Thesis (PhD)-- Stellenbosch University, 2013. / ENGLISH ABSTRACT: Mathematics teacher development is a source of national and international concern. This study describes how primary school mathematics teachers develop didactisation practices. In considering how teachers could develop, so that student learning is optimised; the concepts of didactisation and the mathematical work of teaching were sourced from existing literature. The concept didactisation is explored and defined; and is incorporated with the concept of mathematical work of teaching. Nine practices were made explicit through this incorporation: active students, differentiation, mathematisation, vertically aligned lessons, access, probe, connect and assess student thinking, and teacher reflection. These nine practices become the framework for the professional development program and the data generation structure. Five primary school teachers were involved in a professional development program that used model-eliciting activities (MEAs) as a point of departure. A modelling perspective to teacher learning was chosen for the professional development program. The methodology followed the principles of design research and from this, a three phase teaching experiment was designed and implemented. The teachers and researcher met for development sessions and teachers were observed in practice at intervals throughout the program. Their developing didactisation practices were documented through a qualitative analysis of the data. It was established that teachers’ didactisation practices did develop during the nine-month program. Furthermore it was found that didactisation practices developed at different rates and consequently, a hierarchy of didactisation practice development is presented. The impact of the program was also gauged through teachers’ changing resources, goals and orientations. These three aspects also evolved over time. The program proposed in this study may be a suitable model to develop in-service and pre-service mathematics teachers. The study contributes to understanding teacher action in a classroom and how teachers can change their own thinking and practice. / AFRIKAANSE OPSOMMING: Die ontwikkeling van wiskundeonderwysers is ‘n bron van nasionale en internasionale kommer. Hierdie studie beskryf hoe die didaktiseringspraktyke van laerskool wiskundeonderwysers met die oog op optimalisering van leer ontwikkel het. In die bestudering van die ontwikkeling van onderwysers met die oog op optimalisering van leer, is die begrippe didaktisering en die wiskundige werk van onderrig (mathematical work of teaching) nagespoor uit bestaande literatuur. Die begrip didaktisering is deeglik ondersoek, gedefinieer en saamgevoeg met die begrip wiskundige werk van onderrig. Nege praktyke is eksplisiet gemaak deur hierdie inkorperering: aktiewe studente, differensiasie, matematisering, vertikaalgerigte lesse, toegang, indringende ondersoek, gekonnekteerdheid en assessering van studente-denke, en onderwyserrefleksie. Hierdie nege praktyke het die raamwerk gevorm vir die professionele ontwikkelingsprogram en die data genereringstruktuur. Vyf laerskool onderwysers was betrokke in ‘n professionele ontwikkelingsprogram waarin model-ontlokkende aktiwiteite (MOA’s) as ‘n vertrekpunt gebruik is. ‘n Modelleringsperspektief is vir onderwyserleer in die ontwikkelingsprogram gekies. Die metodologie volg die beginsels van ontwerpnavorsing waarna ‘n drie-fase onderrig-eksperiment ontwerp en in werking gestel is. Die navorser en die onderwysers het byeengekom vir ontwikkelingsessies; die onderwysers is op ‘n gereelde basis tydens die program besoek om hul onderwyspraktyk waar te neem. Hul ontwikkelende didaktiseringspraktyke is gedokumenteer en die data is kwalitatief geanaliseer. Onderwysers se didaktiseringspraktyke het wel gedurende die negemaande program ontwikkeling getoon. Hierdie didaktiseringspraktyke het egter teen verskillende tempo’s ontwikkel en daarom kon ‘n hierargie van die ontwikkeling van didaktiseringspraktyke saamgestel word. Die impak van hierdie program op onderwysers se veranderende hulpbronne, doelstellings en oriëntasies is ook gemeet. Die drie aspekte het in hierdie nege maande verder ontwikkel. Die voorgestelde program in hierdie studie mag moontlik ‘n gepaste model wees om indiens en voornemende wiskundeonderwysers te ontwikkel. Die studie lewer ‘n bydrae tot ‘n beter begrip van onderwyserhandelinge in ‘n klaskamer, asook hoe onderwysers hul eie denke en praktyke kan verander.

Thesis (PhD)-- Stellenbosch University, 2013. / ENGLISH ABSTRACT: The teaching and learning of algebra in the middle school grades in Lesotho is dominated by the mechanistic approach where learners are drilled on procedures for solving certain types of problems in algebra, without making any connection to the experience learners had with arithmetic. This is one of the sources of learners’ difficulties in mathematics. Research indicates that use of spreadsheets such as Excel has a potential of bridging the gap between arithmetic and algebra and thus enhancing the teaching and learning of algebra, making it meaningful to the learners. The study sought to answer the question: How do Grade 9 learners in Lesotho experience teaching and learning of algebra through Excel spreadsheets? The research commenced with a literature review that was followed by the empirical study. The theories of instrumental genesis and instrumental orchestration were identified as the framework for the investigation. Instrumental genesis is the process in which learners develop facility with the artifact as they use it towards achieving lesson objectives; technical (conceptual, mechanical) and personal (attitudes, behavior and preferred learning styles) aspects of learners’ experiences were identified. Instrumental orchestration is the steering of learners’ instrumental genesis by the teacher and the manner in which this process is carried out, depends on the teacher’s Technological Pedagogical Content Knowledge (TPCK). The research was a multi-case study following a mixed-methods approach, where both qualitative and quantitative methods were used. The empirical study was conducted in two schools in Lesotho. In each school, fifteen learners volunteered participation. The investigation was done through classroom teaching by me as the researcher. The focus was on what challenges learners encountered and how they benefited from their “spreadsheets algebra” learning experience. Data were collected through classroom observations where field notes were recorded and an observation schedule was used by the researcher and the Assistant Observer respectively. A questionnaire was also administered to all learner participants after the whole teaching period. Six learners, representative of high, medium and low performances in class, were also interviewed with a goal of finding out their experiences. The Assistant Observer was also interviewed to reduce the bias that may result from to the researcher studying her own practice The study found that learners experiences with learning algebra through spreadsheets, comprised of both challenges and benefits. The challenges encountered by learners could be classified into those that were school-based and those that were instruction-based. The school-based challenges related to inadequate physical structures and lack of well-functioning equipment in the computer laboratories. Instruction-based challenges encountered by learners were both technical and personal. Technical challenges related to the physical manipulation of the artifact and the lack of understanding of concepts involved, where the spreadsheets meet the algebra. Personal challenges related to learners’ attitude and behavior towards use of spreadsheets for algebra teaching and learning. While school authorities could address some of the challenges, it was found that both the teacher and learners could initiate strategies that could be used to overcome the instruction-based challenges. Teaching strategies such as “technical-demo”, “explain-the-screen”, “discuss-the–screen”, “link-screen-board” and “spot-and-show” (building on learners’ responses), and the general organization of classroom environment were helpful in orchestrating algebra learning within the spreadsheets. It was also found that use of spreadsheets had both cognitive and affective values for the learners. Even though use of spreadsheets may benefit both teachers and learners in algebra teaching and learning, implementation of the practice would require critical considerations in terms of teacher preparation and infra-structural improvements in the schools. / AFRIKAANSE OPSOMMING: Die onderrig en leer van algebra in die middelbare skoolgrade in Lesotho word oorheers deur die meganiese benadering waarvolgens leerders gedril word in prosedures om oplossings vir sekere tipe algebraprobleme te vind en die ervaring wat leerders in rekenkunde opgedoen het, nie daarmee in verband gebring word nie. Dit is een van die oorsake waarom leerders met wiskunde sukkel. Navorsing toon dat die gebruik van sigblaaie soos Excel moontlik die gaping tussen rekenkunde en algebra kan oorbrug, en dat die onderrig en leer van algebra daardeur kan verbeter, wat dit sinvol vir leerders sal maak. Die studie was daarop gemik om ’n antwoord op die volgende vraag te vind: Hoe ervaar graad 9-leerders in Lesotho die onderrig en leer van algebra deur middel van Excel-sigblaaie? Die navorsing het met ’n literatuuroorsig begin en is deur ’n empiriese studie opgevolg. Die teorieë instrumentele genese en instrumentele orkestrasie is uitgewys as die raamwerk vir die ondersoek. Instrumentele genese is die proses waarvolgens leerders bedrewenheid in die produk ontwikkel namate hulle dit gebruik om lesdoelstellings te bereik; tegniese (konseptuele, meganiese) en persoonlike (ingesteldheid, gedrag en voorkeurleerstyle) aspekte van leerders se ervarings is geïdentifiseer. Instrumentele orkestrasie is die stuur van leerders se instrumentele genese deur die onderwyser; en die wyse waarop hierdie proses uitgevoer word, hang van die onderwyser se Tegnologiese Pedagogiese Inhoudskennis (TPCK) af. Die navorsing het ’n meervoudige gevallestudie gebruik en ’n gemengde metodebenadering is gevolg, terwyl beide kwalitatiewe en kwantitatiewe metodes gebruik is. Die empiriese studie is in twee skole in Lesotho uitgevoer. Vyftien leerlinge uit elke skool het vrywillig deelgeneem. Die ondersoek is by wyse van klaskameronderrig deur my as die navorser gedoen. Daar is gefokus op die uitdagings wat leerders teëgekom het en hoe hulle by die “sigbladalgebra”-leerervaring gebaat het. Data is aan die hand van klaskamerwaarnemings versamel, waar veldnotas afgeneem is en ’n waarnemingskedule onderskeidelik deur die navorser en die assistentwaarnemer gebruik is. ’n Vraelys is na die volle onderrigtydperk by al die leerderdeelnemers afgeneem. Onderhoude is gevoer met ses leerders, wat hoë, medium en lae prestasies in die klaskamer verteenwoordig, met die doel om hulle ervarings te bekom. ’n Onderhoud is ook met die Assistentwaarnemer gevoer om vooroordeel, deurdat die navorser moontlik haar eie praktyk kon bestudeer, te verminder. Die studie het bevind dat leerders se ervarings met die leer van algebra deur middel van sigblaaie uitdagings sowel as voordele inhou. Die uitdagings wat leerders teëgekom het, kan onderskeidelik as skoolgebaseerde uitdagings en onderrig-gebaseerde uitdagings geklassifiseer word. Die skoolgebaseerde uitdagings hou verband met onvoldoende fisiese strukture en ’n gebrek aan behoorlik funksionerende toerusting in die rekenaarlaboratoriums. Die onderrig-gebaseerde uitdagings vir leerders is beide tegnies en persoonlik. Tegniese uitdagings hou verband met die fisiese manipulering van die produk en ’n gebrek aan begrip ten opsigte van die betrokke konsepte, naamlik waar die verband tussen die sigblaaie en algebra bestaan. Persoonlike uitdagings hou verband met leerders se ingesteldheid en gedrag rakende die gebruik van sigblaaie by die onderrig en leer van algebra. Alhoewel die skoolowerhede bepaalde uitdagings kan aanspreek, is daar bevind dat onderwysers sowel as leerders strategieë kan aanvoor om onderrig-gebaseerde uitdagings te oorkom. Onderrigstrategieë soos “tegniese demo”, “verduidelik die skerm”, “bespreek die skerm”, “koppel-skerm-bord” en “vind-en-wys” (wat op leerders se reaksie voortbou), asook die algemene organisasie van die klaskameromgewing, dra tot die orkestrasie van algebraonderrig met die sigblaaie by. Daar is ook bevind dat die gebruik van sigblaaie kognitiewe sowel as affektiewe waarde vir die leerders inhou. Selfs al sou onderwysers en leerders voordeel uit die gebruik van sigblaaie in die onderrig en leer van algebra kon trek, sou die implementering van die praktyk kritiese oorwegings moet geniet ten opsigte van onderwyservoorbereiding en infrastruktuurverbetering by die skole.

Thesis (PhD)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: The aim of this study was to investigate and describe the current state of Afrikaans FAL teaching in selected Gr 4 – 6 classes in Western Cape urban schools. This was done by presenting an overview of the literature relevant to FAL teaching and FAL curricula as well as the results from questionnaires and semi-structured interviews with Gr 4 – 6 teachers. This study does not offer a quick-fix solution to the problems in the Afrikaans FAL classrooms, but the researcher believes that the findings will highlight the daily challenges Afrikaans FAL teachers have to face and that all role players will become actively involved in improving the state of Afrikaans FAL teaching in the Western Cape. The relevance of this study lies in the national drive towards the promotion of multilingualism among the general population and especially in education. Feedback from student teachers returning from practice teaching indicated that not enough time is allocated by schools for the instruction of Afrikaans FAL; greatly differing methodologies as well as teaching and learning materials are being used in Afrikaans FAL classes; and learner and teacher Afrikaans proficiency varies from class to class. A theoretical framework for language teaching and learning, a literature study pertaining to first additional language teaching nationally and internationally, and an analysis of South African FAL curricula support the research. Constructivism, social constructivism and teacher knowledge were identified as the underpinning theories for language teaching and learning. The literature study provides an overview of all the major methodologies relevant to FAL teaching and the researcher concluded that there is no single method or approach that will ensure effective FAL teaching, but that teachers should implement an eclectic approach to achieve the best results. This study used a mixed methods approach to generate empirical data; 125 questionnaires, completed by Grade 4 – 6 Afrikaans FAL teachers, provided the quantitative data. For the qualitative strand of the study, semi-structured interviews were conducted with 17 Afrikaans FAL teachers. The data of these interviews were used as triangulation, to confirm or disconfirm and elaborate on the results of the questionnaires. One of the major factors impacting negatively on the teaching and learning of Afrikaans FAL is the negative attitude towards Afrikaans among many learners and their parents. Teachers are not equipped with teaching strategies and techniques to manage these negative attitudes. The results show that many teachers still follow a teacher-centred approach to teaching, which is an indication that learners are not given enough opportunities to develop their communicative competence through interaction with others. Another factor affecting Afrikaans FAL teaching negatively is that not all schools implement the prescriptions of the various language policies and curricula as they should. The study also investigated the use of appropriate and relevant learning and teaching support materials (LTSM) in the FAL classroom. The results showed that most teachers still mainly use the textbook as teaching resource. There is a dire need for appropriate Afrikaans LTSM for FAL. The expectation is that, in the age of technology we find ourselves in today, learners’ interest will be stimulated through the use of technological teaching aids. Teachers should therefore have access to, and use, a variety of media and technological teaching aids and be able to integrate them effectively into their language teaching. The findings of the study revealed teachers are caught up in traditional language teaching methods and strategies which do not contribute to the enhancement of learners’ proficiency in the target language. The study also closely examined the different types of knowledge that a language teacher should have. The results showed that the teachers’ knowledge of the curriculum, language policies, language teaching and learning theories as well as methodologies is extremely limited. Therefore a new method or approach is needed, which is why this study recommends that the HEIs and the WCED ensure that initial teacher training programmes and in-service training workshops are upgraded and adapted in order to prepare the teachers adequately to implement the prescribed curriculum using appropriate methodologies and strategies. / AFRIKAANSE OPSOMMING: Die doel van hierdie studie was om die huidige stand van Afrikaans Eerste Addisionele Taal (EAT) onderrig in gekose Gr 4 – 6 klasse in Wes-Kaapse stedelike skole te ondersoek en beskryf. Dit is gedoen by wyse van die aanbieding van ’n oorsig van die betrokke literatuur oor EAT-onderrig en EAT-kurrikula, asook die resultate van vraelyste en semigestruktureerde onderhoude met Gr 4 – 6-onderwysers. Hierdie studie bied nie ’n kitsoplossing vir die probleme in die Afrikaans EAT klaskamers nie, maar die navorser glo dat die bevindinge die uitdagings wat Afrikaans EAT-onderwysers daagliks trotseer, sal beklemtoon en dat alle rolspelers aktief betrokke sal raak om die stand van Afrikaans EAT onderrig in die Wes-Kaap te verbeter. Die toepaslikheid van hierdie studie is gesetel in die nasionale klem op die bevordering van meertaligheid onder die algemene pupliek en veral in die onderwys. Studente terugvoer na die praktiese onderwys dui daarop dat nie genoeg tyd aan die onderrig van Afrkaans EAT in skole bewillig word nie en dat daar ‘n groot verskeidenheid onderrigmetodes en onderrig- en leerhulpmiddels in Afrikaans EAT-klasse aangewend word. Leerders en onderwysers se vaardigheid in Afrikaans wissel ook van klas tot klas. Die navorsing is ondersteun deur ‘n teoretiese raamwerk van taalonderrig en –leer, ‘n literatuurstudie van eerste addisionele taalonderrig, nasionaal sowel as internasionaal, asook ‘n analise van Suid-Afrikaanse EAT-kurrikula. Konstruktivisme. sosio-konstruktivisme en onderwyser kennis is geïdentifiseer as die teoretiese begronding vir taalonderrig en –leer. Die literatuurstudie gee ‘n oorsig van al die belangrike en relevante EAT-metodieke. Die navorser het tot die gevolgtrekking gekom dat nie een enkele metode of benadering effektiewe EAT-onderrig kan verseker nie, maar dat onderwysers ‘n eklektiese benadering behoort te volg om die beste resultate te verseker. Hierdie studie het ‘n gemengde navorsingsbenadering gevolg ten einde empiriese data te genereer. Die kwantatiewe data is ingesamel by wyse van talle vraelyste wat deur Gr 4 – 6 Afrikaans EAT-onderwysers voltooi is. Semi-gestruktureerde onderhoude is met 17 Afrikaans EAT-onderwysers gevoer ten einde die kwalitatiewe data in te samel. Die data van hierdie onderhoude is gebruik as tri-angulasie om die resultate van die vraelyste te bevestig of te weerspreek. Een van die belangrikste faktore wat die onderrig en leer van Afrikaans EAT negatief beïnvloed, is baie leerders en ouers se negatiewe houding teenoor Afrikaans. Onderwysers is nie toegerus met die nodige onderrigstrategieë en –tegnieke om hierdie negatiewe houdings aan te spreek nie. Die resultate wys daarop dat baie onderwysers steeds ’n onderwyser-gerigte benadering volg wat daartoe lei dat die leerders nie genoegsame geleentheide kry om hulle kommunikatiewe vaardighede by wyse van interaksie met ander te ontwikkel nie. ‘n Ander faktor wat Afrikaans EAT-onderrig negatief beïnvloed, is die feit dat nie alle skole die voorskrifte van die verskillende taalbeleide en kurrikula implementeer soos van hulle verwag word nie. Hierdie studie het ook die gebruik van gepaste en relevante onderrig- en leerondersteuningsmateriaal in die EAT-klaskamer ondersoek. Die resultate het daarop gedui dat die meeste onderwysers nog steeds die handboek as belangrikste onderrighulpmiddel gebruik. Daar bestaan ‘n geweldige behoefte na gepaste Afrikaanse onderrig- en leermateriaal vir EAT. In die tegnologiese era waarin ons ons bevind, bestaan die verwagting dat leerders se belangstelling gestimuleer sal word deur die gebruik van tegnologiese onderrigmateriaal. Onderwysers behoort dus toegang te hê tot en ‘n wye verskeidenheid media en tegnologiese onderrigmateriaal te kan gebruik en in staat wees om dit suksesvol te integreer in hulle taalonderrig. Die bevindinge van hierdie studie het gewys dat onderwysers vasgevang is in tradisionele taalonderrigmetodes en –strategieë wat nie bydra tot die ontwikkeling van die leerders se vaardigheid in die teikentaal nie. Die verskillende soorte kennis waaroor ‘n taalonderwyser behoort te beskik is ook onder die vergrootglas geplaas. Die resultate het getoon dat die onderwysers se kennis van die kurrikulum, taalbeleid, taalonderrig en –leerteorieë en metodieke uiters beperk is. Daar is dus ‘n behoefte aan ‘n nuwe metode of benadering en daarom beveel hierdie studie aan dat onderwyseropleidingsprogramme en indiensopleidingswerkswinkels opgegradeer en aangepas word deur die Hoëronderwysinstellings en die WKOD om te verseker dat onderwysers voldoende voorberei word om gepaste metodieke en strategieë toe te pas in die implementering van die voorgeskrewe kurrikula.

Thesis (PhD)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: The purpose of the study was to investigate The effect of the Problem-Based Learning Problem Based Learning (PBL) approach in the teaching of composition and inverse functions in a foundation programme. PBL is a philosophical approach to teaching and learning where problems drive the learning. The study was important because it was trying to find out if PBL can improve students’ performance in compositions and inverses of functions at the bridging course for undergraduate mathematics at Oshakati Campus. The study intended to come up with a PBL model suitable for FP mathematics in the teaching of compositions and inverses of functions. The study was done on Science Foundation students who are registered for FP. Eighty students were randomly selected from the foundation students registered for the 2013 academic year. The students were randomly assigned into the experimental and the comparison groups of 40 each. In this study the comparison group of the Foundation students was predominantly taught through the traditional lecture approach while the experimental group was predominantly taught using a hybrid PBL approach. The study also attempted to establish the students’ perceptions with regard to the relevance of inverses and compositions of functions as a concept in a topic that determines their academic destination. It also attempted to ascertain how the PBL approach could best be implemented in order to improve FP students’ understanding of inverses and composition of functions; how Bridging course for undergraduate mathematics (FP) students experience the PBL approach in the teaching and learning of inverses and composition of functions compared to those who are taught using the lecture method and how FP students’ performance on inverses and composition of functions as a result of their PBL experience compare to those who are taught using the lecture method. This study used the concurrent nested mixed methods (qualitative and quantitative) research designs. A quasi experimental design was adopted through the administration of a pre-post-test on experimental and comparison groups. The other designs or methods included a questionnaire survey, focus group interviews, non-participant lesson observation and a group research project on compositions and inverses of functions. The experimental group was then mainly taught through a hybrid PBL approach while the comparison group mainly through the lecture approach for a period of three months. The findings of this research study showed that experimental group students performed significantly better in the overall results analysis but there were no significant differences in performance between the two groups for some Hypothetical Learning Trajectory (HLT) domains on compositions and inverses of functions. It is recommended that PBL should be implemented in the other foundation programme subjects. However, the role of the conventional teaching approaches cannot be undermined in the teaching and learning of compositions and inverses of functions since the students who were taught using this method also improved their performances, and as such these conventional teaching approaches should be used together with PBL in order to get the best results on FP students’ mathematics performance. This study recommends further research on how PBL can be implemented in other FP subjects. This study also recommended that PBL should be implemented right at the beginning of the year when the FP students start their classes in the foundation programme. / AFRIKAANSE OPSOMMING: Die doel van die studie was om die effek van die probleemgebaseerde leer (PBL) benadering in die onderrig van die samestelling en inverse funksies in 'n Stigting program te ondersoek. PBL is 'n filosofiese benadering tot onderrig en leer waar probleme ry die leer. Die studie is belangrik omdat dit probeer het om uit te vind of PBL kan studente se prestasie in komposisies en inverses van funksies te verbeter by die Stigting Program op Oshakati-kampus. Die studie bedoel om vorendag te kom met 'n PBL model wat geskik is vir fondament in die onderrig van komposisies en inverses van funksies. Die studie is gedoen op Science Foundation studente by Oshakati-kampus van die Universiteit van Namibië. Tagtig studente is lukraak gekies uit die fondament studente wat geregistreer is vir die 2013 akademiese jaar. Die studente is ewekansig toegewys in die eksperimentele en die vergelyking groepe van 40 elk. In hierdie studie is die vergelyking groep van die Stigting studente is hoofsaaklik geleer word deur die tradisionele lesing benadering terwyl die eksperimentele groep was hoofsaaklik geleer met behulp van 'n hibriede PBL benadering. Die studie het ook probeer om vas te stel uit wat die studente se persepsies met betrekking tot die toepaslikheid van inverses en komposisies van funksies is soos 'n konsep in 'n onderwerp wat bepaal hul akademiese bestemming. Dit het ook probeer om vas te stel hoe die PBL benadering kan die beste om FP studente se begrip van inverses en samestelling van funksies te verbeter geïmplementeer word; hoe FP studente die PBL benadering in die onderrig en leer van inverses en samestelling van funksies in vergelyking met diegene wat geleer is met behulp van die lesing metode en hoe FP studente se prestasie op inverses en samestelling van funksies as 'n gevolg van hul PBL ervaring vergelyk met dié wat geleer is met behulp van die lesing-metode. Hierdie studie gebruik om die konkurrente geneste gemengde metodes (kwalitatiewe en kwantitatiewe) navorsing ontwerpe. 'N quasi eksperimentele ontwerp is aangeneem deur die administrasie van 'n pre-na-toets op eksperimentele en vergelyking groepe. Die ander ontwerpe of metodes het 'n vraelys opname, fokusgroeponderhoude, nie-deelnemer leswaarneming, en 'n groep navorsingsprojek oor komposisies en inverses van funksies. Die eksperimentele groep is dan hoofsaaklik geleer deur middel van 'n kruising PBL benadering terwyl die vergelyking groep hoofsaaklik deur die lesing benadering vir 'n tydperk van drie maande. Die bevindinge van hierdie navorsing het getoon dat die eksperimentele groep studente uitgevoer aansienlik beter in die algehele resultate analise, maar daar was geen betekenisvolle verskille in prestasie tussen die twee groepe vir 'n paar MTT gebiede op komposisies en inverses van funksies. Die studie het ook bevind dat PBL aan die begin van die jaar reg geïmplementeer moet word wanneer die FP studente begin hul klasse in die fondament program. Dit word aanbeveel dat PBL in al die ander fondament program vakke moet geïmplementeer word. Tog kan die rol van die konvensionele onderrig benaderings nie ondermyn word in die onderrig en leer van komposisies en inverses van funksies, en as sodanig die konvensionele onderrig benaderings moet saam met PBL word gebruik om die beste resultate op FP studente se wiskunde prestasie te kry . Hierdie studie beveel aan verdere navorsing oor hoe PBL in 'n ander fondament program vakke geïmplementeer kan word.

Thesis (MEd)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Research has confirmed the educational value of mathematical modelling for learners of all abilities. The development of modelling competencies is essential in the modelling approach. Little research has been done to identify and develop the mathematising modelling competency for specific sections of the mathematics curriculum. The study investigates the development of mathematising competencies during the modelling of number pattern problems. The RME theory has been selected as the theoretical framework for the study because of its focus on mathematisation. Mathematising competencies are identified from current literature and developed into models for horizontal and vertical (complete) mathematisation. The complete mathematising competencies were developed for number patterns and mapped on a continuum. They are internalising, interpreting, structuring, symbolising, adjusting, organising and generalising. The study investigates the formulation of a hypothetical trajectory for algebra and its associated local instruction theory to describe how effectively learning occurs when the mathematising competencies are applied in the learning process. Guided reinvention, didactical phenomenology and emergent modelling are the three RME design heuristics to form an instructional theory and were integrated throughout the study to comply with the design-based research’s outcome: to develop a learning trajectory and the means to support the learning thereof. The results support research findings, that modelling competencies develop when learners partake in mathematical modelling and that a heterogeneous group of learners develop complete mathematising competencies through the learning of the modelling process. Recommendations for additional studies include investigations to measure the influence of mathematical modelling on individualised learning in secondary school mathematics. / AFRIKAANSE OPSOMMING: Navorsing steun die opvoedkundige waarde van modellering vir leerders met verskillende wiskundige vermoëns. Die ontwikkeling van modelleringsbevoegdhede is noodsaaklik in 'n modelleringsraamwerk. Daar is min navorsing wat die identifikasie en ontwikkeling van die bevoegdhede vir matematisering vir spesifieke afdelings van die wiskundekurrikulum beskryf. Die studie ondersoek die ontwikkeling van matematiseringsbevoegdhede tydens modellering van getalpatrone. Die Realistiese Wiskundeonderwysteorie is gekies as die teoretiese raamwerk vir die studie, omdat hierdie teorie die matematiseringsproses sentraal plaas. Matematiseringsbevoegdhede vanuit die bestaande literatuur is geïdentifiseer en ontwikkel tot modelle wat horisontale en vertikale (volledige) matematisering aandui. Hierdie matematiseringsbevoegdhede is spesifiek vir getalpatrone ontwikkel en op ‘n kontinuum geplaas. Hulle is internalisering, interpretasie, strukturering, simbolisering, aanpassing, organisering en veralgemening. Die studie lewer die formulering van ‘n hipotetiese leertrajek vir algebra, die gepaardgaande lokale onderrigteorie en beskryf hoe effektiewe leer plaasvind wanneer die ontwikkelde matematiseringsbevoegdhede volledig in die leerproses toegepas word. Die RME ontwikkellingsheuristieke, begeleidende herontdekking, didaktiese fenomenologie en ontluikende modellering, is geïntegreer in die studie sodat dit aan die uitkoms van ‘n ontwikkelingsondersoek voldoen. Die uitkoms is ‘n leertrajek en ‘n beskrywing hoe die leerproses ondersteun kan word. Die analise het tot die formulering van ‘n lokale-onderrig-teorie vir getalpatrone gelei. Die resultate van die studie kom ooreen met navorsingsbevindings dat modelleringsbevoegdhede ontwikkel wanneer leerders deelneem aan modelleringsaktiwiteite, en bewys dat ‘n groep leerders met gemengde vermoëns volledige matematiseringsbevoegdhede ontwikkel wanneer hulle deur die modelleringsproses werk. 'n Aanbeveling vir verdere navorsing is om die uitwerking van die modelleringsperspektief op individuele leer in hoërskool klaskamers te ondersoek.

Thesis (PhD)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: The special needs community is in the midst of a philosophical and physical shift from a segregated system to an integrated system, not only in placement, but more importantly, in terms of learning and affording learners with special needs access to mainstream curricular materials. Mathematical modelling, or challenging mathematics problems solved in small groups, is part of the Australian mainstream curriculum. The purpose of the study was to investigate the way special needs learners learn mathematics from a modelling learning environment. To do this, it was necessary to identify the critical characteristics of the best practice in teaching and learning for learners with special needs, and the critical features of modelling. One theory of learning that has the capacity to promote special needs learners' interaction with mathematical modelling is Feuerstein’s theory of Structural Cognitive Modifiability. A hypothetical learning trajectory was designed for special needs learners at middle school according to general design principles from theory, which was adapted to the learning characteristics of the class. The learning environment comprised of three challenging modelling tasks, together with recommended implementation and support conditions in the classroom. Specifically, the research sought to investigate the ways in which special needs educators can support the higher reasoning processes of special needs students during modelling through design in general, and through mediation specific to each learner. The research took the form of a qualitative study, combining the phases of design-based research with a multiple case study approach. Three cases were analysed in depth. Empirical data were collected through a range of qualitative methods, which included data from student files, field observations, video and audio recordings, focus group interviews with students, and the input of various collaborators across the different phases of planning, design, implementation, and revision. Data were coded and analysed inductively according to emerging patterns and themes. Findings suggest that the use of modelling was successful when implemented with certain characteristics defined in the literature, and that it enabled learners to learn mathematics and also to develop additional outcomes such as social skills and language. During this study, learners' higher-order reasoning was supported through dynamic assessment and subsequent mediation. / AFRIKAANSE OPSOMMING: Die onderwysgemeenskap vir leerders met spesiale behoeftes bevind hulle in die middel van filosofiese en fisiese verskuiwings van 'n geskeide sisteem na 'n geïntegreerde sisteem. Dit omvat die plasing van leerders, maar meer belangrik ook die bemoontliking van toegang van hierdie leerders tot hoofstroom kurrikulêre materiale. Wiskundige modellering, en uitdagende wiskundeprobleme wat deur leerders in klein groepies opgelos word, is deel van die Australiese hoofstroomkurrikulum. Die doel van die studie was om die wyse te ondersoek waarvolgens leerders met spesiale behoeftes wiskunde in 'n modelleringsomgewing leer. Dit is gedoen deur die belangrike kenmerke van beste praktyk vir onderrig en leer in spesiale onderwys, asook die kritiese kenmerke van modellering, te vind. Een leerteorie wat die interaksie van leerders met spesiale behoeftes met wiskunde bevorder, is Feuerstein se teorie van Strukturele Kognitiewe Modifieerbaarheid. 'n Hipotetiese leertrajek was ontwerp vir leerders met spesiale behoeftes op middelskoolvlak. Empiriese data is deur 'n reeks kwalitatiewe aksies: data van studentelêers, veldwaar-nemings, video en klankopnames, fokusgroeponderhoude met studente, asook die insette van verskeie medewerkers oor die verskillende fases van beplanning, ontwerp, implementering en hersiening gegenereer. Die spesifieke leerkenmerke van hierdie leerders volgens algemeen-teoretiese en lokaalgekontekstualiseerde ontwerpbeginsels is nagekom. Die leertrajek het bestaan uit drie uitdagende modelleringsprobleme met aanbevole implementering en ondersteuningsriglyne in die klaskamer. Die navorsing het spesifiek gesoek na wyses waarop hierdie leerders se hoër beredeneringsvaardighede deur hul onderwysers, volgens elkeen se eie behoefte gedurende modellering, deur ontwerp in die algemeen en mediasie in die besonder, ondersteun kan word. Die navorsing, 'n kwalitatiewe studie, was gekombineer met fases van ontwikkelingsgebaseerde ontwerp wat uitgespeel het in 'n veelvuldige gevallestudiebenadering. Drie gevalle is in diepte ondersoek. Data was induktief gekodeer en geanaliseer volgens ontluikende patrone en temas. Bevindinge wys uit dat die gebruik van modellering suksesvol was wanneer die implementering volgens spesifieke kenmerke in die literatuur was. Dit het leerders instaat gestel om wiskunde te leer asook om addisionele uitkomste soos sosiale vaardighede en taal te ontwikkel. In hierdie studie is hoër-orde denke ondersteun deur dinamiese assessering en voortspruitende mediasie.

Thesis (MEd (Curriculum Studies))--University of Stellenbosch, 2009. / The research was done in a period of curriculum transformation and reform in South Africa, a period during which teachers attended many interventions by service providers other than the education department to sharpen their skills. In this study, I investigate teacher professional development and professional development programmes using two case studies, MTN-SUNSTEP (Mobile Telecommunications Network – Stellenbosch University Schools Technology in Electronics Programme) and SWAP (Schools Water Project). These INSET programmes were intended to assist with the improvement of science teaching in terms of the new curricula suggested for the country. The study proceeds by way of two case studies within an interpretive paradigm. The purpose of the study is to evaluate the two programmes based on the criteria developed through a literature review and to understand and give meaning to the process of teacher professional development that might have occurred in the programmes. This process is framed by my research question, “can resource kits enable professional development in science teaching”. Data was collected at various stages by way of field notes, questionnaires, interviews and observations at workshops. Teachers generally found the processes useful but do indicate dissatisfaction with certain areas of the programmes. Issues that emerged from the study were that teacher development processes need to be organized over a long period, the development of teacher networks needs to be nurtured, and support at school level was essential to ensure implementation. An important insight was that the research, once again, emphasized a key aspect of effective teacher development programmes. Effective programmes require a team consisting of three agents that bring their own expertise to the process namely, teachers, district officials and university staff. This team has to take into account the factors that ensure effective teacher professional development and should occur within the parameters of accepted professional development practices and learning support materials play an important role in supporting this process.

Thesis (MEd) -- University of Stellenbosch, 2009. / ENGLISH ABSTRACT: Learning and the development of numeracy in the foundation phase are regarded as highly important. In this critical period learners attain the fundamental knowledge that is needed for future learning and development. A serious problem within South African schools is that a great number of foundation phase learners show difficulties in numeracy. Various reasons can be provided for these difficulties, for example, learners do not receive good teaching, they hold negative attitudes towards numeracy, they encounter language barriers, and so forth. It is important that the learners’ difficulties in numeracy are attended to immediately and effectively, to be able to minimise the gaps between the learners’ functioning level and the level the learner is suppose to be on. The Western Cape Education Department (WCED) laid bare their concern with the current problem in numeracy when they launched the WCED Literacy and Numeracy Strategy 2006-2016. As part of this the Metropole North Education and Management Development Centre (EMDC) requested and implemented an intervention programme for grade 2 and grade 3 learners. Part of this problem is that many foundation phase educators lack the necessary knowledge to identify the learners with difficulties and especially to attend to these difficulties. The goal of this research study was to investigate the realization of an early intervention programme implemented over a limited period of time of ten weeks in grade 2 and grade 3 classrooms. The current study aimed at providing educators with resources to offer learning support to their disadvantaged learners within the Numeracy classroom. This study has implications for the way in which educators approach and teach numeracy, as well as for the type of learning support the educators provide their disadvantaged learners with. The intervention programme is based on a problem centered approach with constructivism as underlying epistemology. This approach views the educator as a facilitator who provides the learners with opportunities to be actively involved in the learning environment to construct knowledge. The importance of word problems and discussion within the Numeracy classroom are emphasized and it is necessary that the educator create the appropriate classroom atmosphere so that this approach can be implemented effectively. Learning is an active process and it is the responsibility of the educator to provide the learners with appropriate learning activities for learning and development to take place. The study made use of programme evaluation as research methodology. Programme evaluation refers to a research goal instead of specific research methods, because various methods can be used which may be quantitative or qualitative in nature. The gathering of data was done through classroom observations by the researcher, informal interviews with participating educators, questionnaires, and semi-structured interviews with the educators at the four schools the researcher monitored intensively during the implementation process of the intervention programme. The gathering of data was an ongoing process and the researcher analysed and interpreted the learners’ work at the end of the implementation process. The researcher checked this analysis by means of the literature review, transcripts and field notes. This formed the basis on which conclusions and recommendations could be made. The conclusions confirmed the importance of early intervention and the efficacy of the problem centered approach within the numeracy classroom. It also confirmed the disadvantages of the training model that was used, that is the Cascade model. With this study it was possible to determine that a ten week intervention programme was too short a period to observe meaningful improvements in learners. Therefore further research should be done on addressing learners’ difficulties and to train and support educators to provide learning support to their learners.

Thesis (MPhil)--University of Stellenbosch, 2006. / ENGLISH ABSTRACT: The Forensic Science Laboratory (FSL) is a component of the South African Police Service (SAPS). The Questioned Document Unit (QDU) is a section within the FSL. It has been practice in the QDU to recruit members of the SAPS for training as Questioned Document Examiners within the FSL. Although the SAPS has a policy on education, training and development, it is not applied. Even after the establishment of the South African Qualifications Authority (SAQA) and the National Qualifications Framework (NQF), the QDU and the rest of the FSL continued their training practices at the workplace outside the outcomes-based paradigm. As part of standard practice, the FSL has taken content experts (forensic analysts) and turned them into trainers. These forensic experts had no training qualifications and little or no facilitation skills. Their knowledge of outcomes-based education (OBE) and adult learning was also either inadequate or non-existent. This shortcoming has influenced the quality of learning in this environment. In 2004 the Forensic Science Laboratory began to give some members an opportunity to be trained as trainers, assessors and moderators of learning. However, this has been a disjointed effort. Generally, learners have had to endure a frustrating period of more than four years of internal training before being certified as competent to act as examiners. Before 1994 the QDU employed mostly white personnel as examiners. Most black personnel still occupy the lower salary levels amongst examiners. There are no black trainers. At present (2006) in the FSL, the tendency is that white personnel hold senior positions and black personnel are juniors. There is covert racial tension among the members. In the QDU, the training manager has always been a trainer as well. In the training environment at the QDU there have been obvious problems, namely – 􀂾���������� poor practice of OBE and adult learning; 􀂾���������� relationship problems between trainer and learners; 􀂾���������� distrust and a lack of communication and dialogue between trainer and learners; and 􀂾���������� underlying racial tension. The action research process on which we (the learners, training manager and I) embarked was aimed at – 􀂾���������� opening dialogue/communication between the training manager and learners; 􀂾���������� increasing learner participation in the process; and 􀂾���������� providing the opportunity for both the learners and the training manager to increase their knowledge of adult learning and OBE. We hoped that by making the entire action research process transparent we could create a platform for the learners and the training manager to build relationships in order to bring about an improvement in learning practice. We used an action research process that included participation by both the learners and the training manager. Change occurs within the action component of the action research process, while the research component is meant to generate knowledge. We used a cyclic method that entailed stages of planning, action, observation and reflection. Continuity was achieved by the reflection stage of one cycle informing the planning stage of the next. The action research process used in this setting has supported the existing theory and assumptions that adult learners want to participate, be involved in decision-making, and learn by doing. It has also shown that they are critically aware. The learning practice at the QDU has improved. The action research process that took place at the unit can serve as a powerful case study for trainers who endeavour to improve practice in other environments. / AFRIKAANSE OPSOMMING: Die Forensiesewetenskapslaboratorium (FWL) is 'n komponent van die Suid-Afrikaanse Polisiediens (SAPD), terwyl die Betwistedokumente-Eenheid (BDE) ʼn afdeling binne die FWL is. Sedert die stigting van die BDE was dit algemene praktyk om lede van die SAPD te werf en hulle binne die FWL as eksaminatore van die BDE op te lei. Alhoewel die SAPD ʼn beleid het ten opsigte van onderwys, opleiding en ontwikkeling, word dit nie toegepas nie. Selfs ná die totstandkoming van die Suid-Afrikaanse Kwalifikasieowerheid (SAKO) en die Nasionale Kwalifikasieraamwerk (NKR), het die BDE en die res van die FWL hul werkgebaseerde opleidingspraktyke buite die paradigma van uitkomsgebaseerde onderwys voortgesit. Die FWL het tot dusver forensiese skeikundiges in opleiers omskep. Hulle het geen kwalifikasies in opleiding gehad nie en hul kennis van uitkomsgebaseerde onderwys (UGO) en volwasseneleer, asook hulle fasiliteringsvaardighede, was onvoldoende. In sommige gevalle het dit geheel en al ontbreek. Hierdie tekortkoming het ʼn nadelige invloed op die gehalte van leer gehad. Onlangs (2004) het die FWL begin om sommige polisielede die geleentheid te bied om as opleiers, assessors en moderators van leer opgelei te word, maar hierdie pogings is nog nooit behoorlik gestruktureer nie. Binne die huidige opset is daar leerders wat meer as vier jaar interne opleiding moes ondergaan voor hulle as bevoeg gesertifiseer is om as opleiers op te tree. Die BDE het in die verlede meesal wit personeel in diens geneem as eksaminatore. Die meeste van die swart eksaminatore in die BDE is op die laer salarisvlakke, en daar is geen swart opleiers nie. Tans (2006) is daar hoofsaaklik wit personeel in die seniorposte in die FWL, met die swart personeel meesal in juniorposte. Daar is onderliggende rassespanning onder die lede. In die BDE was die opleidingsbestuurder nog altyd ook ʼn opleier. Die volgende probleme is in die opleidingsomgewing van die BDE geïdentifiseer: 􀂾��������� swak praktyk t.o.v. UGO en volwasseneleer; 􀂾��������� troebel verhoudings tussen die opleier en die leerders; 􀂾��������� wantroue en gebrekkige kommunikasie en dialoog tussen die opleier en die leerders; en 􀂾��������� onderliggende rassespanning. Die aksienavorsingsproses wat ons (ek, die leerders en die opleidingsbestuurder) aangepak het was daarop gemik om – 􀂾��������� dialoog/kommunikasie tussen die opleidingsbestuurder en die leerders te vestig; 􀂾��������� leerderdeelname in die proses te verhoog; en 􀂾��������� vir beide die leerders en die opleidingsbestuurder die geleentheid te bied om hul kennis van volwasseneleer en UGO uit te brei. Deur die hele aksienavorsingsproses deursigtig te maak, het ons gehoop om vir alle rolspelers ʼn geleentheid te skep om verhoudinge te bou ten einde ʼn verbetering in die leerpraktyk teweeg te bring. ʼn Aksienavorsingsproses is aangewend wat deelname deur beide die leerders en die opleidingsbestuurder ingesluit het. In aksienavorsing vind verandering binne die aksiekomponent van die proses plaas, terwyl die navorsingskomponent daarop gemik is om kennis vir die deelnemers – en as deel van die proses self – te genereer. Ons het ʼn sikliese metode gebruik wat beplanning, handeling, waarneming en refleksie behels het. Kontinuïteit is verseker deurdat die refleksiestadium van een siklus die basis gelê het vir die beplanningstadium van die volgende. Die aksienavorsingsproses wat in hierdie opset gebruik is, het die bestaande teorie en aannames ondersteun dat volwasse leerders wil deelneem, dat hulle by besluitneming betrokke wil wees, dat hulle wil leer deur te doen, en dat daar ‘n groter kritiese bewussyn is. Die leerpraktyk aan die BDE het verbeter. Die aksienavorsingsproses aan die BDE kan ʼn betekenisvolle gevallestudie wees vir diegene wat poog om hul praktyk in ander omgewings te verbeter.

Thesis (MEd (Curriculum Studies)--University of Stellenbosch, 2007. / Culinary studies is a relatively young and unfamiliar field of study which engages the application of life and natural sciences, business and technology in a food-specific environment. The growth in the demand for training, re-training and continuing professional development in the culinary arts can be attributed to factors such as enhanced culinary programmes at high school level, an increased level of education and expectation about food in general, growth in culinary tourism and highly polished food magazines or other publications. The celebrity status of high-profile chefs and a stronger economy that enables more people to explore fine dining and gourmet food can be added to the list of factors that influence the interest in culinary arts as a profession. The number of qualified culinary professionals has increased over the last 10 years in South Africa, providing a large corps of people with a good understanding of this field of study and a need for continuing professional development opportunities. This study was a response to the challenges of the demand for continuing professional development opportunities from this growing body of culinary professionals. The demand for continuing professional development opportunities in culinary studies is becoming more complex and challenging for both learner and teacher. Although institutions that provide training in hospitality are also in the business of culinary education, their perspectives and focus are quite diverse. Culinary education can therefore be improved through a deeper understanding of the curriculum development process combined with the expectations of both the industry and the individual. A fundamental shortcoming in the field of culinary education is often that persons with limited expertise in the field of curriculum studies bear the primary responsibility for addressing curriculum challenges. The development of curricula for professional development in culinary studies subsequently happens at the expense of understanding the curriculum. The primary aim therefore of this study was to identify and propose a curriculum framework for continuing professional development in culinary arts. This framework might contribute to providing a curriculum foundation, credibility to the broad hospitality industry and specifically to culinary studies, as well as to the securing of some uniformity of standards over the spectrum of culinary qualifications. A scientifically validated situational analysis was executed by means of focus group discussions, personal interviews, curriculum comparisons and an electronic mail questionnaire survey, which mainly generated qualitative data. These techniques were used in triangulation as research instruments to investigate the needs for continuing professional development in culinary arts and the availability of curricula to address these needs. The greatest challenges in culinary studies were reflected in the complexity and multidisciplinary nature of this relatively undefined industry and field of study. The intricate relation, interaction, collaboration and contexts between secondary schools, various culinary training institutions, industry and culinary professionals were investigated. The data generated confirmed the need for training and development opportunities to improve the professional status of culinary professionals in South Africa The promotion of a change in direction for culinary studies development in South Africa could ensure future growth as a discipline cognitively and professionally, in line with international standards, procedures and practices. The range of challenges and changes facing the industry varies from social issues to the educational issues of qualifications and credible accreditations, which are addressed in the proposed curriculum framework. The lack of more empirical research in this field of study is an indication that both students and professionals should be encouraged to do the relevant research and that higher education institutions should provide the opportunities and structures for such research.