Mathematical Concept Construction through Abstraction: in the View of APOS and AiC Theory

Abstract

The cognition study about understanding mathematical concepts brings to abstraction, which
is transforming mathematical knowledge into new concepts on the higher level. Although
there are several theories on abstraction, the theory of Abstraction in Context (AiC) and
APOS are the two most widely studied and comprehensively developed. This paper examines
the abstraction, in the view of APOS theory and AiC, since it plays an essential role in
forming mathematical knowledge. The research methods is literature, and the results show
that both of these theory has developed from fundamental epistemology to the abstraction
process, pedagogical implementation, and research methods. Exploration and comparison to
APOS and AiC showed that both theories provide a robust theoretical foundation for
constructors. The abstraction process in APOS theory occurs through mental mechanisms
that give schema as the final result, while AiC goes through RBC epistemic actions that
produce the construct. Both APOS and AiC agreed that abstraction does not appear by itself,
and they suggested group learning to foster abstraction. In particular, APOS recommends the
ACE strategy. APOS and AiC also provide research methods to guide educators and
researchers working with abstractions. Both of their approaches need the identification of
how a concept is formed theoritically.