What is the educational philosophy that best supports the idea of Reconstructionism?
I think that you are looking for an educational philosophy that stresses the transformative nature of learning and reality if one is embracing Reconstructionism. Consider that the idea of reconstructionism is one that demands that individuals embrace the idea of a "better society." This helps to enhance the notion that the classroom is a realm for transformation. Students and teachers have to be committed to the idea that learning can be conducted so that individuals are able to possess the insight to transform reality into what should be as opposed to what it is.
In this light, I think that you could find a Constructivist classroom as being able to accomplish such a purpose. The Constructivist learning philosophy demands that students interact with the content and the classroom setting in a social context, enabling them to see the need and possibility as transformation as something real. The Constructivst classroom views learning as a reciprocal process, existing between the shared dialogue between teacher and learner. This is transformative in the notion of education and learning, helping to open dialogue in the most formative of learning stages and thereby seeking to create a "better society." Additionally, the facilitation role of the teacher helps the student to construct their own notion of learning and understanding it as primarily social, being able to have applicability to the world outside the classroom, echoing the calls of Reconstructionism.