Editing Function (section)

==Core findings from mathematics education research==
Students conceptual difficulties with respect to functions have been investigated at least since the 1980s. These findings have led to the development of APOS Theory<ref name=":0" /> which is a general constructivist theory explaining how mathematical understanding works and develops. It models what might happen in a person's mind when one tries to learn a mathematical concept. It postulates that a person's understanding of a mathematical topic develops as the person constructs or reconstructs certain mental structures and organizes them into schemas. The mental structures proposed by APOS theory are actions, processes, objects and schemata; hence the acronym APOS.  

In case of functions in mathematics, different mental structures are required depending on the developmental stage of understanding. At the action stage, a function is seen merely as an instruction that specifies how the function value (output) is to be calculated. In the process stage, a function is seen as a process that assigns an output to every possible input. This process can be thought of without the need to execute or know the instructions specified in the function's calculation rule/algorithm. At the object stage, the function becomes an object, which in turn can itself be affected or manipulated by processes (e.g. differentiation). Going even further, functions can be organized as objects with other mathematical processes or objects to form larger schematas (e.g. function spaces). 

Note that there are two core transitions in a person's conceptualization of function: from action to process stage (referred to as interiorization in APOS) and from process to object stage (referred to as encapsulation). In terms of [[Treshold concept|treshold concepts]] both transitions can be considered as thresholds. This indicates the cognitive complexity of the function concept since acquiring it requires multiple (re-)conceptualizations.