Abstract
M.Ed. (Mathematics in Education)
In spite of the efforts of the South African government, the Gauteng Department of Education and many business and private funders to place a high emphasis on mathematics performance, the mathematics achievement of South African learners is still less than desirable. In fact, the results of the three Third International Mathematics and Science Study (TIMSS) (Howie, 2001, 2003) reports in 1995, 1999 and 2003 reported South African learners as the lowest performing in those tests; performing well below the international average amongst the countries that participated. The Southern African Consortium for Monitoring Quality 2004 and Center for Development in Education 2004, 2010 and 2011 reports results show similar results (Moloi, 2005). The research study sought to get a deep understanding of why learners1 continue to perform poorly, and what the factors are which contribute to poor performance. While there are a number of factors responsible for the poor performance, one of the least researched areas is answered examination scripts. This research entailed a detailed error analysis of four items of the 2008 mathematics paper 1 senior certificate examination scripts, to see the trends and patterns of written responses with regards to the types of errors made by learners.
The study was aimed at investigating South African Grade 12 learners‘ errors exhibited when solving quadratic equations, quadratic inequalities and simultaneous equations. Findings of this investigation shed light of the kind of knowledge learners bring into their learning experiences and this knowledge affects how they encode and later retrieve new information learned (Davis, 1984). While the study was not a mixed methods one, the data was analysed quantitatively using frequency counts and qualitatively by studying selected learners‘ solution of examination tasks. The study also identified common errors in the learners‘ work. The four items analysed in the study comprised of questions from three important areas of algebra namely: quadratic equations, quadratic inequalities and simultaneous equations. The scripts were analysed for carelessness, conceptual and procedural errors. The learner misconceptions were discovered in learners‘ work; v
these comprised the notions of equality and inequality, the construct of the variable, order of operations, factorisation, and solution of equations instead of inequalities. From this, the researcher noted that learners' learning difficulties are usually presented in the form of errors they show. Not all the errors that learners had are the same; some errors in procedures can simply be due to learners' carelessness or overloading working memory (Davis, 1984). Some errors in procedures can be caused by faulty algorithms or "buggy algorithms". Other errors can have certain conceptual basis and can be termed as ‗misconceptions‘.
The results obtained indicated a number of error categories under each conceptual area, namely, quadratic equations and inequalities and simultaneous equations. Some errors emanated from misconceptions. Under the conceptual areas indicated above, the main reason for misconceptions seemed to be the lack of understanding of the basic concepts including numbers and numerical operations; functions; the order of operations; equality; algebraic symbolism; algebraic equations, expressions and inequalities; and difference between equations, expressions and inequalities. The abstract nature of algebraic expressions posed many problems to learners such as understanding or manipulating them according to accepted rules, procedures, or algorithms. Inadequate understanding of the uses of the equal sign and its properties when it is used in an equation was a major problem that hindered learners from solving equations correctly. The main difficulty in inequalities was manipulating the inequalities correctly and converting the inequality to an equation. Recommendations to the mathematics educational community based on this research were made.