I'd like to know whether any form a certain hypothesis about the learning of higher mathematics has entered the mathematical or educational literature. I'll frame the hypothesis here but not defend it since this is not a blog-in-disguise; likewise I'm not soliciting debate.
This hypothesis opposes to some degree the common shibboleth which holds that "mastering abstraction" constitutes the single major plateau which undergraduate mathematics students must, but often do not, scale.
For the sake of making the distinction, I'll first flesh out what I mean by "mastering abstraction." Generally speaking, abstraction means reducing to essentials. So mastering mathematical abstraction breaks into two major challenges: learning modelling and learning formal work. Modelers must first know how to decide what they may safely ignore and then how to select or construct formal systems that adequately capture what remains. With a formal system in hand, getting answers requires skill with its internal operation, sometimes despite the loss of intuition that arises from distance to the original situation. Of course a feedback cycle often arises -- "answers" from formal work can demand systemic revision of the formalism.
On the the current hypothesis, namely that something else constitutes the major glass ceiling for advance mathematics students. I'll call that something else cognitive platonization. (If someone else has already coined a better name I'd like to know!) So cognitive platonization occurs when mathematicians confer objecthood on the collection of some or all configurations of a known object. Examples abound: taking all solutions of certain differential equations as elements of a vector space, forming (iterated) power sets and cumulative hierarchies in set theory, studying state spaces in dynamical systems, moduli spaces in geometry, homology and cohomology groups or Stone-Cech compactifications in topology. Like abstraction, cognitive platonization often induces a loss of intuition due to distance from the original situation, but I contend a different sort of distance. Abstraction involves reasoning away from a picture you may feel afraid to lose; cognitive platonization involves reasoning on the way to a picture you may fear will never congeal.
As an aside, I chose the name because some radical philosophers challenge the very "existence" of just these sort of things I see students struggling to comprehend.
I'd like to know several things:
1) Does the challenge of teaching cognitive platonization (known by whatever name) have a theoretical literature?
2) Does cognitive platonization have a practical literature, meaning materials aimed directly at students, perhaps at the (American) college sophomore level?
3) Do any books from the popular science genre frame this issue and do a good job at communicating its essentials to a wide-audience?
4) What testable implications of the hypothesis can anyone suggest? Might success or failure with, say, abstract algebra or measure theory correlate with a student's response to tasks, otherwise unrelated to that subject matter, that indicate their ability or willingness to embrace this process of conferring objecthood? If so, what sort of tasks?
Final note: I'm asking here because most mathematics education research looks at K-12 teaching and learning, or perhaps calculus. Almost all writing about teaching higher mathematics comes from practicing mathematicians.
