The subject of teacher competences or knowledge has been a key issue in mathematics education reform. This study attempts to identify and analyze teacher competences necessary in the orchestration of a problem-centred approach to dynamic geometry teaching and learning. The advent of dynamic geometry environments into classrooms has placed new demands and expectations on mathematics teachers. In this study the Teacher Development Experiment was used as the main method of investigation. Twenty third-year mathematics major teachers participated in workshop and microteaching sessions involving the use of the Geometer’s Sketchpad dynamic geometry software in the teaching and learning of the geometry of triangles and quadrilaterals. Five intersecting categories of teacher competences were identified: mathematical/geometrical competences, pedagogical competences, computer and software competences, language and assessment competencies. / Mathematics Education / M. Ed. (Mathematics Education)

The study explores Mathematics educators’ use of students’ out-of-school experiences in the teaching of Transformation Geometry. This thesis focuses on an analysis of the extent to which students’ out-of-school experiences are reflected in the actual teaching, textbook tasks and national examination items set and other resources used. Teachers’ teaching practices are expected to support students’ learning of concepts in mathematics. Freudenthal (1991) argues that students develop their mathematical understanding by working from contexts that make sense to them, contexts that are grounded in realistic settings. ZIMSEC Examiners Reports (2010; 2011) reveal a low student performance in the topic of Transformation Geometry in Zimbabwe, yet, the topic has a close relationship with the environment in which students live (Purpura, Baroody & Lonigan, 2013). Thus, the main purpose of the study is to explore Mathematics teachers’ use of students’ out-of-school experiences in the teaching of Transformation Geometry at secondary school level. The investigation encompassed; (a) teacher perceptions about transformation geometry concepts that have a close link with students’ out-of-school experiences, (b) how teachers are teaching transformation geometry in Zimbabwe’s rural secondary schools, (c) the extent to which students’ out-of-school experiences are incorporated in Transformation Geometry tasks, and (d) the extent to which transformation geometry, as reflected in the official textbooks and suggested teaching models, is linked to students’ out-of-school experiences. Consistent with the interpretive qualitative research paradigm the transcendental phenomenology was used as the research design. Semi-structured interviews, Lesson observations, document analysis and a test were used as data gathering instruments. Data analysis, mainly for qualitative data, involved coding and categorising emerging themes from the different data sources. The key epistemological assumption was derived from the notion that knowing reality is through understanding the experiences of others found in a phenomenon of interest (Yuksel & Yildirim, 2015). In this study, the phenomenon of interest was the teaching of Transformation Geometry in rural secondary schools. In the same light, it meant observing teachers teaching the topic of Transformation Geometry, listening to their perceptions about the topic during interviews, and considering how they plan for their teaching as well as how students are assessed in transformation geometry. The research site included 3 selected rural secondary schools; one Mission boarding high school, a Council run secondary school and a Government rural day secondary school. Purposive sampling technique was used carefully to come up with 3 different types of schools in a typical rural Zimbabwe. Purposive sampling technique was also used to choose the teacher participants, whereas learners who sat for the test were randomly selected from the ordinary level classes. The main criterion for including teacher participants was if they were currently teaching an Ordinary Level Mathematics class and had gained more experience in teaching Transformation Geometry. In total, six teachers and forty-five students were selected to participate in the study. Results from the study reveal that some teachers have limited knowledge on transformation geometry concepts embedded in students’ out-of-school experience. Using Freudenthal’s (1968) RME Model to judge their effectiveness in teaching, the implication is teaching and learning would fail to utilise contexts familiar with the students and hence can hardly promote mastery of transformation geometry concepts. Data results also reveal some disconnect between teaching practices as espoused in curriculum documents and actual teaching practice. Although policy stipulates that concepts must be developed starting from concrete situations and moving to the abstract concepts, teachers seem to prefer starting with the formal Mathematics, giving students definitions and procedures for carrying out the different geometric transformations. On the other hand, tasks in Transformation Geometry both at school level and the national examinations focus on testing learner’s ability to define and use procedures for performing specific transformations at the expense of testing for real understanding of concepts. In view of these findings the study recommends the revision of the school Mathematics curriculum emphasising pre-service programmes for teacher professional knowledge to be built on features of contemporary learning theory, such as RME theory. Such as a revision can include the need to plan instruction so that students build models and representations rather than apply already developed ones. / Curriculum and Instructional Studies / D. Ed. (Curriculum Studies)

CAPES / Apresentamos neste trabalho alguns paradoxos lógico-matemáticos, como o paradoxo de Galileu, e também alguns paradoxos geométricos, como os paradoxos de Curry, de Hooper e de Banach-Tarski. Empregamos os paradoxos de Curry e de Hooper para motivar o estudo de conceitos de Geometria e de Teoria dos Números, tais como área, semelhança de triângulos, o Teorema de Pitágoras, razões trigonométricas no triângulo retângulo, o coeficiente angular da reta e a sequência de Fibonacci, e organizamos atividades lúdicas para a sala de aula no Ensino Fundamental e no Ensino Médio. / We present in this work some logical-mathematical paradoxes, as Galileo's paradox, and also some geometric paradoxes, such as Curry's paradox, Hooper's paradox and the Banach-Tarski paradox. We employ the Curry and Hooper paradoxes to motivate the study of concepts of Geometry and Number Theory, such as area, triangle similarity, Pythagorean Theorem, trigonometric ratios in the right triangle, angular coefficient of the line, and Fibonacci sequence, and we organize recreation activities for the classroom in Elementary and High School.

Euclidean Geometry is a challenging topic for most of the learners in the secondary schools. A qualitative case study explores the reasoning skills of ninth graders in the proving of congruent triangles in their natural environment. A class of thirty-two learners was conveniently selected to participate in the classroom observations. Two groups of six learners each were purposefully selected from the same class of thirty-two learners to participate in focus group interviews. The teaching documents were analysed. The Van Hiele’s levels of geometric thinking were used to reflect on the reasoning skills of the learners. The findings show that the majority of the learners operated at level 2 of Van Hiele’s geometric thinking. The use of visual aids in the teaching of geometry is important. About 30% of the learners were still operating at level 1 of Van Hiele theory. The analysed books showed that investigation help learners to discover the intended knowledge on their own. Learners need quality experience in order to move from a lower to a higher level of Van Hiele’s geometry thinking levels. The study brings about unique findings which may not be generalised. The results can only provide an insight into the reasoning skills of ninth graders in proving of congruent triangles. I recommend that future researchers should focus on proving of congruent triangles with a bigger sample of learners from different environmental settings. / Mathematics Education / M. Ed. (Mathematics Education)

This study examined “problem solving skills in circle geometry concepts in Euclidean Geometry. This study was necessitated by learners’ inability to perform well with regards to Euclidean Geometry in general and Circle Geometry in particular. The use of naturalistic observation case study research (NOCSR) study was employed as the research design for the study. The intervention used for the study was the teaching of circle geometry with Polya problem solving instructional approach coupled with social constructivist instructional approach. A High School in the Northern Cape Province was used for the study. 61 mathematics learners (grade 11) in the school served as participants for the first year of the study, while 45 mathematics learners, also in grade 11, served as participants for the second year of the study. Data was collected for two consecutive years: 2018 and 2019. All learners who served as participants for the study did so willingly without been coerced in any way. Parental consent of all participants were also obtained. The following data were collected for each year of the research intervention: classroom teaching proceedings’ video recordings, photograph of learners class exercises (CE), field notes and the end-of-the- Intervention Test (EIT). Direct interpretations, categorical aggregation and a problem solving rubric were used for the analysis of data. Performance analysis and solution appraisal were also used to analyse some of the collected data. It emerged from the study that the research intervention evoked learners’ desire and interest to learn circle geometry. Also, the research intervention improved the study participants’ performance and problem solving skills in circle geometry concepts. Hence, it is recommended from this study that there is the need for South African schools to adopt the instructional approach for the intervention: Polya problem solving instructional approach coupled with social constructivist instructional approach, for the teaching and learning of Euclidean geometry concepts. / Mathematics Education / M. Sc. (Mathematics Education)

Researchers conceptually recommend inquiry-based learning as a necessary means to alleviate the problems of learning but this study has embarked on practical implementation of inquiry-based facilitation and learning in Euclidean Geometry. Inquiry-based learning is student-centred. Therefore, the teaching or monitoring of inquiry-based learning in this study is referred to as inquiry-based facilitation. The null hypothesis discarded in this study explains that there is no difference between inquiry-based facilitation and traditional axiomatic approach in teaching Euclidean Geometry, that is, H0: μinquiry-based facilitation = μtraditional axiomatic approach. This study emphasises a pragmatist view that constructivism is fundamental to realism, that is, inductive inquiry supplements deductive inquiry in teaching and learning. Participants in this study comprise schools in Tshwane North district that served as experimental group and Tshwane West district schools classified as comparison group. The two districts are in the Gauteng Province of South Africa. The total number of students who participated is 166, that is, 97 students in the experimental group and 69 students in the comparison group. Convenient sampling applied and three experimental and three comparison group schools were sampled. Embedded mixed-method methodology was employed. Quantitative and qualitative methodologies are integrated in collecting data; analysis and interpretation of data. Inquiry-based-facilitation occurred in experimental group when the facilitator probed asking students to research, weigh evidence, explore, share discoveries, allow students to display authentic knowledge and skills and guiding students to apply knowledge and skills to solve problems for the classroom and for the world out of the classroom. In response to inquiry-based facilitation, students engaged in cooperative learning, exploration, self-centred and self-regulated learning in order to acquire knowledge and skills. In the comparison group, teaching progressed as usual. Quantitative data revealed that on average, participant that received intervention through inquiry-based facilitation acquired inquiry-based learning skills and improved (M= -7.773, SE= 0.7146) than those who did not receive intervention (M= -0.221, SE = 0.4429). This difference (-7.547), 95% CI (-8.08, 5.69), was significant at t (10.88), p = 0.0001, p<0.05 and represented a large effect size of 0.55. The large effect size emphasises that inquiry-based facilitation contributed significantly towards improvement in inquiry-based learning and that the framework contributed by this study can be considered as a framework of inquiry-based facilitation in Euclidean Geometry. This study has shown that the traditional axiomatic approach promotes rote learning; passive, deductive and algorithmic learning that obstructs application of knowledge in problem-solving. Therefore, this study asserts that the application of Inquiry-based facilitation to implement inquiry-based learning promotes deeper, authentic, non-algorithmic, self-regulated learning that enhances problem-solving skills in Euclidean Geometry. / Mathematics Education / Ph. D. (Mathematics, Science and Technology Education)

The study investigated the strategies used in teaching geometry in primary six as well as the perception of teachers on geometry vocabulary teaching, how geometry vocabularies were taught and, lastly, how the teaching of geometry vocabulary influenced primary six learners’ performance in geometry. The Van Hiele Theory of geometrical thinking and the Constructivist Theory of learning guided the study. The study conveniently sampled 250 primary 6 learners and 7 primary 6 mathematics teachers from three privately-owned primary schools in the Greater Accra Region of Ghana. It combined quantitative and qualitative approaches, using O1–X–O2 design. Data collection instruments were 5-point Likert type scale questionnaires (one for teachers, one for learners), a pre-test and post-test of basic geometry, and a semi-structured one-on-one audio-recorded interview of a selected number of learners and all seven teachers. An intervention was carried out in-between the pre-test and post-test, where the researcher taught geometry vocabulary to participants. Quantitative data were analysed using tables, charts, and simple tests while the qualitative analysis involved the transcription of interviews that were coded, categorised and themed. The study found that geometry vocabularies were not taught and that the most commonly used strategy for teaching geometry was the drawing of 2-D shapes and models of 3-D objects on the board. The pre-test and post-test scores were analysed using a paired t-test and the results indicated that the intervention had a positive effect. The qualitative and quantitative results confirmed that the teaching of geometry vocabulary improved learners’ performance in geometry. The study developed a prototype lesson plan for teaching 3-D objects, a geometry vocabulary activity sheet, a sample assessment for prisms and pyramids and recommends a curricular reform to inculcate the teaching of geometry vocabulary in the curriculum with a geometry vocabulary list for learners in each year group, as contribution to knowledge in mathematics education. The study recommends further research to investigate the effect of geometry vocabulary teaching on learners’ performance in geometry across all year groups in the primary school. / Dyondzo a yi lavisisa maendlelo lawa ya tirhisiwaka ku dyondzisa geometry ya tidyondzo ta le hansi ta ka ntsevu, mavonelo ya vadyondzisi eka madyondziselo ya marito ya geometry, tindlela leti tirhisiweke ku dyondzisa marito ya geometry xikan’we ni ndlela leyi madyondziselo ya marito ya geometry ya khumbheke matirhelo ya vadyondzi va tidyondzo ta le hansi ta ka ntsevu. Dyondzo ya ndzavisiso yi leteriwile hi ehleketelelo ra Van Heile ra maehleketelelo ra ndlela ya geometry ni ndlela yo dyondzisa leyi pfumelelaka vadyondzi ku vumba vutivi ku nga ri ntsena ku teka vutivi ku suka eka mudyondzisi. Dyondzo ya vulavisisi yi hlawurile vana va 250 va tidyondzo ta le hansi ta ka ntsevu na 7 wa vadyondzisi va tnhlayo ta tidyondzo ta le hansi ta ka ntsevu kusuka eka swikolo swinharhu swo ka swi nga ri swa mfumo e Greater Accra etikweni ra Ghana. Yi hlanganisile qualitative na quantiutative aapproach, yi tirhisa O1–X–O2 design. Switirhisiwa swo hlengeleta data a swi ri swivutiso hi muxaka wa 5-point scale(yin’we ya vadyondizi, yin’we ya vadyondzi), xikambelwana xo rhanga na xo hetelela xa geometry ya masungulo, xikan’we na nkandziyiso wa mburisano wa vanhu vambirhi eka nhlayo ya vadyondzi ni vadzyondzisi hinkwavo va nkombo. Ntirho wo nghenelerisa wu endliwile exikarhi ka xikambelwana xo rhanga ni xo hetelela laha mulavisisi a nga dyondzisa marito ya geometry eka vanhu lava ngheneleleke. Quantitative data yi hleriwile hi ku tirhisa matafula, ti charts ni swikambelwana swo olova kasi vuhleri bya qualitative byi nghenise kutsariwa ka miburisano leyi hundzuluxiweke yi nyika tinhlamuselo leti tumbeleke. Leti vekiweke hi ku ya hi mintlawa ni maendlelo ya tona. Dyondzo ya ndzavisiso yi kume leswaku marito ya geometry a ya dyondzisiwanga ni leswaku maendlelo yo toloveleka ya ku dyondzisa geomeyry i ya drawing ya xivumbeko xa 2-D ni mfanekiso wa nchumu wa 3-D eka bodo. Mbuyelo wa Xikambelwana xo sungula na xo hetelela wu hleriwile hi ku tirhisa t-test (xikambelwana xa T) lexi hlanganisiweke naswona mbuyelo wu komba leswaku maendlelo himkwawo ya vile ni xiave lexinene. Mbuyelo wa Qualitative na Quantitative wu tiyisisile leswaku ku dyondzisiwa ka marito ya geometry swi antswisa matirhelo ya vadyondzi eka dyondzo ya geometry. Dyondzo ya vulavisisi yi antswisile kumbe ku kurisa prototype lesson plan ya ku dyondzisa 3-D objects, sheet ya migingiriko ya marito ya geometry na ku bumabumela circular reform ku dyondzisa madyondziselo ya marito ya geometry eka kharikhulamu leyi ng na nxaxamelo wa marito ya geometry ya vadyondzi eka ntlawa wa lembe na lembe, ta ni hi mpfuneto wa vutivi eka dyondzo ya tinhlayo. Dyondzo ya vulavisisi yi bumabumela leswaku vulavisisi byi ya emahlweni ku lavisisa xiave xa madyondziselo ya marito ya geometry eka matirhelo ya vadzyondzi eka geometry eka malembe ni mintlawa hinkwayo exikolweni xa le hansi. / Thuto ye e nyakisisitse ditsela tseo di somiswago go ruteng ga geometry go mphato wa bo tshelela, temogo ya barutisi go ruteng tlotlontsu ya geometry, tsela yeo ditlotlontsu tsa geometry di rutilwego ka gona go akaretswa le, sa mafelelo, ka mokgwa wo thuto ya tlotlontsu ya geometry e tutueditsego mabokgoni a barutwana ba mphato wa bo tshelela go dithuto tsa geometry. Thuto ya van Hiele ya geometrical thinking le ya constructivist theory of learning di hlahlile thuto ye. Thuto ye e somisitse ga bonolo mohlala wa barutwana ba 250 ba mphato wa 6 le barutisi ba dipalo ba supa ba go ruta mphato wa 6 go tswa dikolong tsa tlase tse tharo tsa go ikema seleteng sa Greater Accra Region of Ghana. Thuto ye e kopantse mekgwa ya bontsi/dipalopalo (quantitative) le boleng (qualitative), go somiswa tlhamo ya O1-X-O2. Didiriswa tsa kgobaketso ya boitsebiso e bile 5-point Likert Type Scale Questionnaire (ye tee ya barutisi, ye tee ya barutwana), moleko wa pele le moleko wa morago wa geometry ya motheo, le poledisano yeo e gatisitswego ya tlhamego ya sewelo (semi-structured) ya barutwana bao ba kgethilwego ga mmogo le barutisi ka moka ba supa. Thekgo e ile ya phethagatswa/fiwa magareng ga moleko wa pele le moleko wa morago moo monyakisisi a rutilego tlotlontsu ya geometry go batseakarolo. Boitsebiso bja bontsi (quantitative data) bo sekasekilwe ka go somisa ditafola, ditshate, le teko e bonolo mola ditshekatsheko tsa boleng (qualitative analysis) di akareditse go ngwalolla dipoledisano tseo di thulagantswego, tsa hlophiwa le go beakanywa ka sehlogo. Thuto ye e itullotse gore ditlotlontsu tsa geometry ga se tsa rutwa ebile mekgwana yeo e somisitswego ya setlwaedi go ruta geometry ebile go thala dibopego tsa 2-D le mehlala ya didiriswa tsa 3-D letlapeng. Dintlha tsa moleko wa pele le moleko wa bobedi di sekasekilwe ka go somisa mokgwa wa go phera moleko wa t (t-test). Dipoelo di supeditse gore thekgo yeo e filwego e bile le khuetso ye botse. Dipoelo tsa bontsi le boleng di netefaditse gore go ruta tlotlontsu ya geometry go kaonafatsa mabokgoni a barutwana dithutong tsa geometry. Nyakisiso ye e tsweleditse lenaneothuto la go dira diteko go ruteng didiritswa tsa 3-D le papetlatshomelo ya tlotlontsu ya geometry gape le go kgothaletsa mpshafatso ya lenaneo-thuto go tsenyeletsa thuto ya tlotlontsu ya geometry ka gare ga lenaneo-thuto gammogo le lelokelelo la tlotlontsu ya geometry ya barutwana go dihlopha tsa mengwageng ka moka. Se e tla ba e le tlaleletso ya tsebo go thuto ya dipalo. Thuto ye e kgothaletsa dinyakisiso tsa go ya pele go nyakolla mafelelo a go ruta tlotlontsu ya geometry go tiro ya, goba dipoelo tsa, barutwana go thuto ya geometry go dihlopha tsa mengwaga ka moka tsa sekolo sa tlase. / Mathematics Education / Ph. D. (Mathematics Education)