What is the name of this subdiscipline in math education?
In one of the comments, Dave L Renfro has a reference to Piaget, whose work was primarily done with younger schoolchildren. With respect to extending this work into the older years of one's education, a potentially good place to look would be APOS Theory, which is due to Ed Dubinsky and collaborators. For an introduction, see APOS: A Constructivist Theory of Learning in Undergraduate Mathematics Education Research (pdf) by Dubinsky and Michael McDonald.
More generally, you might search out additional information on APOS theory using google scholar.
As pertains to your description:
they can compute with concrete numbers but are not able to think in terms of e.g. functions on natural numbers and come up with a general solution
These students are probably at either the Action level, or, if they are carrying out these computations in an algorithmic, procedural way, then they may be at the Processes level. (The final two letters stand for Objects and Schemas; more can be found in the first link, above, as well as in its references.)
