

Concrete operations
Around age 7, as children begin to master conservation problems, they move into Piaget's third stage of development. The concrete operations stage lasts until age 11 or 12 and is marked by increases in symbolic thinking. Children in this stage understand relational terms (for example, big, bigger, and biggest) and make considerable use of abstract conceptual categories. For example, if you give preoperational children a bunch of toys and ask them to sort them, they will typically sort the toys by color and sizephysical properties of the toys. In contrast, if you give the same bunch of toys to children who are in the concrete operations stage, they will sort the toys by function. Trucks, cars, and planes will go in one pile because they are forms of transportation. Pans, measuring cups, and spatulas will go in another because they are all used in cooking. Similarly, children in this stage become increasingly aware of the constraints that physical reality imposes. The story of Santa Claus becomes harder for them to believe as they begin to think about the logistics that would be involved in delivering toys around the world in a single night with chimneys as the main port of entry.   
While the cognitive development of children in the concrete operations stage resembles the cognitive development of adults in many ways, children in this stage still have difficulty dealing with the most abstract forms of thinking and problem solving. Math problems, for example, are easier if they are presented as concrete word problems than if they are presented in purely abstract terms. For example, solving the problem "How many cows does a farmer own if he inherited two, bought ten more, and then sold three?" is easier than solving "2X + 10X  3X = ?"
Formal operations
Around the age of 12, most children transition into the stage of formal operations. In this stage, cognitive problem solving occurs at a highly abstract level and closely resembles the problemsolving strategies that adults use. Children in the formal operations stage are able to reason logically and symbolically. The hallmark of this progress is the emergence of deductive reasoning. In deductive reasoning, conclusions are drawn on the basis of an initial set of premises. For example:
 All snails have shells.
 No other animal has a shell.
 This animal has a shell.
 Therefore, this animal must be a snail.
Deductive logic is highly abstract, in part because it is reducible to completely symbolic terms. For example, the snail shell problem can be restated as "All X have Y. Only X have Y. Z has Y. Therefore, Z is X." Children in the formal operations stage are also able to engage in hypothesis testing (e.g., if X is true, then Y should also be true), another adultlike reasoning strategy.



