Piaget's Stages of Cognitive Development

Children go through a series of cognitive developments, marked by stages. These stages were not pinpointed by a psychologist, however. They were determined by developmental biologist Jean Piaget. His stages of development can be helpful in determining what reasoning power you can expect a child of a certain age to have. For example, kids under the age of seven think very differently than a 12 year old.

Piaget determined that infants from birth to 24 months old experience the sensory motor stage. At this age, children experience the world around them through their sense of touch. They touch everything, and their reflexive behaviors become more complex until they are more intentional. Towards the end of this stage, the child develops problem-solving sequences.

From the ages of 2 to 7, children experience the preoperational period, according to Piaget. Although speech remains egocentric, the child begins to use verbal representation. The speech becomes less egocentric with time. During this stage, the child also begins to grasp symbolism and transductive reasoning. Additionally, the child doesn't grasp principles, just simple do's and don'ts for guiding their behavior.

From the ages of 7 to 11, children have an easier time grasping organized and logical thought. Children are able to grasp multiple classification tasks, and logical sequences. Children can also sort objects into logical categories (rather than superficial ones like color) and grasp reversibility (like reversible operations in math).

Children can also grasp conservation during this stage. They are able to learn that an object isn't always the way it appears. Kids also develop the ability to imagine "what ifs" (what if 'x' happens, what if 'y' happens, etc.).

Additionally, Piaget determined that children's operational thought is limited during this stage. They can engage in logical thought about an object only if they can manipulate it. Once a child progresses to the next stage, they can grasp logical thought about an object without the object ever being present.

Piaget determined that children next experience the formal operations stage. This happens from the ages of 11 to 15. Kids' thoughts become more abstract during this stage. Along with abstract thought comes formal logic. At this stage kids are able to generate multiple hypotheses and determine the outcome of each. In mathematical studies, kids can grasp proportion and algebra.

During this final stage, kids can also grasp propositional problems. Comparison and evaluation of verbal assertions, without comparing those assertions to something in "real life", is possible.

Sources:
http://www.childdevelopmentinfo.com/development/piaget.shtml
http://ehlt.flinders.edu.au/education/DLiT/2000/Piaget/stages.htm