I'm using the word constructivist as it is used in this paper, not in the sense used in logic. The abstract should be sufficient to understand at least roughly what the model is. The important part, more or less, is that students construct their own questions and pursue them, rather than being dictated and tested upon a standardized curriculum.
Anecdote. When I was in college, I took a course on videogame culture. Many of the lectures consisted of a discussion of relevant current events, usually involving most of the audience, but pushed along by the professor when things stood still. Out of class, we wrote papers about topics of our choice, and there was a community blog that we posted on about other relevant issues. We were required to read, post, and comment in certain quotas every week, but many students became very engaged and posted more. I learned a lot in this class.
I have been fantasizing for some time about conducting a mathematics course with a similar approach. In my fantasy, my students post questions and answers to our own private StackExchange, exploring their own ideas about pre-calculus algebra in a mathematically meaningful way. One student makes up a family of polynomials and explores its properties in a series of well-presented questions, inspiring others to write answers, as well as further questions expanding on his. One student derives a cubic formula. Gosh, while we're at it, maybe another student asks what would happen if commutativity was dropped, stumbling inadvertently into some group theory. During face-to-face meetings, we look over everyone's work as a group, highlighting the best progress and supporting those who are stuck. It's a pretty rad dream, alright.
The thing is, I think this could even almost work with upperclassmen math majors, given the right tweaking and incentives. But what I would really love is to find a way to bring a constructivist learning model to lower level coursework, since many of the students are so unengaged by the conventional style. The main problem is that I suspect many of these students have insufficient skill to participate, despite the course being college level. I wonder if there is a way to adapt the model for this purpose. Could a partial integration solve this issue? Would it be impossible to achieve the literacy required to pass a traditional final exam?