Tricky stuff, isn't it? I'm probably not going to be all that helpful, despite the fact I'm currently finishing a doctoral class called "Theories of Mathematics and Science Learning." Here are a couple brief but hopefully helpful thoughts:

1. Constructivism is a learning theory, not a teaching theory. I think you get that, but you muddy the water a bit at the top of the post when you refer to it as a "methodology." Constructivism itself says nothing about what a classroom should look like - it's just how we learn, regardless of the environment we're in.

2. I've never heard of "discovery" as a learning theory. It seems to be a term commonly used to describe instruction that is more student-centered or inquiry-based. Because constructivism says knowledge is something constructed in the mind of the learner, and not transmitted, discovery is seen to have advantages over a transmission-based style of instruction.

3. The way we label, distinguish, and use theory varies from person to person. A few people have made good attempts to categorize major themes (Cobb, 2007; Sfard, 1998) and on my to-do list is to blog about those when the semester is over.

4. Can you say more about what you mean by a "socially just" form of mathematics education? Are you talking about a type of mathematics education with a focus on equity?

From the little I understand of it, it sounds like you're thinking of radical constructivism, where everything is subjective -- what I know, what you learn if I try to teach you what I know, and what I think you've learned when I try to teach you what I know. None of it has any certainty.

While constructivism can be seen as pedagogy-neutral, it seems to me that a key element for a teacher should be to give himself/herself opportunities to observe student learning. Once you think knowledge is constructed in the mind of the learner (regardless of the particular nuances of the theory), monitoring that construction helps you guide the student. This means the actual act of teaching and learning could vary greatly, so long as you're not expecting some perfect transmission of knowledge to happen without further intervention.

When you step away from the "knowledge as acquision" metaphor to the "knowledge as participation" metaphor (Sfard, 1998), then how a student participates becomes key, because participation *is* learning. (We've gone a theoretical step beyond constructivism by this point. There's no "thing" to construct!) If you're teaching students mathematics, then there needs to be a special importance placed on what kinds of participation are valued.

Raymond, I disagree that Sfard's "participation is learning" is a theoretical step beyond constructivism. What she seeks is to reinscribe mathematics not as "knowledge" but as "participation." This is a fine move; it buys some cache in thinking about what is mathematics. It is not a more robust theory for knowing however. In this way, I do not see it as a theoretical step beyond; it does not supersede Constructivism. It does not account for what Constructivism can, plus "something" in the way that Constructivism supersedes other theories for knowing/learning, such as behaviorism, information processing, etc.

Papert did some amazing work with computer programming. Primarily with the logo language and Turtle graphics. His book Mind Storms, printed in the 80s is still revolutionary. As for me, I think the most important part of discovery based learning is that I want the students to discover why these laws or formalities are actually important. Yes, we want them to agree with the community at large, but an attempt to refute the larger community can only bring about deeper understanding of what the community is stating as a truth. This, the end result will either be acceptance as a result of understanding, or innovation, as a result of understanding.