A considerable amount of research shows that it is both possible and plausible to teach young children mathematics. However, there is less agreement regarding the content and the framing of such teaching. In this symposium we will present and discuss possibilities with teaching young children mathematics through challenging problem solving. Based on empirical studies of children 3-6 years old we will imply that it is both possible and plausible to teach young children mathematics through challenging problem solving. This since the children in the presented studies both learn a lot of mathematics and enjoy the activities they are involved in.