The idea of developmentally-appropriate behaviors, tasks and goals was first developed by Piaget. It was probably the first time ever that anyone had stated that individuals have specific stages of development which enable (and sometimes disable) the possibility to perform a task. For this reason, differentiated instruction based on developmental ability and appropriate tasking is one of those things we definitely owe Jean Piaget in modern education. It is also important to consider that his theories have persisted through time and are still accepted in the 21st century as innovative.
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.
I think that you could probably focus on much from Piaget that has modern relevance in teaching and learning. His presentation of the stages of learning concept holds powerful implications on the practices of teaching and learning. On one hand, Piaget presents education as something that is not strictly linear. It is contingent on stages of development, and maturation of learner. I think that this is powerful because it creates the idea that assessment must take this into account. High stakes standardized testing might be invalidated, to an extent, under this idea because it fails to account for the growth and stage maturation of students. At the same time, Piaget places a great deal of understanding and emphasis in how the student organizes and makes sense of information. This reflects how teaching and learning are independent processes and how the teacher must pay attention to this idea of individual harnessing in the process of instruction.