Radical Constructivism, Education, and Mathematics

A constructivist approach to educating would involve letting students construct their own knowledge. Educators often simply tell their students "the facts" and expect them to memorize formulas without understanding them. A constructivist would let their students invent their own version of knowledge through experimentation. For example, being told that pi=3.14, and that the circumference of a circle equals its diameter times pi, will not help a student really understand the meaning of pi. However, some educators let their students discover the meaning of pi by giving them a string and a cylinder and having them realize that the circumference is 3.14 times greater than the diameter. This is constructivist because the students are constructing their own knowledge; they aren't simply being told what pi is. However, since all students would theoretically construct the same ideas about pi, are they really inventing their own knowledge, or are they discovering something about an objective universe? If the latter is true, then this approach to education is not constructivist, since radical constructivists believe there is no objective reality or universal truth. So, is mathematics discovered or created? How does this fit into radical constructivism?

Many people would agree that the world is somewhat, if not completely, constructed. Given the many conflicting ideas, perspectives, and versions of truth, some may conclude that there is no objective reality. Radical constructivists believe that individuals construct their own version of reality through cognitive processes. However, I think there are other ways for knowledge to be constructed. Groups of people can construct aspects of their reality. There are many social institutions that have been constructed by humanity as a whole, not just one individual person. These social constructs do not, however, reflect any objective reality. This is similar to radical constructivism because the knowledge and ideas constructed do not reflect any empirical data. What is the relationship between social constructs and cognitive constructs of knowledge?