Mental representations of mathematical objects and relations in the first grades

The importance of adequate external and internal (mental) representations both for mathematical understanding and for generalization is shown with examples taken from early mathematics in pre-school education and from primary mathematics in schools. Regarding relations between numbers and special aspects of addition and subtraction, in the main part it is discussed whether or to which extent referring to actions with concrete materials and to children’s every-day life experience might be a learning obstacle and not helpful for children’s insight and for their ability to generalize mathematical concepts. In addition, alternative ways for classroom practice are discussed