This is a start, just to get the juices flowing. More is forthcoming, including a few good references.
First I'll respond to Mike and Richard's comments, and then I'll use those as a springboard for my own (which will probably be in another post). Finally, as a response to the much-needed reminder from K-12 teachers regarding our purpose, I'll attempt to relate my points to classroom practice (also probably in a later post).
Mike cautions against bringing religion into the discussion. I tentatively agree with the intent, although I'll caution against forgetting that religion can ever really be out of the discussion. As Calvin of comic strip fame says, "Math is a religion." There's a lot of humor here, and a lot of truth.
On to Richard. "Reform trajectories/orientations are based not so much on math education research as on general and popular cognitive psychological theories." Unquestionably true 20 years ago, arguably true ten years ago. Now, and fortunately, things are changing. Lampert, Ball, and Voigt - to name but three biggies - have developed their thorough and careful theories based on classroom practice, and this trend seems to be on the rise.
"Many [math educators] are willing to jump on the latest reform bandwagon without substantially questioning the philosophical issues." An indictment of math ed, education in general, and society in general. This is not a defense but an observation; we have a responsibility, too often shirked, to pay attention and to think about what we're doing. That this is often not done is indeed bothersome, but the Standards neither created this phenomenon nor claim to attempt to address it. This is the responsibility of the individual.
"The constructivist ideology seems to assume that there is no ultimate reality 'out there' that we can adequately or accurately model, that we can merely make it up as we go along." Here's the meat of Richard's argument. And later: "Do we want students to learn math as it is used by others, or mostly to develop their own private concepts and procedures? Do we want students to build 'on the shoulders of giants,' or primarily on their own individualized schema?" Richard has set up here a series of false dichotomies. In fact, it is possible to maintain as noncontradictory the notions that (a) there is in fact no ultimate reality, and (b) we have a responsibility to make sure that our kids do a lot more than just make it up as they go along.
This noncontradiction is at the heart of the arguments of the constructivists whom Richard would find "worth talking to;" he would probably find that most people in math ed today subscribe to this viewpoint in one form or another, even if it is simply as a reaction against the metaphor of the mug and jug, i.e., that we as teachers hold the jug of mathematical knowledge from which we are to fill the conceptual mugs of our students. If you say that there's more to it than that, then you will probably quickly recognize that at some level, we have to acknowledge that people understand things in different ways, that different doesn't necessarily mean wrong, and that there can be meaningful mathematical communication even within a context in which not every participant can or would explain things exactly the same way as every other participant.
I'll elaborate on this and other ideas later; for now I'll point out that there are situations in which the radical constructivism of which Richard speaks so disparagingly has been demonstrated. In a study by DiSessa, Hammer, Sherin, and Kolpakowski, students "invented" graphing in a form very similar to the standard Cartesian coordinate system. They were trying to decide how best to represent change over time, and "created" this graphing technique as one of several and with some guidance from their teacher. As the study itself points out, the situation in which this very constructivist learning took place was not a typical one: there were intensive after-school sessions with the students, intensive planning and interpretation sessions among the teacher and the university faculty involved, more-than-typical days allotted for the activities, and fewer-than-typical students involved. It was a good example of an extreme situation; most of the constructivists out there would not hold this up as an example of what they claim students do or can do. Recall that I offered this as an example of *radical* constructivism, not the run-of-the-mill constructivism that is equally sound theoretically and more sound pragmatically.
Again, elaboration, resources, and an attempt to restore the relationship between theory and practice which ought to characterize this list are forthcoming. Just want to test the waters and make sure Ron's excellent questions don't languish in the archives.
Kreg A. Sherbine | To doubt everything or to believe Graduate Student | everything are two equally convenient Vanderbilt University | solutions; both dispense with the email@example.com | necessity of reflection. -H. Poincare