Piaget held that knowledge is constructed by the development of the human organism.  Thus, the individual must learn how to learn; a rich, supportive environment and a guiding teacher are key to this individual development of the student.  Thus, such programs as Head-Start for culturally deprived children have evolved from the theories of Piaget.

Until Piaget came along, most people just expected kids to soak up knowledge.  When he began studying how they learn, understand, and remember based on his conversations with children and his daily interactions with them, his discoveries changed the attitudes (not instantly, but over time) of educators and their approaches to subject matter.  Check out this link for more info:

http://www.learningandteaching.info/learning/piaget.htm

I have to agree with #2 - Piaget is now famous and mentioned at least in all educational programmes that teach people how to teach. His key contribution to our understanding of pedagogy focuses on cognitive development and how we learn, absorb, process and internalise facts. Thus his psychological contribution into the area of learning and education is absolutely crucial and has helped shaped the educational models that we have in today's society.

I think Piaget's biggest claim in the area of psychological and therefore educational change was his focus on children in the early childhood stages.  Before Piaget, most psychologists were in the habit of studying children who were old enough to answer questions.  Piaget suggested that behavioral patterns begin in the brain, and children can think before they can speak.  His study focused on infants and very young children (namely, his own at first).

Next, Piaget began to shift away from a focus on environmental factors as the motivators for behavior.  He identified essentially four stages of cognitive development that he suggested every human goes through, and provided approximate time frames for each.  Through these stages of cognitive development teachers of early childhood (and parents of infants) were able to approach learning, discipline, and communication in a completely different way.

Mainly, from birth until about age 7, children are only in the sensory-motor and pre-operational stages.  This means in pre-school, kindergarten and 1st grades, most students are unable to think abstractly.  Every thought is driven by what they can see and touch.  Cause and effect are very difficult concepts.  Memorization of facts and skills does not connect with understanding.

Basically, Piaget can be credited with the fairly recent movement in early childhood education from a teacher-centered approach to a student-centered approach.  Classrooms today are much more interactive, full of hand-held manipulatives, and much of the education happens through self-discovery and play time.  Because at the early stages, children are inherently ego-centric, curriculums were developed to move from the child, outward.  Learning is built on previous knowledge that actually grows with the child's cognitive ability to understand.

How did Jean Piaget redefine knowledge?

In addition to the stages mentioned, Piaget redefined knowledge by suggesting that some people get stuck in certain stages of development.  For instance, a mentally disabled person may not advance to the abstract thinking stage.  This leads to the idea that people can have intellectual ages as well as chronological ones.

How did Jean Piaget redefine knowledge?

Piaget redefined knowledge by determining that (1) knowledge is developed in four invariant, hierarchical and universal stages and (2) children are not cognitively able to perform some tasks of logic and deduction, which academic opinion assumed they could perform, until they reached age 11 or older. The first stage is infancy to two years of age. The second stage is two years of age to seven years of age. The third stage is between seven years of age and eleven years. The fourth and last stage begins at eleven years of age or older and continues until after adolescence.

The first stage (0 - 2 years old) is the sensorimotor development stage in which the infant is believed to understand the world only through physical actions within or upon the physical world. Actions in stage one are founded upon inborn instincts that are utilized in expanded tasks. The second stage (2 -7 years old) is the perceptual or intuitive stage which follows language acquisition and in which events are believed to become independent of physical action and result in mental images.

The third stage (7-11 years old) develops concrete operational intelligence during which class and relationship comparisons emerge and recognition of multiple salient characteristics supplants object recognition based on single salient characteristics (e.g., if a hot-dog looks like more when cut up, then it is more). The fourth stage (11 years and older) develops formal operational intelligence in which logic, reasoning, deduction, hypothetical thinking and thoughts of time beyond the present emerge. Piaget's theory asserts that stage four components of formal operational intelligence are not possible for children prior to the invariant, hierarchical, universal fourth stage, which is a concept that redefined the previous assumption that children could engage in logical, hypothetical, deductive reasoning at early ages.

How did Jean Piaget redefine knowledge?

You can basically ask what was Piaget's contribution to the field of education. Every specific element you mention in the question falls under the umbrella of Pedagogy (in my humble opinion the BEST field one could ever study).  The answer to that much simpler question is that he was one of the first proponents of developmentally-minded tasking for students by levels of ability. Back in the day there was no way anyone would even consider the idea of differentiation based on skill. Back in his time, it was all rote learning, memory, and dry facts. His contribution was so huge that is still accepted today as one of the most effective ways to teach and one of the most productive ways to implement instruction at a general and specific level.