Table 2 This shows examples of what work represents stages of APOS Theory

Pre-action stage

This is when a student is not yet at an action stage, where he/she is still underdeveloped to learn a concept.

For example, a student who cannot recall the derivative of tan x.

Action stage

A transformation is first conceived as an action, when it is a reaction to stimuli which an individual perceives as external. For example, a student who requires an explicit expression to think about the derivative of a function.

For example, to find the derivative of f(x) = sin x, a student who can do little more than perform the action f′(x) = cos x is considered to have an action understanding of the derivative of a function.

Process stage

A process is a mental structure that performs the same operation as the action but wholly in the mind of the individual. Specifically, a student can imagine performing transformation without executing each step explicitly.

For example, a student can perform the derivative of the function f(x) = cos² x by rewriting this as f(x) = cos x ⋅ cos x and apply the product rule to find the derivative.

Object stage

If one becomes aware of a process stage in totality,for example, when a student can find the derivative of a function by applying various actions and processes, then we say she/he is at an object stage. This could be a student being able to see a function as the composite of two functions.

For example, to find the derivative of f(x) = tan² x ²requires application of a chain rule by applying the power rule first and the derivative of a tangent function and then lastly, the derivative of x ²; the answer is \( \begin{array}{l}{f}^{\prime }(x)=2 \tan {x}^2\cdot { \sec}^2{x}^2\cdot 2x\\ {}\Rightarrow {f}^{\prime }(x)=4x \tan {x}^2{ \sec}^2{x}^2.\end{array} \)

Schema stage

If one is able to apply various actions, processes and objects that need to be organised and linked as a coherent framework.

For example, to find the derivative of \( y={x}^3{e}^{2x+3}\sqrt{ \cos x}. \)