One important task that children learn during the concrete operational stage is to arrange things in order according to one attribute, such as size or weight. Logical inferences such as this are not possible until the stage of concrete operations, during which children develop the ability to make two mental transformations that require reversible thinking. The first of these is inversion (+A is reversed by -A), and the second is reciprocity (A=A).
            A final ability that children acquire during the concrete operational stage is class inclusion; re-creating a relationship between a part and the whole. Concrete operational thought is decentered; a child can now focus on two classes simultaneously. While the differences between the preoperational and concrete operational stages are dramatic, concrete operational children still do not think like adults. They are very much rooted in the world as it is, and have difficulty with abstract thought.

Formal Operations

Children who attain the formal operation stage are capable of thinking logically and abstractly. They can also reason theoretically. Piaget stated that although the children would still have to revise their knowledge base, their way of thinking was actualized.

There is an introduction of formal thought and logical assumptions. Formal reasoning connects assumptions, propositions, hypotheses, i.e., relationships in which one does not necessarily believe, but which one admits in order to see where the consequences lead. Mathematical, scientific and logical reasoning have their basis here.

Problem-solving games, stories, movies, plays, and cartoons are important in formal operations play. Arts and crafts become more exact, realistic, and detailed. Peer approval is important. Teamwork, group cohesion, and skill in sports become important. Fluid knowledge is being developed as well as crystal knowledge.

It is not agreed that all children in all cultures develop formal operations. Children in agricultural and pre-agricultural settings do not have the same needs and appear to function well without obtaining this stage (Berk, 2000).