Constructivists suggest that we cannot be certain of any absolute truth but that people construct, or create, "knowledge" based on their experiential reality (or their interactions). "Truth" or "knowledge" is a social construct attained when people agree on particular mental models that appear to be consistent with our collective experiential reality. However, this social agreement does not necessarily imply "universal truth." For instance, people once believed that the world was flat (collective social agreement) but, as they discovered, this turned out not to be a "truth."
Discovery learning is based in a different belief about knowledge and truth. This theory posits that there is a certain body of knowledge available and that teachers can help students "come to know" (or discover) this knowledge by implementing well-designed tasks in the classroom.
As it pertains to mathematics, I believe constructivists would suggest that the collective set of rules, procedures, and beliefs we call "mathematics" are not, in fact, universal truths but rather mutually agreed upon constructions from the mathematics community. I have some questions about how the implications of this theory for the teaching of mathematics:
1. Social constructivism posits that we rely on other people to both challenge and confirm our ways of knowing. When they have been mutually agreed upon, they become our ontological reality (our truth). What happens when students agree upon an ontological reality that is different from that of the teacher or that of the mathematics community at large? Whose reality is deemed correct in this instance?
2. Constructivist learning theory suggests that "learning" happens when an existing mental model is confronted with an experience that does not conform. The learner must then revise their mental model to accommodate this cognitive conflict. This suggests that the role of the teacher is to present students with situations that create this cognitive conflict (or disequilibrium) in students. Doesn't this imply that the teacher is pressing on students to rewrite their mathematical ways of knowing in a way that is more consistent with that of the mathematical community? Isn't this, in effect, discovery learning?
3. The questions above seem to indicate that it is impossible to create a socially just form of mathematics education that involves topics about which there is already a socially agreed upon "reality" (content standards). Because there is already an agreed upon reality, the students' interaction with them will always be measured against, and subject to, this greater authority. What, then, is the purpose of mathematics education and how can it be socially just?
Please, help me "sort this out..."