Made available in DSpace on 2016-04-27T16:58:56Z (GMT). No. of bitstreams: 1 Marcos de Miranda Paranhos.pdf: 6182763 bytes, checksum: 2e98801c415e63f4e40730adcf71a33b (MD5) Previous issue date: 2009-09-23 / The aim of this work is to present fundamental ideas of differential and integral calculus and its applications in solving problems. As a teacher of calculus, I see my trajectory and by exchanging experiences with other professionals, a common sense about the mechanization of techniques and low student achievement in relation to the ideas and applications so significant that the calculation might provide. Reflecting, experiencing and informing me about this issue, I think much of this problem in a limited way with which we have presented these ideas in our classes. Every teacher develops along its trajectory ways to represent the ideas you want to convey and that is the essence of pedagogical reasoning. In that sense, I understood that every idea must be transformed to be taught and it was this aspect that directed this work. Inspired by the possibility of using software in the teaching of Mathematics and didactically based on "Dialectic Tool-Object" and "Game Tables" by Régine Douady, I performed this work that consists of a sequence of activities, divided into six modules, where basic ideas about derivative, integral and optimization functions are presented by means of software and GeoGebra Winplot. The strings are made to functions with one and two variables, can be developed along with the student or be provided only by the teacher. I hope with this work is expanding the size that most students have the Calculus and its applications, besides stimulating the use of technological resources as tools for large capacity in interpreting and solving problems / O objetivo deste trabalho é apresentar idéias fundamentais do Cálculo Diferencial e Integral e suas aplicações na resolução de problemas. Como professor de Cálculo, constato pela minha trajetória e pela troca de experiências com outros profissionais da área, um senso comum a respeito da mecanização de técnicas e do baixo aproveitamento dos alunos com relação às idéias e aplicações tão significativas que o Cálculo poderia lhes proporcionar. Refletindo, experimentando e me informando sobre essa questão, penso que grande parte dessa problemática está na forma limitada com que temos apresentado essas idéias em nossas aulas. Todo professor desenvolve ao longo de sua trajetória formas de representar as idéias que deseja transmitir e essa é a essência do raciocínio pedagógico. Nesse sentido, acredito que toda idéia compreendida deve ser transformada para ser ensinada e foi esse aspecto da questão que direcionou esse trabalho. Inspirado pela possibilidade do uso de softwares no ensino do Cálculo e fundamentado didaticamente na Dialética Ferramenta-Objeto e o Jogo de Quadros de Régine Douady, realizei este trabalho que consiste de uma seqüência de atividades, divididas em seis módulos, em que as idéias básicas sobre derivada, integral e otimização de funções são apresentadas por meio dos softwares Geogebra e Winplot. As seqüências são feitas para funções com uma e duas variáveis, podendo ser desenvolvidas juntamente com o aluno ou ser apenas apresentadas pelo professor. Espero com esse trabalho estar ampliando a dimensão que a maioria dos estudantes tem do Cálculo e de suas aplicações, além de estimular o uso de recursos tecnológicos como ferramentas de larga capacidade na interpretação e resolução de problemas

The widespread use of Dynamic Geometry Software (DGS) is raising many interesting questions and discussions as to the necessity, usefulness and meaning of proof in school mathematics. With these questions in mind, a didactical sequence on the topic “Conics” was developed in a teacher education course tailored for pre-service secondary math methods course. The idea of the didactical sequence is to introduce “Conics” using a concrete manipulative approach (paper folding) then an explorative DGS-based construction activity embedding the need for a proof. For that purpose, the DGS software serves as an intermediary tool, used to bridge the gap between the physical model and the formal symbolic system of proof. The paper will present an analysis of participants’ geometric thinking strategies, featuring proof as an embedded process in geometric construction situations.

In our presentation we will show how technology enables us to improve the teaching and learning of linear functions at the middle school level. Through various classroom activities that involve technology such as dynamic geometry software, graphing calculators and Excel, students explore functions and discover basic facts about them on their own. Students then work with real life data and on real life problems to draw graphs and form linear models that correspond to given situations as well as draw inferences based on their models. Participants will receive complete classroom materials for the unit on linear functions.

Este trabalho traz uma proposta de atividades, a serem desenvolvidas em sala de aula, com o objetivo de introduzir o conceito de derivadas para os alunos da primeira série do Ensino Médio. Antes das atividades, estão presentes algumas breves pesquisas. O histórico da presença de tópicos do Cálculo Diferencial e Integral no Ensino Médio no Brasil, assim como a análise de alguns livros didáticos, serve para mostrar como o assunto já foi e está sendo tratado no país. Também são exibidos aspectos sobre o Ensino Médio na Alemanha e nos Estados Unidos, países onde o cálculo está presente na Escola Secundária, embora de formas bastante diferentes. Um capítulo sobre a preparação adequada para as aulas também foi incluído, uma vez que a simples inserção da derivada poderia causar problemas de tempo para o cumprimento do cronograma e não trazer os resultados esperados. São necessários algum grau de adequação dos conteúdos ministrados e de cooperação com professores de Física. As atividades visando o ensino dos conceitos iniciais de derivada são motivadas por um problema físico de movimento. O foco é dado na intuição e na visualização de gráficos, para que haja uma melhor compreensão dos conceitos envolvidos. A utilização de um software de geometria dinâmica é requerida em boa parte do tempo, como importante ferramenta de apoio pedagógico / This paper presents a proposal of activities to be developed in the classroom, with the goal of introducing the concept of derivative for students in the first grade of secondary school. Before the activities, some brief researches are presented. The historical presence of the topics of Differential and Integral Calculus in brazilian High Schools, as well as the analysis of some textbooks, serves to show how it has been and is being treated in the country. Aspects of the High School are also shown in Germany and the United States, countries where the calculus is present in High School, though in quite different ways. A chapter about the proper preparation for these classes was also included, since the simple insertion of the derivative could cause problems for meeting the schedule and could not bring the expected results. Some degree of adequacy of the contents and cooperation with Physics teachers are needed. The activities aiming at teaching the initial concepts of derivatives are motivated by a physical problem of motion. The focus is given on intuition and visualization of graphs, so there is a better understanding of the concepts involved. The use of a dynamic geometry software is required for much of the time, as an important tool for pedagogical support

Text in Afrikaans / Real empowerment of school learners requires preparing them for the age of technology. This empowerment can be achieved by developing their higher-order thinking skills. This is clearly the intention of the proposed South African FET National Curriculum Statements Grades 10 to 12 (Schools). This research shows that one method of developing higher-order thinking skills is to adopt an integrated curriculum approach. The research is based on the assumption that an integrated curriculum approach will produce learners with a more integrated knowledge structure which will help them to solve problems requiring higher-order thinking skills. These assumptions are realistic because the empirical results of several comparative research studies show that an integrated curriculum helps to improve learners' ability to use higher-order thinking skills in solving nonroutine problems. The curriculum mentions four kinds of integration, namely integration across different subject areas, integration of mathematics with the real world, integration of algebraic and geometric concepts, and integration into and the use of dynamic geometry software in the learning and teaching of geometry. This research shows that from a psychological, pedagogical, mathematical and historical perspective, the theme conic sections can be used as an integrating factor in the new proposed FET mathematics curriculum. Conics are a powerful tool for making the new proposed curriculum more integrated. Conics can be used as an integrating factor in the FET band by means of mathematical exploration, visualisation, relating learners' experiences of various parts of mathematics to one another, relating mathematics to the rest of the learners' experiences and also applying conics to solve real-life problems. / Mathematical Sciences / D.Phil. (Wiskundeonderwys)

NASCIMENTO, Eimard Gomes Antunes . Avaliação do software geogebra como instrumento psicopedagógico de ensino em geometria. 2012.113f. Dissertação (Mestrado em Educação) – Universidade Federal do Ceará, Faculdade de Educação, Programa de Pós-Graduação em Educação Brasileira, Fortaleza-CE, 2012. / Submitted by Maria Josineide Góis (josineide@ufc.br) on 2012-07-06T12:04:55Z No. of bitstreams: 1 2012_Dis_EGANascimento.pdf: 1109485 bytes, checksum: 2166a7f6639ae4cd68fe3dc82b971e3a (MD5) / Approved for entry into archive by Maria Josineide Góis(josineide@ufc.br) on 2012-07-09T12:44:46Z (GMT) No. of bitstreams: 1 2012_Dis_EGANascimento.pdf: 1109485 bytes, checksum: 2166a7f6639ae4cd68fe3dc82b971e3a (MD5) / Made available in DSpace on 2012-07-09T12:44:46Z (GMT). No. of bitstreams: 1 2012_Dis_EGANascimento.pdf: 1109485 bytes, checksum: 2166a7f6639ae4cd68fe3dc82b971e3a (MD5) Previous issue date: 2012 / This current study presents the assessment on Geogebra free software for teaching geometry, as a psycho-pedagogical approach, highlighting resources that facilitate integration and use of program with professors and students’ learning topics. The software can be applied from earlier primary-school students to college ones as well as distance education programs by using internet. Geogebra is a free dynamic mathematic software and multiple-platform for all learning levels, combining geometry, algebra, tables, plotting, graphics, statistics and one-system calculation. The geometry presented was guaranteed and assigned as dynamic geometry and interactive (GDI) that is it, a computer implementation such as: rulers, drawing, square, calipers. Those tools provide motion by keeping construction properties in a static state though, in addition, dynamic motions in order to transforming the computer into a mathematics laboratory where one can perform several technological practices. The study about teaching and learning on geometry and algebra configures itself by means of a descriptive research with exploiting characteristics and almost experimental. The established steps follow as: - re-conception of teaching and learning of mathematics; re-dimension of theories and researches on evaluation. The study of case about assessment of programs on near-by qualitative methodology, in which data and the interface between several presentation of data analysis. We have concluded that the software is innovator and well measured by students and professors who have already used that technology for acquiring knowledge on mathematics and consequently for using in algebra and geometry. Currently, the program is in a long run a great insight and well known throughout. / O estudo tem por objetivo avaliar o software livre Geogebra para o ensino aprendizagem de geometria, como uma ferramenta psicopedagógica, com destaque aos recursos que facilitam a integração e uso do programa com os conteúdos curriculares, professores e alunos. O software pode ser aplicado desde as séries inicias do ensino fundamental, em estudos universitários e em cursos a distância via internet. O Geogebra é um software sobre matemática dinâmica, gratuito e de multi-plataforma para todos os níveis de ensino, que combina geometria, álgebra, tabelas, gráficos, estatística e cálculo em um único sistema. A geometria que apresenta foi validada e designada como Geometria Dinâmica e Interativa (GDI), isto é, uma implementação computacional das tecnologias usadas, tais como: régua, compasso, esquadro e transferidor os quais permitem que os modelos construídos, apesar de estáticos, sejam movidos mantendo as propriedades da construção, acrescenta ainda movimentos dinâmicos e transforma o computador em um laboratório matemático, onde se pode executar várias práticas tecnológicas. O estudo sobre o ensino-aprendizagem da geometria e álgebra se configura por meio de uma pesquisa descritiva de caráter exploratório e quase-experimental. O trabalho desvela a realidade empírica por meio da metodologia quali-quantitativa, na qual se coteja os dados e a interface entre as variáveis qualitativas e quantitativas, mediante os princípios do estudo de caso único com apresentação da análise dos dados. Conclui-se que o programa, além de inovador, é bem aceito pelos alunos e professores que se apropriam desta tecnologia para a aquisição de saberes e conhecimentos em matemática e, por via de consequência, da geometria e álgebra. Atualmente, o programa está em grande expansão e bem divulgado em todo no mundo.

Este trabalho traz uma proposta de atividades, a serem desenvolvidas em sala de aula, com o objetivo de introduzir o conceito de derivadas para os alunos da primeira série do Ensino Médio. Antes das atividades, estão presentes algumas breves pesquisas. O histórico da presença de tópicos do Cálculo Diferencial e Integral no Ensino Médio no Brasil, assim como a análise de alguns livros didáticos, serve para mostrar como o assunto já foi e está sendo tratado no país. Também são exibidos aspectos sobre o Ensino Médio na Alemanha e nos Estados Unidos, países onde o cálculo está presente na Escola Secundária, embora de formas bastante diferentes. Um capítulo sobre a preparação adequada para as aulas também foi incluído, uma vez que a simples inserção da derivada poderia causar problemas de tempo para o cumprimento do cronograma e não trazer os resultados esperados. São necessários algum grau de adequação dos conteúdos ministrados e de cooperação com professores de Física. As atividades visando o ensino dos conceitos iniciais de derivada são motivadas por um problema físico de movimento. O foco é dado na intuição e na visualização de gráficos, para que haja uma melhor compreensão dos conceitos envolvidos. A utilização de um software de geometria dinâmica é requerida em boa parte do tempo, como importante ferramenta de apoio pedagógico / This paper presents a proposal of activities to be developed in the classroom, with the goal of introducing the concept of derivative for students in the first grade of secondary school. Before the activities, some brief researches are presented. The historical presence of the topics of Differential and Integral Calculus in brazilian High Schools, as well as the analysis of some textbooks, serves to show how it has been and is being treated in the country. Aspects of the High School are also shown in Germany and the United States, countries where the calculus is present in High School, though in quite different ways. A chapter about the proper preparation for these classes was also included, since the simple insertion of the derivative could cause problems for meeting the schedule and could not bring the expected results. Some degree of adequacy of the contents and cooperation with Physics teachers are needed. The activities aiming at teaching the initial concepts of derivatives are motivated by a physical problem of motion. The focus is given on intuition and visualization of graphs, so there is a better understanding of the concepts involved. The use of a dynamic geometry software is required for much of the time, as an important tool for pedagogical support

Made available in DSpace on 2016-06-02T20:02:47Z (GMT). No. of bitstreams: 1 3058.pdf: 5697362 bytes, checksum: 2088aeefa0205fb3bbcbd92c178b8cf9 (MD5) Previous issue date: 2010-04-08 / This study presents the construction of a virtual learning environment on topics of plane analytic geometry and its application in classrooms of public high schools in the city of Barra Bonita, São Paulo, Brazil. The environment, implemented in Moodle platform of Distance Education is supported by dynamic geometric viewers idealized by GeoGebra. It approaches points in the Cartesian plane, the distance between two points, midpoint, barycentric, angular coefficient, alignment condition of three points, equation of a line, forms of equation of straight lines, relative positions of two straight lines and perpendicular straight lines. This virtual learning environment contains activities in two units implemented using the tools of Moodle, including WEB page, Lesson, Forum and Questionnaire, always founded in the learning theories of Piaget, Vygotsky and Ausubel. All of this seeks a construction of knowledge through the interactivity with the approached topics, in an autonomous way and respecting the self-paced learner. It also aims the understanding of the concepts in a practical and dynamic way, not limited to the dissemination of the information in a pre-defined static way. In the dynamic of such application, the virtual theoretical activities are followed by practical issues incorporating the content studied, with frequent assessments in the form of discussion forums, quizzes and tests with questions randomly selected from a database of the virtual environment. Despite the difficulties, the first unit can be successfully completed verifying a significant learning of the use of dynamic geometry simulators, with the students showing greater interest and focus on the approached issues. / Este trabalho apresenta a construção de um ambiente virtual de aprendizagem sobre tópicos de geometria analítica plana e sua aplicação em turmas do ensino médio público da cidade de Barra Bonita, interior de São Paulo. O ambiente, implementado na plataforma Moodle de Educação a Distância, é apoiado em visualizadores geométricos dinâmicos idealizados no GeoGebra. Aborda pontos no plano cartesiano, distância entre dois pontos, ponto médio, baricentro, coeficiente angular, condição de alinhamento de três pontos, equação de uma reta, formas de equação de retas, posições relativas de duas retas e retas perpendiculares. Contém atividades em duas unidades implementadas através das ferramentas do Moodle, entre elas Página WEB, Lição, Fórum e Questionário, sempre buscando embasamento nas teorias de aprendizagem de Piaget, Vygotsky e Ausubel. O ambiente busca a construção do conhecimento através da interatividade com os temas abordados, de forma autônoma e respeitando o ritmo do aluno aprendiz, objetivando sempre a compreensão dos conceitos de forma prática e dinâmica, sem se restringir à disseminação de informações de forma estática pré-definida. Na dinâmica de aplicação as atividades virtuais teóricas são seguidas de questões práticas envolvendo o conteúdo estudado, com avaliações constantes na forma de fóruns de discussão, simulados e provinhas com perguntas aleatoriamente escolhidas de um banco de questões auto-corrigíveis do ambiente. Apesar das dificuldades encontradas, a primeira unidade pode ser concluída com sucesso verificando uma aprendizagem significativa a partir da utilização dos simuladores de geometria dinâmica, com os alunos mostrando maior interesse e concentração nos assuntos abordados.

Les outils technologiques sont omniprésents dans la société et leur utilisation est de plus en plus grande dans les salles de classe. Du côté de l'enseignement et de l'apprentissage des mathématiques, ces outils se sont vu attribuer des rôles qui ont évolué avec les années. Les rôles de soutien, de visualisation et d'enrichissement des contenus en sont des exemples. Une utilisation des outils technologiques dans l'enseignement s'accompagne d'apports pragmatiques et épistémiques potentiels, mais comporte également des limites et des risques. Il s’avère important d’examiner le rôle accordé à l’outil technologique dans les activités qui le mobilisent. Puisque le raisonnement mathématique fait partie d'une des compétences visées à l’école (MELS, 2006) et que les futurs enseignants semblent accorder moins d'importance à la validation et la preuve comme composantes de ce raisonnement (Mary, 1999), nous émettons l'hypothèse qu'une séquence d'activités montrant la complémentarité de la preuve et des explorations tirant parti de la technologie pourrait aider les futurs enseignants à mieux saisir ces enjeux. La présente recherche s’appuie sur l'ingénierie didactique pour développer et valider une séquence d'activités intégrant le logiciel GeoGebra. Cette séquence d'activités a été conçue dans les buts suivants : initier les futurs enseignants en mathématiques au secondaire à un logiciel de géométrie dynamique et leur donner l'occasion de voir des activités mathématiques utilisant la technologie et visant le développement du raisonnement, par l’articulation de l’exploration et de la preuve. Le cadre théorique sur lequel repose cette recherche intègre des éléments de l'approche anthropologique (Chevallard, 1992, 1998, 2003) et de l'approche instrumentale (Vérillon et Rabardel, 1995; Trouche, 2000, 2003, 2007; Guin et Trouche, 2002). Certaines idées sur les constructions robustes et molles (Soury-Lavergne, 2011), la distinction figure/dessin (Laborde et Capponi, 1994) et le réseau déductif (Tanguay, 2006) ont servi de repères dans la construction de la séquence d'activités. Cette recherche s'est déroulée au cours de l'hiver 2016 dans une université québécoise, dans le cadre d’un cours de didactique de la géométrie auprès de futurs enseignants au secondaire en mathématiques. Un questionnaire pré-expérimentation a été rempli par les participants afin de voir leurs connaissances préalables sur les programmes, les outils technologiques ainsi que leurs conceptions au sujet de l'enseignement et de l'apprentissage des mathématiques. Par la suite, les étudiants ont expérimenté la séquence d'activités et ont eu à se prononcer sur les connaissances mises en jeu dans chacune des activités, l’opportunité de son utilisation avec des élèves du secondaire, et les adaptations perçues nécessaires pour sa réalisation (s'il y a lieu). Des traces écrites de leur travail ont été conservées ainsi qu'un journal de bord au fur et à mesure du déroulement de la séquence. En triangulant les diverses données recueillies, il a été constaté que la séquence, tout en contribuant à l’instrumentation des participants au regard du logiciel utilisé, a eu chez certains d’entre eux un impact sur leur vision du développement du raisonnement mathématique dans l’enseignement des mathématiques au secondaire. L’analyse des données a mis en lumière la place accordée au raisonnement par les futurs enseignants, les raisonnements mobilisés par les étudiants dans les diverses activités ainsi que des indices sur les genèses instrumentales accompagnant ces raisonnements ou les induisant. Suite à l’analyse de ces données et aux constats qui en découlent, des modifications sont proposées pour améliorer la séquence d’activités. / Technological tools are ubiquitous in society and their use is growing in the classroom. In mathematics education, these tools have been assigned roles that have evolved over the years: support, visualization, content enrichment. The use of technological tools in education comes with potential pragmatic and epistemic contributions, but also has limitations and risks. We must therefore examine at the activity level the role technology should play. Mathematical reasoning is one of the competencies aimed by school (MELS, 2006) and future teachers seem to place less emphasis on validation and proving processes as components of this reasoning (Mary, 1999). We hypothesize that a sequence of activities showing the complementarity of the proving processes with explorations leveraging technology could help future teachers better understand these issues. This research is based on didactical engineering to develop and validate a sequence of activities with GeoGebra software. The sequence of activities has been designed to: introduce pre-service secondary mathematics teachers to dynamic geometry software and give them the opportunity to see mathematical activities using technology that aim at developing mathematical reasoning and proof. The theoretical framework on which this research is based integrates elements of the anthropological theory of the didactic (Chevallard, 1992, 1998, 2003) and of the instrumental approach (Vérillon and Rabardel, 1995; Trouche, 2000, 2003, 2007; Guin and Trouche, 2002). Some ideas on robust and soft constructions (Soury-Lavergne, 2011), the distinction between figure and drawing (Laborde and Capponi, 1994) and the deductive network (Tanguay, 2006) served as benchmarks in the construction of the sequence of activities. This research took place at a Quebec university during the winter of 2016, in a geometry didactics course for pre-service secondary mathematics teachers. A preliminary questionnaire was given to the participants to capture their prior knowledge of programs, technological tools and conceptions about mathematics teaching and learning. Subsequently, the students experienced the sequence of activities and had to decide on the knowledge involved in each activity, the relevance of its use with high school students, and the perceived adaptations necessary for its realization (if considered). Written traces of their work have been kept as well as a diary as the sequence unfolds. By triangulating the various data collected, it was found that the sequence, while contributing to the instrumentation of the participants with regard to the software used, had, for some of them, an impact on their vision of the development of mathematical reasoning in mathematics education at secondary level. The analysis of the data highlighted the place given to the reasoning by the future teachers, the reasonings mobilized by the students in the various activities and also signs of the instrumental geneses inducing these reasonings and accompanying them. Following the analysis of these data and the findings that follow, modifications are proposed to improve the sequence of activities.

This paper describes some results of a research study on conjecturing and proving in a dynamic geometry environment (DGE), and it focuses on particular cognitive processes that seem to be induced by certain uses of tools available in Cabri (a particular DGE). Building on the work of Arzarello and Olivero (Arzarello et al., 1998, 2002; Olivero, 2002), we have conceived a model describing some cognitive processes that may occur during the production of conjectures and proofs in a DGE and that seem to be related to the use of specific dragging schemes, in particular to the use of the scheme we refer to as maintaining dragging. This paper contains a description of aspects of the theoretical model we have elaborated for describing such cognitive processes, with specific attention towards the role of the dragging schemes, and an example of how the model can be used to analyze students’ explorations.