A study of the development of mathematical knowledge in a geogebra-focused learning environment

Pfeiffer, Cerenus (2017-12)

Thesis (PhD)--Stellenbosch University, 2017

Thesis

ENGLISH ABSTRACT : This study was about a GeoGebra-focused learning environment in which students could develop mathematical knowledge. It was done during an intervention with a convenient sample of 48 Science and Mathematics bridging programme students at Stellenbosch University (SciMathUS). Students in the SciMathUS year have to improve their Mathematics marks in order to qualify for admission into Science, Technology, Engineering and Mathematics (STEM) oriented programmes the following year at any tertiary institution in South Africa. The teacher-researcher felt that these students needed to be exposed to a hybridisation of the student-centred approach and teacher as facilitator of learning approach which had the potential to enhance conceptual and deeper understanding of Mathematics. GeoGebra was a pivotal teaching tool in the teacher-researcher’s interactive lecturing. Students were afforded the opportunity to engage individually, or in groups, with the learning material on transformations of functions and circle geometry by making use of GeoGebra. The prepared tasks in the different learning trajectories used in this research were guided by a social constructivist view of learning, Realistic Mathematics Education (RME), the Van Hiele theory, the Theory of Instrumental Genesis (TIG) and the Technological Pedagogical and Content Knowledge (TPACK) framework for technology integration. A learning route for the students in the instruction of transformations of functions and circle geometry was established through the creation of learning trajectories intertwined with transformation geometry, straight line, triangle and quadrilateral geometry, respectively. The methodology used was a mixed methods exploratory case study. The quantitative part of the study was a pretest-posttest experimental design. This study utilised pre- and post-intervention questionnaires, pre- and post-tests for transformations of functions and circle geometry, observations, and in-depth and focus group interviews. The results of the quantitative and qualitative data were triangulated, with a higher priority given to analysis of the qualitative data to answer the research questions. The results from both quantitative and qualitative data showed that the pedagogical processes involved in using GeoGebra were more than just technology. They showed that real-life activities, guidance, exploration and interaction were important RME principles to be borne in mind when using GeoGebra. The results also underlined the fact that students had certain preferences on how GeoGebra could be used as a pedagogical tool. The qualitative and quantitative results also revealed that GeoGebra afforded students an opportunity to better understand transformations of functions, circle geometry and general solutions of trigonometric equations. The qualitative content analysis of pre- and post-test for both transformations of functions and circle geometry showed that students moved to higher levels of abstraction. Students attributed this to the instrumentation processes of GeoGebra, i.e. visual affordance, potentialities and enablements. Moreover, the results revealed that exploring with GeoGebra made the teaching and learning of mathematics more fun. The students utilised hand movements to articulate their ideas with a classmate, or the teacher-researcher. During Guide-and-explain orchestrations students explained to their group members, or teacher-researcher, how they understood the intended mathematics from the activities and in this way used informal reasoning (horizontal mathematising) and then moved within the discussions to formal reasoning (vertical mathematising). The results also showed that GeoGebra afforded the students an opportunity to acquire physical and logico-mathematical knowledge. Challenges observed whilst the students were working with GeoGebra in the computer lab and based on students’ responses in the interviews, were constraints posed by syntax and menu commands. A small percentage of students found the different teaching and learning approach to be challenging at times. A few others could not see the intended mathematics from activities with GeoGebra, or felt that it further confused them, or that this approach required more thinking. Consequently it is recommended that GeoGebra should be part of a teacher’s arsenal to teach mathematical concepts that lend themselves to technology integration.

AFRIKAANSE OPSOMMING : Hierdie studie handel oor ʼn GeoGebra-gefokusde leeromgewing omgewing waarin studente wiskundige kennis kan ontwikkel. Dit is deur middel van 'n intervensie met 'n gerieflike steekproef van 48 Wetenskap- en Wiskunde-studente in die Universiteit van Stellenbosch se oorbruggingsprogram (SciMathUS) uitgevoer. Studente in die SciMathUS-jaar moet hulle Wiskunde-punte verbeter om die volgende jaar toelating te verkry tot Wetenskap-, Tegnologie-, Ingenieurswese- en Wiskunde georiënteerde programme (STEM) aan enige tersiêre instelling in Suid-Afrika. Die onderwyser-navorser is van mening dat hierdie studente aan 'n hibriede van student-gesentreerde leer en leer waarin die onderwyser as fasiliteerder optree, blootgestel moet word. Dit het die potensiaal om konseptuele en dieper begrip van wiskunde te bevorder. GeoGebra is ʼn belangrike tegnologiese onderrighulpmiddel in die onderwyser-navorser se interaktiewe lesaanbiedingbenadering. Studente is 'n geleentheid gebied om met leermateriaal oor transformasies van funksies en sirkel-meetkunde met behulp van GeoGebra individueel, of in groepe, te werk. Die voorbereide take in die onderskeie leertrajekte wat in hierdie navorsing gebruik is, is begelei deur 'n sosiaal-konstruktivistiese benadering van leer, Realistiese Wiskunde-Onderwys (RWO), die Van Hiele-teorie, die Teorie van Instrumentele Genesis (TIG) en die Tegnologiese, Pedagogiese en Inhoudskennis (TPACK) raamwerk vir die integrasie van tegnologie. Die transformasies van funksies en sirkel-meetkunde se onderrig is op ʼn leerroete met behulp van 'n verskeidenheid van leertrajekte geplaas en is met transformasiemeetkunde reguitlyn-, driehoek- en veelhoekmeetkunde, verweef. Die metodologie was 'n gemengde metode verkennende gevallestudie. Die kwantitatiewe deel van die studie was 'n voortoets-natoets eksperimentele ontwerp. Hierdie studie het voor- en na-intervensie vraelyste, voor- en na-toetse vir transformasies van funksies en sirkelmeetkunde, waarnemings, en in-diepte en fokusgroep-onderhoude gebruik. Die resultate van die kwantitatiewe en kwalitatiewe data is getrianguleerd om die navorsingsvrae te beantwoord. Hoër prioriteit is aan die ontleding van die kwalitatiewe data gegee. Die resultate van beide kwantitatiewe en kwalitatiewe data het getoon dat die pedagogiese prosesse verbonde aan die gebruik van GeoGebra meer as net tegnologies van aard was. Dit het getoon dat kontekstuele aktiwiteite, begeleiding, ontdekking en interaksie belangrike RWO-beginsels is wat in gedagte gehou moet word wanneer GeoGebra gebruik word. Die resultate het voorts die feit benadruk dat studente sekere voorkeure gehad het oor hoe GeoGebra as 'n pedagogiese instrument gebruik moet word. Die kwantitatiewe en kwalitatiewe resultate het vervolgens daarop gedui dat GeoGebra studente 'n geleentheid gebied het om beide transformasies van funksies, sirkelmeetkunde en algemene oplossings van trigonometriese vergelykings beter te verstaan. Die kwalitatiewe ontleding van inhoudskennis in die voor- en na-toetse vir beide transformasies van funksies en sirkelmeetkunde het bewys dat studente na hoër vlakke van abstraksie beweeg het. Studente skryf dit toe aan die instrumentasie-prosesse van GeoGebra, onder meer visuele voorstelling, potensialiteite en instaatstelling. Uit die resultate het dit ook geblyk dat die ondersoek met GeoGebra die onderrig en leer van wiskunde meer pret maak. Studente gebruik handbewegings om hulle idees aan 'n klasmaat, of die onderwyser-navorser, oor te dra. Tydens die begelei- en verduidelik-orkestrasie het die studente aan hulle groeplede, of die onderwyser-navorser, verduidelik hoe hulle die wiskunde van die aktiwiteite verstaan het en hoe hulle informele beredenering (horisontale matematisering) gebruik het om tydens die gesprekke deur te beweeg tot formele redenasie (vertikale matematisering). Die resultate het getoon dat GeoGebra die studente die geleentheid gebied het om fisiese en logies-wiskundige kennis te verkry. Studente wat met GeoGebra in die rekenaarlokaal besig was, het uitdagings ervaar met sintaksis en menu-opdragbeperkings. Studente se terugvoering het hierdie aspek bevestig. ʼn Klein persentasie studente het die gebruik van verskillende onderrig- en leerbenaderings soms uitdagend gevind. ʼn Paar ander kon nie die beoogde wiskunde in die aktiwiteite met GeoGebra raaksien nie en het gevoel dat dit hulle meer verward maak. Soms het hulle besef dat die benadering meer dinkwerk vereis. Derhalwe word aanbeveel dat GeoGebra deel moet uitmaak van ʼn onderwyser se arsenaal van hulpmiddels wat hom/haar spesifiek leen tot die gebruikmaking van tegnologie vir die onderrig van wiskundige konsepte.

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