In this multiple-case study, I developed cases describing three students‘ (Mary, Ryan and David) solution methods for algebra word problems and investigated the effect of technology on their solution methods by making inferences about their preferences for visual or nonvisual solutions. Furthermore, I examined the students‘ solution methods when presented with virtual physical representations of the situations described in the problems and attempted to explain the effect of those representations on students‘ thinking preferences. In this study, the use of technology referred to the use of the dynamic software program Geogebra. Suwarsono‘s (1982) Mathematical Processing Instrument (MPI) was administered to determine their preferences for visual and nonvisual thinking. During the interviews, students were presented with paper-andpencil-based tasks (PBTs), Geogebra-based tasks (GBTs) and Geogebra-based tasks with virtual physical representations (GBT-VPRs). Each category included 10 algebra word problems, with similar problems across categories. (i.e., PBT 9, GBT 9 and GBT-VPR 9 were similar). By investigating students‘ methods of solution and their use of representations in solving those tasks, I compared and contrasted their preferences for visual and nonvisual methods when solving problems with and without technology. The comparison between their solutions of PBTs and GBTs revealed how dynamic software influenced their method of solution. Regardless of students‘ preferences for visual and nonvisual solutions, with the use of dynamic software students employed more visual methods when presented with GBTs. When visual methods were as accessible and easy to use as nonvisual methods, students preferred to use them, thus demonstrating that they possessed a more complete knowledge of problem-solving with dynamic software than their work on the PBTs. iii Nowadays, we can construct virtual physical representations of the problems in technology environments that will help students explore the relationships and look for patterns that can be used to solve the problem. Unlike GBTs, GBT-VPRs did not influence students‘ preferences for visual or nonvisual methods. Students continued to rely on methods that they preferred since their preferences for visual or nonvisual solutions regarding GBT-PRs were similar to their solution preferences for the problems on MPI that was administered to them to determine their preferences for visual or nonvisual methods. Mary, whose MPI score suggested that she preferred to solve mathematics problems using nonvisual methods, solved GBT-VPRs with nonvisual methods. Ryan, whose MPI score suggested that he preferred to solve mathematics problems using visual methods, solved GBT-VPRs with visual methods. David, whose MPI score suggested that he preferred to solve mathematics problems using both visual and nonvisual methods, solved GBT-VPRs with both visual and nonvisual methods.

The purpose of this thesis is to investigate opportunities to learn (OTL) algebra by grade ten learners at three Catholic secondary schools in South Africa. Performance in mathematics is poor and is a great cause for concern. Despite the government’s effort to make education open and available to all, underperformance has continued among the black majority who were previously marginalised in the former regime. This thesis focuses on the OTL which are afforded learners who are given the chance to attend classes. This thesis met its aims through an extensive review of related literature and the implementation of practical research. The latter was carried out through case studies conducted in three schools where lessons were observed and interviews conducted with the respective teachers. Literature on how OTL mathematics are created is lacking in South Africa. Real OTL still needs to be created if the expected level of performance is to be achieved. The research produced a number of key findings: the learners were given the right to attend class but were subjected to different OTL, learning to convert within and between the different registers of representation of algebraic concepts is necessary to provide learners with OTL, it is not enough for learners to master certain facts and procedures, and learning is enhanced if the means to make the conversion necessary for concept building is developed and the OTL provided. The teacher’s approach influences the way OTL are realised and utilised by learners. The main conclusion drawn from this research is that the OTL afforded the grade ten learners were not the same and that different chances to make conversion within and between registers of representation of algebra concepts were given. Giving the teachers guidelines without expounding the meaning of specific terms such as ‘convert’ leaves gaps in their practices and results in some learners receiving adequate OTL and others not. This research argues for a more involved capacity building programme for in-service teachers to acquaint them with the expected learner-centred approaches to lesson delivery as well as familiarise them with the terminology used in defining terms in the syllabus. / Educational Studies / D. Ed. (Curriculum Studies)

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 (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.

The purpose of the current study was to discern the effects of three latent constructs – self-efficacy, academic engagement, and student-teacher relationships on Algebra I achievement among ninth-grade African American male students. A nationally representative sample from the High School Longitudinal Study of 2009 (HSLS09) was used in the study. Study participants were 697 African American males enrolled in ninth grade in the fall of 2009 across the United States. Structural Equation Modeling (SEM) analytical procedures were performed to test the hypothesized relationships of Bandura’s social cognitive theory (SCT) theoretical assumptions. The results indicate that the three latent variables directly or indirectly were related to Algebra I achievement among ninth grade African American male students. Moreover, the results revealed that self-efficacy and student-teacher relationships constructs had direct significant impact on Algebra I academic performance; nonetheless, the relationships were not strong. These two latent variables had small effect sizes of 5% and 1%, respectively. Combined, self-efficacy, academic engagement, and student-teacher relationships explained only 8% of the variance in the Algebra I achievement among African American males across the United States (R2=.08). The magnitude effect of these factors on Algebra I achievement was minimal. Overall, these findings suggest that the self-efficacy and student-teacher relationships latent variables had a negligible effect as predictors of Algebra I academic success among ninth grade African American male students. A summary of the results are presented and future research is recommended.

The purpose of this thesis is to investigate opportunities to learn (OTL) algebra by grade ten learners at three Catholic secondary schools in South Africa. Performance in mathematics is poor and is a great cause for concern. Despite the government’s effort to make education open and available to all, underperformance has continued among the black majority who were previously marginalised in the former regime. This thesis focuses on the OTL which are afforded learners who are given the chance to attend classes. This thesis met its aims through an extensive review of related literature and the implementation of practical research. The latter was carried out through case studies conducted in three schools where lessons were observed and interviews conducted with the respective teachers. Literature on how OTL mathematics are created is lacking in South Africa. Real OTL still needs to be created if the expected level of performance is to be achieved. The research produced a number of key findings: the learners were given the right to attend class but were subjected to different OTL, learning to convert within and between the different registers of representation of algebraic concepts is necessary to provide learners with OTL, it is not enough for learners to master certain facts and procedures, and learning is enhanced if the means to make the conversion necessary for concept building is developed and the OTL provided. The teacher’s approach influences the way OTL are realised and utilised by learners. The main conclusion drawn from this research is that the OTL afforded the grade ten learners were not the same and that different chances to make conversion within and between registers of representation of algebra concepts were given. Giving the teachers guidelines without expounding the meaning of specific terms such as ‘convert’ leaves gaps in their practices and results in some learners receiving adequate OTL and others not. This research argues for a more involved capacity building programme for in-service teachers to acquaint them with the expected learner-centred approaches to lesson delivery as well as familiarise them with the terminology used in defining terms in the syllabus. / Educational Studies / D. Ed. (Curriculum Studies)

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)

Smartphone technology can be integrated in mathematics education, since most learners immerse in this technology. Teachers should be facilitators of confined learning in traditional classrooms, including digital facilitators. The aim of the study was to determine the effect of integrating smartphone technology on Grade 10 learners’ achievement in algebra. The study focused on two Grade 10 classes from different township schools in the Gauteng Province. The study was conducted during the first term of 2016. A quasi-quantitative experimental research design was used. Smartphones were integrated in algebra teaching and learning, using the Microsoft Math online application: https://www.math.microsoft.com. Despite challenges encountered during the study, the results indicated a positive impact on learners’ achievements in algebra, amongst experimental group learners. A need exists for future research in integrating smartphone technology in mathematics education. / Mathematics Education / M. Ed. (Mathematics Education)

The study investigated errors made by Grade 9 learners when simplifying algebraic expressions. Eighty-two (82) Grade 9 learners from a rural secondary school in Limpopo Province, South Africa participated in the study. The sequential explanatory design method which uses both quantitative and qualitative approaches was used to analyse errors in basic algebra. In the quantitative phase, a 20-item test was administered to the 82 participants. Learners’ common errors were identified and grouped according to error type. The qualitative phase involved interviews with selected participants. The interviews focused on each identified common error in order to establish the reasons why learners made the identified errors. The study identified six (6) common errors in relation to simplifying algebraic expressions. The causes of these errors were attributed to poor arithmetic background; interference from new learning; failure to deal with direction and operation signs; problems with algebraic notation and misapplication of rules. / Mathematics Education / M. Ed. (Mathematics Education)

The aim of this study was to explore the effects of a problem-solving heuristic instructional method on Grade 6 learners’ achievements in algebra. Two main theories inspired the design of this teaching method, namely the modelling and modelling perspective, and action, process, object, schema (APOS) theory. Modelling and modelling perspectives guided the development of modelling-eliciting activities used in the teaching method and the APOS theory guided the sequence of activities used to develop Grade 6 learners’ conceptions in algebra. The impact of the problem-solving heuristic instructional method was investigated with 198 Grade 6 learners from four different primary schools in the Zululand district of Kwazulu-Natal that were conveniently sampled. A mixed-method approach was used in this study and a hypothesis was formulated to investigate the effects of the teaching method on the learners’ achievements in algebra. The qualitative component consisted of a pre-intervention class observation of mathematics lessons of all four mathematics educators in the schools used for this study. The design and implementation of the problem-solving heuristic instructional method and the quantitative component employed non-equivalent control group design with pre-test and post-test measure. The main instruments for data collection were an observation schedule to document sequence of events in the classroom during the class observation, a standardized achievement test in algebra used to measure effects of the problem-solving heuristic instructional method and modelling-eliciting activities used as a medium of interaction between learners and the researcher during the implementation of the problem-solving heuristic instructional method. Findings from the class observation indicated all four schools made use of comparable traditional methods of instruction. The implementation of the problem-solving instructional method gave insights into how a problem-solving heuristic instructional method can be developed and used in Grade 6 algebra lessons, and the factors that could influence learners’ conceptual development in algebra. The findings from the quantitative component supported the initial hypothesis that improved scores in algebra are achieved through participation in the problem-solving heuristic instructional method. Quantitative data was analysed using the t-test, analysis of covariance, Johnson-Neyman (J-N) technique and the effect size. / Mathematics Education / D. Phil. (Mathematics Education)