Thesis (M. Ed. (Mathematics Education)) -- University of Limpopo, 2017 / The purpose of this study was to explore the role of folding back in enriching grade 10 students’ reasoning in geometry. Although various attempts are made in teaching and learning geometry, evidence from several research studies shows that most learners struggle with geometric reasoning. Hence, this study came as a result of such learners’ struggles as shown in the literature as well as personal experiences. The study was a constructivist teaching experiment methodology that sought to answer the following research questions: How does folding back support learners’ interaction with geometric reasoning tasks during the lessons? How does a Grade 10 mathematics teacher use folding back to enrich student reasoning in geometry? The teaching experiment as a research design in this study was found useful in studying learners’ geometric reasoning as a result of mathematical interactions in their learning of geometry. Therefore, it should be noted that the teaching experiments were not conducted as an attempt to implement a particular way of teaching, but rather to understand the role of folding back in enriching learners’ reasoning in geometry. As a referent to the teaching experiment methodology, the participants in this study were 7 grade 10 mathematics learners’ sampled from a classroom of 54 learners. These seven learners did not necessarily represent the whole class in accordance with the purpose of the study. This requirement was not necessary in determining rigour in teaching experiments. Instead interest was on “organising and guiding [teacher-researchers] experience of learners doing mathematics” (Steffe & Thompson, 2000, p. 300). Furthermore, the participants were divided into two groups while working on the learning activities. Participants were further encouraged to share ideas with each other as they solved the learning activities. Data was collected through video recording while learners were working on mathematical learning activities. The focus was on the researcher-teacher – learners and learners-learners interactions while working on geometric reasoning learning activities. Learning activities and observations served as subsets of the video data. Learners were encouraged to share ideas with each other as they v solved the learning activities as recommended by Steffe and Thompson (2000). Likewise, in order to learn the learners’ mathematics, the researcher could teach and interact with learners in a way that encourage them to improve their current thinking (Steffe & Thompson, 2000). In analysing data, the study adopted narrative analysis. The researcher performed verbatim transcription of the video recordings. Subsequently, information-rich interaction for each mathematical learning activity, where folding back was observed was selected. The selections of such information-rich interactions were guided by Martin’s (2008) framework for describing folding back. The main findings of the study revealed that in a learning environment where folding back takes place, learners’ reasoning in geometry is enriched. The researcher-teacher’s instructional decisions such as discouraging, questioning, modelling and guiding were found to be effective sources through which learners fold back. The findings also revealed that learners operating at different layers of mathematical understanding are able to share their geometry knowledge with their peers. Moreover, the findings indicated that in a learning environment where folding back takes place, learners questioning ability is enriched. Based on the findings of the study, the recommendations were that Mathematics teachers should create a learning environment where learners are afforded the opportunity to interact with each other during geometry problem solving; such is a powerful quest for folding back to take place. / Research Chair Developmental Grant at the University of Limpopo

Thesis (MEd)--Stellenbosch University, 2001. / ENGLISH ABSTRACT: The study of three-dimensional boxes is widely regarded as a suitable topic for entry-level geometry. In order to inform teaching and curriculum design this study takes children's intuitive knowledge of boxes as point of departure to research the following aspects: • the meaning children assign to boxes • children's ability to create and manipulate mental images • children's strategies to solve a three dimensional construction problem • possible age and gender differences • the role of measurement • possible links with Van Hiele's thought levels Revised clinical interviews were conducted with groups of six to ten children. Each child could select an object from a set provided by the researcher and had to build a box from cardboard into which the object could fit. The problem-solving processes were captured on video. The children's boxes, verbal utterences and overt manipulation of the materials as well as the empirical referent objects were analised. The researcher made use of grounded theory procedure to analise and categorise the data. The research indicates that young children are not necessarily aware of the shape and structural properties of three-dimensional boxes. It is also evident that children's ability to represent properties of shape of boxes through language, drawing and hand movements does not necessarily indicate adequate understanding of the structural properties of boxes to enable them to build a box from two-dimensional materials. The research further indicates that aspects of representation that are judged to be based on lack of knowledge of conventions in a medium such as drawing, may have deeper intuitive and conceptual roots. An alternative view of childrens spatial/geometric thought on VanHiele level 0 is described based on their assignment of meaning to geometric tasks. / AFRIKAANSE OPSOMMING: Drie-dimensionele houers word allerweë as geskikte konteks vir aanvangsmeetkunde-onderrig beskou. In hierdie studie word kinders se intuïtiewe begrip van alledaagse houers of bokse as vertrekpunt gebruik om die volgende aspekte te ondersoek met die oog daarop om inligting te bekom vir kurrikulering: • die betekenis wat kinders aan bokse (as meetkundige objekte) toeken • kinders se vermoë om 'n denkbeeld te skep en te manipuleer • kinders se strategieë om 'n drie-dimensionele konstruksieprobleem op te los • moontlike ouderdoms- en geslagsverskille • die rol wat meting speel wanneer 'n drie-dimensionele konstruksieprobleem opgelos word • moontlike ooreenkomste met Van Hiele-denkvlakke 'n Taakgerigte ondersoek (revised clinical interview) is met groepies van ses tot tien kinders op 'n slag gedoen. Elke kind kon 'n voorwerp kies uit 'n versameling en moes 'n houer of boks uit karton bou waarin die voorwerp kon pas. Die probleemoplossingsproses is met 'n videokamera afgeneem. Die kinders se voltooide houers, taaluitinge en overte hantering van die empiriese verwysingsvoorwerpe en die materiaal waarmee hulle gewerk het, is as data ontleed. Die navorser het gebruik gemaak van die analitiese prosesse wat beskryf word as begronde teorie-ontwikkeling om die data te ontleed en te kategoriseer. Die navorsmg hat aangetoon dat jong kinders nie noodwending bewus is van die vorm- en strukturele eienskappe van drie-dimensionele houers nie. Verder is dit duidelik dat kinders se voorstellings van veral vormeienskappe van houers deur middel van taal, handbewegings en tekeninge, nie impliseer dat hulle die nodige begrip van die struktuur het om so 'n houer uit tweedimensionele materiaal te kan vervaardig nie. Die navorsing het verder aan die lig gebring dat aspekte van kinders se ruimtelike kennis wat voorheen as aanduiding van gebrekkige kennis van konvensies beskou is, moontlik dieper konseptuele wortels het. 'n Alternatiewe beskouing van kinders se ruimtelik-meetkundige denke op Van Hiele Vlak 0 word ook voorgestel.

published_or_final_version / abstract / Education / Master / Master of Education

published_or_final_version / abstract / Education / Master / Master of Education

published_or_final_version / abstract / Education / Master / Master of Education

published_or_final_version / abstract / Education / Master / Master of Education

This thesis presents some results in triangle geometry discovered using dynamic software, namely, Geometer’s Sketchpad, and confirmed with computations using Mathematica 9.0. Using barycentric coordinates, we study geometric problems associated with the triangle of reflections T of a given triangle T, yielding interesting triangle centers and simple loci such as circles and conics. These lead to some new triangle centers with reasonably simple coordinates, and also new properties of some known, classical centers. Particularly, we show that the Parry reflection point is the common point of two triads of circles, one associated with the tangential triangle, and another with the excentral triangle. More interestingly, we show that a certain rectangular hyperbola through the vertices of T appears as the locus of the perspector of a family of triangles perspective with T, and in a different context as the locus of the orthology center of T with another family of triangles. / Includes bibliography. / Thesis (M.S.)--Florida Atlantic University, 2014. / FAU Electronic Theses and Dissertations Collection

This dissertation examined the changes of Geometry Education in New York State in connection with the differences in the New York State Mathematics Standards (1999, 2005, 2011). As a result of this analysis, a theoretical framework was created to support teachers in making the shift from teaching towards the 2005 learning standards to teaching towards the goals of the Common Core Standards (2011). Once created, the theoretical framework served as the basis of the development of a collection of problems on various topics in geometry used by teachers in their geometry classrooms. This document can be found in the Appendix of this dissertation. As seen in the past, curriculum, standards, and assessment are all intertwined and reflect one another. In order to bridge the gaps and explore relationships between these components, this research compares the various New York State Mathematics Standards to determine differences in topical coverage as well as an analysis of the New York State Geometry Regents examinations under the 2005 standards and Common Core Standards. Additionally, the research builds on these results and also analyzes select New York State Regents Examination questions in specific topics. This study used the information gathered to create a collection of problems based on certain principles to support teachers in adequate preparation of students for the Common Core Geometry Course. Teachers found the principles provided to be very useful in creating their own problems for additional topics, and found the collection of problems to be very helpful in the teaching and learning of geometry to their students.

The purpose of this research was to identify if 1) there is a difference in student achievement between students who use the GeoLeg manipulative tool and students who use a traditional compass, protractor, and ruler on the same geometry unit; 2) there is a difference in student achievement between the genders between those who use the GeoLeg manipulative tool and those students who do not; and 3) there is a relationship between identified learning styles and student achievement on a geometry unit posttest after using the GeoLeg manipulative tool. There were 317 students in the study. The research found that students using the GeoLeg manipulative tool produced significantly better student performance on a posttest in this particular school setting. Although these results cannot be generalized to other school sites, it is plausible that these results could generalize to school sites whose demographics are similar. The research findings revealed that there was no statistically significant difference between male and female students within the treatment group. The significant finding is that the GeoLeg manipulative tool appears to work equally well with both genders. None of the learning styles, as identified by the Honey and Mumford Learning Styles Questionnaire, were correlated with student posttest score achievement on the tested geometry unit. In addition, there was no evidence to suggest that a student's learning style moderates the effectiveness of the use of the GeoLeg manipulative tool. There is no evidence to suggest that the effectiveness of the GeoLeg manipulative tool is any different depending upon the student's gender or learning style. The results of this research provide strong support for the use of the GeoLeg manipulative tool for improving student performance. Further research is needed to confirm these results in similar and different populations. / by Phyllis Pacilli. / Thesis (Ed.D.)--Florida Atlantic University, 2010. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2010. Mode of access: World Wide Web.

Orientador: Maurício Compiani / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Geociências / Made available in DSpace on 2018-09-11T21:22:04Z (GMT). No. of bitstreams: 1 Barbosa_MagaliAndrade_M.pdf: 2558420 bytes, checksum: eba0ccdd876f993077950ab641ff4427 (MD5) Previous issue date: 2013 / Resumo: Neste trabalho se pretendeu analisar quais conhecimentos geométricos foram apropriados pelos alunos que participaram dos estudos do lugar/ambiente em um projeto com temática ambiental. O objetivo principal é investigar o papel mediador do professor no planejamento de atividades contextualizadas sobre conhecimentos geométricos em uma microbacia urbana. Além disso, se discutiu sobre nossa atuação como professora na elaboração de atividades, na aplicação e nas reflexões de conhecimentos produzidos a partir de estudos do lugar/ambiente do entorno da escola. As atividades desenvolvidas neste trabalho fazem parte de uma pesquisa mais ampla intitulada: "Projeto Ribeirão Anhumas na Escola", na qual participávamos de um subgrupo na Escola Estadual Adalberto Nascimento, que incluía Matemática (professora dessa pesquisa), Geografia, Português e Artes. Por essa característica multidisciplinar do projeto, procurou-se estabelecer conexões entre o conteúdo de geometria e com as atividades interdisciplinares por meio de temas geocientíficos que permitiram a articulação com a realidade vivida pelo aluno. Desenvolveu-se uma concepção de contextualização, que entende que a produção de conhecimento é fruto do espaço cultural em que se situam os sujeitos, que busca tratar de temas e problemas reais da vida dos estudantes situando o raciocínio do aluno a partir de um conceito que seja uma forma mais elementar daquele tema ou problema considerado, mediando à aprendizagem e o desenvolvimento desses alunos da estrutura de pensamento elementar para atingir outra estrutura mais elevada. A pesquisa é de cunho qualitativo, fundamentada metodologicamente como pesquisa-ação colaborativa, em especial no que tange ao planejamento dos problemas de ensino-aprendizagem, à formulação e aplicação das atividades e ao acompanhamento dos processos de aprendizagem dos alunos. Este projeto, por sua vez, compõe-se com essa metodologia e trata especificamente da reflexão de uma professora e da aprendizagem contextualizada da geometria sob uma abordagem histórico-cultural (apoiado nos trabalhos de Vygotsky e de Freitas, 2002). Há um tratamento mais específico da aprendizagem e ampliação dos conhecimentos matemáticos à luz da matematização horizontal e vertical (Luccas e Batista, 2011) e das dimensões dialética horizontal e vertical (Compiani, 2007 e 2013). Descreve-se o conjunto de atividades realizadas em três grandes momentos da pesquisa. No entanto, para a interpretação mais detalhada, utilizam-se três atividades do primeiro momento, três atividades do segundo momento e do terceiro momento uma das atividades e seis questões trabalhadas no 2º trabalho de campo, considerado como importante no ensino de geometria para mostrar o raciocínio do aluno com relação à dinâmica da microbacia urbana. Concluiu-se que, as reflexões da prática pedagógica foram contribuições adquiridas no período de formação do professor no projeto que ao se trabalhar com o lugar/ambiente na construção de conhecimentos geométricos foi fundamental estabelecer a relação escola-vida com o intuito de aprimorar a aprendizagem e desenvolver o raciocínio espacial e racional do aluno no seu processo de formação cognitivo, social e afetivo / Abstract: This paper aims to analyze geometrical knowledge which were appropriated by the students who participated in the studies of the place / environment on a project with environmental issues. The main objective is to investigate the mediating role of the teacher in planning activities contextualized knowledge about geometry in an urban watershed. In addition, they discussed about our role as teacher-building activities in the application of knowledge and reflections produced from studies of place / environment around the school. The activities developed in this paper are part of a broader study entitled: "Project Ribeirão Anhumas School", in which we took part of a subgroup in the State School Adalberto Birth, which included Mathematics (teacher of this research), Geography, Arts and Portuguese. For this feature multidisciplinary project, we tried to establish connections between the geometry content and activities through interdisciplinary geoscientific themes that allowed the articulation with the reality experienced by the student. Developed a concept of contextualization, which means that the production of knowledge is a result of the cultural space in which are located the subjects, which seeks to address the issues and problems of real life students situating the student's reasoning from a concept that is a more elementary that topic or problem considered mediating the learning and development of these students thought elementary structure to achieve another higher structure. The research is a qualitative, grounded methodologically as collaborative action research, particularly in relation to the planning of the problems of teaching and learning, the formulation and implementation of activities and the monitoring of students learning processes. This project, in turn, is composed with this methodology and specifically deals with the reflection of a teacher and contextualized learning geometry in a cultural-historical approach (based on the works of Vygotsky and Freitas, 2002). There is a more specific treatment of learning and extension of mathematical knowledge in light of horizontal and vertical mathematization (Luccas and Batista, 2011) and the horizontal and vertical dimensions dialectic (Compiani, 2007 and 2013). Describes the set of activities performed in three major moments of the research. However, for a more detailed interpretation, we use three activities the first time, three activities of the second phase and the third phase of the activities six issues worked on the 2nd field work, considered important in teaching geometry to show student's reasoning with regard to the dynamics of urban watershed. Concluded that the reflections of pedagogical practice contributions were acquired during teacher training in design when working with the place/environment in the construction of geometrical knowledge was essential to establish a school- life in order to enhance learning and develop spatial reasoning and rational student in the process of forming cognitive, social and emotional / Mestrado / Ensino e Historia de Ciencias da Terra / Mestra em Ensino e História de Ciências da Terra