Linguistic Pointers to Students' Understanding in Introductory Algebra: A Cognitive Approach

Abstract

Teachers use "extremely subtle pragmatic interpretive judgements [...] regularly in the course of mathematics teaching and learning..." (Pimm, 1987, p.167). The form of their discourse – the coherence, the structure and modality, characteristics of natural language in use – indicates the commitment of students to the truth-value of their statements. Hence, the listener might infer the extent of students' confidence in their understanding. In this study, linguistic features were identified that could be aligned with the conceptual growth of students in the context of introductory algebra. The aim was to devise a model that provided explicit, objective evidence to support the subtle, interpretive judgements made by teachers. Secondary students in Years 8 and 9 (13-15 year olds) from three schools in a NSW regional centre (N=222) participated in the study. The study consisted of two phases of data collection. The first was the collection of quantitative data from students' responses to a survey (test) of 40 algebra items drawn from the algebra syllabus for the first four years of secondary schooling in NSW. Survey data provided information about algebra concepts, and conceptual development demonstrated by the students, through Rasch modelling of the responses and an analysis of errors. The Rasch model indicated items and students clustered around significantly different estimates of, respectively, difficulty and ability. Clustering indicated groups of items requiring similar levels of conceptual development to be addressed successfully, and the corresponding groups of students who demonstrated this development. End-points of clusters indicated where conceptual change was necessary for further success on items, and the students who could achieve this.