The roles of reification and reflective abstraction in the development of abstract thought: Transitions from arithmetic to algebra

This study utilized a psychological constructivist perspective to examine the transitions that students make from arithmetic to algebra in the context of problems, that from the expert's perspective, involve the concept of linear inequality. Unstructured interviews were used to gather data that were used to develop an explanation concerning student understanding. Thirteen college students were interviewed individually and asked to solve nine related tasks. The interviews were videotaped and the protocols were analyzed to document student conceptions. Five case studies were used to develop and substantiate an explanation regarding students' transitions from arithmetic to algebra. Cifarelli's (1988) levels of reflective abstraction and Sfard and Linchevski's (1994) theory of reification provided a framework for this explanation. This paper discusses an integration of Cifarelli and Sfard's constructs. Students who completed a transition to algebra operated at higher levels of reflective abstraction than students who were unable to complete such a transition. Operating at higher levels of reflective abstraction enabled these students to consider concepts as both processes and abstract objects. Developing this ability was found to be critical in achieving a transition to using algebraic methods.