When dealing with Piaget's affects on classrooms, you cannot leave out that Piaget basically started the idea of the "child-centered" classroom.
This change in American and European education started in the 70s and 80s and was most prominent in the early childhood grades. Education up until this point focused mostly on teaching kids to sit still and listen to and model themselves after the teacher. Typical classrooms included students in straight rowed desks, sitting, reading, writing, and reciting. Piaget's theories influenced education as we most often see it today, even up through high school classrooms: students engaged in self-motivated learning, problem-solving, curiosity, and educational "play."
Student-centered classrooms put the burden and responsibility for learning on the student, rather than the teacher. Because of this, the environment is more conducive to engaging multiple learning styles. While students build an educational framework independently they also learn how to engage with others and build ideas ideas interactively.
If you haven't used it already, a great resource is Conversations with Jean Piaget.