The purpose of this investigation was to evaluate the effect of two modes of exposition of tenth-grade incidence geometry on logically evaluated problem solving ability. To achieve this purpose two classes of tenth-grade geometry students were chosen to be the experimental and control groups. The two treatments, which were of nine class hours duration per group, and were both taught by the investigator, involved the use of a set theoretic symbolic-nonrepresentational mode for the experimental group, and a pictorial-representational mode for the control group. The content of the treatments was Euclidean incidence geometry. At the termination of the treatment a criterion test was administered to both groups. The criterion test was composed of two types of problems- Type NR problems, which were believed to be most successfully solved by a symbolic-nonpictorial analysis, and Type R problems, which were believed to be most successfully solved by a pictorial analysis. Two hypotheses, of null form, were considered: that the mean scores of both groups on Type NR problems would be equal and that the mean scores of both groups on Type R problems would be equal. Both hypotheses were tested by means of an appropriate t-statistic at the .05 level of significance. Analysis of the data indicated that both null hypotheses were not to be rejected. A difference in means on Test NR of the control over experimental group was observed at the .20 level of significance. The implication of the analysis of the data and the restrictions imposed by the limitations of the study is that the pictorial-representational exposition was as effective as the experimental symbolic-nonrepresentational exposition for Type NR problems and for Type R problems. Since the pictorial-representational mode of exposition is generally considered standard practice in the teaching of tenth grade geometry it should be continued for the present. / Education, Faculty of / Curriculum and Pedagogy (EDCP), Department of / Graduate

This dissertation is intented to explore the similarities and differences of pedagogical characteristics in geometry though analyzing the contents of the junior Secondary school (grade 7-9) mathematics textbooks of England, Hong Kong and Taipei. By studying the topic of “Relationships for angles at a point, angles on a line, vertically opposite angles, angles associated with a transversal cutting parallel lines and perpendicularity” of the textbooks, the textbooks of the three places have demonstrated how they organize the contents to achieve the various aims with different means and different emphasis. For English textbooks, their contents are characterized by using geometric intuition for verifying the properties of geometric objects and the theorems. On the contrary, for Taipei textbook, their contents are characterized by the extensive use of deductive reasoning and proof for verifying the properties of geometric objects and the theorems. Lastly, for Hong Kong textbooks, their contents are characterized by encouraging conjecturing through various explorative activities. For developing students’ cognitive thinking in geometry, the progress of developing students’ level of sophistication in deductive reasoning is depended upon their age for England while Hong Kong and Taipei do not. Process of visualization is commonly used in the three countries for enhancing the reasoning process and various apprehensions of geometric objects. On the other hand, Hong Kong and Taipei textbooks tend to apply more on the process of reasoning in the teaching activities while English textbooks apply more on the process of construction. / published_or_final_version / Education / Master / Master of Education

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)

A research report submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg in partial fulfilment of the requirements for the degree of Master of Science, 2017 / This research explores the importance of Self- Directed Metacognitive questioning in the solving of Euclidean Geometry problems by grade 11 learners. A quasi-experiment was carried out at an urban school with fifty eleventh grade learners. Most researches in Mathematics Education aimed at unveiling socio-economic factors that hamper mathematics learning. In this research, I suggested that strategies that target the learners' metacognitive development can assist in addressing poor mathematics achievement. Metacognition (thinking about thinking) makes learners drivers of their own cognitive processes so that they can become better doers of the subject The research answered the question: Does teaching of metacognition to Further Education and Training (FET) learners help them to become better learners of Euclidian Geometry? This question was broken down into the following sub-questions: What is the effect of the use of Self -Directed Metacognitive (SDM) questions on the confidence level and preparedness of learners in the learning of Euclidean Geometry? To what extent does purposeful teaching and learning of metacognitive skills yield positive results in answering Euclidean Geometry questions? Metacognitive skills helped learners to perform better in problem-solving. This result agrees with previous researchers and is also consistent with the results of earlier investigations showing that achievement in mathematics can be raised through instruction enriched with metacognitive activity. Where previous research dealt with metacognitive training in an implicit manner this research on metacognitive training was done explicitly and it resulted in improved mathematics performance by learners. / XL2017

Geometry is an essential branch in mathematics that helps students learn to grasp their environment and leverage that grasp into abstract understanding and reasoning. There has been an observable decrease in geometrical content in secondary education curricula, and particularly a “puzzling scarcity” in three-dimensional geometry, which has led to a decline in students’ geometrical abilities, spatial thinking and deductive reasoning abilities. This study addresses this issue by scrutinizing the enacted curriculum standards and the most influential textbooks related to three-dimensional geometry in two prominent countries, the US and China, both of which embrace the interplay of both conventional and innovative practices. This qualitative study used both content analysis and cross-cultural comparison methods to inquire about and to understand the current situation of three-dimensional geometry in high school. I focused on probing the communication types, objects, concepts, and spatial thinking abilities related to three-dimensional geometry in the standards and texts. To understand spatial abilities, I synthesized a spatial thinking abilities framework with six attributes and used this framework to exam the affordance of these abilities in the texts and requirements in the standards. The result and analysis reveal the details of each text and standards individually and offer an examination of the alignment between the standards and texts. The comparison of the two countries’ different approaches also sharpens the understanding of the issue. I also worked to unveil students’ multiple ways of making sense of geometry concepts by two geometry learning models, Piaget’s model and van Hiele’s model, as well as spatial thinking abilities.

Prospective teachers enter teacher education programs with previously formed conceptions of geometry and its teaching. These conceptions help them make sense of new information about teaching, their roles as teachers, and their translation of mathematics into learning activities. The purpose of this study was to investigate the relationships among preservice teachers' conceptions of geometry, conceptions of teaching geometry and classroom practices. Ten preservice mathematics teachers completed a card sort task with an interview. They also participated in a videotape task which consisted of viewing three experienced geometry teachers on videotape. Four of these preservice teachers were observed eight times each during their professional internship experience. All interviews and observations were videotaped and transcribed for data analysis. Results of this study indicated a complex relationship between the preservice teachers' conceptions of geometry and conceptions of teaching geometry. The preservice teachers could not discuss their conceptions of geometry without discussing the teaching of geometry. Their conceptions about geometry and their belief that geometry was linear, in nature were so strong that these views became connected with their views of teaching geometry. Clearly, the preservice teachers' conceptions of geometry influenced their conceptions of teaching geometry and the teaching of subject matter influenced the preservice teachers' conceptions of geometry as well. The relationship between the preservice teachers' conceptions of geometry and their classroom practices was directly influenced by the textbooks used. They believed geometry was ordered according to the textbook and their classroom practices also followed the textbook. The relationship of the preservice teachers' conceptions of geometry teaching to classroom practices indicated that what the preservice teachers said they believed and what they did in the classroom were not always consistent. Their beliefs about teaching geometry rarely emerged in their classroom practices. Finally, these preservice teachers had an overwhelming concern with classroom management. This concern governed their thinking about teaching. / Graduation date: 1996

published_or_final_version / Education / Master / Master of Education

Students nowadays are relatively confident in directly applying geometrical theorems and theories. Nevertheless, it has been a common phenomenon that students are not confident in constructing geometric proofs. They lack the confidence and sufficient experience and knowledge in conducting deductive geometrical proofs. To some students, they treat proofs simply as another type of examination questions which they can tackle by repeated drillings. Students make use of straightedges and compasses to construct different geometry figures in geometric constructions. Through geometric constructions, we can train our prediction and logical thinking skills when investigating the properties of geometric figures. Geometric constructions provide students with hands-on experience to geometry learning which requires students to have more in-depth thinking. This is an empirical study on the implementation of geometric construction workshops among junior secondary students in Hong Kong. Results have shown that students enjoyed the construction tasks during the workshops. Analysis has implied that geometric constructions help improve students’ ability in constructing geometric proofs and to raise their interests in geometry learning. / published_or_final_version / Education / Master / Master of Education

Geometry learning at the junior secondary level should focus on the connection between students’ intuitive, spatial thinking and formal, deductive reasoning. It is crucial for students at this stage to develop abstract concepts based on knowledge of and reasoning with geometric properties. Dynamic geometry tools are promising resources in classroom teaching for enriching students’ experience in geometry. Studies about students’ learning with dynamic geometry tools usually focus on the long-term development of tool use in the contexts of problem solving and explorations. Data are mostly obtained from interviews and observations with students working in small groups or individually. This study begins with an original approach. It aims at understanding students’ perceptions of geometric properties in simple dynamic figures without assuming any prior experience of students with dynamic geometry tools. A special web-based platform was developed to capture students’ results of dragging in dynamic figures. Quantitative and qualitative data were obtained for analysis from this platform and task-based interviews respectively. In a set of 8 tasks designed in this study, students were asked to drag free points in pre-constructed figures to create examples of geometric configurations with parallel lines. Results of the tasks were collected through the web-based platform from 1,589 secondary 1 to 4 Hong Kong students in 11 schools of different backgrounds. In the next stage, 24 secondary 2 students from another 6 schools took part in task-based interviews, enabling detailed observations and analysis of their reasoning. The basic assumption of this design is that students working on the tasks will generate examples of geometric figures that reflect their understanding of relevant geometric properties. Their results could then be analyzed in the form of collective and personal example spaces with critical dimensions of variation to be identified (Watson & Mason, 2005). The findings of the study indicate how participating students vary in their ways of discerning critical geometric properties in dynamic figures. In particular, the results reflect students’ general limited awareness of basic geometric properties, such as equal opposite sides or angles, while manipulating and interpreting pre-constructed figures. Graphical representations of collective example spaces, developed in this study, provide useful means for revealing dimensions of variation in examples generated. It is hoped that the findings and method of the study can inform classroom teaching with dynamic geometry tools that capitalizes on variation in students’ perceptions. / published_or_final_version / Education / Doctoral / Doctor of Philosophy

The main focus of the research was to find out the causes of a poor performance in euclidean geometry especially in a grade eleven class. An easier way to find that information was to investigate the techniques that educators who are teaching grade eleven are following when they teach euclidean geometry. The necessary data was therefore collected from the educators as well as learners who were in grade eleven. This study is guided by the constructivist's VIew. The theoretical framework of this research is based on the ideas of theorists like Piaget, Vygotsky and other authors who conform to constructivism. Changes that affected the education system of South Africa due to the adoption of the new constitution were also visited. A shift from the traditional way of teaching and an Outcomes Based Education system, as a recommendation by the National Curriculum Statement was highlighted. The data was collected through both interviews and questionnaires. The semi-structured interviews of three educators from three participating schools were audio taped. In each school one educator was interviewed and six learners were given questionnaires to answer. The above gave a total of eighteen learners and three educators. Written responses from learners and audio taped responses from educators were kept and analyzed. The interview was focused on the techniques that educators employ in their teaching of euclidean geometry in grade eleven. The questionnaires administered to learners were aimed at confirming the responses from the educators. It is envisaged that the educators participated in the study can provide enough information which can assist in correcting the teaching approach in euc1idean geometry. The findings show that the conditions under which educators teach contribute to their methods of teaching euclidean geometry. The testing system and the focus on better results by the education department proved to be the main determining factors of the methods that educators resort to when they teach learners. It also came up from this study that some learners do not take mathematics out of their will. Their parents or the school forces them to take mathematics. Those who like to take mathematics are constantly discouraged by comments of educators who deem mathematics as a subject responsible for bringing down the pass rate of the school. The above diminishes the love of mathematics to learners and euclidean geometry becomes the section that suffers the most. Suggestions and recommendations aimed at improving the teaching and learning of the euclidean geometry have been made. / Thesis (M.Ed.)-University of KwaZulu-Natal, 2007.