Persistent Errors in Indices: A Cognitive Perspective

Abstract

This thesis explores the understanding of indices shown by students in their senior years of secondary schooling. Quantitative and qualitative data are used to investigate how students perceive, and work with, an index as a raised number in arithmetic and in algebra. Earlier research focused largely on identifying errors students make. However, little has been done to explore the thinking being applied when those errors are made. ... The form of understanding being applied was researched in terms of students' use of routine strategies as against accessing underlying concepts. This was done within the framework of 'relational' and 'instrumental' understanding as proposed by Skemp (1978). Students were found, almost invariably, to apply understanding which was instrumental in nature. The way in which the persistent errors emerge from these routine strategies was explored within the concepts of 'connections' (Shevarev 1946) or 'frames' (Davis 1984). Under these constructs, errors result when students adopt approaches which, while correct for certain other items of similar appearance, ignore important aspects of the question. Particular 'connections' or 'frames' were identified for the operations previously mentioned, namely: multiplication and division of expressions involving numerical bases; raising expressions to a power; simplifying algebraic fractions involving indices; and, taking the square root of expressions involving indices.