Learner Errors on Calculus Tasks in the NSC Examinations: Towards an Analytical Protocol for Learner Perturbable Concepts in Introductory Differentiation

By Judah P. Makonye.

Published by The Learner Collection

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Article: Print $US10.00
Article: Electronic $US5.00

Calculus is an important topic in the mathematics curriculum at Further Education and Training (FET) level. Yet mathematics lecturers in higher education institutions (see for example Engelbrecht, Harding & Potgieter, 2005) report that many students arrive at university exhibiting low levels of calculus competency. Recognising this problem, this paper analyses students’ perturbable concepts made in responding to National Senior Certificate Examination (NSC) calculus items. The foci of the paper were to identify and categorise students’ perturbable concepts in order to formulate an error analysis protocol providing a language of description for the errors and misconceptions. The theoretical framework for the study draws from concept image and concept definition constructs (Tall and Vinner, 1981). Similarly the conceptual framework for analysis of errors draws from Donaldson (1963) formulations of structural errors and executive errors as well as Hirst (2003) categorisation of errors as due to pseudo-linearity, procedural extrapolation and equation balancing. The perturbable concepts observed in scripts have been coded into different categories and some vignettes of prototype errors are included in this paper. Content analysis was done on 1000 scripts from the 2008 South African Mathematics Paper 1 examination. The findings of the study show that a single student often commits several different errors even in answering a single item. These errors are sometimes mutually exclusive but they are often inter-dependent and inter-related. Some are systematic and theory like and common across different students but most are random, fragmented, and idiosyncratic. The study also found that many misconceptions in solving calculus problems are caused by student confusion with meanings of terms such as function, gradient, limit, tangent, intercept et. cetra. Despite these difficulties, it was highly disturbing that as much as 30% of the students seemed completely ignorant of calculus concepts or techniques as there were not any traces of these ideas in their responses. The study suggests an analytical protocol for student calculus errors grounded from these findings. Recommendations for the wider mathematics education community are made.

Keywords: Differentiation, Perturbable Concepts, Analysis

The International Journal of Learning, Volume 18, Issue 6, pp.339-358. Article: Print (Spiral Bound). Article: Electronic (PDF File; 1.287MB).

Dr. Judah P. Makonye

Student, Witwatersrand University, South Africa


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