APOS theory – Mathematics and Computing Pedagogy

The APOS model is a constructivist theory of how learning mathematics might take place and was developed by Ed Dubinsky and others, as an attempt to explain Piaget’s concept of reflective abstraction in learning, with particular focus to mathematical learning.

APOS theory – Actions, Processes, Objects and Schemas – can be used in the assessment of success or failure of students on mathematical tasks with the specific mental constructions they may or may not have made. In this approach, a theoretical approach is made of each topic to be learned. Mental structures are specified and it is proposed that if a student constructs them, the learner will be able to learn the concept in question. Instruction is designed to focus on getting students to make the proposed mental structures and use them to learn the concept in question.

Dubinsky has proven that by replacing the lecture method with constructive, interactive methods involving computer/mathematical activities and cooperative learning the amount of meaningful learning that takes place, can radically improve, and that “Experience, theory and research all point to the fact that verbal explanations that do not relate to the students prior experience are quite ineffective” (Dubinsky, 1989).


The theoretical framework and the adoption of APOS theory to a computer programming module, in encouraging active learning was investigated. In utilizing APOS in computing classes, students have diverted from studying for end of term exam, to a pattern of continuous learning and having discussions with friends about the computing assignments and how to solve them.


Dubinsky, E., 1989, On Teaching Mathematical Induction II. Journal of Mathematical Behaviour. Vol 8: pp.285-304.


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