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.
Best Answer
The platonization you are looking at seems related to the idea of reification that appears in mathematics education literature (roughly the compression of a mathematical process to a mathematical object). Note: This is clearly distinct from what you are asking about, but shares the feature of bestowing objecthood on something in a way that requires the shift to (what is at first) a radically different viewpoint.
A few things on this that may be helpful can be found here.
Particularly, here are a couple of papers on reification.