M.Ed. / This study focuses on problems of adjustment which black pupils in Soweto (may) experience in the transition from primary to secondary school as far as mathematics is concerned. This problem is not investigated in vacuo, but is located within a definite context. This context consists of children of a distinct population group, who have reached a distinct level of development; who come from a distinct home background and who are taught a distinct subject by a distinct teacher in the service of a distinct education authority with a distinct teaching policy. As background to the problem a contextual and research literature survey is directed towards an empirical investigation. After the literature survey an empirical study focusing on standard 6 pupils was made. In this empirical investigation, various research methods were used in order to illuminate the purpose and hypothesis of this investigation. * Emphases is placed on determining the standard of mathematics amongst black pupils in the standard 6 experimental group by using a three-pronged approach: 1) The standard 6 pubils completed the HSRC diagnostic mathematics test for standard 5 and 6 in order to determine their competence in the basic functions of addition, subtraction, mUltiplication and division. 2) In order to determine the mathematical proficiency of standard 6 pupils the TOAM computer system was used. 3) To determine any possible deterioration in mathematical competency in standard 6 pupils in their transition from primary to secondary school, the same mathematics examination written at the end of standard 5, was written after three months in standard 6. * In order to attempt to determine the factors influencing the problems experienced by black pupils compelled to move from standard 5 to standard 6, a questionnaire probing socio-economic and general school situation was completed by all standard 6 pupils in the eight secondary schools used from this investigation. Questionnaires completed by teachers responsible for standard 6 mathematics teaching in the secondary schools were used to determine opinions concerning both the-standard of mathematics amongst standard 6 pupils and the problem of adjustment to the secondary School.

This investigation examined Chinese children's development of place-value numeration concepts from ages 3 through 9, compared the development of place-value understanding of these Chinese children with that of American and Genevan children whose performances had been described in the literature, and examined the influence of adult assistance during Chinese children's performances on some of the place-value tasks.

The Mathematical Quality of Instruction (MQI) is an important feature of a classroom mathematics lesson. Before teachers can strengthen the MQI of their own lessons, however, teachers must first be able to notice MQI-related features of instruction. The purpose of this dissertation is to investigate intermediate grades teachers’ MQI-related noticing as they go through a ten-week research-based MQI professional development (PD) program. Specifically, this dissertation is guided by two research questions: (1) How do individual teachers’ noticing of MQI-related features of instruction shift as they go through an MQI-focused professional development program? and (2) How do teachers approach the task of noticing students’ engagement in mathematical practices? To address the first research question, I developed a multi-level framework out of a priori noticing categories in combination with data from 73 PD teachers’ individual typed responses to two video clips of mathematics instruction, in order to characterize degrees of individual teachers’ noticing of MQI-related features of instruction. I then used a mathematized version of the framework along with non-parametric statistical analyses to explore shifts in the teachers’ noticing of MQI-related features of instruction at three different time-points throughout the PD. I also illuminated the nature of these shifts by exploring snapshots of two teachers’ MQI Noticing journeys as they progressed through the PD. To address the second research question, I used thematic analyses of transcripts of one group of five teachers’ discussions in order to identify noticing perspectives exhibited by the group as they engaged in the specific task of discussing potential instances of students’ engagement mathematical practices during the PD sessions. The findings highlight and explicate the variation in teachers’ MQI Noticing, and it establishes that shifts in such noticing are possible as teachers progress through research-based MQI PD.

The purpose of this study was to examine my own practice of teaching place value and the influence virtual manipulatives had, in addition to physical manipulatives, on place value understanding of my second grade students. I wanted to see how adding a base-ten computer applet might better meet the needs of all learners while also meeting the needs of today’s technological classroom. Through this study, I found that both physical and virtual manipulatives helped students acquire place value concepts. I found that virtual manipulatives had features that engaged students in a way that increased their mathematical language, increased students’ ability to represent more conceptual understanding of composing and decomposing numbers, and express enthusiasm towards mathematics. A pretest and posttest revealed that students’ academic performance increased. While research on virtual manipulatives and mathematical achievement is fairly recent, this study offers insight to other classroom teachers and the research community.

No summary available / Mathematical Sciences/Teacher Education / D.Phil.

This study investigates to what extent educators’ pedagogical content knowledge affects learners’ achievement in quadratic functions. The components of pedagogical content knowledge (PCK) examined are: (i) mathematical content knowledge (MCK), (ii) knowledge of learners’ conceptions, and misconceptions, and (iii) knowledge of strategies. The participants were seventeen mathematics educators and ten learners from each educator’s class. The sample of educators was a convenient sample, while the sample of learners was selected by means of random sampling. A mixed method design was used to execute the study. Data about educators’ MCK, and knowledge of learners’ misconceptions were collected by means of a questionnaire. An interview was used to gather data about educators’ knowledge of strategies. Data on learners’ achievements and misconceptions was collected by means of a questionnaire. Descriptive statistics were used to describe the effect of each component of the educators’ PCK on learners’ achievements. The result indicates that the achievement of learners who are taught by educators who have strong PCK is higher than the achievement of learners who are taught by educators who have weak PCK. / Mathematical Sciences / M. Ed. (Mathematics Education)

Thesis (M.Ed.)--University of South Africa, 2002.

M. Tech. Education / The study investigated the extent to which Grade 11 learners were able to formulate Mathematics and Mathematical Literacy problems from a given soccer context and to determine the extent to which the problems formulated by learners reflect the given soccer context. Engaging learners in the activity of task formulation within the context of soccer was an attempt to enable the learners to appreciate the mathematics concepts can be extracted from familiar or everyday situations.

Thesis (PhD)--Stellenbosch Universityh, 1995. / ENGLISH ABSTRACT: This study investigates the influence of readability of mathematics examination questions on achievement. The aim of any mathematics examination is to assess whether the aims of a specific mathematics programme have been realized. Readability factors that unnecessarily prevent a clear understanding of questions could jeopardize this aim. The important issue is, therefore, whether there are indeed readability factors in mathematics examination questions that cause comprehension problems for students and, if there are, do they hav~ any effect on test scores? The issue of readability is of even greater importance for second language readers. In the South Mrican context, the reading problems of second language readers are of particular importance as most students at school are second language learners. An important question would therefore be: What readability factors cause comprehension difficulties for second language students, especially those whose mother tongue is not kindred to English? Furthermore, what is the influence of cultural factors on readability? This study provides answers to these and other related questions for mathematics text at senior secondary school level. Protocol analysis was used to ascertain what readability problems are experienced by students when reading examination questions in mathematics. Three different language groups, comprising 17 -18-year-old students, were used in the study: English First Language students and two groups who had English as a second language. One second language group had Mrikaans as first language while the other group comprised Mrican students whose mother tongue is unrelated to English. A framework was developed to analyse the protocols and it comprised five categories: unfamiliar vocabulary structural problems obscure information visualization difficulties non-verbal factors Mter the protocol study, students were asked to adapt the examination questions to a more comprehensible form. Students' adaptations addressed lexical, syntactical, discourse and non-verbal factors. Most of the readability problems identified in the literature study were verified in the empirical study. However, the empirical study generated additional readability problems that are mainly restricted to mathematics text and relate to nonverbal factors like mathematical expressions. During the last phase of the empirical study a composite test was used to test the hypothesis that improved readability of the common language used in mathematics examination questions' will improve achievement. Nine socalled "word problems" from previous examination papers were set in three different versions: original, adapted and non-verbal. The hypothesis was confirmed in a number of important cases. A significant finding of the study was, therefore, that readability factors not only influence the comprehension of mathematics examination questions, but also have a marked influence on students' achievement levels. The results of the empirical study are reported quantitatively as well as qualitatively. Other results include the following: Not only second language students, but also first language students experienced a variety of readability problems. All three language groups demonstrated the same level of competency on the non-verbal versions. When comparing test scores of the verbal versions, differences in achievement levels between the different language groups were often caused by linguistic and cultural factors. Cultural thought patterns, typical of a mother tongue but absent in a second language, were often a source of comprehension difficulties for second language readers. This study has led to certain conclusions for teaching and examination practice. For example, factors influencing the readability of ordinary English should be considered with other factors when writing mathematics examination questions. Furthermore, the distinctly different reading needs of second language students suggest that examination papers be set, so that the language needs of second language learners are accommodated. Guidelines for writing more readable examination questions were developed and are presented as a readability checklist. Suggestions for further research include the investigation of the influence of readability on achievement in authentic examination conditions. / AFRIKAANSE OPSOMMING: Hierdie studie ondersoek die invloed van leesbaarheid van wiskunde eksamenvrae op prestasie. Die doel van enige wiskunde eksamen is om vas te stel of die doelwitte van 'n spesifieke wiskunde program bereik is. Leesbaarheidsfaktore wat die volledige begryp van vraestelle onnodig belemmer, kan die bereiking van hierdie doel verhinder. Dit is dus belangrik om vas te stel of daar wel leesbaarheidsfaktore in wiskunde eksamenvrae bestaan wat vir leerlinge begripsprobleme veroorsaak en, indien wel, of hulle enige effek op toetspunte het. Vir tweedetaallesers is die kwessie van leesbaarheid van nog groter belang. In die SuidAfrikaanse opset is die leesprobleme van tweedetaal lesers 'n uiters aktuele saak aangesien die meeste skoolleerlinge tweedetaal leerders is. Belangrike vrae is dus: Watter faktore veroorsaak leesbaarheidsprobleme vir tweedetaalleerlinge, veral diegene met 'n nie-verwante moedertaal en, watter invloed het kultuur op leesbaarheid? Hierdie ondersoek bied antwoorde op hierdie en ander verwante vrae ten opsigte van wiskunde eksamenvrae op senior sekondere vlak. Protokol analise is gebruik om vas te stel watter leesbaarheidsprobleme ondervind word wanneer leerlinge wiskunde eksamenvrae lees. Drie verskillende taalgroepe, bestaande uit 17 -18-jarige leerlinge, het aan die ondersoek deelgeneem: Engels Eerstetaalleerlinge en twee groepe wat Engels as tweede taal gehad het. Een van die tweedetaal groepe het Afrikaans as eerste taal gehad terwyl die ander groep bestaan het uit Afrikane wie se moedertaal nie aan Engels verwant is nie. 'n Raamwerk is ontwikkel om die protokolle te analiseer en het uit die volgende vyf kategoriee bestaan: onbekende woordeskat strukturele probleme onduidelike inligting visualiseringsprobleme nie-verbale probleme Gedurende die tweede fase van die ondersoek is leerlinge gevra om die vrae tot 'n meer verstaanbare vorm aan te pas. Leerlinge se aanpassings het leksikale, sintaktiese, diskoers en nie-verbale faktore aangespreek. Sommige leesbaarheidsprobleme wat in die literatuurstudie gei'dentifiseer is, is in die empiriese ondersoek geverifieer. Die empiriese ondersoek het egter addisionele leesbaarheidsprobleme uitgelig wat meerendeels wiskundig van aard is en verband hou met nie-verbale faktore soos wiskunde uitdrukkings. Gedurende die laaste deel van die empiriese ondersoek is 'n samegestelde toets gebruik om die volgende hipotese te toets: Verbeterde leesbaarheid van die gewone taal wat in wiskunde eksamenvrae gebruik word, sal die prestasie van leerlinge verb et er. N ege sogenaamde woordsomme is op verskillende maniere gestel: oorspronklik, aangepas en nie-verbaal. Die hipotese is in 'n hele aantal belangrike gevalle bevestig. Een van die bevindinge van die ondersoek was dus dat leesbaarheidsfaktore nie slegs be grip ten opsigte van wiskunde eksamenvrae bei:nvloed nie, maar ook 'n beduidende invloed op prestasie het. Die resultate van die empiriese ondersoek word kwalitatief sowel as kwantitatief weergegee. Ander resultate sluit die volgende in: Nie slegs tweedetaal leerlinge nie, maar ook eerstetaal leerlinge het 'n verskeidenhied van leesbaarheidsprobleme ondervind Al drie taalgroepe het op die nie-verbale weergawe dieselfde bekwaamheidsvlak getoon. Verskille in die prestasievlakke tussen die verskillende taalgroepe op die verbale weergawes is baiekeer deur taalkundige en kulturele faktore veroorsaak Kulturele denkpatrone wat tipies is van 'n leerling se moedertaal, maar nie in die tweedetaal voorkom nie, het dikwels tot begripsprobleme by tweedetaal lesers gelei. Hierdie ondersoek het sekere gevolgtrekkings v1r o~derrig- en eksamenpraktyk. Byvoorbeeld, faktore wat die leesbaarheid van gewone Engels bei:nvloed, behoort saam met ander faktore in ag geneem te word wanneer wiskunde eksamenvraestelle opgestel word. Verder is daar duidelike aanduidings dat aparte vraestelle vir eerste- en tweedetaal leerlinge opgestel moet word sodat die taalbehoeftes van tweedetaal leerlinge in ag geneem kan word. Riglyne vir die skryf van meer leesbare eksamenvrae is saamgestel en as 'n oorsiglys aangebied. Voorstelle vir verdere navorsing sluit in die ondersoek na die invloed van leesbaarheid op prestasie in ware eksamen omstandighede.

Thesis (PhD)--Stellenbosch University, 2015. / ENGLISH ABSTRACT: In this study, the pedagogical knowledge of Foundation Phase teachers is explored (and unpacked) in order to obtain insight into their understanding of the teaching and learning of mathematics. The teacher’s knowledge is explored, as it is one of the most important variables that impacts on what is done in the classroom. The exploration is undertaken against the background of the very poor overall achievement of learners in the national systemic evaluations and in international assessment studies, which is currently a cause for great concern. This has resulted in different nation-wide intervention programmes that are aimed at improving teacher performance and effectiveness. In this study, the teacher is the focal point of the intervention. Problem-based learning (PBL), which is well-regarded as being one of the best examples of a constructivist learning environment, is introduced to a group of 15 Foundation Phase teachers. The study is an unpacking of the Foundation Phase teachers’ pedagogical knowledge and beliefs regarding, and practices in, the teaching and learning of mathematics, as well as in the use of PBL as a vehicle for the teaching and learning of mathematics. The unpacked knowledge can be used to address the challenges that are related to the improvement of the teaching and learning of mathematics in the Foundation Phase. A combination of qualitative and quantitative research methods, including questionnaires, interviews, lesson observation, and workshops, were used to explore the teachers’ current pedagogical knowledge, beliefs and practices with regard to problem-solving. It was also used to expose the teachers to PBL as an alternative approach to teaching and learning mathematics in the Foundation Phase. The study provides a body of knowledge on the Foundation Phase teachers’ pedagogical knowledge, practices and beliefs regarding the teaching and learning of mathematics in general, and approaches to problem-solving in particular, thus providing insights into some of the factors that might lie behind learner outcomes. Study findings indicate that the majority of teachers’ daily mathematical teaching culture was deep-rooted in the traditional approach (direct transmission). This approach was characterised by the teachers concerned focusing on the following of rules and procedures, and on doing demonstrations on the chalkboard, whereafter the learners were encouraged to practise what they had learned by asking them to do pen-and-paper calculations. The lessons were generally not structured to develop critical thinking and reasoning skills. In instances where the teachers created learner-centred activities that were conducive to the development of such skills, deep-rooted traditional approaches manifested themselves in the way in which the teachers showed the learners how to solve their given problem at the earliest signs of any difficulty in doing so was exhibited by the learners. In so doing, the majority of the teachers, despite initially creating learning opportunities by posing problems to their learners, they soon snatched away the selfsame opportunities from them. This was because they did not allow sufficient time for the learners to grapple with a problem, and to engage in critical thinking. After exposure to PBL, the educators were able to implement PBL so effectively that they could address the problems related to low learner achievement in mathematics, as reflected in the international assessment studies, and in the national systemic evaluations within the current South African context. / AFRIKAANSE OPSOMMING: In hierdie studie is die pedagogiese kennis van Grondslagfase-onderwysers ondersoek ten einde insig te verkry in hulle begrip van die onderrig en leer van wiskunde. Die onderwysers se kennis is ondersoek aangesien dit een van die belangrikste veranderlikes is wat 'n invloed het op dit wat in die klaskamer uitgevoer word. Die ondersoek is onderneem teen die agtergrond dat die algehele prestasie van leerders in die nasionale sistemiese evaluerings en internasionale assesseringstudies uiters swak en 'n bron van groot kommer was. Dit het gelei tot verskillende intervensieprogramme wat gemik is op die verbetering van onderwyserprestasie en -doeltreffendheid. In hierdie studie is die onderwyser die fokuspunt van die intervensie. Probleem-gebaseerde leer (PBL), wat beskou word as een van die beste voorbeelde van 'n konstruktivistiese leeromgewing, is aan 'n groep van 15 onderwysers in die Grondslagfase gebring. Die studie was ‘n poging om nuwe kennis te identifiseer ten opsigte van Grondslagfase-onderwysers se pedagogiese geloof en praktyke in die onderrig en leer van wiskunde, en die gebruik van PBL as 'n middel vir die onderrig en leer van wiskunde – kennis wat gebruik kan word om die verwante uitdagings aan te spreek ter verbetering van die onderrig en leer van wiskunde in die Grondslagfase. 'n Kombinasie van kwalitatiewe en kwantitatiewe navorsingsmetodes, wat vraelyste, onderhoude, les-waarneming en werkswinkels ingesluit het, is aangewend om die onderwysers se huidige pedagogiese sienings en praktyke met betrekking tot probleemoplossing grondig te ondersoek en dan voort te gaan om die onderwysers bloot te stel aan PBL as alternatiewe benadering tot onderrig en leer van wiskunde in die Grondslagfase. Die studie het bevind dat die meerderheid van die onderwysers se huidige onderrigkultuur een was wat diep gewortel is in die tradisionele benadering van onderrig en leer van wiskunde (direkte oordrag): dit is gekenmerk deur die onderwysers se onderrig van wiskunde deur te fokus op reëls en prosedures, demonstrasies aan die klas op die swartbord en leerders dan te laat oefen deur pen- en papierberekeninge te doen. Dié het die meerderheid van die onderwysers daagliks gedoen. Lesse is oor die algemeen nie gestruktureer om kritiese denke en beredenering te ontwikkel nie. In gevalle waar die onderwysers leerder-gesentreerde aktiwiteite geskep het wat weens hulle ontwerp bevorderlik is vir die ontwikkeling van kritiese denke en redenasie, het die diepgewortelde, tradisionele benaderings hulself gemanifesteer in die feit dat die onderwysers, met die eerste aanduiding dat die leerders sukkel, hulle te hulp gesnel het en die leerders gewys het hoe om die probleem op te los. Met dié optrede het die meerderheid van die onderwysers aanvanklik leergeleenthede geskep (deur probleme aan hulle leerders voor te hou), maar dit spoedig dan weer weggeraap weens die feit dat hulle nie genoegsame tyd toegelaat het vir hulle leerders om met idees te worstel en deel te hê aan kritiese denke nie. Blootstelling van die opvoeders aan PBL het aan die lig gebring dat opvoeders in die Grondslagfase PBL doeltreffend kan implementeer om probleme rondom lae leerder prestasie in wiskunde aan te spreek wat in internasionale assesseringstudies en in die nasionale sistemiese evaluerings binne die huidige Suid -Afrikaanse konteks weerspieël word.