This article presents findings of a study on elementary algebra in South Africa, Eastern Cape Province. The study analyses the learners' performances in elementary algebra and explores the mistakes made by the learners. The subjects of this study consisted of 136 grade 10 high school mathematics learners who wrote a test on elementary algebra made up of twelve questions. The study used four cognitive levels demand assessment tasks suggested in the TIMSS study (1999). The test consisted of two knowledge questions, four routine procedural questions, four complex procedural questions and two problem-solving questions. In data analysis, descriptor code keys were used. When the learners' overall performances were considered for all twelve questions, the results showed that learners conceptual understanding of simplifying Elementary algebra was severely flawed. The results of the study showed that conceptual, language, procedural and application errors hinder learners' appropriate understanding as well as the application of algebra. Notable among the four errors were conceptual and procedural errors that seemed to permeate every attempt learners made to simplify algebra. The investigation went further into pedagogy that focuses on recognized learning errors and misconceptions in algebra could assist the learners in overcoming their errors. The research included the collection of data aimed at (1) identifying the learners misconceptions and errors in the algebra fractions (2) a pedagogical intervention that discusses the identified learners errors (3) a post-test to gauge the learning gains. It is noted that there are possibilities that some of the types of learning errors in elementary algebra that can be mitigated with the use of the pedagogical intervention. However, there are some of the intervention that still persists, such as cancellations error, despite the intervention put in place.

Keywords: Learners' errors, misconceptions, and algebraic fractions.

Introduction

Mathematical concepts such as algebra is a very abstract branch of mathematics. It is the quotient of two algebraic expressions (Petit., Laird, Marsden & Ebby, 2016). Algebra is one branch of mathematics that gives a challenge to most grade 10 learners. Most topics in grade 10 depend on this topic. Thus, any learner who does not have algebraic knowledge is bound to perform poorly in mathematics. This research help learner to understand algebra and its application in other fields such as economics, science and information technology. Computations of algebra usually involve coordination of other mathematical concepts such as an equation, factorization, division, and exponents at grade 10 level. The challenge for most teachers is to acknowledge the necessary fact that understanding algebra concepts form an important base for mathematical learning in general and to adopt this as a foundation for teaching mathematics to learners (Fennell, 2007). Various research works have been carried out to establish problems and difficulties associated with learning and teaching algebra. Kieran and Yerushalmy, (2004) investigated how various applications in graphic calculators and computer can boost learning of elementary algebra. For instance, Essien and Setati (2006) researched on equal signs in textbooks. They realized almost all textbook use equal signs as operational symbols rather than relational symbols. This has adverse consequences to grade 10 learners when they start learning algebra. This finding does not concentrate on errors and misconceptions. And if it mentions, it does not give how to solve the errors. This research is to establish that learner errors and misconceptions in algebra are to trigger an inquiry; as learning and teaching resource to help students master algebra in terms of its errors.

Misconceptions and errors

Research has established that students misconception in the knowledge of algebra has led to students use of wrong procedures in solving algebraic problems. (Anderson, 1989; Van Lehn and Jones 1993). It is a common mistake to use incorrect procedure when learning algebra (Sebrechts, Enright, Bennett and Martin, 1996), and naturally this triggers accurate solutions to problems ( Booth and Koedinger, 2010, p.571) Welder (2012) explains that to understand something means to assimilate it into an appropriate schema (cognitive structure). The assimilation of information or ideas into an inappropriate scheme makes assimilation of later ideas more difficult and, in some cases, even impossible (Van de Walle, Karp & Bay Williams, 2013).

Welder (2012) cites another obstacle to understanding: the belief that one already understands fully. Learners are very often unaware that they have misconstrued a concept until they have to put it into practice. Often, when a teacher gives a class several similar problems for the learners to do, it is only then that the teacher finds out that a number of learners cannot solve them. Those learners thought that they had understood but, instead, all they retained was just a collection of memorized rules and facts (Naseer & Hassan; 2014). According to Li (2006) and Luneta (2008), learner errors are symptoms of misconstrued concepts.

Drews (2008:18) defines misconception as a misapplication of a rule, an over- or under- generalization and they are stubborn to remedy. Learners tend to be attached to their misconceptions because they actively constructed them and they give them smart solutions (Li, 2006). Research on misconceptions suggests that strategies such as repeating a lesson to emphasize a point do not help learners who have acquired an alternative conception or misconceived concepts in mathematics (Ada & Kurtulus, 2010; Idris, 2011).

Generally, misconceptions manifest through errors (Luneta, 2008). Luneta and Makonye (2010:35) define errors as a mistake, slip, blunder or inaccuracy and deviation from accuracy. Errors are also said to be indicators or symptoms of some misconceptions. Swan (2001:150) also asserts that errors could be the result of careless or misinterpretation of symbols or text. These definitions imply that most errors are an indication of misconceptions and these misconceptions are solid concepts that the learner was supposed to have understood (Luneta, 2013). Therefore, some misconceptions and errors concern students misinterpretation of the previous knowledge as opposed to new concepts. For example, consider the meaning of the sign known to the learners. Symbols like (-) and (+), implying subtraction or merely instructing to bring together. (Booth, 1984). Another misconception in dealing with algebraic operations is cancellation error. Some learners tempt to cancel when they see common variables in the denominator and numerator even when the variable is not really a common factor in both the denominator and the numerator (Demby, 1997, p.42). another very common problem amongst learners concerns closure, in this most learners, find difficulties in accepting expressions as final answers after simplifying the algebraic expression. For instance, some students cannot accept to leave their final answer as X + 5, instead would prefer to join it as 5X, which is an error. Hence, diagnosing and analyzing learners work critically has become an essential activity for teachers because error analysis often reveals misconceptions learners have (Bush and Karp, 2013; Idris, 2011; Luneta and Makonye, 2010 and Olivier, 1989). Also in this research, it suggests methodology to apply in order to help the student in the learning situations.

Background to the study

This study was compelled by our experiences of teaching mathematics for over 10 years. As grade 12 mathematics examination markers we noticed some of the difficulties learners had with mathematical concepts such as simplification of algebraic expressions. The algebra concept is an integral part of both General Education and Training (GET) and Further Education and Training (FET) band curricula. The Curriculum and Assessment Policy (CAPS) for mathematics emphasizes that learners should be able to simplify algebraic expressions with confidence (2013:10), yet algebra is one of the most challenging concepts for learners to grasp (Kaplan, 2007). It is essential to note also that knowledge of algebra helps in simplifying and solving mathematical tasks in topics such as functions, calculus, trigonometry, financial mathematics and others (Fennell, 2007). Essentially it means good knowledge of algebra is crucial for success in mathematics education (Brown & Quinn, 2006). Competency in algebra is basic in understanding mathematics ( Usiskin, 2004). Globally, incompetency in algebra is regarded as a major barrier to studying mathematics for most children (Mhakure, Jacobs and Julie, 2014) because it forms a language in mathematics is written in and spoken on ( Ramirez, Yuen, and Ramey, 1991, p.3)

Statement of the problem

It is hoped that the results of the study could be used to assist teachers to address learners errors and misconceptions on algebra and probably provide appropriate instructional approaches to address the errors.

The Department of Basic Educations (2013:136) National Diagnostic Report points out a concern that the matriculants performance is quite dismal when it comes to simplification of algebraic expressions. The chief examiners reports in both mathematics first and second papers also outlined that matriculants consistently have difficulties with simplifying expressions and solving equations (DBE, 2013). According to the National Diagnostic Report (DBE, 2013), although algebraic expressions and equations cover 30% in both grade 10 and 11, and 17% of the grade 12 curriculum, learners find simplifying algebraic expressions difficult. The report adds that many of the candidates were "discredited" because of their inability to simplify algebraic expressions. Kaplan (2007) also asserts that learners find algebraic expressions difficult to learn probably because it demands an understanding of other concepts such as an exponent, factorization, and division. For instance, a grade 10 learner answered:

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Vignette A: Example of possible misconception. Source: Author

The learner (21) expressed the numerator as (2x2y)(2x2y) and wrongly factorized 2x as the common factor from both the numerator and the denominator. The principle of factorization with algebraic fractions applies when there is an operation sign plus or minus in the expression. This is a practice based reason for the study. The results of the study can be used to assist teachers to address learners misconceptions and errors in algebraic fractions and provide instructional approaches to address mathematical errors.

The following are seven other examples of misconceptions and errors that were commonly committed by grade 10 learners;

Another learner, the study of Otten, Figueras, and Males (2008) a grade 10 student committed 4x

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2x

Such and many other related errors were found to be common in examination scripts, test papers, and classwork. It is for this reason that I embarked on this study to understand the root causes of the problem and how best they could be addressed. These errors and misconceptions motivated this study to identify and categorize errors which are a result of misconceptions formed when learners study tasks involve algebraic fractions. It is my hope that such an inquiry will contribute to the effective teaching and learning of algebraic fractions at secondary schools.

Purpose of the study

The purpose of the study was to explore learners ideas when simplifying algebraic expressions and to help learners to improve their performance in simplifying algebraic fractions by suggesting to teachers possible ways to minimize errors and misconceptions. The first objective of the...

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