Learners’ errors and misconceptions associated with common fractions

**Authors:**Mdaka, Basani Rose**Date:**2012-11-02**Subjects:**Mathematics - Study and teaching , Fractions (Mathematics)**Type:**Mini-Dissertation**Identifier:**uj:7310 , http://hdl.handle.net/10210/8049**Description:**M.Ed. , This research aimed to explore errors associated with the concept of fractions displayed by Grade 5 learners. This aim specifically relates to the addition and subtraction of common fractions. In order to realize the purpose of the study, the following objective was set: To identify errors that learners display when adding and subtracting common fractions. The causes which led to the errors were also established. Possible ways which can alleviate learners’ misconceptions and errors associated with them were also discussed. The study was conducted at Dyondzo (Fictitious name) Primary School, Vhembe District in Limpopo Province. The constructivist theory of learning was used to help understand how learners construct their meanings of newly acquired knowledge. It was a qualitative study where most of the data and findings were presented with think descriptions using descriptive analysis techniques. A group of forty nine learners was selected purposively within two classes of Grade 5 to write the class work, home work and test on addition and subtraction of fractions. Learners were interviewed and so were two teachers. The five teachers also completed a questionnaire of five questions to supplement the interviews. The study found that learners made a number of errors in the addition and subtraction of fractions, including conceptual errors, carelessness errors, procedural errors and application errors. This finding supports findings that primary school children experience difficulties when learning the concept of fractions.**Full Text:**

**Authors:**Mdaka, Basani Rose**Date:**2012-11-02**Subjects:**Mathematics - Study and teaching , Fractions (Mathematics)**Type:**Mini-Dissertation**Identifier:**uj:7310 , http://hdl.handle.net/10210/8049**Description:**M.Ed. , This research aimed to explore errors associated with the concept of fractions displayed by Grade 5 learners. This aim specifically relates to the addition and subtraction of common fractions. In order to realize the purpose of the study, the following objective was set: To identify errors that learners display when adding and subtracting common fractions. The causes which led to the errors were also established. Possible ways which can alleviate learners’ misconceptions and errors associated with them were also discussed. The study was conducted at Dyondzo (Fictitious name) Primary School, Vhembe District in Limpopo Province. The constructivist theory of learning was used to help understand how learners construct their meanings of newly acquired knowledge. It was a qualitative study where most of the data and findings were presented with think descriptions using descriptive analysis techniques. A group of forty nine learners was selected purposively within two classes of Grade 5 to write the class work, home work and test on addition and subtraction of fractions. Learners were interviewed and so were two teachers. The five teachers also completed a questionnaire of five questions to supplement the interviews. The study found that learners made a number of errors in the addition and subtraction of fractions, including conceptual errors, carelessness errors, procedural errors and application errors. This finding supports findings that primary school children experience difficulties when learning the concept of fractions.**Full Text:**

Access to mathematics through a primary language : implications for curriculum policy and classroom practice

**Authors:**Nkosi, Elijah Raymond**Date:**2012-06-06**Subjects:**Mathematics teaching , Instruction in home language , Swazi language , SiSwati language , Bilingual education**Type:**Mini-Dissertation**Identifier:**uj:2484 , http://hdl.handle.net/10210/4939**Description:**M.Ed. , The report presents an account of how learners and teachers use siSwati as their home language to improve access to mathematics in bilingual classrooms where the language of learning and teaching is not the home language of the learners and teachers. The practices on two grade 6 teachers in two different schools in Mpumalanga Province, where the home language of both the learners and the teachers is siSwati were studied. The study revealed that while teachers succeeded in translating their non-mathematical discourses in English into siSwati they struggled to effectively translate their mathematical discourses in English into siSwati. Where they managed to translate their mathematical discourse in English into siSwati, access to mathematics improved.**Full Text:**

**Authors:**Nkosi, Elijah Raymond**Date:**2012-06-06**Subjects:**Mathematics teaching , Instruction in home language , Swazi language , SiSwati language , Bilingual education**Type:**Mini-Dissertation**Identifier:**uj:2484 , http://hdl.handle.net/10210/4939**Description:**M.Ed. , The report presents an account of how learners and teachers use siSwati as their home language to improve access to mathematics in bilingual classrooms where the language of learning and teaching is not the home language of the learners and teachers. The practices on two grade 6 teachers in two different schools in Mpumalanga Province, where the home language of both the learners and the teachers is siSwati were studied. The study revealed that while teachers succeeded in translating their non-mathematical discourses in English into siSwati they struggled to effectively translate their mathematical discourses in English into siSwati. Where they managed to translate their mathematical discourse in English into siSwati, access to mathematics improved.**Full Text:**

Effective teaching methods used by teachers to teach grade 11 quadratic equations in the context of South African schools of Limpopo Province

**Authors:**Makgakga, Sello William**Date:**2012-06-08**Subjects:**Effective teaching , Effective teaching methods , Quadratic equations , Grade 11 students , Mathematics Study and teaching**Type:**Thesis**Identifier:**uj:8743 , http://hdl.handle.net/10210/5095**Description:**M.Ed. , This dissertation is about the instructional approaches used by teachers to teach Grade 11 quadratic equations, errors learners made and misconceptions they possessed. The main topics that I had focused on were solving quadratic equations by factoring, completing a square and using quadratic formula. The intention was to observe teachers’ teaching approaches in quadratic equations and diagnosed types of errors learners displayed and misconceptions they possessed in quadratic equations. Literature review had served as a secondary source of information that was relied upon for the research study. Sources such as scholarly books, government documents, dissertations, professional journals and electronic resources were used to gather the information pertinent to the research topic. Review was also done on how teachers teach quadratic equations, learners’ learning of quadratic equations and teachers teach and learners learn mathematics. This study is action research under qualitative research paradigm in which the information collected was analyzed through thick description and not statistically. Pre-test, self and post test evaluation methods are discussed of quadratic equations by factorization, completing a square and using quadratic formula. Learners were tested on factoring, completing a square and using quadratic formula. In addition to the learners’ class exercises and home work, these scripts were also analysed for errors and misconceptions. Collected data is presented that helped to address errors and misconceptions learners displayed in solving quadratic equations and teachers’ teaching methods and approaches. Data was collected from schools in the neighborhood and the school I was attached as a mathematics teacher. In all schools, five teachers’ three lessons were observed which added up to a total of fifteen. All five teachers were interviewed as well as five learners in each school. Interviews were analyzed by comparing what their teaching approaches with the types of learners’ errors and misconceptions. In classroom observations, Indicator Evaluation Form adopted from Luneta (2006) was used to collect data as well as analyzing it. Questionnaires were prepared for both teachers and learners for interviews.**Full Text:**

**Authors:**Makgakga, Sello William**Date:**2012-06-08**Subjects:**Effective teaching , Effective teaching methods , Quadratic equations , Grade 11 students , Mathematics Study and teaching**Type:**Thesis**Identifier:**uj:8743 , http://hdl.handle.net/10210/5095**Description:**M.Ed. , This dissertation is about the instructional approaches used by teachers to teach Grade 11 quadratic equations, errors learners made and misconceptions they possessed. The main topics that I had focused on were solving quadratic equations by factoring, completing a square and using quadratic formula. The intention was to observe teachers’ teaching approaches in quadratic equations and diagnosed types of errors learners displayed and misconceptions they possessed in quadratic equations. Literature review had served as a secondary source of information that was relied upon for the research study. Sources such as scholarly books, government documents, dissertations, professional journals and electronic resources were used to gather the information pertinent to the research topic. Review was also done on how teachers teach quadratic equations, learners’ learning of quadratic equations and teachers teach and learners learn mathematics. This study is action research under qualitative research paradigm in which the information collected was analyzed through thick description and not statistically. Pre-test, self and post test evaluation methods are discussed of quadratic equations by factorization, completing a square and using quadratic formula. Learners were tested on factoring, completing a square and using quadratic formula. In addition to the learners’ class exercises and home work, these scripts were also analysed for errors and misconceptions. Collected data is presented that helped to address errors and misconceptions learners displayed in solving quadratic equations and teachers’ teaching methods and approaches. Data was collected from schools in the neighborhood and the school I was attached as a mathematics teacher. In all schools, five teachers’ three lessons were observed which added up to a total of fifteen. All five teachers were interviewed as well as five learners in each school. Interviews were analyzed by comparing what their teaching approaches with the types of learners’ errors and misconceptions. In classroom observations, Indicator Evaluation Form adopted from Luneta (2006) was used to collect data as well as analyzing it. Questionnaires were prepared for both teachers and learners for interviews.**Full Text:**

The role of logical principles in proving conjectures using indirect proof techniques in mathematics

**Authors:**Van Staden, Anna Maria**Date:**2012-08-28**Subjects:**Logic, Symbolic and mathematical , Proof theory**Type:**Thesis**Identifier:**uj:3369 , http://hdl.handle.net/10210/6769**Description:**M.Ed. , Recently there has been renewed interest in proof and proving in schools worldwide. However, many school students and even teachers of mathematics have only superficial ideas on the nature of proof. Proof is considered the heart of mathematics as individuals explore, make conjectures and try to convince themselves and others about the truth or falsity of their conjectures. There are basically two categories of deductive proof, namely proof by direct argument and indirect proofs. The aim of this study was to examine the structural features common to most of the mathematical proofs for formalised mathematical systems, with the emphasis on indirect proof techniques. The main question was to investigate which mathematical activities and logical principles at secondary school level are necessary for students to become proficient with proof writing. A great deal of specialised language is associated with reasoning. Such words as axiom, theorem, proof, and conjecture are just some of the terms that students must understand as they engage in the proof-making task. The formal aspect of mathematics at secondary school is extremely important. It is inevitable that students become involved with hypothetical arguments. They use among others, proofs by contradiction. Furthermore, necessary and sufficient conditions are related to theorems and their converses. It is therefore apparent that the study of logic is necessary already at secondary school level in order to practise mathematics satisfactorily. An analysis of the mathematics syllabus of the Department of Education has indicated that students should use indirect techniques of proof. According to this syllabus students should be familiar with logical arguments. The conclusion which is reached, gives evidence that students’ background in logic is completely lacking and inadequate. As a result they cannot cope adequately with argumentation and this causes a poor perception of what mathematics entails. Although proof writing can never be reduced to a mechanical process, considerable anxiety and uncertainty can be eliminated from the process if students are exposed to the principles of elementary logic and techniques. Mathematics educators and education researchers have reported students’ difficulties with mathematical proof and point out the conflict between the nature of this essential mathematical activity and current approaches to teaching it. This recent interest has led to an increased effort to teach proof in innovative ways.**Full Text:**

#### The role of logical principles in proving conjectures using indirect proof techniques in mathematics

**Authors:**Van Staden, Anna Maria**Date:**2012-08-28**Subjects:**Logic, Symbolic and mathematical , Proof theory**Type:**Thesis**Identifier:**uj:3369 , http://hdl.handle.net/10210/6769**Description:**M.Ed. , Recently there has been renewed interest in proof and proving in schools worldwide. However, many school students and even teachers of mathematics have only superficial ideas on the nature of proof. Proof is considered the heart of mathematics as individuals explore, make conjectures and try to convince themselves and others about the truth or falsity of their conjectures. There are basically two categories of deductive proof, namely proof by direct argument and indirect proofs. The aim of this study was to examine the structural features common to most of the mathematical proofs for formalised mathematical systems, with the emphasis on indirect proof techniques. The main question was to investigate which mathematical activities and logical principles at secondary school level are necessary for students to become proficient with proof writing. A great deal of specialised language is associated with reasoning. Such words as axiom, theorem, proof, and conjecture are just some of the terms that students must understand as they engage in the proof-making task. The formal aspect of mathematics at secondary school is extremely important. It is inevitable that students become involved with hypothetical arguments. They use among others, proofs by contradiction. Furthermore, necessary and sufficient conditions are related to theorems and their converses. It is therefore apparent that the study of logic is necessary already at secondary school level in order to practise mathematics satisfactorily. An analysis of the mathematics syllabus of the Department of Education has indicated that students should use indirect techniques of proof. According to this syllabus students should be familiar with logical arguments. The conclusion which is reached, gives evidence that students’ background in logic is completely lacking and inadequate. As a result they cannot cope adequately with argumentation and this causes a poor perception of what mathematics entails. Although proof writing can never be reduced to a mechanical process, considerable anxiety and uncertainty can be eliminated from the process if students are exposed to the principles of elementary logic and techniques. Mathematics educators and education researchers have reported students’ difficulties with mathematical proof and point out the conflict between the nature of this essential mathematical activity and current approaches to teaching it. This recent interest has led to an increased effort to teach proof in innovative ways.**Full Text:**

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