Error patterns and underlying misconceptions of fractions among 15 grade 6 learners from a Public School in Gauteng

Abstract

This study aimed to identify common error patterns concerning fractions for Grade 6 learners and the misconceptions associated with procedural, conceptual and factual errors, as well as subtraction and multiplication at a public school in Orange Farm, Gauteng. The study was based on the ideas of constructivism to understand learners’ reasoning in developing constructions in the concept of fractions. Recent studies on errors and underlying misconceptions about fractions have indicated that most primary school learners struggle with fractions, affecting learners’ future performance at the secondary level and their mathematics experience. The study used a qualitative case study approach to identify the errors made by Grade 6 learners. This approach provided a platform to investigate how Grade 6 learners solve fractions-related problems, allowing the researcher to uncover learners' misconceptions when dealing with fractions. The researcher chose Grade 6 since it is the exit level for the Intermediate Phase. The annual teaching plan, stipulated by the Curriculum Policy Statement policy of the Department of Basic Education (DBE) specifically for Grade 6 learners, focuses on the three operations: addition, subtraction and multiplication. Fifteen learners were purposively sampled and given a set of ten questions based on the operations mentioned above. The study’s main data sources were a written test and interviews. The data collected during the written test and interviews were analysed using a qualitative approach and discussed using constructivism as a theoretical framework. Learners were allowed to explain their responses based on their incorrect answers. The results show that learners lack basic mathematical skills in fractions. They believe that numerator and denominator are the same; they add or subtract both denominators and numerators. When multiplying, learners add the numerators and denominators together and again, due to a lack of conceptual understanding of the multiplication of fractions, learners are challenged to deal with the multiplication of mixed fractions. Integers are multiplied separately from mixed fractions. In addition, some learners don’t understand how they obtained their respective answers. Constructivism values prior knowledge as a basis for developing new knowledge. Teachers should not take for granted what has been covered before learners have dealt with fractions, as this can lead to misunderstandings. Therefore, reviewing the previous information before starting a new lesson is recommended. Another suggested measure is that learners should understand the difference between numerator and denominator and the correct applications of the three operations. The study recommends that appropriate teacher development training in specific mathematics topics be done at the primary school level, as this will lay a foundation for better transmission, not only on the basics of mathematics but also general mathematical knowledge.

Thuto ena e ne e ikemiseditse ho hlwaya mekgwa e tlwaelehileng ya diphoso tse mabapi le dikarolo tsa barutuwa ba Kereiti ya 6 le menahano e fosahetseng e amanang le diphoso tsa mokgwa, mohopolo le nnete ho tlatseletsa, ho ntsha le ho atisa dikarolwana sekolong sa mmuso Orange Farm, Gauteng. Thuto ena e ne e ipapisitse le mehopolo ya kaho ya tsebo ho utlwisisa menahano ya barutuwa ho ntshetsapele meaho mohopolong wa dikarolwana. Diphuputso tsa moraorao tse mabapi le diphoso le menahano e fosahetseng ya motheo mabapi le dikarolwana di bontshitse hore boholo ba barutuwa ba dikolo tsa pele ba sokola ka dikarolwana, tse amang tshebetso ya kamoso ya barutuwa sehlopheng sa sekondari le boiphihlelo ba bona ba dipalo. Patlisiso e sebedisitse mokgwa wa ho batlisisa wa maemo ho lemoha diphoso tseo entsweng ke barutuwa ba Kereiti ya 6. Mokgwa ona o fane ka sethala sa ho batlisisa hore na barutuwa ba Kereiti ya 6 ba rarolla mathata a amanang le dikarolwana jwang, ho dumella mofuputsi ho sibolla menahano e fosahetseng eo barutuwa ba nang le yona kapa ba e bontshang ha ba sebetsana le dikarolwana. Thuto ena e tla fana ka kenyelletso e ntle ka ho fetisisa ya algebra maemong a phahameng le a FET. Mofuputsi o kgethile Kereiti ya 6, kaha ke boemo ba ho tswa ho Mokgahlelo o Mahareng. Leano la ho ruta la selemo le selemo, le hlalositsweng ke pholisi ya Setatamente sa Pholisi ya Kharikhulamo la Lefapha la Thuto ya Motheo (DBE) ka ho qolleha bakeng sa barutuwa ba Kereiti ya 6, le tsepamisitse maikutlo hodima mesebetsi e meraro: ho eketsa, ho tlosa le ho atisa. Barutuwa ba leshome le metso e mehlano ba ile ba etswa disampole ka sepheo se nepahetseng mme ba fuwa dipotso tse leshome tse ipapisitseng le tshebetso e boletsweng ka hodimo. Mehlodi e meholo ya dintlha tsa thuto e ne e le teko e ngotsweng le dipuisano. Dintlha tse bokelletsweng nakong ya hlahlobo e ngotsweng le dipuisano di ile tsa hlahlojwa ka mokgwa wa boleng mme tsa tshohlwa ka mokgwa wa kaho ya tsebo e le moralo wa teori. Barutuwa ba ile ba dumellwa ho hlalosa dikarabo tsa bona ho latela dikarabo tsa bona tse fosahetseng. Diphetho di bontsha hore barutuwa ba haellwa ke tsebo ya mantlha ya dipalo ka dikarolwana. Ba dumela hore dipalo le dinomineitha di a tshwana; ba eketsa kapa ba fokotsa ka bobedi di- dinomineitha le dipalo. Ha ba atisa, barutuwa ba kopanya dipalo le di- dinomineitha mmoho, hape, ka lebaka la ho hloka kutlwisiso ya moelelo wa katiso ya dikarolwana, barutuwa ba phephetswa ho sebetsana le katiso ya dikarolwana tse tswakilweng. Dinomoro di ngatafaditswe ka thoko ho dikarolwana tse tswakilweng. Ho feta viii moo, barutuwa ba bang ha ba utlwisise hore na ba fumane dikarabo tsa bona tse fapaneng jwang. Kaho ya tsebo e nka tsebo ya pele e le motheo wa ho ntlafatsa tsebo e ntjha. Matitjhere ha ba tshwanela ho nkela fatshe se ileng sa akaretswa pele ha barutuwa ba sebetsana le dikarolwana, ka hobane sena se ka baka ho se utlwisise hantle. Ka hona, ho kgothaletswa ho hlahloba boitsebiso bo fetileng pele o qala thuto e ntjha. Tekanyo e nngwe e sisintsweng ke hore barutuwa ba lokela ho utlwisisa phapano pakeng tsa dinomoro le dinomineitha le tshebediso e nepahetseng ya ditshebetso tse tharo. Patlisiso e tsitsinya hore kwetliso ya ntlafatso e nepahetseng ya hlabollo ya matitjhere dihloohong tse tsebahalang tsa dipalo e etswe boemong ba sekolo sa mantlha, ka ha sena se tla theha motheo wa phetiso e ntle, e seng feela motheong wa thuto ya dipalo empa le tsebo ya kakaretso ya dipalo.

Die doel van hierdie studie was om algemene foutpatrone rakende breukpatrone vir Graad 6-leerders en die wanopvattings wat verband hou met konseptuele en feitelike foute by optel, aftrek en vermenigvuldiging van breuke by ’n openbare skool in Orange Farm, Gauteng, te identifiseer. Die studie is gebaseer op die idees van konstruktivisme om leerders se redenasie in die ontwikkeling van konstruksies in die konsep van breuke te verstaan. Onlangse studies oor foute en onderliggende wanopvattings oor breuke het aangedui dat die meeste laerskoolleerders met breuke sukkel wat leerders se toekomstige prestasie op sekondêre vlak en hulle wiskunde-ervaring beïnvloed. Die studie het ’n kwalitatiewe gevallestudiebenadering gebruik om foute wat deur Graad 6-leerders gemaak is, te identifiseer. Hierdie benadering het ’n platform verskaf om te ondersoek hoe Graad 6-leerders breukverwante probleme oplos, wat die navorser in staat gestel het om die wanopvattings wat leerders het of openbaar wanneer hulle met breuke te doen het, te ontbloot. Die studie sal die beste inleiding tot algebra in die Senior en VOO-fases verskaf. Die navorser het Graad 6 gekies aangesien dit die uittreevlak vir die Intermediêre Fase is. Die jaarlikse onderrigplan bepaal deur die Kurrikulumbeleidsverklaring van die Departement van Basiese Onderwys (DBO) spesifiek vir Graad 6-leerders, fokus op die bewerkings: optel, aftrek en vermenigvuldiging. Vyftien leerders is in ’n doelbewuste steekproef gebruik en het ’n stel van tien vrae gekry wat gebaseer is op die bogenoemde bewerkings. Die studie se hoofdatabronne was ’n geskrewe toets en onderhoude. Die data wat tydens die geskrewe toets en onderhoude ingesamel is, is met behulp van ’n kwalitatiewe benadering ontleed en bespreek deur ’n teoretiese raamwerk van konstruktivisme. Leerders is toegelaat om hulle antwoorde op grond van hulle verkeerde antwoorde te verduidelik. Die resultate toon dat leerders nie basiese wiskundige vaardighede in breuke het nie. Hulle glo dat teller en noemer dieselfde is; hulle tel beide noemers en tellers op of af. Wanneer daar vermenigvuldig word, tel leerders die tellers en noemers bymekaar en, weer eens, as gevolg van ’n gebrekkige konseptuele begrip van die vermenigvuldiging van breuke, word leerders uitgedaag om die vermenigvuldiging van breuke te hanteer. Heelgetalle word afsonderlik van gemengde breuke vermenigvuldig. Daarbenewens verstaan sommige leerders nie hoe hulle hulle onderskeie antwoorde gekry het nie. Konstruktivisme besef die waarde van vorige kennis as ’n basis vir die ontwikkeling van nuwe kennis. Onderwysers moet nie dit wat gedek is as vanselfsprekend aanvaar voordat leerders met breuke te make het nie, aangesien dit tot misverstande kan lei. Daarom word daar aanbeveel dat vorige inligting hersien word voordat ’n nuwe les begin word. Nog ’n voorgestelde maatstaf is dat leerders die verskil tussen die teller en die noemer en die korrekte toepassings van die drie bewerkings moet verstaan. Die studie beveel aan dat toepaslike onderwyserontwikkelingsopleiding in spesifieke wiskunde-onderwerpe op laerskoolvlak gedoen word, aangesien dit ’n grondslag lê vir beter oordrag, nie net oor die basiese beginsels nie, maar ook algemene wiskundige kennis.