Pre-university students square-root from squared things: A commognitive account of apparent conflicts within learners' mathematical discourses

This study explores the grasp of square roots among eleven students in a remedial mathematics course, with a special focus on where the same student generated apparently conflicting responses. Building on the commognitive framework, the analysis distinguished between routines that individual students consistently implemented in situations where roots “stood alone” and where they were incorporated in more compound exercises, where roots were extracted from square numbers and from squared radicands, where roots were applied to monomials and binomials, and where parameters named with different letters were involved. Differences were found in routines’ degree of objectification, procedures, and tasks. These differences are explained with a theoretical account, suggesting that what may seem as a conflict within a student’s discourse could be a sensible difference of actions taken in situations that this student construed as different. The contribution of this study to the body of knowledge on teaching and learning of roots and to commognitive research is discussed.