (5) connections and networking; and (6) logistics, obtained through the grounded methodological approach in a theoretical model. The suggested model should also adequately represent the development of task, task-format and LE in it. Additionally, a descriptive case study for illustrating the theoretical model for design and analysis of potentials of tablet apps for elementary geometry is also offered. The proposed model may be meaningful for teachers' decision making when selecting and implementing touchpad-apps in their instructional practices but also for developmental surveying of existing apps, their re-designs and further novel designs involving identified potentials.