APOS Theory

Interiorization turns actions into processes and encapsulation turns processes into objects; encapsulation is the hardest step to teach.

APOS Theory describes mathematical understanding as built from actions, processes, objects and schemas. An action is an explicit transformation of an already understood mathematical object, guided step by step by external instructions. When an action is fully internalized it becomes a process, which the learner can carry out without external instruction, skipping steps and imagining it without specific inputs. This shift is called interiorization. When the learner can think of the whole process as one thing and put further structure on it, the process becomes an object, and this shift is called encapsulation. Encapsulation is considered the most difficult step in APOS-based instruction, and the structures are not necessarily gone through in a linear progression.