Exploration of Grade 11 learners’ mental constructions and difficulties in learning and solving trigonometric equations: a case of one school in Umlazi district = Ukuhlolwa kwezincazelokuzakhela engqondweni kanye nezinkinga zokufunda nokuxazululwa kwama-equation esifundo seminxamithathu kubafundi beBanga 11: ucwaningo kwesinye sezikole esifundeni saseMlazi.

Abstract

Trigonometry is a particularly challenging area of mathematics for high school learners. This study investigated South African Grade 11 learners' mental constructions and difficulties when learning and solving trigonometric equations. Participants were selected from an after-school mathematics programme that they attended on a voluntary basis. Data was collected using an activity sheet and semi-structured interviews. Baseline data was collected using an activity sheet from 17 learners Grade 10 learners; data was collected from the same learners a year later using another activity sheet with a different set of problems. Semi-structured interviews were conducted with 7 learners to probe their responses on the activity sheet. Most of the learners were found to be unable to make the necessary mental constructions to solve trigonometric problems at the Grade 11 level. Dubinsky’s (1991) constructivist APOS theory, which describes how learners construct their knowledge of mathematics concepts in stages characterised by action, process, object and schema, was used to analyse the mental constructions of learners. During both phases of data collection, most learners were found to rely on explicit step-by-step calculations to solve problems, indicating that they were operating at the action stage; a smaller number were able to do some of the steps mentally without writing them out, indicating that they had advanced to the process stage. No evidence was found of learners having advanced to the object or schema stages. Moreover, the findings showed that, while the learners perform procedures correctly, they applied rules without giving reasons. Piaget and Garcia’s (1989) triad mechanism were used to analyse the difficulties that hindered learners’ mental construction of concepts. Learners’ difficulties included incorrect conceptions of the equal sign, overgeneralization of rules, and failure to integrate algebra concepts into their construction of trigonometric concepts. Based on the findings, the study recommends that teachers reinforce basic algebraic skills—such as collecting like and unlike terms, using brackets, and addition and subtraction of algebraic terms—before introducing trigonometric concepts. Teachers are urged to explore different methods for teaching trigonometric equations to enable learners to construct knowledge effectively, such as collaborative learning and differentiated classroom activities.

Iqoqa.

Isifundo seminxamithathu siyingxenye eyinselelo enkulu ezibalweni ze-mathematics kubafundi bamabanga aphezulu. Lolu cwaningo lwahlola izincazelokuzakhela engqondweni kanye nezinkinga uma abafundi beBanga 11 befunda nalapho bexazulula izinkinga zesifundo seminxantathu eNingizimu Afrikha. Ababambiqhaza bakhethwa kulabo abasohlelweni lokufunda izibalo emva kokuphuma kwesikole abaziyela kulo ngokuzinikela. Kwasetshenziswa ishidi lemisebenzi kanye nezingxoxo ezingahlelelwe ukuze kuqoqwe imininingo. Imininingo eqoqwa ekuqaleni kocwaningo yatholakala kubafundi abayi-17 beBanga le-10 kusetshenziswa ishidi lemisebenzi; imininingo yaqoqwa kubo labo bafundi ngonyaka olandelayo kusetshenziswa ishidi leminye imisebenzi elinezibalo ezihlukile. Kwabanjwa izingxoxo ezingahleliwe nabafundi abayi-7 ukuthola izimpendulo zabo mayelana neshidi lemisebenzi. Iningi lalaba bafundi kwatholakala ukuthi abakwazi ukuqhamuka nezincazelokuzakhela emakhanda ukuze bakwazi ukuxazulula izinkinga zesifundo seminxantathu ezingeni lomsebenzi weBanga 11. Ukuhlaziya izincazelokuzakhela ezingqondweni zabafundi kwasetshenziswa injulalwazi ye-constructivist APOS kaDubinsky (1991) echaza ukuthi abafundi balwakha kanjani ulwazi lwabo lwemiqondomsuka yezibalo ngokwamazinga/ngokwezigaba ahambisana nokwenza, inqubo, okufundwayo/okuxazululwayo, kanye ne-schema. Kulezi zigaba ezimbili zokuqoqwa kwemininingo kwatholakala ukuthi abafundi abaningi bathembela ekubaleni isinyathelo ngasinye ukuze baxazulule inkinga. Lokhu kukhombisa ukuthi basezingeni lokwenza; abafundi abambalwa bakhombisa ukukwazi ukulandela izinyathelo ezithile ngekhanda bengabhalanga phansi, okukhombisa ukuthi babethuthukile kancane sebesezingeni lenqubo. Abutholakalanga ubufakazi obuveza ukuthi abafundi base bethuthukele emazingeni okufundwayo noma e-schema. Okunye futhi ukuthi imiphumela yakhombisa ukuthi abafundi balandela izinyathelo ngendlela efanele; basebenzisa imithetho ngaphandle kokunikeza izizathu. ukuhlaziya izinkinga ezithikameza ukwakha ezingqondweni izincazelokuzakhela zemiqondomsuka. Izinkinga zabafundi zazihlanganisa ukungaqondisisi ngendlela uphawu lokulingana, imithetho yokucacisa ngokwedlulele, kanye nokwehluleka ukuhlanganisa imiqondomsuka ye-algebra nokusebenzisana kwayo neyesifundo seminxantathu. Ngokwemiphumela yocwaningo kuphakanyiswa ukuthi othisha baqinise amakhono e-algebra ayisisekelo – njengokuhlanganisa amatemu afanayo kanya nangafani, ukusetshenziswa kwabakaki, kanye namatemu e-algebra okuhlanganisa nokusuza – ngaphambi kokuqalisa ngemiqondomsuka yesifundo seminxantathu. Bayanxuswa othisha ukuba bahlole izindlela ezihlukahlukene zokufundisa ama-equation esifundo seminxantathu ukuze abafundi bakwazi ukuzakhela ulwazi ngendlela efanele. Lezi zindlela kungaba ukufunda ngokubambisana noma ukunikezwa kwemisebenzi ehlukene ngokwamazinga abafundi.