What are the educational applications of Piaget's cognitive theories of learning?

Piaget's theories of development and cognition have direct applicability to education.  He posited that there are stages of intellectual development that every child must go through, not necessarily on the same schedule, but always in the same order. He also posited that children construct their learning by incorporating knowledge into their existing knowledge. This is why his theories are taught in education courses.  We need to understand these stages and how children learn, so we can capitalize on that understanding in the way best suited to each student.

Essentially, children move from not being able to make any distinction between themselves and the world around them, to being able to incorporate concrete experiences and sensations as separate from themselves, to understanding the stability and constancy of the world, to processing language as representational, to being able to understand abstractions. In each of these stages, the child is gaining a greater and deeper understanding of the world, with more "content" internalized to allow the child to take in even more.  The child begins to categorize the world, and new information can be more easily "fit" into these categories, and new and more subtle distinctions can be made. A child, for example, may learn that while napkins made of paper can be thrown away, books, also made of paper, cannot be thrown away.  A child who has not learned about constancy will report that there is less or more water if water is poured into a container that is a different size or shape.  All of our understanding of this, from Piaget, informs our planning and teaching, in particular, up to and including adolescence, at which time most children are at the highest stage of Piaget's development ladder. 

So, for example, a pre-school or early education teacher is going to know that the constancy of a quantity is something that his or her students is not capable of understanding. A lesson based upon this would be pointless, since children are unlikely to be at the stage of development in which they can process this concept. Teaching a fourth grader algebra is a foolish idea, since algebra involves abstract thinking.  To the degree that language represents abstractions, teaching a vocabulary of abstractions is a foolish idea before a certain stage. Before children can categorize or see patterns, expecting them to do so makes no sense.  But the more a teacher can offer any student the experiences of the concrete world, what we now call hands-on learning, the more the student brings cognitively to the table when the higher stages begin.  This is one of the reasons that children in poverty do so poorly in school, in fact. They have not had as many concrete experiences that prepare them to learn. Their parents speak to them about 80% less than middle-class and wealthy parents speak to their children, even before they begin school. So teachers must make up for this deficiency, and the theories of Piaget are one key to doing so.