In John Dewey's Experience and Education, he proposes a qualitative measure for an experience (please simply understand this in common usage for the purposes of this question) to be educative. Namely, an experience is educative to the extent that it opens the possibility for further experiences.
Consider the following interesting quotation:
The belief that a genuine education comes about through experience does not mean that all experiences are genuinely or equally educative. Experience and education cannot be directly equated to each other. For some experiences are miseducative. Any experience is miseducative that has the effect of arresting or distorting the growth of further experience. An experience may be such as to engender callousness; it may produce lack of sensitivity and of responsiveness. Then the possibilities of having richer experience in the future are restricted. Again, a given experience may increase a person's automatic skill in a particular direction and yet tend to land him in a groove or rut; the effect again is to narrow the field of further experience. An experience may be immediately enjoyable and yet promote the formation of a slack and careless attitude; this attitude then operates to modify the quality of subsequent experiences so as to prevent a person from getting out of them what they have to give. Again, experiences may be so disconnected from one another that, while each is agreeable or even exciting in itself, they are not linked cumulatively to one another. Energy is then dissipated and a person becomes scatter- brained. Each experience may be lively, vivid, and "interesting," and yet their disconnectedness may artificially generate dispersive, disintegrated, centrifugal habits. The consequence of formation of such habits is inability to control future experiences. They are then taken, either by way of enjoyment or of discontent and revolt, just as they come. Under such circumstances, it is idle to talk of self-control.
Reuben Hersh's position that Pragmatism provides a decent philosophy of Mathematical Practice leads one to wonder whether the overarching goal of mathematics education ought to aim to train students to find highly educative mathematical experiences (i.e. good problems and questions?)
Despite the heavy preamble above, I'd like some references to the math education literature that develops connections with Dewey's above point of view. I am aware of, and very much in favor of, IBL methods in mathematics education. I'd be interested in papers related to problem-posing. I am particularly interested in studies of activities aimed at teaching students to evaluate a problem or direction of inquiry for fecundity.
To clarify this last point a bit, I wonder if anyone has studied the ability of students to work on problems with a view toward seeing how many distinct interesting ideas they generate, rather than with an eye toward solving the problems. This sort of evaluation is somewhat essential to the practice of (pure) mathematics. The ability to find problems that generate interesting mathematics and abandon those problems that don't should be a central emphasis of an experience (inquiry?)-based approach to mathematics education. Of course this depends on individual student experience and maturity…but it seems that this is something that should be actively assessed as a primary goal of instruction. In fact, the ability of a student to perform this selection in a way closer to a professional mathematician may be deemed a major criterion for measuring "mathematical maturity". Dewey's above quote and the related discussion may reinforce this point.
