Reflections on APOS theory in Elementary and Advanced Mathematical Thinking

What are the processes by which we construct mathematical concepts? What is the nature of the cognitive entities constructed in this process? Based on the theories of cognitive construction developed by Piaget for younger children, Dubinsky proposed APOS theory to describe how act ions become interiorized into processes and then encapsulated as mental objects, which take their place in more sophisticated cognitive schemas. He thus takes a method of construction hypothesised in (elementary) school mathematics and extends it to (advanced) college/university mathematics. In this paper I respond to Dubinsky’s theory by noting the need for cognitive action to produce cognitive structure, yet questioning the primacy of action before object throughout the whole of mathematics. Biological underpinnings reveal cognitive structures for object recognition and analysis. I use this to suggest that APOS theory has already shown its strength in designing undergraduate mathematical curricula but question its universal applicability, in particular in geometry, and, more interestingly, in the formal construction of knowledge from definitions to deductions in advanced mathematical thinking.