Constructivism offers a sharp contrast to this view. Knowledge is actively created or invented by the child, not passively received from the environment. It is a relation that the child superimposes on a set of objects. This relation is constructed by the child by reflecting on actions performed on numerous sets of objects, such as contrasting the counting of sets having four units with the counting of sets having three and five units. Rather they invent new ways of thinking about the world.
Children create new mathematical knowledge by reflecting on their physical and mental actions.
Ideas are constructed or made meaningful when children integrate them into their existing structures of knowledge. No one true reality exists, only individual interpretations of the world. These interpretations are shaped by experience and social interactions. Thus, learning mathematics should be thought of as a process of adapting to and organizing one's quantitative world, not discovering preexisting ideas imposed by others. Mathematical ideas and truths, both in use and in meaning, are cooperatively established by the members of a culture.