APOS Theory: Use of Computer Programs to Foster Mental Constructions and Student’s Creativity

Abstract

According to Piaget, the root of all intellectual activity is reflective abstraction. In this context, mathematical creativity arises through students’ abilities to make reflective abstractions. Considering that reflective abstraction is the main premise of APOS Theory, the theory provides a theoretical tool to guide the development of instruction that supports mathematical creativity. The letters that make up the acronym—A, P, O, S—represent the four basic mental structures—Action, Process, Object, Schema—that an individual constructs as he or she reflects on and reorganizes content in coming to understand a mathematical concept. Much of the instruction that involves the application of APOS Theory has been delivered using the ACE Teaching Cycle, a lab-oriented pedagogical approach that facilitates collaborative activity within a computer environment (programming and/or dynamic). The letters that make up the acronym—A, C, E—represent the three components of a pedagogical cycle—Activities, Classroom Discussion, Exercises—that facilitate reflection and collaboration. Numerous studies have demonstrated the efficacy of this approach when applied to the teaching and learning of a variety of mathematical topics at the elementary, secondary, and collegiate levels. We illustrate this with a description of instruction for the topics of cosets, infinite repeating decimals, and slope. To introduce these examples, we provide a brief overview of APOS theory with all its components in the context of learning the concept of function. Opportunities for development of mathematical creativity are emphasized throughout the entire chapter.