The mathematical situations of play and exploration introduced here have been developed as an empirical research instrument for the longitudinal study "erStMaL" (early Steps in Mathematics Learning). They are designed as situations that allow children and a guiding adult to construct situation-related knowledge in common dialogue processes and in mutual conditions. In addition, the situations focus on the activity of the persons involved, as well as the meaning of artefacts as objects of different cultural worlds. Besides relevant theoretical aspects for the development of the mathematical situations of play and exploration, exemplary analyses are introduced to clarify the conceptual aims.