Musings on Three Epistemic Algebraic Students

Abstract

An epistemic subject is “that which is common to all subjects at the same level of development, whose cognitive structures derive from the most general mechanisms of the co-ordination of actions” (Beth & Piaget, 1966, p. 308). Epistemic algebraic students are dynamic models of students’ ways and means of operating that are taken to characterize a stage in the development of students’ algebraic activity and that can be used to communicate purposefully with actual algebraic students. The purpose of this paper is to present the author’s working, and sometimes second-order, models of three epistemic algebraic students, and to generate comments and questions about learning trajectories for these students. These three epistemic students are distinct in that they are operating with three qualitatively different multiplicative concepts, which are based on how students create and coordinate units of units. The paper focuses on relationships between students’ fractional knowledge and algebraic reasoning in order to contribute to understanding how students’ arithmetical and quantitative reasoning can be a basis for algebraic thinking.