This diploma thesis called Estimations in Mathematics at Elementary School is created as an auxiliary teaching material. It acquaints the readers with the concept of mathematical estimation and presents its basic divisions. It also provides graphic instructions how to make estimations and it contains plenty of problems which serve as a systematic practicing and training of the ability of estimating. In addition, the form of individual problems shows spheres in which estimating is convenient to develop. At the end of the work there are presented research-oriented problems with the need of estimation.

Submitted by Daniele Amaral (daniee_ni@hotmail.com) on 2016-09-15T16:05:03Z No. of bitstreams: 1 DissWCCJ.pdf: 1969639 bytes, checksum: cbe6746c0279668ba7ff3b8de72d8caf (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-16T19:51:53Z (GMT) No. of bitstreams: 1 DissWCCJ.pdf: 1969639 bytes, checksum: cbe6746c0279668ba7ff3b8de72d8caf (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-16T19:51:57Z (GMT) No. of bitstreams: 1 DissWCCJ.pdf: 1969639 bytes, checksum: cbe6746c0279668ba7ff3b8de72d8caf (MD5) / Made available in DSpace on 2016-09-16T19:52:02Z (GMT). No. of bitstreams: 1 DissWCCJ.pdf: 1969639 bytes, checksum: cbe6746c0279668ba7ff3b8de72d8caf (MD5) Previous issue date: 2015-08-21 / Não recebi financiamento / Cryptography is a fascinating topic, concerning the practical point of view, and it is useful to access bank accounts, e-mails and social networks. According to this perspective, this study aimed to show how simple it may be to make use of cryptography. It was proved, through the work performed, that it is possible for students from elementary and high school to encrypt and decrypt messages using R.S.A. Cryptography, which involves the concept of creating public keys and private keys for the encryption of messages. A simple but powerful activity on the utility view was assigned to show students from the 9th grade of elementary school and from the 1st and 2nd years of high school from a private school in São Paulo State how interesting and pleasurable the use of Mathematics can be. Such activities are not usually present in the traditional didactic materials. The theoretical tools, containing theorem and corollaries, as well as their demonstrations, which mathematically justify the validity of techniques and the algorithm used on R.S.A. Cryptography will be presented. The activities do not assume sophisticated prerequisites and can be applied in the classroom, in real situations, so that the students can appreciate the beauty of Mathematics. / A criptografia é um assunto fascinante do ponto de vista prático; é útil para acessar contas bancárias, e-mails e redes sociais. Segundo esta perspectiva, esta dissertação baseou-se em mostrar o quão simples pode ser a utilização da criptografia. No trabalho realizado foi mostrado que é possível, para alunos do Ensino Fundamental e Médio, codificar e decifrar mensagens utilizando a criptografia - RSA, a qual envolve o conceito da criação de chaves públicas e chaves privadas para a codificação de mensagens. Uma atividade simples, porém importante do ponto de vista utilitário, foi aplicada para mostrar aos alunos do 9º ano do Ensino Fundamental, da 1ª e 2ª séries do Ensino Médio de uma escola particular do interior paulista para mostrar o quão interessante e prazerosa pode ser a utilização da Matemática. Tais atividades não constam usualmente nos materiais didáticos tradicionais. Nesta dissertação será apresentado o ferramental teórico, contendo teoremas e corolários, assim como suas demonstrações, os quais justificam, matematicamente, a validade das técnicas e do algoritmo utilizados na criptografia – RSA. As atividades não pressupõem pré-requisitos sofisticados, podendo ser aplicadas em sala de aula, em situações reais, para que os alunos apreciem a beleza da Matemática.

Submitted by Irene Nascimento (irene.kessia@ufpe.br) on 2016-09-19T19:21:41Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Dissertação Priscila Lima.pdf: 4417746 bytes, checksum: 83b0ecf6ed6d9f3ab9800feb7f2911c8 (MD5) / Made available in DSpace on 2016-09-19T19:21:41Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Dissertação Priscila Lima.pdf: 4417746 bytes, checksum: 83b0ecf6ed6d9f3ab9800feb7f2911c8 (MD5) Previous issue date: 2016-02-29 / Esta pesquisa tem o objetivo de analisar práticas de ensino de Matemática de três professores que atuam noCiclo de Alfabetização. Os professores sujeitos desta investigação participaramem 2014, do programa de formação continuada no âmbito do Pacto Nacional pela Alfabetização na Idade Certa (PNAIC) e atuamna rede pública do município deRecife/PE. Para isso, fizemos a identificação dos conhecimentos pedagógicos geral, pedagógico do conteúdo e do currículo mobilizados por estes professores caracterizando as formas de organização de ensino presentes nas aulas de Matemática. O pressuposto teórico sobre o qual nos reclinamos são as teoriasda base do conhecimento do professor sobrevindas das pesquisas de Lee S. Shulman. Esta pesquisa nos levou também a identificar como a Matemática é abordada em sala de aula, a escolha e uso dos recursos didáticos (com destaque:livro didático, o quadroe o caderno) e o conhecimento presente no raciocínio pedagógico doprofessor.A coleta de dados ocorreu por meio de observações de aulas, registros no Caderno de Observação e entrevistas semiestruturadas.A partir dos dados coletados, sinalizamos que a prática do professor é singulare tem sido pouco guiada pelos elementos que deveriam ser norteadores da atividade docente:planejamento e currículo. Percebemos que o professor precisa desenvolver em base do conhecimento denovas estratégias que ultrapassem a priorização do ensino de outras áreas de conhecimento em detrimento da Matemática, a maior ênfase em um bloco de conteúdos e o trabalho que perceba e envolva a Matemática além da que é trabalhada formalmente e explicitamente. / This research's objective is to analyze mathematics teaching techniques of three Literacy Cycle teachers. These teachers, the research subjects, participated during 2014 of a continued education program under the Pacto Nacional pela Alfabetização da Idade Certa (PNAIC) andare acting in public schools in the city of Recife/Pe. To accomplish it, we identified the general pedagogical knowledge, pedagogical content knowledge, and curriculum knowledge mobilized by polyvalent teachers that act in the Literacy Cycle characterizing the way teaching is organized in mathematic classes. The theoretical assumption in which we are inclined is that the teacher's knowledge base is that of Lee S. Shulman’s research. The field work was done through class observations, records in the observation notepad, and semi-structured interviews. This research also leadedus to identify how mathematics is explored inside the classroom, the didactic resources use and choice (with feature: the book, board and notebook) and pedagogical reasoning behind teacher knowledge. Based on the data collected, we noticed teacher’spractice is unique and has not been as guided as it should be by the teacher’sactivity elements: planning and curriculum. We realized teacher needs to developthroughknowledge base new strategies to surpass: other knowledge areas teaching prioritization but math, focus better in one specific content block and the mathematics perception and involvement worked mostly formally and in an explicit way.

Submitted by MARCIA ROVADOSCHI (marciar@unifra.br) on 2018-08-20T16:54:25Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertacao_GabrielDeOliveiraSoares.pdf: 3563656 bytes, checksum: 8232661e148b570fa94209b37941261c (MD5) / Made available in DSpace on 2018-08-20T16:54:25Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertacao_GabrielDeOliveiraSoares.pdf: 3563656 bytes, checksum: 8232661e148b570fa94209b37941261c (MD5) Previous issue date: 2018-01-03 / This qualitative research has as general objective to analyze the limit concept of a function in a point, presented by students of two undergraduate mathematics teaching courses, as well as its strategies of solving questions, in the light of the Theory of the Three Worlds of Mathematics. Firstly, was made an analysis of the introduction of the concept of limit in Calculus textbooks, based on the theoretical framework. Then, a mapping of articles on the ideas of David Tall was done, as well as a search of dissertations and theses that were based on these ideas. In the last phase, a test on limits of functions was applied to students of mathematics teaching courses, who had already studied the discipline of Calculus I and were interviewed professors of this discipline in both courses. The analysis of the textbooks made it possible to verify that, in the introductory chapters, there is a work that advocates features of the symbolic and embodied worlds, especially the first, with some aspects of the formal world. In relation to the interviews with the teachers, they affirmed that begin with graphical examples and tables of functions values to introduce the concept and only then try to reach the formal definition, crossing the Three Worlds of Mathematics. In the test applied to the students, it was concluded that most of them use the natural language to conceptualize limit, presenting characteristics of the embodied world, with some symbolic elements, but without achieving a development compatible with the formal axiomatic world. / Esta pesquisa, de caráter qualitativo, tem como objetivo geral analisar o conceito de limite de uma função em um ponto, apresentado por estudantes de dois cursos de Licenciatura em Matemática, bem como suas estratégias de resolução de questões, à luz da Teoria dos Três Mundos da Matemática. Primeiramente, foi realizada uma análise da introdução do conceito de limite em livros didáticos de Cálculo, com base no quadro teórico. Em seguida, foi feito um mapeamento de artigos sobre as ideias de David Tall, bem como uma busca de dissertações e teses que se basearam nessas ideias. Na última etapa, foi aplicado um teste sobre limites de funções a alunos que já haviam cursado a disciplina de Cálculo I e foram entrevistados docentes dessa disciplina nos dois cursos. A análise dos livros didáticos possibilitou verificar que, nos capítulos introdutórios, há um trabalho que preconiza características dos Mundos Simbólico e Corporificado, especialmente do primeiro, com alguns aspectos do Mundo Formal. Em relação às entrevistas com os professores, estes declararam partir de exemplos gráficos e tabelas de valores de função para introduzir o conceito e só depois tentam chegar à definição formal, perpassando os Três Mundos da Matemática. Em se tratando do teste aplicado aos estudantes, concluiu-se que a maior parte dos alunos utiliza a linguagem natural para conceituar limite, apresentando características do Mundo Corporificado, com alguns elementos simbólicos, mas sem atingir um desenvolvimento compatível com o Mundo Axiomático Formal.

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

The purpose of this study was to investigate the relationship between individual student differences and academic success in three pedagogical methods (traditional classroom, computer-aided instruction (CAI) in an on-campus setting, and CAI in a distance education setting) for developmental mathematics classes at the community college level. Locus of control, math anxiety and learning style were the individual differences examined. Final grade, final exam score and persistence were the indicators of success. The literature review focused on developmental mathematics, pedagogical techniques and variables contributing to academic performance. Two parallel research populations consisted of 135 Beginning Algebra students and 113 Intermediate Algebra students. The Rotter I-E Locus of Control Scale, the Abbreviated Mathematics Anxiety Rating Scale, the 4MAT Learning Type Measure, and an instrument to gather demographic data were used. It was the conclusion of this study that the instructional methods were not equal with respect to achievement. In Beginning Algebra, the CAI students received significantly higher final grades than did the traditionally taught students. In Intermediate Algebra traditional students scored significantly higher on the final exam than did the CBI students. There were more students persisting than expected in traditionally taught Beginning Algebra and no significant difference in attrition in Intermediate Algebra. There was no significant prediction of achievement in Beginning Algebra. For Intermediate Algebra math anxiety was a significant predictor for final exam percentage and locus of control was a significant predictor for final grade percentage. Only the instructional method contributed significantly to the prediction of attrition. While these findings are statistically significant, they account for only a small part of student success. However, the results had implications for the future. In particular, further study should be given to the question of whether CAI, and its associated expenses, is prudent for developmental mathematics instruction.

Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Humanities, School of Education, 2013 / Mathematics-for-teaching (MfT) is complex, multi-faceted and topic-specific. In this study, a Financial Mathematics course for pre-service secondary mathematics teachers provides a revelatory case for investigating MfT. The course was designed and taught by the author to a class of forty-two students at a university in South Africa. Eight students, forming a purposive sample, participated as members of two focus tutorial groups and took part in individual and group interviews. As an instance of insider research, the study makes use of a qualitative methodology that draws on a variety of data sources including lecture sessions and group tutorials, group and individual interviews, students’ journals, a test and a questionnaire. The thesis is structured in two parts. The first part explores revisiting of school mathematics with particular focus on compound interest and the related aspects of percentage change and exponential growth. Four cases are presented, in the form of analytic narrative vignettes which structure the analysis and provide insight into opportunities for learning MfT of compound interest. The evidence shows that opportunities may be provided to learn a range of aspects of MfT through revisiting school mathematics. The second part focuses on obstacles experienced by students in learning annuities, their time-related talk, as well as their use of mathematical resources such as timelines and spreadsheets. A range of obstacles are identified. Evidence shows that students use timelines in a range of non-standard ways but that this does not necessarily determine or reflect their success in solving annuities problems. Students’ use of spreadsheets reveals that spreadsheets are a powerful tool for working with annuities. A key finding with regard to teachers’ mathematical knowledge, and which cuts across both parts of the thesis, is the importance of being able to move between compressed and decompressed forms of mathematics. The study makes three key contributions. Firstly, a framework for MfT is proposed, building on existing frameworks in the literature. This framework is used as a conceptual tool to frame the study, and as an analytic tool to explore opportunities to learn MfT as well as the obstacles experienced by. A second contribution is the theoretical and empirical elaboration of the notion of revisiting. Thirdly, a range of theoretical constructs related to teaching and learning introductory financial mathematics are introduced. These include separate reference landscapes for the concepts of compound interest and annuities

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

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

Submitted by Roberta D´Angela Menduni Bortoloti (robertamenduni@yahoo.com.br) on 2016-07-20T14:39:35Z No. of bitstreams: 1 tese_FIM_Roberta.pdf: 154195498 bytes, checksum: 7914e88c9de15de25d87beef14a3d99f (MD5) / Approved for entry into archive by Maria Auxiliadora da Silva Lopes (silopes@ufba.br) on 2016-07-21T14:35:10Z (GMT) No. of bitstreams: 1 tese_FIM_Roberta.pdf: 154195498 bytes, checksum: 7914e88c9de15de25d87beef14a3d99f (MD5) / Made available in DSpace on 2016-07-21T14:35:10Z (GMT). No. of bitstreams: 1 tese_FIM_Roberta.pdf: 154195498 bytes, checksum: 7914e88c9de15de25d87beef14a3d99f (MD5) / UESB / Apresentamos uma matemática para o ensino como um modelo para o ensino do conceito de proporcionalidade. Este modelo permite reunir uma variabilidade de formas de comunicar o conceito de proporcionalidade e (re)apresentá-la por meio de uma estrutura teórica que organiza seus modos de ocorrência. O objetivo geral da pesquisa foi a construção de um modelo de uma matemática para o ensino do conceito de proporcionalidade, no qual identificamos diferentes modos de comunicar o conceito em questão, utilizando três fontes: artigos científicos, um grupo de professores e livros didáticos de matemática. Três objetivos específicos foram propostos para que se alcançasse o objetivo geral. O primeiro consistiu em construir uma matemática para o ensino do conceito de proporcionalidade a partir de uma revisão sistemática de literatura, identificando e sintetizando estudos. Fundamentamos os dois outros objetivos no método qualitativo, sendo o segundo o de construir uma matemática para o ensino do conceito de proporcionalidade a partir de um grupo com professores da educação básica e, o terceiro objetivo construir uma matemática para o ensino do conceito de proporcionalidade a partir de livros didáticos de matemática da educação básica. A justificativa para a escolha do método qualitativo encontra-se na construção do modelo por meio do que é comunicado como proporcionalidade, seja por professores da educação básica ou autores de livros didáticos de matemática. Inspirados em Brent Davis, recorremos ao Estudo do Conceito como dispositivo investigativo para a produção dos diferentes usos do conceito de proporcionalidade. A apropriação que fizemos desse dispositivo, entrelaçado às definições teóricas dos trabalhos desenvolvidos pela pesquisadora Anna Sfard, se constituiu em instrumento de análise e estratégia de modelagem teórica. Os resultados mostraram uma diversidade de realizações do conceito de proporcionalidade, distribuída em três cenários, formando, assim, um modelo teórico para o ensino do conceito de proporcionalidade. No primeiro cenário, o conceito de proporcionalidade foi relatado como razão e realizou-se como taxa, escala, divisão, probabilidade, razão trigonométrica, porcentagem, divisão e quotização proporcionais, vetor e intervalos musicais. No segundo, ele foi descrito pela igualdade entre razões a partir do uso da regra de três, da divisão proporcional de segmentos e da porcentagem. No último cenário, esse conceito foi apresentado como taxa de variação de uma função, podendo ser identificada como uma constante de proporcionalidade, um fator-escala, um coeficiente angular ou uma declividade. / ABSTRACT We present Mathematics for the teaching as a model for the teaching of the proportionality concept. This model allows to gather a variability of ways of communicating the proportionality concept and (re) introduce it through a theoretical structure that organizes its ways of occurrence. The general objective of the study was the building of a model of Mathematics for the teaching of the proportionality concept. We have identified three different ways to communicate this concept, through the use of three sources: scientific papers, a group of teachers and mathematics textbooks. There were proposed three specific objectives in order to achieve the general objective. The first one was to build Mathematics for the teaching of the proportionality concept from a systematic review of literature, through the identification and syntheses of the studies. We have founded the two other objectives in the qualitative method, being the second one to build Mathematics for the teaching of the proportionality concept through a group with Elementary School teachers, and the third one to build Mathematics for the teaching of the proportionality concept through textbooks of Mathematics in Elementary School. The reason for the choice of the qualitative method can be found in the building of the model through the way of what has been taught as proportionality, has it been done by Elementary School teachers or authors of mathematics textbooks. Being inspired by Brent Davis, we used the Concept Study as an investigative tool for the production of the different uses of the proportionality concept. The appropriation that we made of this tool, together with the theoretical definitions of the work by the researcher Anna Sfard, were used in the analysis and strategy of theoretical modeling. The results showed diversity for the proportionality concept that had been distributed in three different landscapes and, this way, creating a theoretical model for the teaching of the proportionality concept. In the first landscape, the proportionality concept was related as ratio and it was hold as rate, scale, division, probability, trigonometric ratio, percentage, proportional division and partition, vector and music intervals. In the second one, it was described through the equality between ratios through the use of the rule of three, the proportional division of segments and percentage. In the last landscape, this concept was presented as a rate of variation of a function and it can be identified as a constant of proportionality, a scale factor, an angular coefficient or a declivity.