The purpose of this study was to determine the relationship between High School learners‟ mathematics-related belief systems and their approaches to mathematics non-routine problem-solving. A mixed methods approach was employed in the study. Survey questionnaires, mathematics problem solving test and interview schedules were the basic instruments used for data collection. The data was presented in form of tables, diagrams, figures, direct and indirect quotes of participants‟ responses and descriptions of learners‟ mathematics related belief systems and their approaches to mathematics problem solving. The basic methods used to analyze the data were thematic analysis (coding, organizing data into descriptive themes, and noting relations between variables), cluster analysis, factor analysis, regression analysis and methodological triangulation. Learners‟ mathematics-related beliefs were grouped into three Learners‟ mathematics-related beliefs were grouped into three categories, according to Daskalogianni and Simpson (2001a)‟s macro-belief systems: utilitarian, systematic and exploratory. A number of learners‟ problem solving strategies were identified, that include unsystematic guess, check and revise; systematic guess, check and revise; trial-and-error; logical reasoning; non-logical reasoning; systematic listing; looking for a pattern; making a model; considering a simple case; using a formula; numeric approach; piece-wise and holistic approaches. A weak positive linear relationship between learners‟ mathematics-related belief systems and their approaches to non-routine problem solving was discovered. It was, also, discovered that learners‟ mathematics-related belief systems could explain their approach to non-routine mathematics problem solving (and vice versa). / Mathematics Education / D.Phil. (Mathematics Education)

The aim of this study was to investigate the comparative effects of Constructivist-Based Teaching Method (CBTM) and the Traditional Teaching Method (TTM) on Grade 11 Mathematics learners’ errors in algebra. The constructivist learning theory (CLT) was used to frame this study. Mainly, CLT was used to influence the design of CBTI to hone participants’ errors in algebra that militate against their performance in Mathematics. The study was conducted in the Mpumalanga Province of South Africa with a four-week intervention programme in each of the two participating secondary schools. Participants consisted of n=78 Grade 11 Mathematics learners and one Grade 11 Mathematics teacher. A non-equivalent control group design consisting of a pre-test and post-test measure was employed. The Grade 11 teacher in the control school employed the TTM while the researcher implemented CBTM in the experimental school. The main aspects of CBTM entailed participants’ construction of their own knowledge from the base of prior knowledge and through group learning approach and exploratory talk in which discussions included argumentation, verbalising explanations, justifications and reflections. Participants in experimental school became familiar with the basic principles of CBTI such as group work, which enhanced the construction of conceptual understanding of algebraic concepts. This reduced most of the errors they commit in algebra and elevated their performance in Mathematics. The principal instruments for data collection consisted of a standardised Algebra Concept Achievement Test and lesson observations. The pre-test was used to determine participants’ initial errors in algebra before the intervention. A post-test was given at the end of intervention to ascertain change in participants’ errors in algebra over a four-week intervention period. Using descriptive and inferential statistical techniques, the study found that participants in experimental school significantly reduced their errors in algebra than those in control school. The study showed that CBTM was a more effective pedagogy that improved the errors Grade 11 learners commit in algebra than the TTM. / College of Education / M. Ed. (Mathematics Education)

The study intends to investigate the nature and cognitive demands of contextual word-problems posed in the FET mathematics examinations of IEB and NSC. The analysis of the mathematization of real-life situations to form contextual word-problems is based on the theory of authentic task situations. The theoretical basis for analyzing mathematics teaching and learning is the Realistic Mathematics Education (RME) theory. Data was obtained using the schedule of mathematization of real-life situations and the schedule of total marks of contextual word-problems and national performance. All contextual word-problems included in the 2008-2013 question papers of IEB and NSC mathematics examinations were analysed. The research revealed that 509 marks out of 1800 marks were allocated to contextual word-problems in IEB examinations; whereas 473 marks out of 1800 marks were allocated to contextual word-problems in NSC examinations. / Mathematics Education / M. Sc. (Mathematical Education)

The aim of this study was to identify types of behaviour manifested by learners during mathematics instruction, and the impact that this behaviour might have on the mathematics performance of learners. The study was conducted in Far East cluster of Johannesburg East District, in the province of Gauteng, South Africa. At the time of the study the Far East cluster of Johannesburg East District consisted of a population of seven public high schools, of which two schools were randomly sampled to participate in the study. Participants consisted of (n=10) Grade 10 mathematics learners, 2 mathematics teachers and 2 heads of mathematics departments (HODs). Data from learners were collected using a set of their assessment scores accumulated over a period of six months, that is, from January 2014 to June 2014 (Section 1.3.3). Also, semi-structured interviews were carried out with learners to determine types of classroom behaviour they perceived to influence their mathematical performance. The aim of documenting learners‟ assessment scores (document analysis) was to determine their average performance in Grade 10 mathematics over a stipulated period. Teachers and HODs completed questionnaires to identify types of classroom behaviour that learners manifested during mathematics instruction. The study followed a qualitative approach with phenomenology research design (Section 3.2). The study identified several types of classroom behaviour that characterized mathematics instruction in both schools, such as making noise and not doing classwork and homework activities. In addition, the study established that forms of behavioural patterns that are manifested by learners during a mathematics instruction influenced their performance in the subject. Huitt‟s (1997) model was used to conceptualize and interpret the results. / Mathematics Education / M. Ed. (Mathematics Education)

O presente trabalho tem por objetivo divulgar e explorar as Provas sem Palavras no âmbito do Ensino Fundamental e Médio para, posteriormente, servir de possível material de consulta por parte de professores, na busca de melhores métodos de explicação sobre importantes assuntos desenvolvidos em sala. Dessa forma, imagina-se que a compreensão dos alunos através da visualização fortalecerá de maneira significativa a aquisição do conhecimento. Para tanto, são sugeridos diversos temas onde a visualização é desenvolvida e explicada, seja através de relato, seja através de demonstrações matemáticas. / The aim of the present work is to disseminate and explore the Proof Without Words in the scope of elementary and high school to be used later as a possible reference material for teachers in the search for better methods of explanation on important subjects developed in the classroom. In this way, it is imagined that students’ comprehension through visualization will significantly strengthen the acquisition of knowledge. In order to do so, several themes are suggested where the visualization is developed and explained, either through storytelling or through mathematical demonstrations.

Computer technology has been used in education for years, and the government budgets large amounts of money to foster technology. However, it is still a debated whether computer technology makes a difference in students' learning outcomes. The purpose of this study is to find if any relationship exists between computer use by teachers and students and the students' academic achievement in math and reading for both traditional populations and English language learner (ELL) tenth graders. Computer use in this study included the computer activities by students and teachers, in terms of the time, frequency, activities types, the places students use computers, teachers' computer activities, and the training teachers received. This study used data gathered from tenth grade students from the dataset Education Longitudinal Study of 2002 (ELS:2002) of the National Center for Education Statistics (NCES). Fifteen thousand, three hundred and sixty-two students were randomly selected to represent all U.S. tenth-graders attending schools in 2002. The findings showed diverse relationships consistent with the literature. Based on the findings, some suggestions were made to teachers and parents about the quality of school work and computer use by students and teachers.

The purpose of this project is to provide a useful model, the Concept Attainment Model in Mathematics, which implicates the concept attainment teaching method based on adjunct content-based instruction by using visuals and manipulatives in order to help ESL students be successful for both substantive content areas.

This study investigated teachers' and learners' perceptions of the causes of poor performance in Mathematics among Grade 9 learners in Durban's Ethusini Circuit of KwaZulu-Natal Province. By using a phenomenographic research design, a sample of 15 Grade 9 learners who take Mathematics, one Grade 9 Mathematics teacher and one Mathematics head of department was selected purposively from three high schools in the Ethusini Circuit. The three schools are one former model C school, one ordinary government school and one private school. Focus group and semi-structured interviews were used to collect data from the participants. The data were analysed by transcribing the focus group and semi-structured interviews; thereafter, the transcriptions were coded into themes and categories. The results of this study indicate that teachers and learners perceive that poor performance in Mathematics in Grade 9 is due to CAPS curriculum challenges, such as overloaded and advanced content, and insufficient time to complete the curriculum. Parental support, English (first additional language) as a language of learning and teaching, methods of teaching (such as the talk and chalk method), and learners' negative attitude towards Mathematics were also cited. From these findings, recommendations for revisiting the CAPS curriculum and other areas of need, as well as topics for future research were suggested. / Dinyakišišo tše di nyakišišitše ka ga maikutlo a barutiši le a baithuti ka ga dilo tšeo di bakago go se šome gabotse ka thutong ya Dipalo gareng ga baithuti ba Kreiti ya 9 ka Sedikothutong sa Ethusini ka Durban ka Phrobentsheng ya KwaZulu-Natal. Ka go šomiša mokgwa wa dinyakišišo wa fenomenokrafiki, sampole ya baithuti ba 15 ba Kreiti ya 9 bao ba ithutelago Dipalo, morutiši o tee wa Dipalo ka go Kreiti ya 9 le morutiši o tee yo e lego hlogo ya lefapha la Dipalo o kgethilwe ka maikemišetšo go tšwa dikolong tše di phagamego tše tharo ka Sedikothutong sa Ethusini. Dikolo tše tharo tše ke sekolo se tee seo se bego se le sa motlolo wa C, se tee sa tlwaelo sa mmušo le se tee ke sekolo sa phraebete. Sehlopha seo se nepišitšwego le dipoledišano tša dipotšišo tšeo di beakantšwego peleng di šomišitšwe go kgoboketša tshedimošo go tšwa go bakgathatema. Tshedimošo e sekasekilwe ka go ngwalolla dipoledišano tša dipotšišo tša sehlopha seo se nepišitšwego le tšeo di beakantšwego peleng; ka morago ga fao, dingwalollo tše di filwe dikhoutu go ya ka merero le magoro a tšona. Dipoelo tša dinyakišišo tše di laetša barutiši le baithuti ba bona gore go se šome gabotse ga baithuti ka thutong ya Dipalo ka go Kreiti ya 9 ke ka lebaka la ditlhohlo tša lenaneothuto la Setatamente sa Melawana ya Tekolo ya Lenaneothuto (CAPS), go swana le diteng tša thuto tšeo di tletšego kudu le tšeo di gatetšego pele kudu, le nako ye e sego ya lekana ya go fetša lenaneothuto. Thekgo ya batswadi, Seisemane (polelo ya gae ya tlaleletšo) bjalo ka polelo ya go ithuta le ya go ruta, mekgwa ya go ruta (ya go swana le mokgwa wa go bolela le go ngwala ka tšhooko letlapeng), le maikutlo ao a sego a loka a baithuti go Dipalo le tšona di filwe bjalo ka dilo tšeo di bakago se. Go dikutollo tše, ditšhišinyo tša go etela leswa lenaneothuto la CAPS le makala a mangwe ao o hlokago šedi, gammogo le dihlogotaba tša dinyakišišo tša ka moso le tšona di ile tša šišinywa. / Lolu cwaningo luphenye imibono yothisha kanye neyabafundi yezimbangela zokungasebenzi kahle esifundweni seziBalo kubafundi beBanga le-9 kuSekhethi yase-Ethusini eThekweni esifundazweni saKwaZulu-Natali. Ngokusebenzisa isakhiwo sokucwaninga esisezingeni eliphezulu, isampula yabafundi abayi -15 beBanga le-9 abathatha iziBalo, uthisha oyedwa weBanga le-9 kanye nomphathi oyodwa womnyango weziBalo bakhethwa ngenhloso ezikoleni ezintathu eziphezulu kuSekhethi yase-Ethusini. Lezi zikole ezintathu esisodwa semodeli C, esisodwa sikahulumeni ojwayelekile kanye nesisodwa isikole esizimele. Iqembu lokugxila kanye nezingxoxo ezihlelwe kahle zisetshenzisiwe ukuqoqa imininingwane evela kubahlanganyeli. Imininingwane yahlaziywa ngokubhalwa kweqembu okugxilwe kulo kanye nezingxoxo ezihlelwe kahle; emva kwalokho, imibhalo ivezwe ngendlela engaqondile ezindikimbeni nasezigabeni. Imiphumela yalolu cwaningo ikhombisa ukuthi othisha kanye nabafundi babona ukuthi ukungasebenzi kahle esifundweni seziBalo kuBanga lesi-9 kungenxa yezinselela yezifundo ezibandakanya isifundo esifundwayo seCAPS, njengokuqukethwe okugcwele kakhulu nokuthuthukile, kanye nesikhathi esinganele sokuphothula izifundo ezibandakanya isifundo esifundwayo. Ukusekelwa kwabazali, isiNgisi (ulimi lokuqala olwengeziwe) njengolimi lokufunda nokufundisa, izindlela zokufundisa (njengeyokufundisa lapho uthisha ebhala ebhodini), kanye nesimo sengqondo esingesihle sabafundi maqondana neziBalo nakho kuveziwe. Ukusuka kulokhu okutholakele, kuphakanyiswe ukuphinda kuphinde kufundwe izifundo ezibandakanya isifundo esifundwayo seCAPS kanye nezinye izindawo ezidingeka, kanye nezihloko zocwaningo lwakusasa ziphakanyisiwe. / Curriculum and Instructional Studies / M. Ed. (Curriculum and Instructional Studies)

The aim of this study was to investigate the effect of context-based problem solving instruction (CBPSI) on the problem solving performance of Grade 10 learners, who performed poorly in mathematics. A cognitive load theory (CLT) was used to frame the study. In addition, CLT was used to: 1) facilitate the interpretation and explanation of participants‟ problem solving performance; and, 2) influence the design of CBPSI to hone participants‟ problem solving skills. The study was conducted in the Gauteng province of South Africa and involved a two-week intervention program in each of the nine participating high schools. Participants consisted of 783 learners and four Grade 10 mathematics teachers. A non-equivalent control group design was employed, consisting of a pre- and post- measure. In addition, classroom observations and semi-structured interviews were conducted with teachers and learners. Teachers employed conventional problem solving instructions in four control schools while the researcher implemented CBPSI in five experimental schools. Instruction in experimental schools entailed several worked-out context-based problem solving examples given to participants in worksheets. The main aspects of CBPSI embraced elements of the effects of self-explanation and split-attention, as advocated by CLT. Due to the design of CBPSI participants in experimental schools became familiar with the basic context-based problem solving tasks that were presented to them through the worked-out example samples. In turn, the associated cognitive load of problem solving tasks was gradually reduced. The principal instrument for data collection was a standardized Functional Mathematics Achievement Test. The pre-test determined participants‟ initial problem solving status before intervention. A post-test was given at the end of intervention to benchmark change in the functionality of CBPSI over a two-week period. Using one-way analysis of covariance (ANCOVA), Analysis of Variance (ANOVA), and other statistical techniques the study found that participants in experimental schools performed significantly better than participants in control schools on certain aspects of problem solving performance. In addition, semi-structured interviews and classroom observations revealed that participants rated CBPSI highly. On the whole, the study showed that CBPSI is an effective instructional tool to enhance the problem solving performance of Grade 10 mathematics learners. / Mathematics Education / D. Phil. (Mathematics, Science and Technology Education)

The study’s purpose was to explore the components of pedagogical content knowledge (PCK) for teaching quadratic function in Grade 10 mathematics classrooms in Mogalakwena district, Limpopo. The study was mainly guided by Shulman’s PCK and it was categorised as content knowledge, knowledge of the curriculum, knowledge of learners and knowledge of teaching strategies. The researcher employed a qualitative case study research design to explore the teaching of three purposefully selected Grade 10 mathematics teachers regarding their knowledge of quadratic function, strategies employed to teach quadratic function and knowledge of Grade 10 mathematics learners in quadratic function. In this study the researcher argued that PCK strands are interwoven, and in order to teach mathematical concepts, such as quadratic function, effectively teachers should employ components of PCK to complement one another. Data were gathered by lesson observation and interviews. Findings revealed that concepts of quadratic function are inefficiently addressed in Grade 10 due to teachers’ lack or inadequacy in some aspects of PCK. Therefore, there is need to develop mathematics teachers’ PCK in the Mogalakwena district to enhance their teaching of Grade 10 quadratic function. / Mathematics Education / M. Ed. (Mathematics Education)