Thesis (M.A. (Mathematics Education)) --University of Limpopo, 2006 / Traditionally, mathematics has been perceived as objective, abstract, absolute and universal subject that is devoid of social and cultural influences. However, the new perspective has led to the perceptions that mathematics is a human endeavour, and therefore it is culture-bound and context-bound. Mathematics is viewed as a human activity and therefore fallible. This research was set out to explore geometrical concepts involved in the traditional circular buildings in Mopani district of Limpopo Province and relate them to the classroom learning in grade 11 classes. The study was conducted in a very remote place and a sample of two traditional circular houses from Xitsonga and Sepedi cultures was chosen for comparison purposes because of their cultural diversity. The questions that guided my exploration were: • Which geometrical concepts are involved in the design of the traditional circular buildings and mural decorations in Mopani district of the Limpopo Province?. How do the geometrical concepts in the traditional circular buildings relate to the learning of circle geometry in grade 11 class?. The data were gathered through my observations and the learners’ observations, my interviews with the builders and with the learners, and the grade 11 learners’ interaction with their parents or builders about the construction and decorations of the traditional circular houses. I used narrative configurations to analyse the collected data. Inductive analysis, discovery and interim analysis in the field were employed during data analysis. From my own analysis and interpretations, I found that there are many geometrical concepts such as circle, diameter, semi-circle, radius, centre of the circle etc. that are involved in the design of the traditional circular buildings. In the construction of these houses, these concepts are involved from the foundation of the building to the roof level. All these geometrical concepts can be used by both educators and learners to enhance the teaching and learning of circle geometry. Further evidence emerged that teaching with meaning and by relating abstract world to the real world makes mathematics more relevant and more useful.

Made available in DSpace on 2014-06-11T19:24:50Z (GMT). No. of bitstreams: 0 Previous issue date: 2008-02-27Bitstream added on 2014-06-13T20:32:18Z : No. of bitstreams: 1 proenca_mc_me_bauru.pdf: 1462830 bytes, checksum: 518c15b861d67ff8aa5024e7c4385c2f (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo da pesquisa foi analisar o conhecimento declarativo de alunos do ensino médio sobre polígnos e poliedros em termos de seus atributos difinidores, das relações subordinadas e supra-ordenadas e de seus exemplos e não-exemplos. A concepção teórica utilizada foi o modelo de formação de conceitos de Klausmeier e Goodwin (1977). Os participantes foram 253 alunos do ensino de uma escola de rede oficial pública de ensino de Bauru, que responderam, na primeira fase, um questionário, uma prova matemática, um testes de atributos definidores, um teste de exemplos e não-exemplos e um teste de relações subordinadas e supra-ordenadas. Na segunda fase, foram selecionados, aleatoriamente, três alunos com média abaixo de cinco pontos e três alunos com média igual ou superior a cinco pontos para participarem de uma entrevista. Os resultados coletados na primeira fase, analisados quantitativamente, mostram que no teste de atributos definidores a nota média foi de 6,03, sendo que não houve diferença significativa entre as séries (p=0,084). No teste de exemplos e não-exemplos a nota média 5,59 refletiu o desempenho dos participantes e não foram encontradas diferenças significativas entre as séries (p=0,057). Em relação ao teste de relações subordinadas e supra-ordenadas a nota média dos participantes foi 5,64, sendo que a nota média obtida pela primeira série diferiu significativamente da nota média obtida pela terceira série (p=0,024). Em relação aos dados da segunda fase, analisados qualitativamente, alguns participantes pensavam, de maneira equivocada, sobre os atributos definidores de polígnos e de poliedros. Os atributos irrelevantes não interferiam na identificação das figuras selecionadas do teste de exemplos e não-exemplos como exemplos polígnos. / The objective of this reseach was to analyse the declarative konowledge of students from Elementary School Teaching about polygons and polyhedrons in terms of their defining atributes, the subordinate relations and supraordinate and their examples and no-exemples. The theoretical conception used was the model of formation from Klausmeier and Goodwin (1977)'s concepts. The participants were 253 students from Elementary School Teaching of one school of the Oficial Public Network from Bauru, who answered, in the first phase one questionnaire, one math test, one test of defining attributes, one test of examples and no-examples and one test of subordinate relations and supraordinate. In the second phase, three students were randomly selected with the average below five points and three students with equal average or higher to five points who participated in an interview. The collected results in the first phase, quantitatively analysed, showed that in the test of defining attributes, the average grade was 6,03, so there was no significative difference among the series (p=0,084). In the test of examples and no-examples the average grade 5,59 reflected the performance of the participants and no significative differences were found among the series (p=0,057). In the relation to the test of subordinate relations ans supraordinate the average grade of the participants was 5,64, so the average obtained by the first grade had no significative difference from the average obtained by the third grade (p=0,024). In relation to the data of the second phase, qualitativety analysed, some participants thought wrongly about the defining attributes of polygons and polyhedrons. The irrelevants attributes did not interfere in the identification of the selected pictures of the test of examples and no-examples as polygons examples.

The purpose with this research paper is to examine the students’ shown knowledge in geometry, with a focus on construction and its concepts, and the educational value and teaching the students got in this area. The students’ homes are used as a starting-point. The students shall, from a self-made drawing of their home and a photograph of it, describe what their home looks like. In this paper, the mathematical concepts the students used will be analyzed and compared with the education they received. The analytical framework is based on Van Hieles levels of knowledge and Blooms Taxonomy. The study was done at a Secondary School in Kenya. Four students were selected and interviewed. The lesson observations were made with the purpose to get an understanding for how the education for these students look like and to get examples on how the teaching is conducted for these students. Finally, interviews with the teachers were carried out. The students show a good knowledge in the national exams. However, the study shows that when the students are supposed to use this particular knowledge outside of the classroom, the students experience difficulties. Mostly, the students encounter problems when they are supposed to estimate measurements. Furthermore, they lack the ability to compare scales. The research also shows that the education for these students is monotone and much time during the lessons is spend either with a teacher lecturing in front of the board or students working with examples in the textbook. According to the Variation Theory, the knowledge of the students should deepen if the objects of learning are varying. This variation is not something the students receive in the present situation. / Syftet är att undersöka några gymnasieelevers visade kunskaper i geometri med fokus på konstruktion och begreppsanvändning samt den undervisning som erbjuds eleverna inom området. Elevernas hem används som utgångspunkt. Eleverna ska utifrån en teckning, som de själva ritat, och ett fotografi beskriva hemmet. De matematiska begrepp som eleverna använder analyseras. Analysverktyget bygger på van Hieles kvalitativa kunskapsnivåer och Blooms Taxonomi. Undersökningen genomfördes på en gymnasieskola i Kenya. Fyra utvalda elever intervjuades. Lektionsobservationer genomfördes i syfte att få förståelse för hur elevernas undervisningssituation ser ut och få exempel på hur undervisningen bedrivs. Slutligen intervjuades två av elevernas lärare. Eleverna har goda kunskaper på nationella prov men undersökningen visar att när dessa kunskaper skall överföras till något utanför lektionssalen stöter eleverna på problem. De har svårt att uppskatta längdenheter och svårt att jämföra skala. Det kommer också fram att deras undervisning är ganska monoton. Mycket tid läggs till att läraren undervisar eleverna framme vid tavlan eller att eleverna jobbar med uppgifter i sin övningsbok. Enligt variationsteorin, som beskrivs i arbetet, skulle elevernas kunskaper ges möjlighet att fördjupas om de geometriska objekt som skall förstås varieras. Denna variation erbjuds inte eleverna i nuläget.

Orientador: Nelson Antonio Pirola / Banca: Odalea aparecida Viana / Banca: Regina Maria Pavanello / Resumo: O objetivo da pesquisa foi analisar o conhecimento declarativo de alunos do ensino médio sobre polígnos e poliedros em termos de seus atributos difinidores, das relações subordinadas e supra-ordenadas e de seus exemplos e não-exemplos. A concepção teórica utilizada foi o modelo de formação de conceitos de Klausmeier e Goodwin (1977). Os participantes foram 253 alunos do ensino de uma escola de rede oficial pública de ensino de Bauru, que responderam, na primeira fase, um questionário, uma prova matemática, um testes de atributos definidores, um teste de exemplos e não-exemplos e um teste de relações subordinadas e supra-ordenadas. Na segunda fase, foram selecionados, aleatoriamente, três alunos com média abaixo de cinco pontos e três alunos com média igual ou superior a cinco pontos para participarem de uma entrevista. Os resultados coletados na primeira fase, analisados quantitativamente, mostram que no teste de atributos definidores a nota média foi de 6,03, sendo que não houve diferença significativa entre as séries (p=0,084). No teste de exemplos e não-exemplos a nota média 5,59 refletiu o desempenho dos participantes e não foram encontradas diferenças significativas entre as séries (p=0,057). Em relação ao teste de relações subordinadas e supra-ordenadas a nota média dos participantes foi 5,64, sendo que a nota média obtida pela primeira série diferiu significativamente da nota média obtida pela terceira série (p=0,024). Em relação aos dados da segunda fase, analisados qualitativamente, alguns participantes pensavam, de maneira equivocada, sobre os atributos definidores de polígnos e de poliedros. Os atributos irrelevantes não interferiam na identificação das figuras selecionadas do teste de exemplos e não-exemplos como exemplos polígnos. / Abstract: The objective of this reseach was to analyse the declarative konowledge of students from Elementary School Teaching about polygons and polyhedrons in terms of their defining atributes, the subordinate relations and supraordinate and their examples and no-exemples. The theoretical conception used was the model of formation from Klausmeier and Goodwin (1977)'s concepts. The participants were 253 students from Elementary School Teaching of one school of the Oficial Public Network from Bauru, who answered, in the first phase one questionnaire, one math test, one test of defining attributes, one test of examples and no-examples and one test of subordinate relations and supraordinate. In the second phase, three students were randomly selected with the average below five points and three students with equal average or higher to five points who participated in an interview. The collected results in the first phase, quantitatively analysed, showed that in the test of defining attributes, the average grade was 6,03, so there was no significative difference among the series (p=0,084). In the test of examples and no-examples the average grade 5,59 reflected the performance of the participants and no significative differences were found among the series (p=0,057). In the relation to the test of subordinate relations ans supraordinate the average grade of the participants was 5,64, so the average obtained by the first grade had no significative difference from the average obtained by the third grade (p=0,024). In relation to the data of the second phase, qualitativety analysed, some participants thought wrongly about the defining attributes of polygons and polyhedrons. The irrelevants attributes did not interfere in the identification of the selected pictures of the test of examples and no-examples as polygons examples. / Mestre

This paper views technologies as human constructs, becoming as such in the objectification process, having a specific function when shared within the social practice. Among the technologies, it is possible to highlight the media inventions for having multiple utilities alongside the accumulated human intelligence in producing and reproducing its material life, modifying the human relationships and nature. The school, considered as a privileged place where knowledge is shared, faces the paradigm to follow the path done by the intellectual human development, besides ensuring the re-flexibility on the actions as we establish in the world. It is pictured, in school, the singular role of teacher and students, in mediated relationships aiming at achieving the human culture. It is important to focus attention on that issue, in order to reflect on the purpose of teaching geometrical concepts, using the mediating software called GeoGebra. Nevertheless, it is not any type of teaching that accepts the majority of formation concepts, as well as by the fact that it is not any mediation that qualifies for such process, under the zone of proximal development (ZPD) perspective, based on the cultural-historical theory. It was performed a qualitative field research focused on the cultural historical perspective among 5th graders of Elementary School attending a private school in Santa Maria/RS. As for the methodological procedures, it was selected the participant-observation, semi-structured interview followed by an opened questionnaire and field journal, besides the elaboration of activities created in the GeoGebra software and digital records. The analysis of content was used for data analysis, therefore being characterized by the different methodological resources. As for the results, it is possible to conclude that the digital tools are an indirect means for human action around him, alongside his presence in the educational context as changing the teaching perspectives. On the other hand, in order to transformations of spontaneous knowledge into scientific knowledge occur, the child needs, undoubtedly, the teacher to act as the organizer of activities to foster learning, promoting collaborative actions to reach their goals. / Este trabalho compreende as tecnologias como construtos humanos, constituindo-se como tal no processo de objetivação, que quando compartilhados, passam a ter uma função específica no interior da prática social. Dentre as tecnologias, as invenções midiáticas se destacam pelas múltiplas utilidades que assumem, e pela história acumulada de inteligência humana em produzir e reproduzir sua vida material, modificando as relações humanas e a própria natureza. A escola, locus privilegiado em transmitir os conhecimentos acumulados historicamente, insere-se, portanto, no paradigma de acompanhar o transcurso do desenvolvimento intelectual humano, e ainda garantir a reflexibilidade sobre as ações que estabelecemos no mundo. No espaço escolar, tem-se a imagem ímpar do professor e estudantes, em relações mediadas no objetivo de apropriação da cultura humana. A este aspecto direcionamos nossos olhares, para a reflexão do processo educativo intencional de formação de conceitos geométricos, utilizando o instrumento mediador software GeoGebra. Haja vista, que não é qualquer ensino que concebe a máxima formação de conceitos, e pela compreensão de que não é qualquer mediação que o qualificam, é que assumimos neste trabalho a zona de desenvolvimento proximal (ZDP), com base na teoria histórico-cultural. A pesquisa é de campo-qualitativa com enfoque sócio-histórico e contou com a participação ativa de estudantes do 5º ano do Ensino Fundamental I, de um colégio privado do município de Santa Maria/RS. Como procedimentos metodológicos, aderimos à observação participante, entrevista semiestrutura com aplicação de questionário aberto e diário de campo, além da elaboração de atividades produzidas com o software GeoGebra e de registros digitais. Como técnica para a análise dos dados, utilizou-se a análise de conteúdo que pelos diferentes recursos metodológicos foram categorizados. Como resultados, destacamos que as ferramentas digitais se constituem um meio indireto de ação do homem com o seu entorno e sua presença no contexto educativo modifica as relações didáticas. Mas que para ocorrer a transformação de conhecimentos espontâneos por parte da criança em conhecimento científico, necessita da presença incontestável do educador com a organização de atividades intencionais de ensino, promovendo ações colaborativas para atingir seus objetivos.

The diploma thesis is focused on pupils attending 4th and 5th grade in primary school. These children are observed for their knowledge of perpendicularity and parallelism and their ability to use it in practice. The theoretical part describes two geometrical terms - perpendicularity and parallelism. Other chapters describe different theories of learning, the process of knowledge construction and also the stages of language development in geometry. Furthermore the thesis explores the topic in state curriculum documents. The geometrical terms and their occurance, frequency and preparative tasks are searched in different textbooks. The practical part includes experiments based on qualitative research. The survey is conducted through a worksheet with different types of tasks which were chosen to test the participants' knowledge of above-mentioned terms. These experiments were recorded and analyzed by means of phenomena that appeared. The phenomena allowed me to answer the questions that were posed in the beginning of the research. Key words: perpendicularity, parallelism, preparative tasks, depth of knowledge, experiment