Finding an effective problem-solving heuristic instructional approach for circle geometry

Abstract

This research study carried out an investigation into finding a contemporary problem- solving instructional approach that will be effective for teaching and learning of mathematics in South African schools, with specific focus on circle geometry. Prior to conducting this study, a retrospection was done into the mathematical practices implemented, in schools in South Africa, by researchers, educational practitioners and stakeholders such as Non-Governmental Organisations. The aforementioned unanimously identified the instructional approaches for teaching and learning of mathematics, particularly, the traditional teaching and learning approach, as problematic and counter-productive, and this might be contributing to poor learners’ performances. In a bid to replace the obsolete traditional approach, the researcher in this study recommended: “teaching thinking skills” and “teaching effective problem-solving instructional approaches” as more appropriate. With regards to teaching thinking skills, the infusion approach (teaching thinking skills, along with content instructions), was highlighted. For teaching effective problem-solving, Polya’s Problem-Solving Model, was investigated. To ensure an effective design and implementation of the proposed problem-solving instructional approach, the APOS theory (ACE teaching cycle) was adopted. Also, the teaching and learning of circle geometry was carried out in a collaborative classroom setting. This proposed instructional approach was tentatively, labelled as “IPAC mathematics problem-solving instructional model’’ or simply, the “IPAC model”. This was an acronym for the four elements of this new approach, namely - the infusion approach, Polya’s approach, and APOS theory in a collaborative learning classroom. Two groups of Grade 11 mathematics learners served as participants for this study: group 1 - 11A had 30 learners (the control group) and group 2- 11B had 32 learners (the experimental group). Data collected methods for this study were: observations of participants in their natural classroom settings, recorded videos, questionnaires, photograph of participants’ work (classwork/homework and standardized tests). This study followed a mixed-method research design, hence, both quantitative and qualitative data analyses procedures were implemented. The quantitative data was analysed by implementing inferential statistics and descriptive statistics, while the APOS theory analysis was used to analyse the qualitative facet of the collected data. During the APOS theory analysis, content analysis was done on participants’ written responses to each of the four standardized tests’ data. The content analysis was carried out on the written responses of participants, from both the control and the experimental groups. The research findings that emanated from this study were the following: that this new method of teaching and learning is valid, practical and effective; there was a statistically significant improvement in the test scores of participants who were taught by the new instructional approach; participants’ conceptual understanding, procedural fluency, strategic competence and mathematical reasoning skills were enhanced; participants’ problem-solving competence improved, during and after the intervention; the IPAC model guided the majority of the participants to operate at the object and schema levels in relation to the APOS theory mental conceptions. Lastly, the ACE teaching instructional approach significantly guided and enhanced participants’ cognitive engagement and development, which ultimately, optimized their problem-solving competence. Based on these research findings, the researcher recommended among others, that the new instructional approach - the IPAC model, should be implemented for teaching and learning of circle geometry in South African schools. The researcher also recommended that cultivation of thinking skills and implementation of effective problem-solving instructional approaches should be prioritized in mathematics classrooms in South Africa. The researcher established from this study that the developed IPAC model will serve as an effective and a reliable pedagogical tool which can address some of the teaching and learning challenges teachers and learners encounter in mathematics classrooms.

Hierdie navorsingstudie het 'n ondersoek gedoen na die vind van 'n kontemporêre probleemoplossende onderrigbenadering wat effektief sal wees vir onderrig en leer van wiskunde in Suid-Afrikaanse skole, met spesifieke fokus op sirkelmeetkunde. Voor die uitvoering van hierdie studie is 'n terugblik gedoen na die wiskundige praktyke wat in skole in Suid-Afrika geïmplementeer is deur navorsers, opvoedkundige praktisyns en belanghebbendes soos nie-regeringsorganisasies. Die instruksionele benaderings vir onderrig en leer van wiskunde, veral die tradisionele onderrig-en-leerbenadering, is eenparig geïdentifiseer as problematies en teenproduktief, en dit kan dalk bydra tot swak leerders se prestasies. In 'n poging om die uitgediende tradisionele benadering te vervang, het die navorser in hierdie studie aanbeveel: "onderrig van denkvaardighede" en "onderrig van effektiewe probleemoplossende onderrigbenaderings" as meer gepas. Met betrekking tot die onderrig van denkvaardig hede, is die infusiebenadering (onderrig van denkvaardighede, tesame met inhoudsinstruksies), uitgelig. Vir die onderrig van effektiewe probleemoplossing is Polya se probleemoplossingsmodel ondersoek. Om 'n effektiewe ontwerp en implementering van die voorgestelde probleemoplossende onderrigbenadering te verseker, is die APOS-teorie (GOS-onderrigsiklus) aanvaar. Die onderrig en leer van sirkelmeetkunde is ook in 'n samewerkende klaskameropset uitgevoer. Hierdie voorgestelde onderrigbenadering is voorlopig, gemerk as "IPAC wiskunde probleemoplossing instruksionele model" of eenvoudig die "IPAC model". Dit was 'n akroniem vir die vier elemente van hierdie nuwe benadering, naamlik - die infusiebenadering, Polya se benadering en APOS-teorie in 'n samewerkende leerklaskamer. Twee groepe graad 11-wiskunde-leerders het as deelnemers vir hierdie studie gedien: groep 1 - 11A het 30 leerders (die kontrolegroep) en groep 2- 11B het 32 leerders (die eksperimentele groep). Data wat ingesamel is metodes vir hierdie studie was: waarnemings van deelnemers in hul natuurlike klaskamerinstellings, opgeneemde video's, vraelyste, foto van deelnemers se werk (klaswerk/huiswerk en gestandaardiseerde toetse). Hierdie studie het 'n gemengde-metode navorsingsontwerp gevolg, dus is beide kwantitatiewe en kwalitatiewe data-ontledingsprosedures geïmplementeer. Die kwantitatiewe data is ontleed deur inferensiële statistiek en beskrywende statistiek te implementeer, terwyl die APOS teorie-analise gebruik is om te analiseer die kwalitatiewe faset van die versamelde data. Tydens die APOS-teorie-analise is inhoudsontleding gedoen op deelnemers se geskrewe antwoorde op elk van die vier gestandaardiseerde toetse se data. Die inhoudsanalise is uitgevoer op die geskrewe reaksie van deelnemers, van beide die kontrole- en die eksperimentele groepe. Die navorsingsbevindinge wat uit hierdie studie voortgespruit het, was die volgende: dat hierdie nuwe metode van onderrig en leer geldig, prakties en effektief is; daar was 'n statisties beduidende verbetering in die toetstellings van deelnemers wat deur die nuwe onderrigbenadering onderrig is; deelnemers se konseptuele begrip, prosedurele vlotheid, strategiese bevoegdheid en wiskundige redenasievaardighede is verbeter; deelnemers se probleemoplossingsbevoegdheid het verbeter, tydens en na die intervensie; die IPAC-model het die meerderheid van die deelnemers gelei om op die objek- en skemavlakke te werk in verhouding tot die APOS-teorie se verstandelike opvattings. Laastens het die GOS-onderrigbenadering die deelnemers se kognitiewe betrokkenheid en ontwikkeling aansienlik gelei en verbeter, wat uiteindelik hul probleemoplossingsbevoegdheid geoptimaliseer het. Op grond van hierdie navorsingsbevindinge het die navorser onder andere aanbeveel dat die nuwe onderrigbenadering - die IPAC-model, geïmplementeer moet word vir onderrig en leer van sirkelmeetkunde in Suid-Afrikaanse skole. Die navorser het ook aanbeveel dat die kweek van denkvaardighede en implementering van effektiewe probleemoplossende onderrigbenaderings in wiskundeklaskamers in Suid-Afrika geprioritiseer moet word. Die navorser het uit hierdie studie vasgestel dat die ontwikkelde IPAC-model sal dien as 'n effektiewe en betroubare pedagogiese hulpmiddel wat sommige van die onderrig- en leeruitdagings wat onderwysers en leerders in wiskundeklaskamers ondervind, kan aanspreek.

Lolu cwaningo luqukethe uphenyo mayelana nokuthola ikhambi elingaxazulula ekutholeni indlela eqondile engaletha imiphumela ewusizo ekufundiseni nasekufundeni kwezibalo ezikoleni zaseMzansi Africa, ezophinde ibhekane ngqo ne circle Geometry. Ngaphambi kokuba kuqale lolu cwaningo, kube nolunye ucwaningo olunzulu olwenziwe ngezinye izindlela esezivele zikhona mayelana nezibalo, ezikoleni zaseMzansi Africa, lwenziwa ngabacwaningi, izifundiswa ezingo ncweti Kanye nezinhlangano ezizimele. Inhlangano ebizwa nge okushiwo ngenhla luhlonze indlela eqondile yokufundisa nokufunda izibalo, ikakhulukazi, indlela ejwayelekile yokwenza, njengezindlela eziyinkinga nezingahambisani, futhi lokhu ngungaba yimbangela ekungenzini kahle kwabafundi. Emkhankasweni wokushintsha lolu hlelo oludala lokwenza olungasasizi, uMhlaziyi kulolu cwaningo uncome ukuthi: “ikhono elufundisa ukuzicabangela” Kanye “nekhono lokufundisa elisebenzayo ukuzixazululela izinkinga” njengendlela okuyiyo efanele. Mayelana nekhono elifundisa ukuzicabangela, indlela eyiqophelo (ikhono elifundisa ukuzicabangela, elihambisana nemigomo equkethwe), luthintiwe. Mayelana nohlelo oluwusizo ekuxazululeni izinkinga, uhlelo luka Polya lokuxazulula izinkinga luphenyiwe. Ukuqinisekisa ukuthi uhlelo olusebenzayo futhi oluzosentsenziswa ekuphakamiseni indlela eqondile enemigomo ekuxazululeni izinkinga yokwenza, i APOS theory (ACE teaching cycle) iyona ekhethiwe. Okunye, uhlelo lokufundisa nokufunda i circle geometry lukhishiwe endleleni ehlanganisayo yokuhlala egunjini lokufunda. Okwamanje Lolu hlelo oluphakanyisiwe lokufundisa, lubekwe njenge “IPAC indlela yezibalo eqondile yokuxazulula izinkinga enemigomo” . Lokhu kuyigama elifinqiwe elakhiwe izinhlamvu ezine kule ndlela entsha ebizwa nge infusion approach, Polya’s approach, Kanye ne APOS theory egunjini lokufunda elihlanganisile. Amaqembu amabili ebanga le shuminanye labafundi bezibalo basentshenzisiwe ukubamba iqhaza kulolu cwaningo: iqembu lokuqala ibanga 11A ebelinabafundi abangu 30 (iqembu labaqondisi) bese iqembu lesibili ibanga 11B ebelinabafundi abangu 32 (iqembu elenzayo). Ucwaningo oluqoqiwe lwalendlela lube kanje: imibono yalaba ebekade bebambe iqhaza egunjini lokufunda obuhleliwe, baqophe amavidiyo, babhala imibuzo, bathatha izithombe zalaba ekade bembambe iqhaza lwalomsebenzi wokubamba iqhaza. (imisebenzi yasegunjini lokufunda/imisebenzi yasekhaya Kanye nokwenza uvivinyo). Lolu phenyo lulandele uhlelo oluxubile okuwuhlelo lokuphenya, yingakho zombili lezi zinhlelo zokuqukethwe nokuseZingeni zokuqoqa uphenyo olwenziwe zisentshenzisiwe. Uhlelo lokuqukethwe lemininingwane lusentshenzisiwe ukuhlaziya ngokusebenzisa uhlelo lokuqoqa okutholakele Kanye nohlelo lokwenza okutholakele, futhi kube kwenziwa ne APOS theory analysis ukuhlaziya okusezingeni eliphezulu zigxenye zonke lwemininingwane eqoqiwe. Ngesikhathi se APOS theory analysis, ukuhlaziywa kokuqukethwe okwenziwe ababambe iqhaza babhale okwenzekile ngesikhathi benza lezi zivivinyo ezine ezibekiwe. Uhlelo lokuhlaziya okuqukethwe lwenziwe labhalwa yilaba kade bebambe iqhaza, kuwo womabili amaqembu , elokuqondisa nelokwenza. Uphenyo olutholakele kulolu hlelo lunje: lolu hlelo lokufundisa nokufunda luyasebenza, luyenzeka, futhi lunomehluko: ngokwezibalo kube nomehluko omkhulu oncono ezibalweni zalabo ekade bebambe iqhaza besebenzisa indlela entsha yemigomo: bonke ekade bebambe iqhaza bathole ithuba lokuthi kuthuthuke amakhono abo ekwazini ukuqonda ukuzicabangela, ekwazini ukwenza izinto ezinomehluko eyinqubomgomo, ukumelana nezindlela eziningi eziphumelelisayo Kanye nekhono lokuqonda izibalo; ikhono lalabo ekade bebambe iqhaza ekuxazululeni izinkinga ngokusezingeni lithuthukile, ngesikhathi nangemuva kokwenza ucwaningo; I IPAC model ukwenzisa abaningi balaba ekade bebambe iqhaza kalula umsebenzi ngokuhlukana kwamazinga kusentsenziswa i APOS theory. Ekugcineni, indlela yokwenza ebizwa nge ACE teaching ikwazile okwenzisa kahle ngokusezingeni eliphezulu futhi yakhuphula labo ebekade bebambe iqhaza yaphinde yabathuthukisa, lokhu okwenze bakwazi ukuba sezingeni lokuphumelela ukuxazulula izinkinga. Ngenxa yalokhu okutholakale kucwaningo, umcwaningi uncome ukuthi kokunye, indlela entsha yokwenza ngemigomo – i-IPAC, kumele isentshenziswe ekufundiseni nasekufundeni i circle geometry ezikoleni zaseMzansi Africa. Umcwaningi uphinde waphakamisa ukuthi ukuthuthukisa ikhono lokuzicabangela nokwenziwa kwezindlela ezisebenzayo zokuxazulula izinkinga kumele zibekwe phambili emagunjini okufunda izibalo eMzansi Africa. Umcwaningi ubeke indlela eseqophelweni eliphezulu eyisisekelo kusukela kwisifundo esenziwe yokuthi i IPAC model iyona esebenza njenge ndlela eyithuluzi elibonakalayo futhi elinemiphumela emihle ethembekile, engakwazi ukubhekana nezinkinga futhi ixazulule izinqinamba zokufundisa nokufunda ezikoleni, lezi othisha nabafundi ababhekana nazo egunjini lokufundela izibalo