I am having a hard time understanding schemes in Piaget's theory of cognitive development so I am omitting them from his theory. I do believe that the main points are here. There are assumptions too by Piaget but I am not writing about it since the word scheme dominated. Keep in mind that the age ranges are averages.

(The man himself.)
1. Sensorimotor Stage (birth to age 2) The book uses the example involving a clown. When a child sees a colorful clown he will reach and try grabbing it until the clown is put in a box. Children focus on what they are doing and seeing at the moment in the sensorimotor stage. Before their first birthday, they make the realization that disappeared objects still exist when children develop object permanence. Also cause-and-effect relationships are formed by realizing that certain actions cause certain results. In the latter half of the second year, children reach symbolic thought, the ability to represent and think about external objects and events in one's mind.
2. Preoperational Stage (age 2 until age 6 or 7) I am confused why the preoperational stage is essentially defined "...children can think about objects and events beyond their immediate view but do not reason in logical, adultlike ways." This is the definition of symbolic thought that occurs in stage one. Anyways a language explosion occurs as children can think about a wide range of objects and events. With this explosion new verbal communication develops but it's limited. Children have preoperational egocentrism, the inability to view other people's perspective and as a result sometimes engage in egocentric speech. Egocentric speech is verbal communication from a child without knowing another child's perspective and knowledge.

Another limit that children have is lack of conservation. If they were showed picture I above they probably would say that both cups have the same amount of water but in picture II that there is more water in cup C because it's taller. Conservation is the ability to recognize that if nothing is added or taken away, the amount stays the same regardless of changes in shape or arrangement.
3. Concrete Operations Stage (age 6 or 7 until age 11 or 12) In the concrete operations stage, adultlike logic appears but is limited to concrete reality. Preoperational egocentrism disappears and children realize other perspectives and opinions. Conservation is able to be done by children as well as deductive reasoning, the process of drawing a true logical inference from information that is true.
4. Formal Operations Stage (age 11 or 12 through adulthood) When logical reasoning processes are applied to abstract ideas and concrete ones too then the student has reached the formal operations stage. Abilities important for science and math occur such as formulating and testing multiple hypothesis and separating and controlling variables. Students now will be able to understand infinity and pi. Along with the mathematics and science understanding is viewing how the world could become better. Formal operational egocentrism may occur, the inability to differentiate abstract logic from other perspectives and from practical considerations. An example of this could be a student hears that the way to end poverty is socialism, lower taxes and capitalism and might think all would work since they can't separate the perspectives.

Another limit that children have is lack of conservation. If they were showed picture I above they probably would say that both cups have the same amount of water but in picture II that there is more water in cup C because it's taller. Conservation is the ability to recognize that if nothing is added or taken away, the amount stays the same regardless of changes in shape or arrangement.
3. Concrete Operations Stage (age 6 or 7 until age 11 or 12) In the concrete operations stage, adultlike logic appears but is limited to concrete reality. Preoperational egocentrism disappears and children realize other perspectives and opinions. Conservation is able to be done by children as well as deductive reasoning, the process of drawing a true logical inference from information that is true.
4. Formal Operations Stage (age 11 or 12 through adulthood) When logical reasoning processes are applied to abstract ideas and concrete ones too then the student has reached the formal operations stage. Abilities important for science and math occur such as formulating and testing multiple hypothesis and separating and controlling variables. Students now will be able to understand infinity and pi. Along with the mathematics and science understanding is viewing how the world could become better. Formal operational egocentrism may occur, the inability to differentiate abstract logic from other perspectives and from practical considerations. An example of this could be a student hears that the way to end poverty is socialism, lower taxes and capitalism and might think all would work since they can't separate the perspectives.