Investigating the beliefs about problem-solving of mathematics teachers at independent secondary schools in South Africa

Willers, Hendrik Stephanus (2019-04)

Thesis (MEd)--Stellenbosch University, 2019.


ENGLISH ABSTRACT : The South African National Curriculum Statement (NCS) for Mathematics calls for teaching that facilitates and promotes problem-solving as a routine activity and which promotes learning and teaching from a constructivist paradigm. This is in line with international curricula. However, teachers have found it difficult to incorporate problem-solving into their teaching and to include problem-solving questions in their school-based assessments, even in privileged school environments. Researchers argue that teachers’ beliefs about the nature of Mathematics and the teaching and learning of Mathematics significantly influence their teaching practices. Teachers who hold a constructivist belief about the teaching and learning of Mathematics have been shown to incorporate problem-solving in their teaching more readily. However, those with traditionalist beliefs find it difficult to include the reformed practices called for by the reformed curricula. This research therefore investigated the beliefs of practising secondary Mathematics teachers about problem-solving, the nature of Mathematics, and the teaching and learning of Mathematics, and how their beliefs related to their implementation of problem-solving activities as required by the NCS. This research is a quantitative case study, augmented by open-ended questions and semistructured interviews. The participants were Mathematics teachers working at predominantly privileged secondary schools in South Africa. As these schools offer a positive teaching and learning environment with qualified and experienced teaching staff, a reasonable expectation would be that the aims and objectives of the reformed curriculum would be met. Data was collected through the use of questionnaires with further data collection taking place in the form of semi-structured interviews. The data from 95 completed questionnaires were analysed, followed by the semi-structured interviews that were used only for the purposes of informing future research, as only two interviews could be completed. The theoretical framework used for the study is based on the three belief dimensions of Mathematics teachers: (i) the nature of the discipline of Mathematics; (ii) the teaching of school Mathematics; (iii) the learning of school Mathematics. Teachers’ beliefs were further categorised on a continuum moving from a traditional belief to a constructivist belief within these three dimensions. In addition, teachers’ beliefs where investigated in relation to their teacher-training qualifications and their academic qualifications in Mathematics. Both these aspects have been shown in prior research to have an influence on a teacher’s beliefs. The research indicated that: (i) the participants hold traditionalist beliefs in general about the nature of Mathematics, while they hold constructivist beliefs about the teaching and learning of Mathematics; (ii) the participants with post-reform qualifications are more likely to hold constructivist beliefs in general than their colleagues with pre-reform qualifications; (iii) the participants with only a Mathematics 1 qualification, and who have both pre- and post-reform professional qualifications, are more likely to have traditionalist beliefs; (iv) the participants (most of whom are qualified and experienced educators) struggle to describe adequately aspects of teaching and learning that involve problem-solving. The study therefore informs schools and teacher-training institutions regarding aspects of teachers’ professional development – that the focus should be on the development of constructivist beliefs which will encourage problem-solving as a routine activity in classroom practice.

AFRIKAANSE OPSOMMING : Die nasionale Suid-Afrikaanse Wiskundekurrikulum, soos ander internasionale kurrikulums, vereis dat die onderrig van probleemoplossing in Wiskunde as ‘n roetine aktiwiteit gefasiliteer word en dat ‘n konstruktiewe onderrigparadigma bevorder word. Onderwysers vind dit egter moeilik om probleemoplossing by hul onderrig te inkorporeer en ook om probleemoplossingstipe vrae by hul skoolgebaseerde asseserings in te sluit - selfs by skole in wat in gunstige omstandighede funksioneer. Navorsers argumenteer dat onderwysers se denkwyses rakende die aard van Wiskunde en die leer en onderrig van Wiskunde, ‘n beduidende invloed op die onderwyser se onderrigpraktyk het. Onderwysers wat ‘n konstruktiewe denkwyse oor die leer en onderrig van Wiskunde het, toon ‘n groter gewilligheid om probleemoplossing by hul onderrig in te sluit. Terwyl dié met tradisionele denkwyses dit moeilik vind om die hervormde praktyke in hul onderrig, soos vereis word deur die hervormde kurrikulums, te inkorporeer. Die studie ondersoek dus sekondêre Wiskunde onderwysers se denkwyses oor probleemoplossing, die aard van Wiskunde en die leer en onderrig van Wiskunde en hoe dit verband hou met die implementasie van probleemoplossingsaktiwiteite soos deur die Nasionale Wiskundekurrikulum vereis word. Die navorsing is ‘n kwantitatiewe gevallestudie wat deur kwalitatiewe ope vrae met semigestruktueerde onderhoude aangevul word. Die deelnemers aan die studies was Wiskunde onderwysers vanaf oorwegend gegoede sekondêre skole in Suid-Afrika. Aangesien die skole oor positiewe onderrig- en leeromgewings met goed opgeleide en ervare onderwysers beskik, is daar ‘n redelike verwagting dat die doelstellings en doelwitte van die hervormde kurrikulum bereik sal word. Data insameling het deur die voltooiing van vraelyste, gevolg deur semi-gestruktueerde onderhoude plaasgevind. Data insameling en analise is deur die gebruik van 95 voltooide vraelyste voltooi, gevolg deur die semi-gestrukueerde onderhoude. Die data van die onderhoude is slegs gebruik om die studie aan te vul en om ‘n basis vir toekomstige studies te bied. Die teoretiese raamwerk wat in die studie gebruik is, is op die drie aspekte van Wiskunde onderwysers se denkwyses gebaseer: (i) Die aard van Wiskunde as ‘n dissipline. (ii) Die onderrig van Wiskunde op skoolvlak. (iii) Die leer van Wiskunde op skoolvlak. Onderwysers se denkwyses word verder op ‘n kontinuum, wat vanaf ‘n tradisionele denkwyse tot ‘n konstruktiewe denkwyse beweeg, gekategoriseer. Onderwysers se denkwyses word verder, in verband met hul onderwyskwalifikasie en akademiese kwalifikasie in Wiskunde, ondersoek. Vorige navorsing het getoon dat beide dié aspekte ‘n onderwyser se denkwyse beïnvloed. Die navorsingstudie het die volgende getoon: (i) Oor die algemeen behou die deelnemers tradisionele denkwyses oor die aard van Wiskunde, maar in teenstelling behou hul konstruktiewe denkwyses oor die leer en onderrig van Wiskunde. (ii) Deelnemers met post-hervorm kwalifikasies is meer geneig om konstruktiewe denkwyses te behou as die pre-hervorm gekwalifiseerde deelnemers. (iii) Deelnemers met Wiskunde 1 as kwalifikasie is meer geneig om tradisionele denkwyses oor beide pre- en post-hervormde professionele gekwalifiseerde deelnemers te behou. (iv) Die deelnemers, wat meestal goedopgeleide en ervare onderwysers uitmaak, vind dit moeilik om aspekte van probleemoplossing binne die leer en onderrig daarvan voldoende te bespreek. Daarom beveel die studie aan dat skole en ander onderwys- en opleidingsinstansies op die aspekte van professionele opleiding fokus wat die onwikkeling van konstruktiewe denkwyses bevorder en probleemoplossing as ‘n roetine aktiwiteit promoveer.

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