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A design based research on students' understanding of quadratic inequalities in a graphing calculator enhanced environment

dc.contributor.advisorNdlovu, M. C.en_ZA
dc.contributor.authorNdlovu, Levien_ZA
dc.contributor.otherStellenbosch University. Faculty of Education. Curriculum Studies.en_ZA
dc.date.accessioned2019-11-24T12:12:45Z
dc.date.accessioned2019-12-11T06:51:53Z
dc.date.available2019-11-24T12:12:45Z
dc.date.available2019-12-11T06:51:53Z
dc.date.issued2019-12
dc.identifier.urihttp://hdl.handle.net/10019.1/107187
dc.descriptionThesis (PhD)--Stellenbosch University, 2019.en_ZA
dc.description.abstractENGLISH SUMMARY : The purpose of this design based research (DBR) study was to investigate the grade 11 students' understanding of quadratic inequalities in a graphing calculator (GC) enhanced mathematics classroom. The study was framed within the pragmatic paradigm which is committed to multiple world-realities. This pragmatic paradigm embraces mixed methods to collect both quantitative and qualitative data on students' understanding of quadratic inequalities and to generate evidence that would guide educational practice. This study consisted of three main research cycles of the teaching experiments i.e., three high schools in Gauteng province and were conducted in phases. A hypothetical learning trajectory (HLT) was developed in the first phase and used for monitoring the hypotheses, assessing the starting point of students’ understanding and formulating the end goals. The instructional activities were created using the heuristics from the guided reinvention, didactical phenomenology and emergent models. The feed-forwards from the first two research cycles helped to improve the HLT leading to a coherent local instructional theory for quadratic inequalities in a GC environment. The findings of the three research cycles were that the use of an integrated approach (graphic and algebraic) proved to be an effective learning strategy for solving quadratic inequalities in a GC mediated classroom. Students were able to visualise and interpret the graphs and their properties (e.g., zeros, intervals, axis of symmetry, concavity and domain) displayed on the screens of the GCs. Students used instrumented action schemes of graphing and tabulating values to develop and reify the concept of quadratic inequalities. Students also led to meaningfully written solution sets of quadratic inequalities using correct interval notations. The results of the pre- and post-tests showed that there was a significant difference in the mean scores, suggesting an improved performance. The effectiveness of the GC use on students’ performance was practically justified by the Cohen’s d effect sizes, which were large in all the three cycles. Secondly the use of real-life mathematical situations involving linear inequalities as the starting points supported the students’ conceptual understanding of quadratic inequalities. The students’ understanding of real-life mathematical situations moved from the referential level to the general level. The use of the GC also enhanced the students’ reasoning and problem solving skills in quadratic inequalities. These skills enabled students to represent real world problems mathematically (horizontal mathematization), solve the problem using the initiated strategies, interpret the model solutions and look back at the adequacy of their solutions. However, a cognitive obstacle for many learners was to help them to develop metacognitive or executive control skill of self-monitoring during problem solving in all three cycles. The use of the GC also afforded the students an opportunity to move from the informal reasoning (horizontal mathematising) to formal reasoning (vertical mathematising). The findings support previous studies in the domain that the use of the GC improves students’ understanding in learning mathematics. The findings of the three cycles permitted to produce evidence-based heuristics such as design principles that might inform the future decisions for learning quadratic inequalities in a flexible GC environment. The main design principle of this study was: Graphically representing quadratic inequalities in a flexible graphing calculator environment. To this end, the focus was to help students become flexible in dealing with quadratic inequalities in the form of symbols, graphs, or contextual problems. Other essential design principles that emerged in these three cycles were a) the training students to use the GC fluently to reduce chances of the limited viewing window for becoming a source for students' misconceptions and b) using the GC cannot address all learning styles, and must be complemented by other traditional methods. It is hoped that the findings of this study will contribute to the research literature on how to effectively teach the topic of quadratic inequalities. Similarly, professional development programmes and workshops for teachers can be conducted at cluster or district level starting with piecemeal group. Furthermore, the findings might be recommended to the textbook or curriculum developers for designing more explorative learning activities with graphing calculators. The results of the three DBR cycles might be added to the likelihood of transferability to other algebraic concepts.en_ZA
dc.description.abstractAFRIKAANSE OPSOMMING : Die doel van hierdie ontwerpgebaseerde navorsing (DBR) was om die graad 11-studente se begrip van kwadratiese ongelykhede in 'n grafiese sakrekenaar-verbeterde wiskundeklaskamer te ondersoek. Die studie is geraam binne die pragmatiese paradigma wat verbind is tot veelvuldige wêreldrealiteite. Hierdie pragmatiese paradigma bevat gemengde metodes om sowel kwantitatiewe as kwalitatiewe gegewens te versamel oor studente se begrip van kwadratiese ongelykhede en om bewyse te genereer wat die onderwyspraktyk kan lei. Hierdie studie het bestaan uit drie hoofnavorsingsiklusse van die onderrigeksperimente, dit wil sê drie hoërskole in die provinsie Gauteng en is in fases uitgevoer. In die eerste fase is 'n hipotetiese leerbaan (HLT) ontwikkel en gebruik vir die monitering van die hipoteses, die beoordeling van die beginpunt van studente se begrip en die formulering van die einddoelwitte. Die onderrigaktiwiteite is geskep deur gebruik te maak van die heuristiek uit die geleide heruitvinding, didaktiese fenomenologie en ontluikende modelle. Die aanvoerders vanaf die eerste twee navorsingsiklusse het gehelp om die HLT te verbeter, wat gelei het tot 'n samehangende plaaslike onderrigteorie vir kwadratiese ongelykhede in 'n GC-omgewing.Die bevindinge van die drie navorsingsiklusse was dat die gebruik van 'n geïntegreerde benadering (grafies en algebraïes) 'n effektiewe leerstrategie was om kwadratiese ongelykhede in 'n GC-bemiddelende klaskamer op te los. Studente kon die grafieke en hul eienskappe (byvoorbeeld nulle, intervalle, simmetrie-as, konkawiteit en domein) wat op die skerms van die GC's verskyn, visualiseer en interpreteer. Studente het instrumentale aksieskemas gebruik om grafieke en tabelleerwaardes te gebruik om die konsep van kwadratiese ongelykhede te ontwikkel en te vernuwe. Studente het ook gelei tot sinvol geskrewe oplossings vir kwadratiese ongelykhede met korrekte intervalnotasies. Die resultate van die voor- en na-toetse het getoon dat daar 'n beduidende verskil in die gemiddelde tellings was, wat dui op 'n verbeterde prestasie. Die doeltreffendheid van die GC-gebruik op studente se prestasie is prakties geregverdig deur die Cohen se d-effekgroottes, wat in al die drie siklusse groot was. Tweedens het die gebruik van wiskundige situasies uit die werklike lewe wat lineêre ongelykhede betrek as vertrekpunte die studente se konseptuele begrip van kwadratiese ongelykhede ondersteun. Die studente se begrip van wiskundige situasies in die werklike lewe het van die referensiële vlak na die algemene vlak beweeg. Die gebruik van die GC het ook die studente se redenasie- en probleemoplossingsvaardighede in kwadratiese ongelykhede verbeter. Hierdie vaardighede het studente in staat gestel om regte wêreldprobleme wiskundig voor te stel (horisontale wiskunde), die probleem op te los met behulp van die geïnisieerde strategieë, die modeloplossings te interpreteer en terug te kyk na die toereikendheid van hul oplossings. 'N Kognitiewe struikelblok vir baie leerders was egter om hulle te help om metakognitiewe of uitvoerende beheersvaardighede van selfmonitering tydens probleemoplossing in al drie die siklusse te ontwikkel. Die gebruik van die GC het ook aan die studente die geleentheid gebied om van die informele redenering (horisontale wiskunde) na formele redenering (vertikale wiskunde) oor te gaan. Die bevindings ondersteun vorige studies op die gebied dat die gebruik van die GC studente se begrip in die leer van wiskunde verbeter. Die bevindings van die drie siklusse is toegelaat om bewysgebaseerde heuristieke te produseer, soos ontwerpbeginsels wat die toekomstige besluite oor kwadratiese ongelykhede in 'n buigsame GC-omgewing kan inlig. Die belangrikste ontwerpbeginsel van hierdie studie was: grafiese voorstelling van kwadratiese ongelykhede in 'n buigsame grafiese sakrekenaaromgewing. Met die oog daarop was die fokus om studente te help om buigsaam te raak in die hantering van kwadratiese ongelykhede in die vorm van simbole, grafieke of kontekstuele probleme. Ander noodsaaklike ontwerpbeginsels wat in hierdie drie siklusse na vore gekom het, was: a) die opleiding van studente om die GC vlot te gebruik om die kanse te verminder dat die beperkte kykvenster 'n bron word vir studente se wanopvattings en b) die gebruik van die GC kan nie alle leerstyle aanspreek nie, en moet aangevul word met ander tradisionele metodes.Die bevindings van hierdie studie kan gebruik word om kennis uit te brei en 'n bydrae te lewer tot die navorsingsliteratuur oor hoe om die onderwerp van kwadratiese ongelykhede effektief te onderrig. Op soortgelyke wyse kan professionele ontwikkelingsprogramme en werkswinkels vir onderwysers op groeps- of distriksvlak aangebied word vanaf 'n groepsverband. Verder kan die bevindings aanbeveel word aan die handboek of kurrikulumontwikkelaars om meer ontdekkende leeraktiwiteite met grafiese sakrekenaars te ontwerp. Die resultate van die drie DBR-siklusse kan moontlik bygevoeg word tot die waarskynlikheid van oordraagbaarheid na ander algebraïese konsepte.af_ZA
dc.format.extentxvii, 357 pages ; illustrations, includes annexures
dc.language.isoen_ZAen_ZA
dc.language.isoen_ZAen_ZA
dc.publisherStellenbosch : Stellenbosch University
dc.subjectGraphic calculatorsen_ZA
dc.subjectQuadratic inequalities -- Study and teaching (Secondary)en_ZA
dc.subjectQuadratic programmingen_ZA
dc.subjectDesign based researchen_ZA
dc.subjectUCTD
dc.titleA design based research on students' understanding of quadratic inequalities in a graphing calculator enhanced environmenten_ZA
dc.typeThesisen_ZA
dc.description.versionDoctoral
dc.rights.holderStellenbosch University


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