A much-cited attempt to analyze Piaget's ideas and carry them further has been given by Ed Dubinsky in <A HREF="http://www.math.wisc.edu/~wilson/Courses/Math903/ReflectiveAbstraction.pdf">Reflective abstraction in advanced mathematical thinking</A> (1991). 

> Reflective abstraction is a concept introduced by Piaget to describe
> the construction of logico–mathematical structures by an individual
> during the course of cognitive development. Two impor- tant
> observations that Piaget made are first that reflective abstraction
> has no absolute beginning but is present at the very earliest ages in
> the coordination of sensori–motor structures and second, that it
> continues on up through higher mathematics to the extent that the
> entire history of the development of mathematics from antiquity to the
> present day may be considered as an example of the process of
> reflective abstraction.
> 
> In the majority of his own work, however, Piaget concentrated on the
> development of mathematical knowledge at the early ages, rarely
> going beyond adolescence. What we feel is exciting is that, as he
> suggested, this same approach can be extended to more advanced topics
> going into undergraduate mathematics and beyond. It seems that it is
> possible not only to discuss and conjecture, but to provide evidence
> suggesting, that concepts such as mathematical induction,
> propositional and predicate calculus, functions as processes and
> objects, linear independence, topological spaces, duality of vector
> spaces, duality of topological vector spaces, and even category theory
> can be analyzed in terms of extensions of the same notions that
> Piaget used to describe children’s construction of concepts such as
> arithmetic, proportion, and simple measurement.