مرکز منطقه ای اطلاع رساني علوم و فناوري - Epistemology and cognitive development of the function concept

Abstract :

The concept of function is considered the most crucial idea in mathematics curricula. It appears in the mathematics curricula from elementary level to undergraduate studies. For instance, basic operations, such as addition, involve the idea of function because they transform elements from IR×IR to IR. The notion of symmetry captures the very essence of the concept; since symmetry line operates as a function transforming (reflecting) entities from one region to another. At the high school level functions are studied within a topic in its own right; it is introduced as a correspondence between the elements of two sets, as a sub-set of a Cartesian product, and as a dependence between two varying quantities. Within a calculus course the concept of derivative and integral takes a function as an argument. At the undergraduate level functions are used to compare abstract mathematical structures; for instance, to illustrate whether or not topologies are homeomorphic. Due to its crucial role and prevailing feature within the mathematics curricula educators suggest that the function concept should be used as a unifying theme in teaching and learning mathematics. This paper aims to provide an extensive literature reviews about the epistemology of the function concept and the cognitive development of this notion. The paper discusses with reference to contemporary learning theories the misconceptions and their sources that the students could develop while learning the functions. In addition, it provides pedagogical suggestion for classroom teachings.