Thesis (PhD (Education))--University of Stellenbosch, 1998. / This dissertation deals with typical, widespread student errors with respect to kinematic graphs as revealed by a literature survey, as well as an own empirical investigation into the nature and extent of these misconceptions. The fact that certain misconceptions turned out to be more widespread than initially believed, has serious consequences for educators' assumptions about students' understanding of graphs in general, as well as their ideas on how to minimise some generally occurring "alternative views on graphs". Students' graphing skills are analysed and described in terms of a number of translations between various representations of physical events involving motion. A special focus is placed on graph transformations, which are translations from one graphical representation to another. It turned out that this provides valuable information about a learner's graphing skills, as well as his understanding of the relevant kinematic quantities and conventions required to make successful transformations.

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Thesis (PhD)--University of Stellenbosch, 2004. / ENGLISH ABSTRACT: This study investigated the impact that use of a microcomputer-based laboratory (MBL), in this case the TRAC PAC and associated software, had on student understanding in relation to common 'alternative learner ideas' and difficulties related to kinematics and kinematic graphs. It was carried out in the South African context, and focussed on subject matter that learners are expected to work with in preparation for the Senior Certificate South African examination. Twenty Grade 12 learners from four different schools participated in the study. Three key questions were investigated: 1. What conceptual difficulties do learners in this context experience in relation to kinematics and kinematic graphs? 2. Does use of the TRAC PAC as a microcomputer-based laboratory contribute to learner understanding of graphs of motion and related concepts? 3. If learning is enhanced using the TRAC PAC, what are some of the 'ways of learning' evident as learners participated in the MBL programme? To answer these questions, the study employed both an empirical quantitative dimension and an ethnographic qualitative dimension. The empirical study involved the use of pre- and post-questionnaires which were administered before and after learners participated in a TRAC PACbased learning programme comprising of six 3-hour learning activities conducted over three days. Overall learner performance on the questionnaires, as well as responses to individual questions, were analysed statistically, as well as through use of an 'item and matrix' analysis technique described by Svec (1999). Chapter 8 of this document reports on this component of the study. The ethnographic component of the study made use of observational data, and transcripts of video and audio recordings of learners as they participated in the learning activities. The data gathered using these techniques was analysed largely through use of a 'verbal analysis' technique described by Chi (1997). Chapter 9 of this document reports on this component of the study. In relation to Research Question 1, the main findings of the study were: • A literature review highlighted common 'alternative learner ideas' identified by other researchers, and these allowed me to group them into four main areas. These are described in Chapter 4 of this report. • The analysis of the questionnaires highlighted 'alternative learner ideas' that the group of learners who participated in this project held. These are described in Chapter 8 of this report. • The analysis of the video and audio transcripts also allowed for the identification of 'alternative learner ideas' held by this group of learners. These are described in Chapter 9 of this report. There was a high degree of commonality between the 'alternative learner ideas' identified through use of these three different sources. Research Question 2 was answered mainly through the empirical study described in Chapter 8 of this report. It was found that the MBL experience generally resulted in an improvement in learner understanding in this area of kinematics and kinematic graphs. More detailed statistical and 'item and matrix' analyses showed that the impact on learner understanding was better in certain areas than in others. The ethnographic study described in Chapter 9 contributed to answeri The degree of learner involvement in learning activities seemed to impact on the effectiveness of the programme. Possible factors impacting on involvement were identified . • 'Alternative learner ideas' were made visible in the context of 'argumentation episodes' and 'discussion and explanation episodes'. Consequently, these formed the contexts in which shifts in understanding were most likely to take place. Key learner behaviours and skills necessary for participation in these episodes are identified, and linked to success and non-success on the programme. Recommendations arrsmg from findings m the study are described m Chapters 8, 9 and 10of this report. / AFRIKAANSE OPSOMMING: Hierdie ondersoek bestudeer gebaseerde laboratorium, III gepaardgaande sagteware, op kinematikagrafieke het. die uitwerking wat 'n hierdie geval die TRAC studente se begrip van mikro-rekenaar PAC en die kinematika en Die ondersoek is III 'n Suid-Afrikaanse konteks uitgevoer en is toegespits op die vakmateriaal wat leerders behoort te beheers ter voorbereiding vir die Suid-Afrikaanse Senior Sertifikaat. Twintig Graad 12' s van vier verskillende skole het aan die ondersoek deelgeneem. Drie sleutelvrae is ondersoek: 1. Watter begripsprobleme ondervind leerders in hierdie verband met betrekking tot kinematika en kinematikagrafieke? 2. Dra die gebruik van die TRAC PAC as 'n mikro-rekenaar gebaseerde laboratorium by tot die leerder se begrip van kinematikagrafieke en verwante begrippe? 3. Indien "leer" deur die gebruik van die TRAC PAC bevorder word, watter "vorme van leer" is waarneembaar as leerders deelneem aan die MBL-program? Beide 'n empiries-kwantitatiewe dimensie en 'n etnografiese kwalitatiewe dimensie is in die navorsing gebruik. Die empiriese ondersoek maak van beide 'n voortoets en 'n na-toets gebruik. Hierdie vrae is aan die leerders voorsien voordat asook nadat hulle aan die TRAC PAC gebaseerde leerprogram deelgeneem het. Die leerprogram het bestaan uit ses leeraktiwiteite, elk drie uur lank, wat oor 'n tydperk van drie dae gedoen is. Die leerders se prestasie/uitslae met betrekking tot die vrae asook hul reaksie op individuele vrae is statisties ontleed, asook met behulp van die 'item en matriks' analitiese tegniek soos deur Svec (1999) beskryf. Hoofstuk 8 van hierdie verslag verwys na hierdie deel van die ondersoek. Die etnografiese komponent van die die ondersoek maak gebruik van waarnemingsdata en transkripsies van band- en video-opnames van leerders verkry tydens hul deelname aan die leeraktiwiteite. Die data so verkry, IS hoofsaaklik geanaliseer deur van die 'n verbale analise-tegniek gebruik te maak soos deur Chi (1999) voorgestel. Hoofstuk 9 van hierdie dokument doen verslag oor hierdie komponent van die ondersoek. Wat Navorsingsvraag 1 betref, IS die hoofbevindings van die studie die volgende: • 'n literatuur-oorsig beklemtoon die algemene alternatiewe leerderopvattings wat deur ander navorsers geidentifiseer is. Dit het my in staat gestelom hulle in 4 hoofareas te groepeer wat ek in hoofstuk 4 van die verslag bespreek. • Die analise van die vraelyste beklemtoon die alternatiewe leerderopvattings van die groep leerders wat aan hierdie proj ek deelgeneem het. Dit word in hoofstuk 8 van hierdie verslag bespreek. • Die analise van die band- en video-opnames het ook bygedra tot die identifikasie van' alternatiewe leerder-idees' wat by hierdie groep leerders voorkom. Dit word in hoofstuk 9 van hierdie verslag bespreek. Daar is 'n groot mate van ooreenkoms ten opsigte van die alternatiewe leerderopvattings wat by hierdie drie verskillende groepe voorkom. Navorsingsvraag 2 is hoofsaaklik beantwoord deur die emprrrese studie wat in hoofstuk 8 van hierdie verslag bespreek word. Daar is bevind dat die MBLondervinding oor die algemeen 'n vebetering in die leerders se begrip ten opsigte van kinematika en kinematikagrafieke tot gevolg gehad het. 'n Meer gedetailleerde statistiese 'item en matriks' -analise het getoon dat die uitwerking op die leerders se begrip in sommige areas beter was as in ander. Die etnografiese studie wat in hoofstuk 9 van hierdie verslag beskryf word, dra by tot die beantwoording van Navorsingsvraag 3. Sleutelbevindings met betrekking tot hierdie vraag sluit onder andere in: • Leerderdeelname aan leeraktiwiteite hou skynbaar verband met die sukses wat hulle in die program behaal. Moontlike faktore wat 'n invloed op deelname kon hê, is geidentifiseer. • Alternatiewe leerderopvattings is In die konteks van 'beredeneringsepisodes ' en 'besprekings- en verduidelikings-episodes' uitgelig. Hierdie "episodes" het die waarskynlikste verband uitgewys waarbinne veranderings van insig/begrip kan plaasvind. Kernleerdergedrag en vaardighede wat noodsaaklik IS vir die deelname aan hierdie episodes is geidentifiseer, en is gekoppel aan 'n leerder se sukses en mislukking tydens deelname aan die program. Aanbevelings wat voortspruit uit die bevindings van die ondersoek word In hoofstukke 8, 9 en 10 van hierdie verslag, bespreek.

The crucial role of representation in mathematical and statistical modeling and problem solving as evident in learners' arrangement and representation of statistical data were investigated with focus points data arrangement, data representation and statistical thinking levels. The representation tasks required learners to arrange and represent data through modeling, focusing on spontaneous representations. Successful transnumeration determine the ultimate success of a representation and the ability to organise data is regarded as critical. Arrangement types increased in sophistication with increased grade level and the hierarchical nature of arrangement types became apparent when regarded in the context of an adapted SOLO Taxonomy framework. A higher level arrangement strategy pointed to a higher SOLO level of statistical thinking. Learners in the two tasks produced a rich variety of representations which included idiosyncratic, unsophisticated responses as well as standard statistical representations. The context of the two tasks, the quantitative versus qualitative nature of the data in the tasks, and the statistical tools or representational skills learners have at their disposal, played an important role in their representations. Well-planned data handling activities develop representational and higher order thinking skills. The variety of responses and different response levels elicited in the two tasks indicate that the nature of the tasks rather than the size of the data set play a conclusive role in data tasks. Multiple representations by an individual were an indication of successful modeling, are effective in problem solving and are associated with good performance. The SOLO model which incorporates a structural approach as well as a multimodal component proved valuable in the analysis of responses. Using this model with accompanying acknowledgement of different problem solving paths and the contribution of ikonic support in the concrete symbolic mode, promote the in-depth analysis of responses. This study contributes to the research in the field of data representation and statistical thinking. The analysis and results led to an integrated picture of Grade 4-7 learners' representation of statistical data and of the statistical thinking levels evident in their representations. / Educational Studies / D. Ed. (Didactics)

The crucial role of representation in mathematical and statistical modeling and problem solving as evident in learners' arrangement and representation of statistical data were investigated with focus points data arrangement, data representation and statistical thinking levels. The representation tasks required learners to arrange and represent data through modeling, focusing on spontaneous representations. Successful transnumeration determine the ultimate success of a representation and the ability to organise data is regarded as critical. Arrangement types increased in sophistication with increased grade level and the hierarchical nature of arrangement types became apparent when regarded in the context of an adapted SOLO Taxonomy framework. A higher level arrangement strategy pointed to a higher SOLO level of statistical thinking. Learners in the two tasks produced a rich variety of representations which included idiosyncratic, unsophisticated responses as well as standard statistical representations. The context of the two tasks, the quantitative versus qualitative nature of the data in the tasks, and the statistical tools or representational skills learners have at their disposal, played an important role in their representations. Well-planned data handling activities develop representational and higher order thinking skills. The variety of responses and different response levels elicited in the two tasks indicate that the nature of the tasks rather than the size of the data set play a conclusive role in data tasks. Multiple representations by an individual were an indication of successful modeling, are effective in problem solving and are associated with good performance. The SOLO model which incorporates a structural approach as well as a multimodal component proved valuable in the analysis of responses. Using this model with accompanying acknowledgement of different problem solving paths and the contribution of ikonic support in the concrete symbolic mode, promote the in-depth analysis of responses. This study contributes to the research in the field of data representation and statistical thinking. The analysis and results led to an integrated picture of Grade 4-7 learners' representation of statistical data and of the statistical thinking levels evident in their representations. / Educational Studies / D. Ed. (Didactics)

One major challenge facing mathematics education in South Africa in general and Limpopo in particular, is learners’ underachievement and lack of motivation to learn the subject. Some studies have shown that one of the topics that learners dread is linear functions. Many teachers also find it difficult to teach the topic effectively. Studies in other parts of the world have advocated the integration of graphing software with the teaching and learning of functions to enhance learners’ learning of mathematics. This study therefore investigated the effect of integrating GeoGebra graphing software into the teaching of linear functions on the achievement of Grade 9 learners. The study was guided by APOS theory which, in accordance with constructivist theories, posits that an individual needs to construct the necessary cognitive structures in order to make sense of mathematical concepts. A total of 127 Grade 9 learners from four schools in a circuit in Mopani district of Limpopo Province participated in the study which followed a pretestpost- test quasi-experimental study design. Two schools, namely B (35 learners) and D (33 learners) formed the experimental groups while school A (31 learners) and school C (28 learners) were the control groups. Data were collected using an achievement test and analyzed using descriptive and inferential statistics. The pretest results showed that the groups were of comparable cognitive abilities. The post-test results showed that there was a significant difference between the mean scores of the experimental groups and control groups. There were also statistically significant differences between group treatment means (p < .05). Bonferroni post-hoc test results showed that there were no statistically significant differences between treatments A and C. The results showed that the learners in the two control groups were of comparable cognitive abilities. The implications of the findings are discussed and recommendations made. / Mathematics Education / M. Sc. (Mathematics Education)

The results of mathematics level 3 have always been a problem at TVET colleges as this hampers the certification rate and the progress of the students to level 4. Students who did not do well in the current subject are not allowed to register that subject in the following level. Even though the students are allowed to progress to level 4 they won’t be certificated for both levels until they pass the remaining subject. The above challenges made the researcher to check during the marking and moderation of November / December examination the course of poor results for mathematics level 3. In the process of checking the researcher discovered that rectangular hyperbola is one of the topics that the students of mathematics level 3 are struggling with. This study therefore focuses on exploring the challenges faced by TVET Level 3 NCV students in understanding the hyperbolic function in mathematics. In addition to the literature review, an empirical investigation based on a qualitative approach and involving semi-structured interviews with the students of a TVET college in North West was conducted to collect data. The analysis of documents relevant to the study was also used as the other method. The study used participatory action research, where the researcher, collaborators and students work alongside each other to collect data and to improve practice and follow the spiral pattern of reflection, analysing the results and adapting the action. The research design and methodology was qualitative. This helped the researcher to understand the challenges students faced in the learning of rectangular hyperbola and also came up with ways to minimise those challenges. The data collection methods used was interviewing using semi-structured questions, pre-test and post-tests. During data collection different interventions (IN1 –IN3) was used depending on the understanding of the students. For ethical consideration, ethical clearance was obtained from UNISA. DHET, the principal of the college, collaborators, parents and students will also give written consent on forms which will be sent out explaining what we envisage. Since research was voluntary, an explanation was given that this was not compulsory and that participation was completely voluntary and that they could withdraw at any time. In this study, various methods to empower students were recommended. Recommendations are also made on what was found in this study, as are recommendations for further study. / Mathematics Education / M. Ed. (Mathematics Education)