The main focus of the study was to explore how the experiences that the learners went through in the Technology class during the construction and design of artifacts, could be used to inform the teaching of Geometry in the mainstream Mathematics classes. It was important to find out how the teaching of Geometry would allow the learners to both reflect and utilize the Geometry they know, as a starting point or springboard for further study of Geometry. Data was collected through observations, structured and semi-structured interviews of a sample of twenty grade 9 learners of Mashesha Junior Secondary School of Margate in KwaZulu Natal. It was collected through observation of drawings and completely constructed double-storey artifacts at different intervals of designing. Observations and notes on every activity done by the learners for example, measurements, comparisons, estimations, scaling, drawings use of symmetry and perspective drawing were kept and analyzed. Data for the interviews was collected in the form of drawings, photographs, transcriptions of video and audiotapes. The observations in particular were looking for the Geometry in finished artifacts. Interviews with the learners were directed at how each learner started drawing a house to the finish. When and how scale drawing, projections, angles made and length preservation were used by the learner, was of utmost importance. It is believed that grade 9 learners of Mashesha have Geometric experiences which can be used to inform the teaching of Geometry in mainstream mathematics. It was found that this experience brought by the learners from the Technology construction of artifacts could cause the learners to find mainstream mathematics interesting and challenging. It is also believed that the use of projective Geometry already employed by the learners can be incorporated in mainstream mathematics so as to improve how learners understand Euclidean Geometry. In this way, it is believed, that the teaching of Geometry will allow the learners to utilize and reflect the Geometry already known to them. This Geometry would therefore be used as a starting point for further study of Geometry. Suggestions for further research and recommendations for the improvement of Geometry teaching and learning have also been made. / Thesis (M.Ed)-University of KwaZulu-Natal, Durban, 2005.

The teaching of geometry has been neglected at the expense of other disciplines of mathematics such as algebra in most secondary schools for Africans in South Africa. The research aimed at establishing the extent to which creativity and divergent thinking enhance the internalisation of geometry concepts using the problem-based approach and on encouraging learners to be creative, divergent thinkers and problem solvers. In the research, Grade 7 learners were guided to discover the meaning of geometric concepts by themselves (self-discovery) and to see concepts in a new and meaningful way for them. This is the situation when learners think like the mathematicians do and re-invent mathematics by going through the process of arriving at the product and not merely learn the product (axioms and theorems), for example, discover properties of two- and three-dimensional shapes by themselves. Furthermore, learners were required to use metaphors and analogies, write poems, essays and posters; compose songs; construct musical instruments and use creative correlations in geometry by using geometric shapes and concepts. They tessellated and coloured polygons and pentominoes in various patterns to produce works of art. Divergent thinking in geometrical problem solving was evidenced by learners using cognitive processes such as, amongst others, conjecturing, experimenting, comparing, applying and critical thinking. The research was of a qualitative and a quantitative nature. The problem-based approach was used in teaching episodes. The following conclusions and recommendations were arrived at: * Geometric shapes in the learner's environment had not been used as a basis for earning formal geometry. * Second language learners of mathematics have a problem expressing themselves in English and should thus be given the opportunity to verbalize their perceptions in vernacular. * Learners should be made to re-invent geometry and develop their own heuristics/strategies to problem solving. * Learners should be trained to be creative by, for example, composing songs using geometric concepts and use geometric shapes to produce works of art, and * Activities of creativity and divergent thinking should be used in the teaching and learning of geometry. These activities enhance the internalisation of geometry concepts. Groupwork should be used during such activities. / Educational Studies / D. Ed. (Didactics)

Spatial visualization--Mathematics and geometry achievement--Cognitive structure--Manipulatives--Lessons for triangles, quadrilaterals, polyhedra, polygons, Eulers Formula, and platonic solids.

published_or_final_version / Education / Master / Master of Education

published_or_final_version / abstract / Education / Master / Master of Education

published_or_final_version / Education / Doctoral / Doctor of Philosophy

This investigation of secondary geometry teachers' decision making in a mathematics curricular reform context examined the following questions: (a) What planning and interactive decisions were secondary geometry teachers making during this time of reform, and (b) what factors influenced the decisions that these teachers made? In addition, comparisons were generated between influential factors identified during a mathematics reform context and the stable context of previous decision making studies. A multi-case study approach involving detailed examination of five geometry teachers' decision making was used. The data collected and analyzed included a questionnaire, interviews, observational field notes, audiotapes and videotapes of classroom instruction, and written instructional documents. Teachers' profiles were created describing geometry and teaching biographies, views toward curricular change, the classroom, planning decisions and influential factors, and interactive decisions and influential factors. Findings were developed by searching for similarities and differences across the sample. Teachers' decisions generated descriptions of their geometry courses. One teacher promoted geometry as a mathematical system using predominantly a lecture approach. The other four teachers advocated a multifaceted view of geometry recognizing geometry as a mathematical system and as a setting for developing communication and problem solving skills. In addition, the four teachers' courses included references to connections between geometry and the real world. These four teachers used a variety of instructional approaches that encouraged students' active involvement in their geometry learning with an emphasis on developing student understanding. Factors influencing teachers' decisions included: (a) past geometry experiences, (b) professional development experiences, (c) articulated course goals, (d) advanced planning decisions, (e) teachers' beliefs, (f) the geometry textbook and other materials, (g) teachers' school settings, and (h) students' needs and actions. Some findings highlighted differences between this study and previous decision making studies. All teachers in this study appeared to be influenced by their beliefs about the nature of geometry as a discipline. Teachers were also influenced by whether they viewed the process of becoming an effective teacher as a life-long process. For four of the teachers, reform agendas were influential as another source of curriculum ideas. / Graduation date: 1997

Ball State University LibrariesLibrary services and resources for knowledge buildingMasters ThesesThere is no abstract available for this thesis.

This study investigated how Grade 10 learners conceptualise an introductory activity to the sine function with The Geometers' Sketchpad. In a study by Blackett and Tall (1991), the initial stages of learning the ideas of trigonometry, are described as fraught with difficulty, requiring the learner to relate pictures of triangles to numerical relationships, to cope with ratios such as sinA = opposite/hypotenuse. A computer approach might have the potential to change this by allowing the learner to manipulate the diagram and relate its dynamically changing state to the corresponding numerical concepts. The learner is thus free to focus on specific relationships, called the principle of selective construction, as stated by Blackett and Tall (1991). The use of this educational principle was put to test to analyse the understanding of Grade 10 learners' introduction to the sine function. Data was collected from a high school situated in a middle-class area of Reservoir Hills (KZN) by means of task-based interviews and questionnaires. Given a self-exploration opportunity within The Geometers' Sketchpad, the study investigated learners' understanding of the sine function only within the first quadrant: A) as a ratio of sides of a right-angled triangle B) as an increasing function C) as a function that increases from zero to one as the angle increases from 0° to 90°. D) as a relation between input and output values E) the similarity of triangles with the same angle as the basis for the constancy of trigonometric ratios. The use of Sketch pad as a tool in answering these questions, from A) to E), proved to be a successful and meaningful activity for the learners. From current research, it is well-known that learners do not easily accommodate or assimilate new ideas, and for meaningful learning to take place, learners ought to construct or reconstruct concepts for themselves. From a constructivist perspective the teacher cannot transmit knowledge ready-made and intact to the pupil. In the design of curriculum or learning materials it is fundamentally important to ascertain not only what intuitions learners bring to a learning context, but also how their interaction with specific learning experiences (for example, working with a computer), shapes or changes their conceptualisation. The new ideas that the learners' were exposed to on the computer regarding the sine function, also revealed some errors and misconceptions in their mathematics. Errors and misconceptions are seen as the natural result of children's efforts to construct their own knowledge, and according to Olivier (1989), these misconceptions are intelligent constructions based on correct or incomplete (but not wrong) previous knowledge. Olivier (1989), also argues that teachers should be able to predict what errors pupils will typically make; explain how and why children make these errors and help pupils to resolve such misconceptions. In the analysis of the learners' understanding, correct intuitions as well as misconceptions in their mathematics were exposed. / Thesis (M.Ed.)-University of KwaZulu-Natal, Durban, 2004.

The main focus of the research was to investigate the understanding of Euclidean Geometry of a group of Grade 12 mathematics teachers, who have been teaching Grade 12 mathematics for ten years or more. This study was guided by the qualitative method within an interpretive paradigm. The theoretical framework of this research is based on Bloom’s Taxonomy of learning domains and the Van Hiele theory of understanding Euclidean Geometry. In national matriculation examination, Euclidean Geometry was compulsory prior to 2006; but from 2006 it became optional. However, with the implementation of the latest Curriculum and Assessment Policy Statement it will be compulsory again in 2012 from Grade 10 onwards. The data was collected in September 2011 through both test and task-based interview. Teachers completed a test followed by task-based interview especially probing the origin of incorrect responses, and test questions where no responses were provided. Task-based interviews of all participants were audio taped and transcribed. The data revealed that the majority of teachers did not posses SMK of Bloom’s Taxonomy categories 3 through 5 and the Van Hiele levels 3 through 4 to understand circle geometry, predominantly those that are not typical textbook exercises yet still within the parameters of the school curriculum. Two teachers could not even obtain the lowest Bloom or lowest Van Hiele, displaying some difficulty with visualisation and with visual representation, despite having ten years or more experience of teaching Grade 12. Only one teacher achieved Van Hiele level 4 understanding and he has been teaching the optional Mathematics Paper 3. Three out of ten teachers demonstrated a misconception that two corresponding sides and any (non-included) angle is a sufficient condition for congruency. Six out of ten teachers demonstrated poor or non-existing understanding of the meaning of perpendicular bisectors as paths of equidistance from the endpoints of vertices. These teachers seemed to be unaware of the basic result that the perpendicular bisectors of a polygon are concurrent (at the circumcentre of the polygon), if and only if, the polygon is cyclic. Five out of ten teachers demonstrated poor understanding of the meaning and classification of quadrilaterals that are always cyclic or inscribed circle; this exposed a gap in their knowledge, which they ought to know. Only one teacher achieved conclusive responses for non-routine problems, while seven teachers did not even attempt them. The poor response to these problems raised questions about the ability and competency of this sample of teachers if problems go little bit beyond the textbook and of their performance on non-routine examination questions. Teachers of mathematics, as key elements in the assuring of quality in mathematics education, should possess an adequate knowledge of subject matter beyond the scope of the secondary school curriculum. It is therefore recommended that mathematics teachers enhance their own professional development through academic study and networking with other teachers, for example enrolling for qualifications such as the ACE, Honours, etc. However, the Department of Basic Education should find specialists to develop the training materials in Euclidean Geometry for pre-service and in-service teachers. / Thesis (M.Ed.)-University of KwaZulu-Natal, Durban, 2012.