Orientadora: Arlete de Jesus Brito / Banca: Cármen Lúcia Brancaglion Passos / Banca: Eugênio Maria de França Ramos / Resumo: Esta pesquisa tem por objetivo compreender em que atividades envolvendo problemas de Criptografia podem auxiliar os alunos na exploração das ideias associadas à função afim. As atividades propostas aos estudantes foram estruturadas na forma de enigmas envolvendo contos baseados no personagem Sherlock Holmes, de Sir Arthur Conan Doyle. As atividades foram desenvolvidas com um grupo de alunos do primeiro ano do Ensino Médio da escola pública E. E. Prof. Mycroft, localizada na cidade de Rio Claro/SP. Este trabalho está fundamentado na Metodologia de Pesquisa Intervenção com enfoque Qualitativo. A metodologia de análise dos dados se aproxima de estudos sócio-culturais conjuntamente com estudos cognitivos permeados por meio da resolução de problemas e investigações matemáticas. Como aporte teórico, a dissertação apresenta uma discussão e reflexão acerca do envolvimento da matemática em relação aos campos de poderes (social, politico, econômico, territorial, entre outros) sempre elencando esse entrelaçamento com a constante evolução da criptografia e sua busca constante de cifras seguras e poderosas. Os dados da pesquisa foram produzidos a partir de observações e anotações em um diário de campo, realizadas pela pesquisadora, filmagens e gravações de áudio dos encontros, entrevistas semiestruturadas e fichas de perguntas das atividades. Como conclusões, observou-se que os alunos desenvolveram atitudes autônomas durante seus processos de aprendizagem, fomentando posturas investigativas. Tais atitudes contribuíram na criação e experimentação de diferentes estratégias de resolução o que refletiu nas explorações e investigações realizadas por eles a respeito das ideias associadas ao conceito de função afim / Abstract: This research aims to understand how activities involving encryption problems can assist students to explore the ideias associated with affine functions. The activities proposed to the students were structured in the form of enigmas involving tales based on the character Sherlock Holmes, by Sir Arthur Conan Doyle. The activities were developed with a group of students in the first year of high school of the public school E. E. Prof. Mycroft, located in Rio Claro/SP. This work is based on Intervention Research Methodology with Qualitative approach. .Data analysis methodology approaches socio-cultural studies in conjunction with cognitive studies influenced by problem solving and mathematical investigations. As a theoretical framework, the dissertation presents a discussion and reflection about the involvement of mathematics in relation to the powers of fields (social, political, economic, territorial, etc.) always listing this entanglement with the constant evolution of cryptography and its constant search of safe and powerful figures. The research data were produced through observation and by notes in a field diary, carried out by the researcher, in addition to semi-structures interviews and questions cards activities. In conclusion, it was observed that the students developed autonomous attitudes during their learning processes, encouraging investigative positions. Such attitudes have contributed in the creation and experimentation of different resolution strategies, which have reflected on the explorations and investigations that they did about the ideas associated with the concept of affine function / Mestre

Acompanha : Guia pedagógico de atividades para o ensino de geometria com a utilização de caleidociclos / No trabalho desenvolvido, analisa-se a realidade da utilização dos caleidociclos como proposta de sequência didática para o ensino e aprendizagem da geometria. A investigação deu-se por meio da realização da Oficina de caleidociclos, a qual foi desenvolvida em quatro encontros, envolvendo alunos da 2ª série do ensino médio do Colégio e Faculdade Modelo – na cidade de Curitiba/PR. A pesquisa, de cunho qualitativo propõe a utilização recursos que diversificassem o desenvolvimento de atividades para o ensino e aprendizagem de Matemática na escola básica. Os dados apresentados no trabalho foram coletados durante o período de realização das oficinas na instituição de ensino, as quais ocorreram durante quatro sextas-feiras no período vespertino, horário contra turno das atividades regulares dos participantes. As atividades realizadas propõem desenvolver e aprimorar o conhecimento de geometria dos estudantes envolvidos nas oficinas, tendo como foco principal a construção dos caleidociclos e visualização de conceitos geométricos existentes neste e em outros materiais explorados com menor ênfase durante os encontros. Ao final dos encontros percebeu-se que o ambiente oferecido para o desenvolvimento das atividades e os trabalhos propostos, auxiliaram no desenvolvimento de novos conceitos referentes a geometria, bem como houve um maior estreitamento na relação docente/discente, possibilitando que as atividades enriqueceram o conhecimento em matemática dos estudantes. Para enriquecimento do trabalho foi desenvolvido um encarte que propõe uma sequência didática para a aplicação da oficina por outros docentes. / At work, we analyze the use of caleidociclos as proposed instructional sequence for teaching and learning of geometrical concepts, developed in the meetings held during the workshop caleidociclos, which involved students from two grade school of the College and College Model - in the city of Curitiba/PR with the intention to propose and use resources that diversify the development of activities for teaching and learning of mathematics in elementary school. The data presented in the study were collected during the implementation of the workshops in the education institution, which occurred for four Fridays in the afternoon, time shift from the regular activities of the participants. The activities proposed to develop and enhance students' knowledge of geometry involved in the workshops, with the main focus of caleidociclos the constructionand visualization of geometric concepts in this and other materials exploited moreemphasis during the meetings. At the end of the meetings it became clear that the environment provided for the development of activities and proposed work, helped in the development of new concepts related to geometry, and there was a greater narrowing of the teacher / student ratio, enabling activities have enriched the knowledge students in mathematics. Following the work there is a proposal for implementing the instructional sequence Caleidociclos Workshop. To enrich the work was a booklet that offers an instructional sequence for the application of the workshop for other teachers.

Acompanha: Caderno pedagógico: Modelagem matemática como estratégia de ensino e aprendizagem: uma perspectiva à luz dos futuros professores de matemática / O presente trabalho objetivou investigar as contribuições que a Modelage Matemática pode propiciar como estratégia de ensino e aprendizagem aos futuros professores de Matemática. A pesquisa direcionou-se para a finalidade de incentivar e possibilitar uma preparação aos futuros professores para compreenderem e trabalharem a Modelagem em sala de aula proporcionando aos mesmos o conhecimento e a experiência com a prática da Modelagem. Para tanto, desenvolveu-se uma proposta de Modelagem Matemática durante um período de 30 horas-aula junto aos sujeitos do 4º ano do Curso de Licenciatura em Matemática, a qual foi aplicada na disciplina de Introdução à Modelagem Matemática pela Universidade Estadual do Norte do Paraná, campus Jacarezinho-PR (UENP-CJ). Nessa proposta, desenvolveram-se várias atividades de Modelagem com os licenciandos objetivando a apresentação da Modelagem Matemática no ensino e também o encorajamento para sua utilização nas práticas profissionais. Essas atividades foram planejadas e analisadas à luz da pesquisa qualitativa, aplicadas de cunho interpretativo, e a coleta de dados se deu por meio da observação, questionário e dados obtidos na aplicação da proposta da Modelagem. Desse modo, a Modelagem Matemática desenvolvida foi satisfatória aos sujeitos da pesquisa, pois proporcionaram contribuições tanto ao entendimento desta estratégia pedagógica quanto às futuras práticas docentes, e estabelecer uma conexão entre a Matemática presente em situações cotidianas e a Matemática escolar. A partir da proposta de Modelagem desenvolvida com os futuros professores, elaborou-se um Caderno Pedagógico cujo objetivo é oferecer aos professores, universitários e pesquisadores subsídios bibliográficos e práticos para desenvolverem a Modelagem Matemática como estratégia de ensino e aprendizagem. Esse material didático se encontra disponível ao acesso público na biblioteca on-line do programa de pós-graduação do presente mestrado. / This study aimed to investigate the contributions that Mathematical Modeling can provide as a strategy of teaching and learning for future Mathematics students. The research was directed toward the purpose of encouraging and enabling a preparation for future teachers so that they may understand and work the Modeling in class giving them the knowledge and experience with the Modeling practice. For this purpose, we developed a proposal for Mathematical Modeling for a period of 30 lesson-hours with the students of the 4th year of the Bachelor of Mathematics, which was applied in the subject of Introduction to Mathematical Modeling at the North Paraná State University, campus in Jacarezinho-PR (UENP-CJ). In this proposal, several modeling activities were developed with the undergraduate students aiming the presentation of the Mathematical Modeling in teaching and also the encouragement for using it in their professional practices. These activities were planned and analyzed in the light of qualitative research, applied in interpretation aspect, and data collection was done through observation, questionnaire and data from the proposed modeling application. Thus, the developed Mathematical Modeling was satisfactory to the research people, because they provided contributions to the understanding of this teaching strategy, as well as to future teaching practices, and it establishes a connection between this Mathematics in everyday situations and school mathematics. From the proposed Modeling developed with future teachers, a Teaching Notebook was developed which aims to provide teachers, university students and researchers some assistance literature and practical to develop the Mathematical Modeling as a teaching and learning strategy. This educational material is available to public in the online library of this master’s degree program.

Esta pesquisa pretendeu investigar de que forma aspectos emocionais influenciam o processo de aprendizagem de Matemática nos anos finais do Ensino Fundamental. Para isso, foi realizada uma pesquisa de cunho qualitativo com alunos de uma escola da rede municipal de educação de Caxias do Sul – RS. O corpus da pesquisa foi constituído a partir das transcrições das entrevistas semiestruturas realizadas com quinze alunos da escola. Os dados foram analisados e discutidos utilizando a Análise Textual Discursiva, desenvolvida por Moraes e Galiazzi. A fundamentação teórica do trabalho apresenta as concepções de Matemática, aprendizagem, educação matemática e aspectos emocionais, tendo Vygotsky como principal referencial teórico. Observou-se que os aspectos emocionais influenciam a aprendizagem, principalmente: por meio do relacionamento com os professores; por meio do relacionamento com os colegas; por meio da influência da família e da sociedade; e por meio de reações físicas e comportamentais. Notou-se a importância de os alunos estarem motivados e interessados em aprender Matemática. Percebeu-se que uma relação emocional positiva entre alunos e professores favorece o processo de aprendizagem, assim como um bom relacionamento com os colegas. Destacou-se a dimensão socializadora da escola e a importância da convivência para o desenvolvimento da criança e do adolescente. Notou-se que a influência da opinião dos familiares reflete na relação que o aluno estabelece com seus estudos e no seu pensamento a respeito da importância da Matemática. Recebeu destaque a percepção de que a Matemática será útil no futuro, em uma profissão ou curso superior. Porém, os alunos tiveram dificuldade para perceber uma utilidade da Matemática no tempo atual que vá além de cálculos simples no mercado. Observou-se também a necessidade de o professor reconhecer as reações emocionais de seus alunos para orientá-los a lidar com suas emoções. Sem pretender esgotar as discussões neste trabalho, pretendeu-se desenvolver uma pesquisa que pudesse contribuir para as reflexões sobre a educação, especialmente para a educação matemática. Inúmeras discussões ainda podem ser feitas a respeito dos aspectos emocionais envolvidos na aprendizagem de Matemática, já que as emoções constituem parte importante de todo processo de aprendizagem. / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, CAPES. / This research aimed at investigating how emotional aspects can influence the learning process of Mathematics in the final grades of Elementary School. With that purpose, a qualitative research was carried out with students at a municipal public school in Caxias do Sul – RS. The corpus of the research was constituted from the transcripts of semi structured interviews with 15 students of the school. Data were analyzed according to Discourse Text Analysis proposed by Moraes and Galiazzi. Theoretical support for the study is based on concepts regarding Mathematics, learning, mathematical education and emotional aspects, having Vygotsky as its major reference. It was observed that emotional aspects do influence learning, especially through the relationship established with teachers, the relationship established with classmates; through the influence coming from their family and from Society, and through physical and behavioral reactions. It was also observed that a positive relationship between students and teachers favor learning process, as well as a good relationship with the classmates. A highlight of the study was the socializing dimension of school and the importance of living with others for children and teenagers’ development. It was noticed that family members’ opinion reflects on the relationship that the student establishes with his studies and on his thoughts about the importance of Mathematics. One perception that outstood was the one related to Mathematics being useful in the future, in a profession or in higher education. However, students had difficulty to recognize utility in Mathematics nowadays which would go beyond simple calculations at the market. It was also noticed that it is necessary that teachers can recognize their students’ emotional reaction so as to orient them on how to deal with their emotions. Without any intention of exhausting the discussions on this study, what was intended was to develop a research that could contribute for the reflections about education, especially mathematical education. A countless number of discussions can still take place concerning emotional aspects involved in learning Mathematics, since emotions are important part of the whole learning process.

M.Ed. (Mathematics Education) / The purpose of this study was to explore teachers’ pedagogical content knowledge (PCK) in the teaching of equivalent fractions in grade 3 learners. The study used a qualitative paradigm, where a case study design was adopted. Five teachers were sampled through a purposive sampling strategy. The data were collected through three methods: observations, interviews and document analysis. Four teachers were observed in their classrooms and were also interviewed, the fifth teacher was only interviewed through an interview schedule. Interview and observation data were analyzed and interpreted using the ladder of inference, which follows a three-rung approach. The documents in the form of learners’ class works, home works, tests and teachers’ lesson plans were analyzed. Learners’ works correlated with their teachers’ preparations and presentations. The findings indicate that teachers lacked PCK in the teaching of fractions in particular, equivalent fractions. Teachers were not competent, fluent and not independent in the teaching of fractions. It is also clear from data collected that fractions are notoriously difficult for teachers to teach and for learners to learn. On the whole, teachers seem to lack requisite skills to contextualize content to the appropriate cognitive levels of their learners. Teachers demonstrated poor content and conceptual knowledge ...

D.Ed. / Adolescents who achieve inadequately in mathematics often experience that the different aids available to assist them on their road to recovery are not sufficient. Most adolescents do not know how to bring about desired change. They are not aware what their problem areas are and how it should be addressed. Due to growing numbers in the classrooms in formal teaching, the teacher cannot assist the student individually on an ongoing basis. Factors within the student and his circumstances may also prevent him from benefiting from assistance. Furthermore, most adolescents in the senior secondary phase, who want to study further, are pressurized by educational bodies for higher marks. Therefore, limited time also becomes an issue. Although several studies have been carried out to investigate this matter and create models, the majority of practical programmes lacked depth and a holistic approach.

M.Ed. / The central focus of the study is the development of guidelines and the creation of a implementation model for the integration of Outcomes Based Education (hereafter referred to as OBE) and Cooperative learning. The purpose of both the guidelines and the implementation model is to assist colleges with the implementation of the Education Strategy of the National Department of Education as proposed by the Minister of Education. A literature study was initiated through the identification of a problem area. A problem statement was formulated, followed by determining a working methodology which was to be followed during the execution of the study. During the study as well as the development stages of the implementation model, both OBE and Cooperative learning remained the central focus. By means of a literature study both the focus points were highlighted by the emphasis on the integration of OBE and Cooperative learning. The literature study was followed by a quantitative empirical study. The study was supported by the development of a qualitative data gathering instrument. The empirical study was conducted at 3 colleges. The purpose of the empirical study was to determine the extend to which lecturers were employing OBE and Cooperative learning in the Mathematics classes. The research data forthcoming from the empirical study were analysed. During this analysis the focus fell on specifically six points. The six points were the following: • What level of understanding do the respondents have of OBE and Cooperative learning? • What level of training have respondents undergone pertaining to OBE and Cooperative learning? • Does the need exist among respondents to implement OBE and Cooperative learning in their Mathematics classes? • Are respondents capable of implementing OBE and Cooperative learning in their Mathematics classes? • How often are OBE and Cooperative learning used by the respondents in the Mathematics classes? • Is there a need among respondents for further training in OBE and Cooperative learning? The analysis was followed by an elaborate discussion of the above mentioned points to ensure the data acquired were understood and correctly interpreted. This allowed the researcher to draw certain conclusions, which in turn, along with personal experience, were used to make recommendations with respect to the integration of OBE and Cooperative learning as teaching approaches in Mathematics classes.

D.Ed. (Subject Didactics) / The lack of explicitly formulated criteria for the writing of a mathematics textbook, has the implication that the author of a mathematics textbook has to establish his own criteria. In doing this the author may achieve his objectives but his objectives may not necessarily be those of the curriculum. The purpose of this study is to develop criteria for the writing of a mathematics textbook. In order to do this the history of mathematics from the time of the ancient Egyptians to the present and the various ways (structure) in which mathematics manifests itself have been discussed. The most prominent theories on how children learn, together with the classroom situation as far as mathematics is concerned have been discussed to determine the implications they have for the writing of a mathematics textbook...

Literature emphasizes how important it is that procedural and conceptual knowledge of mathematics should be learned in integration. Yet, generally, the learning and teaching in mathematics classrooms relies heavily on isolated procedures. This study aims to improve teaching and learning of partitive and quotitive division, moving away from isolated procedural knowledge to that of procedures with their underlying concepts through the use of manipulatives, visual representation and questioning. Learning and teaching lessons were designed to teach partitive and quotitive division both procedurally and conceptually. The study explored the roles these manipulatives, visual representations and questioning played toward the conceptual learning of partitive and quotitive division. It was found that manipulatives and iconic visualization enhanced learning, and this could be achieved through scaffolding using a questioning approach. It was concluded that manipulatives and iconic visualization need to be properly planned and used, and integrated with questioning to achieve success in the learning of procedural and conceptual knowledge.

Research indicates that many children finish primary school in Southern Africa still reliant on inefficient counting strategies. This study extends the research of the South African Numeracy Chair project to early mathematics intervention with Grade 2 learners. It investigated the possible adaptation of the Mathematics Recovery programme to facilitate learner progression in early arithmetic strategies. This study aimed to investigate the possibility of adapting the Mathematics Recovery programme for use in a whole class setting, and to research the effectiveness of such an adapted programme. This study also aimed to investigate the extent of the phenomenon of unit counting and other early arithmetic strategies used in the early years in Zambia. This study was conducted from an emergent perspective. A review of the literature indicated that children who become stuck using unit counting face later mathematical difficulties, and that teacher over-emphasis on unit counting in the early years of schooling may be a contributing factor. This study used a qualitative design research methodology that consisted of a preparation phase, teaching experiment and retrospective analysis. The context of this teaching experiment was a seven week after-school intervention with a class of Grade 2 learners aged seven to eight in a rural Zambian primary school. Data collection and analysis focused on video recordings of a sample of 6 learners. The experimental teaching content focused on the Early Arithmetic Strategies aspect of the Mathematics Recovery programme. Although limited by time and research focus, this study found that all learners made some progress in early arithmetic strategies, and indicates that the Mathematics Recovery programme has potential for adaptation for early intervention in whole class teaching to address the mathematical education challenges in Zambia and beyond. This study also found that unit counting predominated in the sample learners, but that strategies were not yet entrenched, indicating this was a suitable age for early intervention. This study makes methodological contributions to a growing body of research into the adaptation of the Mathematics Recovery in Southern African contexts and suggests avenues for possible further research.