The aim of this study was to investigate the relationship between teachers' instructional practices in terms of specific areas of focus pertaining to the teaching and learning of geometry described in literature and, their learners' levels of geometry thinking as elaborated in the Van Hiele theory. A review of literature on the development of geometry understanding was conducted to frame what is meant by 'teachers' instructional practices' as they pertain to the teaching and learning of geometry in this study. These instructional practices are understood to include the appropriate allocation of time for the facilitation of geometry concept development, the use of concrete apparatus, the use of relevant and level appropriate language as well as the use of level appropriate geometry activities. The structure of the curriculum in terms of its content and opportunity for conceptual progression was also considered. Literature reveals continuing discourse regarding the levels of thinking described in the Van Hiele theory, and even though there is no consensus regarding the nature of the levels and that assessing learners' levels of thinking remains difficult and inconclusive, it is generally accepted that the Van Hiele test is a reliable measure in assessing learners' levels of geometry thinking. An exploratory case study design was chosen for this study. The phenomenon being explored is the teaching and learning of geometry in the Foundation and Intermediate Phases of a particular private school. In order to do this, the teachers' timetables and Work Schedules were analysed to determine how much time was allocated to the instruction of Mathematics in general and for the instruction of geometry in particular. These documents also yielded data regarding the type of geometry experiences included in the implemented curriculum. The learners' level of geometry understanding according to the Van Hiele theory was assessed using an instrument designed by Usiskin (1982). This assessment was facilitated by the researcher in the learners' home class and happened in June after six months of instruction in a particular grade level. Data regarding the teachers' perception of geometry and the best method to facilitate the learning of geometry was gathered through a teacher's questionnaire. The teachers were requested to facilitate geometry lessons, which were digitally recorded by the researcher. Each grade level (0-5) was regarded as a sub-unit and analysed as the case for that grade level. The data was then assimilated to present the case of geometry teaching and learning in the Foundation and Intermediate Phases in the school. The findings report that when juxtaposed alongside research, geometry instructional practices in this school, compare favourably with regards to the teachers' professed and observed practice of using concrete aids and tasks that engage the learners actively in developing geometry insight. There is also evidence that these instructional practices support progression through the levels however the shortfall of time allocated to facilitating this progression and the lack of conclusive data regarding the language used and the types of experiences may justify further research into whether this progression is satisfactory. Copyright / Dissertation (MEd)--University of Pretoria, 2011. / Science, Mathematics and Technology Education / unrestricted

This study sought to (a) investigate the effect of Van Hiele theory-based instruction on Grade 11 students’ geometric proofs learning achievement, (b) explore students’ views on their geometry learning experiences, and (c) develop a framework for better teaching and learning of Grade 11 Euclidean geometry theorems and non-routine geometric proofs. The study is based on Van Hiele’s theory of geometric thinking. The research involved a convenience sample of 186 Grade 11 students from four matched secondary schools in the Capricorn district of Limpopo province, South Africa. The study employed a sequential explanatory mixed-methods design, which combined quantitative and qualitative data collection methods. In the quantitative phase, a non-equivalent groups quasi-experiment was conducted. A Geometry Proof Test was used to assess students’ geometric proof construction abilities before and after the teaching experiment. Data analysis using non-parametric analysis of covariance revealed that students from the experimental group of schools performed significantly better than their counterparts from control group schools. In the qualitative phase, data were collected using focus group discussions and students’ diary records. Results revealed that the experimental group students had positive views on their geometry learning experiences, whereas students from the control group of schools expressed negative views towards the teaching of Euclidean geometry and geometric proofs in their mathematics classes. Based on the quantitative and qualitative data findings, it was concluded that in addition to organizing instruction according to the Van Hiele theory, teachers should listen to students’ voices and adjust their pedagogical practices to meet the expectations of a diverse group of students in the mathematics class. A framework for better teaching and learning of Grade 11 Euclidean geometry theorems and non-routine geometric proofs was thus developed, integrating students’ views and Van Hiele theory-based instruction. The study recommends that teachers should adopt the modified Van Hiele theory-based framework to enhance students’ mastery of non-routine geometric proofs in secondary schools.

Syftet med denna systematiska litteraturstudie var att sammanställa aktiviteter som beskrevs i forskning om polygoner i geometriundervisning samt analysera vilken form av förståelse dessa ämnade utveckla. Respektive aktivitet analyserades utefter van Hieles beskrivning av olika nivåer av geometriskt tänkande för att svara på frågan vilken förståelse som aktiviteten ämnade utveckla. Resultatet visade att det fanns fyra huvudkategorier av aktiviteter och att den största kategorin utgjordes av laborativa aktiviteter vilka i huvudsak ämnade utveckla van Hieles nivå två för geometriskt tänkande. De tre resterande kategorierna varierade i vilken nivå de ämnade utveckla mellan van Hieles nivå 1–4. / Polygons are included in mathematics education for grades 4–6. The aim of this study was to compile activities described in previous research on polygons through a systematic literature review and analyze what kind of geometric thinking they intended to develop. The activities were individually analyzed through a comparison with van Hiele’s levels about how geometric thinking is developed. The results showed that the research mainly contained explorative tasks which aimed to develop van Hiele’s level two of geometric thinking. Other activities that were found varied in what level of geometric thinking they developed.

Summaries in Afrikaans and English / Outcomes-Based Education forms the foundation of South Africa's revised national education curriculum, known as The Revised National Curriculum Statement Grades R-9. The Van Hiele Model, developed for the teaching and learning of geometry, has already been used successfully in the school curricula of the Netherlands and Russia. Strong support for this theory exists in leading countries like America and Europe. This study has investigated the possible relationship between the Van Hiele theory and the "Revised National Curriculum Statement Grades R-9" and through critical evaluation found a great compatibility between them. Both are giving strong support for the spiral approach to mathematics education. It is recommended that all geometry learning programmes and teaching units be moulded on the Van Hiele Model. / Suid-Afrika se hersiene nasionale onderwyskurrikulum, die "Revised National Curriculum Statement Grades R-9" word deur uitkomsgebaseerde onderwys gefundeer. Die Van Hiele Model vir die onderrig en leer van meetkunde, is suksesvol in skoolkurrikula van Nederland en Rusland toegepas en geniet wye steun in lande soos Amerika en Europa. Hierdie navorsing het die moontlike verband tussen die Van Hiele raamwerk en die nuwe kurrikulum ondersoek en deur kritiese evaluering baie sterk versoenbaarheid tussen die "Revised National Curriculum Statement Grades R-9" en die Van Hiele Model gevind. Verder is ook aangetoon dat beide sterk gebruik maak van die spiraalbenadering in wiskundeonderwys. Aanbevelings in die verband is dat leerprogramme en onderrigeenhede wat die kurrikulum vir meetkunde in skole implementeer op die Van Hiele Model geskoei word. / Mathematical Sciences / M. Sc. (Mathematics Education)

Traditionally, geometry at school starts on a formal level, largely ignoring prerequisite skills needed for formal spatial reasoning. Ignoring that geometry conceptualisation has a sequential and hierarchical nature, causes ineffective teaching and learning with a long lasting inhibiting influence on spatial development and learning. One of the current reform movements in mathematics education is the appropriate use of dynamic computer technology in the teaching and learning of mathematics. Concerning mathematics education, the lecturers may involve the introduction of both dynamic computer technology and mathematics in meaningful contexts that will enable interplay between the two. Pre-service mathematics teachers (PMTs) can be encouraged to become actively involved in their learning and, therefore, less frustrated in their study orientation in mathematics. Therefore, such learning environments may be essential to enhance the conceptual understanding of PMTs. To be able to reach their eventual learners, PMTs' own conceptual understanding of geometry should be well developed. When PMTs have conceptual understanding of a mathematical procedure, they will perceive this procedure as a mathematical model of a problem situation, rather than just an algorithm. This study aimed at investigating the effect of a technologically enhanced learning environment on PMTs' understanding of geometry concepts and their study orientation in mathematics, as prerequisite for deep conceptualisation. A combined quantitative and qualitative research approach was used. The quantitative investigation employed a pre-experimental one-group pre-test post-test design. A Mayberry-type test was used to collect data with regard to PMTs' conceptualisation of geometry concepts, while the Study Orientation in Mathematics (SOM) questionnaire was used to collect data with regard their study orientation in mathematics. The qualitative investigation employed phenomenological interviews to collect supplementary information about the participating PMTs' experiences and assessment of the influence of the use of the dynamic software Geometer's Sketchpad (GSP) on their learning and conceptualisation of geometry concepts. During post-testing the participating group of PMTs achieved practically significantly higher scores in the Mayberry-type test, as well as in all fields of the SOM questionnaire. Results seem to indicate that PMTs gained significantly in the expected high levels of conceptualisation, as well as high degrees of acquisition of those levels during the intervention programme. The main conclusion of the study is that a technologically enhanced learning environment (such as GSP) can be successfully utilised to significantly enhance PMTs' conceptualisation and study orientation, as prerequisite for deep conceptualisation, in geometry. / Thesis (M.Ed.)--North-West University, Potchefstroom Campus, 2006

Cette recherche se propose de tester en CM2 et en 6ème un nouveau cadre théorique construit à partir d'une part de la théorie des paradigmes géométriques et de la théorie des niveaux de van Hiele. La géométrie à l'école primaire est essentiellement une géométrie spatio-graphique (G1) où les objets sont les représentants d'objets physiques et les validations perceptives. Le niveau de van Hiele que l'élève doit maîtriser à la fin du CM2 est celui de l'identification-visualisation (N1). La géométrie au collège vise à être une géométrie proto-axiomatique (G2) où les objets sont théoriques et les validations de type hypothético-déductif. L'élève doit alors maîtriser le niveau de déduction informelle de van Hiele (N3). <br />Le cadre théorique mis à l'épreuve dans cette étude, propose que le niveau d'analyse (N2) de van Hiele soit une « zone de tuilage » entre les paradigmes géométriques G1 et G2.<br />Des élèves de CM2 et de 6ème ont répondu aux mêmes questions à propos des triangles particuliers, des quadrilatères particuliers, du cercle. <br />L'analyse des réponses a permis de montrer qu'un élève de CM2 ou de 6ème ne pouvait être caractérisé par un mode de fonctionnement dans un paradigme unique ou un seul niveau de van Hiele. Selon l'activité il peut fonctionner dans un paradigme ou dans un autre et témoigner de différents niveaux de van Hiele. Le niveau N2 d'analyse de la théorie de van Hiele se confirme comme la « zone de tuilage » entre les deux paradigmes géométriques. Des activités mettant en évidence ce niveau de van Hiele dans l'un ou l'autre des deux paradigmes permette d'instaurer une continuité dans l'enseignement de la géométrie au passage de l'école primaire vers le collège.

Thesis (M.Ed.)--North-West University, Potchefstroom Campus, 2006.

This study aimed to determine the effects of drama based learning on seventh grade students&rsquo / achievement (angles and polygons, circle and cylinder), attitudes and thinking levels in geometry compared to the cooperative learning. The study was conducted on four seventh grade classes from two public elementary schools in the same district in the 2006-2007 academic year, lasting seven and a half week (30 lesson hours). The data were collected through angles and polygons (APA) / and circle and cylinder achievement (CCA) tests, the van Hiele geometric thinking level test (POSTVHL), geometry attitude scale (PRE-POSTGAS). The quantitative analyses were carried out by using Multivariate Analysis of Variance (MANOVA). The results showed that drama based learning had a significant effect on students&rsquo / angles and polygons achievement, circle and cylinder achievement, van Hiele geometric thinking level compared to the cooperative learning. However, attitude findings regarding the attitudes revealed that there is not a significant difference according to the geometry attitudes of drama group and cooperative group after treatment. Both the two instructional methods supported active participation, created cooperative working environment, included daily life examples and gave the chance to classroom communication. On the other hand, drama group students&rsquo / significantly better performance was attributable to the make belief plays and improvisations of daily life examples included in drama activities.

This study aimed to investigate the effects of using dynamic geometry software while teaching by guided discovery compared to paper-and-pencil based guided discovery and traditional teaching method on sixth grade students&rsquo / van Hiele geometric thinking levels and geometry achievement. The study was conducted in one of the private schools in Ankara and lasted six weeks. The sample of the study consisted 47 sixth grade students in the school. The present study was designed as pretest-posttest control group quasi-experimental study.In order to gather data, Geometry Achievement Test (GAT) and Van Hiele Geometric Thinking Level Test (VHL) were used. At the end of the research, the data were analyzed by means of analysis of covariance. The results of the study indicated that there was a significant effect of methods of teaching on means of the collective dependent variables of the sixth grade students&rsquo / scores on the POSTVHL after controlling their PREVHL scores, and there was a significant effect of methods of teaching on means of the collective dependent variables of the sixth grade students&rsquo / scores on the POSTGAT after controlling their PREGAT scores.

Traditionally, geometry at school starts on a formal level, largely ignoring prerequisite skills needed for formal spatial reasoning. Ignoring that geometry conceptualisation has a sequential and hierarchical nature, causes ineffective teaching and learning with a long lasting inhibiting influence on spatial development and learning. One of the current reform movements in mathematics education is the appropriate use of dynamic computer technology in the teaching and learning of mathematics. Concerning mathematics education, the lecturers may involve the introduction of both dynamic computer technology and mathematics in meaningful contexts that will enable interplay between the two. Pre-service mathematics teachers (PMTs) can be encouraged to become actively involved in their learning and, therefore, less frustrated in their study orientation in mathematics. Therefore, such learning environments may be essential to enhance the conceptual understanding of PMTs. To be able to reach their eventual learners, PMTs' own conceptual understanding of geometry should be well developed. When PMTs have conceptual understanding of a mathematical procedure, they will perceive this procedure as a mathematical model of a problem situation, rather than just an algorithm. This study aimed at investigating the effect of a technologically enhanced learning environment on PMTs' understanding of geometry concepts and their study orientation in mathematics, as prerequisite for deep conceptualisation. A combined quantitative and qualitative research approach was used. The quantitative investigation employed a pre-experimental one-group pre-test post-test design. A Mayberry-type test was used to collect data with regard to PMTs' conceptualisation of geometry concepts, while the Study Orientation in Mathematics (SOM) questionnaire was used to collect data with regard their study orientation in mathematics. The qualitative investigation employed phenomenological interviews to collect supplementary information about the participating PMTs' experiences and assessment of the influence of the use of the dynamic software Geometer's Sketchpad (GSP) on their learning and conceptualisation of geometry concepts. During post-testing the participating group of PMTs achieved practically significantly higher scores in the Mayberry-type test, as well as in all fields of the SOM questionnaire. Results seem to indicate that PMTs gained significantly in the expected high levels of conceptualisation, as well as high degrees of acquisition of those levels during the intervention programme. The main conclusion of the study is that a technologically enhanced learning environment (such as GSP) can be successfully utilised to significantly enhance PMTs' conceptualisation and study orientation, as prerequisite for deep conceptualisation, in geometry. / Thesis (M.Ed.)--North-West University, Potchefstroom Campus, 2006