<p>U doktorskoj disertaciji je prezentovano pedago&scaron;ko istraživanje koje se odnosi na teorijsko i eksperimentalno ispitivanje i proučavanje efekata primene kognitivno-vizuelnog pristupa zasnovanog na primeni grafičkih reprezentacija funkcija u obradi nastavnih sadržaja iz oblasti funkcija i njihovih primena. Na osnovu teorijskih principa kognitivno-vizuelnog pristupa, osmi&scaron;ljeni su i kreirani originalni didaktičko-metodički modeli za realizaciju sadržaja iz oblasti: matematičke analize (funkcije i izvod funkcije); problema sme&scaron;e (problemi me&scaron;anja rastvora); linearne funkcije i njene primene na re&scaron;avanje problema (problemi iz oblasti algebre, fizike i realnog okruženja). Ključni aspekti predloženog metodičkog pristupa su: vizuelizacija, vi&scaron;estruke reprezentacije, konceptualna znanja, postepeni prelazak ka vi&scaron;im nivoima apstrakcije u razvoju matematičkog m&scaron;ljenja, kao i planska i sistematska upotrebu računara sa odgovarajućom softverskom podr&scaron;kom. Okosnicu kognitivno-vizuelnog pristupa u obradi sadržaja matematičke analize i značajnu novinu u izučavanju izvoda funkcije predstavljaju zadaci sa grafičkim sadržajima i/ili zahtevima. Prikazane su metodičke mogućnosti ovih zadataka u kreiranju multireprezentativnog okruženja za učenje čija je polazna tačka formiranje i razvoj grafičkog razumevanja matematičkih koncepata. Takođe je izložena mogućnost klasifikacije zadataka sa grafičkim sadržajima i/ili zahtevima po različitim kriterijumima, na osnovu koje je data tipologija i razrađena metodika njihove primene u nastavnoj praksi.Sprovedeno je empirijsko istraživanje u okviru koga je u periodu od 2013. do 2015. godine realizovano pet pedago&scaron;kih eksperimenata sa paralelnim grupama u kojima su učestvovala 642 ispitanika (222 učenika četvrtog i 120 učenika prvog razreda gimnazije prirodno-matematičkog smera, 180 studenata osnovnih studija hemije i 120 studenata prve godine osnovnih studija fizike). Za potrebe istraživanja osmi&scaron;ljeni su i izrađeni nastavni materijali koji su omogućili učenicima/studentima eksperimentalnih grupa da izučavaju funkcije i njihove primene u multireprezentativnom i vizuelno-dinamičkom računarskom okruženju. Instrumenti primenjeni u istraživanjima bili su inicijalni i finalni testovi znanja. Evaluacija ostvarenih efekata primene kognitivno-vizuelnog pristupa, izvr&scaron;ena je poređenjem postignuća učenika/studenata eksperimentalnih (kognitivno-vizuelnih) i kontrolnih (tradicionalnih) grupa na testovima znanja. Na osnovu rezultata sprovedenih pedago&scaron;kih eksperimenata, utvrđeno je da primena kognitivno- -vizuelnog pristupa zasnovanog na grafičkom i dinamičkom prikazu funkcije u prezentovanju matematičkih sadržaja i re&scaron;avanju problema, u računarskom okruženju, ima značajan uticaj na kvalitet znanja učenika/studenata i ostvarenost optimalnih rezultata u učenju i razumevanju nastavnih sadržaja iz oblasti funkcija i njihovih primena na re&scaron;avanje problema.</p> / <p>In the doctoral dissertation a pedagogical research related to theoretical and experimental study of effects of applying a cognitive-visual approach based on the use of graphic representations of functions in interpretation of teaching contents from the field of functions and their applications is presented. On the basis of theoretical principles of cognitive-visual approach, original didacticmethodical models for the realization of contents were designed and created in the field: of mathematical analysis (functions and function derivative); problem of mixture (problems of mixing solutions); linear functions and its applications to solve problems (problems in the field of algebra, physics and the real environment). Key aspects of the proposed methodical approach are: visualization, multiple representations, conceptual knowledge, gradual transition to higher levels of abstraction in the development of mathematical thinking, as well as the planned and systematic use of computers with appropriate software support. The background of cognitive-visual approach in interpretation of mathematical analysis content and a significant novelty in the study of function derivative represent the problems with graphical contents and/or requirements. The methodological possibilities of these problems in creation of a multi-representative learning environment whose starting point is the formation and development of a graphical understanding of mathematical concepts are presented. The possibility of problems classification with graphical contents and/or requests according to different criteria is presented, on the basis of which, the typology is proposed and methodology for their application in teaching practice is elaborated. During the period 2013-2015 an empirical research was&nbsp; conducted within which five pedagogical experiments were realized with parallel groups involving a total of 642 respondents (222 students of the fourth and 120 first-year high school students major in science, 180 students of bachelor chemistry studies and 120 students of the&nbsp; first year of bachelor physics studies). For the purpose of the research, teaching materials that enabled students of the experimental groups to study the functions and&nbsp; their applications in a multi-representative and visual-dynamic computing environment were created. The instruments used in the research were initial and final tests of knowledge. The realized effects of the application of cognitive-visual approach was evaluated by comparing the achievements of high school students/ students of experimental (cognitive-visual) and control (traditional) groups on knowledge tests. Based on the results of the conducted pedagogical experiments, it has been established that the application of the cognitive-visual approach based on the graphical and dynamic presentation of the function in teaching mathematical contents and in problem solving in a computer environment has a significant impact on the quality of high school students&rsquo;/students&rsquo; knowledge and on the achievement of optimal learning outcomes and understanding of teaching contents from the field of functions and their application to solving problems.</p>

After the students in high school have learned to integrate a function, the calculation of the volume of a solid of revolution, like a rotated parabola, is taken as a good applied example. The next step is to calculate the volume of an object of reality which is interpreted as a solid of revolution of a given function f(x). The students do all these calculations in the same way and get the same result. Consequently the teachers can easily decide if a result is right or wrong. If the students have learned to work with a graphical or CAS calculator, they can calculate the volume of solids of revolution in reality by modelling a possible fitted function f(x). Every student has to decide which points of the curve that generates the solid of revolution can be taken and which function will suitably fit the curve. In Austrian high schools teachers use GeoGebra as a software which allows you to insert photographs or scanned material in the geometric window as a background picture. In this case the student and the teacher can control if the graph of the calculated function will fit the generating curve in a useful way.

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.

Made available in DSpace on 2016-06-02T20:02:50Z (GMT). No. of bitstreams: 1 4002.pdf: 7792183 bytes, checksum: f1d845672bf0e825f73ed21d4b2c4b4e (MD5) Previous issue date: 2011-08-30 / This work is the development, implementation and evaluation of a Virtual Learning Environment to support teaching trigonometry Curriculum Proposal according to the State of São Paulo, with strong use of dynamic geometry viewers. That proposal, implemented since 2008 with the name São Paulo Faz Escola, brings a series of actions to improve of education in face of national assessments of São Paulo, and the international quality indicators established by the Ministry of Education. The teaching material from these actions required a recycling of the practice, focusing on learning skills and abilities. Within this overall picture we develop teaching sequences with the use of ICT (Information and Communication Technologies) to show the student as the protagonist of the construction of knowledge. The implementation of the environment occurred in the first grade of Public Hight School, Newton Prado, during the 4th school grade of 2010 and had part of its five weekly classes held in the computer lab. We believe that these results will undoubtedly contribute to the improvement and development of basic education, especially math. / Este trabalho expõe o desenvolvimento, aplicação e avaliação de um Ambiente Virtual de Aprendizagem de apoio ao ensino de trigonometria segundo a Proposta Curricular do Estado de São Paulo, cujo uso de visualizadores de geometria dinâmica é forte e pedagogicamente promissor. Tal Proposta, implantada a partir de 2008 com o nome São Paulo Faz Escola, traz uma série de ações que visam melhorar a situação da educação paulista frente às avaliações nacionais, internacionais e aos indicadores de qualidade estabelecidos pelo Ministério da Educação. O material didático proveniente destas ações exigiu uma reciclagem das práticas exercidas, focando a aprendizagem nas competências e habilidades. Dentro deste panorama, procuramos desenvolver sequências didáticas com o uso de Tecnologias de Informação e Comunicação (TIC) que evidenciassem o aluno como protagonista do processo de construção de conhecimento. A aplicação do ambiente se deu nas primeiras séries do ensino médio da Escola Estadual Newton Prado que, durante o 4° bimestre letivo de 2 010, tiveram parte de suas cinco aulas semanais realizadas no laboratório de informática. Consideramos que os resultados apresentados poderão contribuir de forma indubitável à melhoria e aperfeiçoamento da educação básica, especialmente, de matemática.

Made available in DSpace on 2016-06-02T20:02:51Z (GMT). No. of bitstreams: 1 4894.pdf: 7272726 bytes, checksum: 44d510efece3e2138b817305459d270d (MD5) Previous issue date: 2012-10-31 / The objective of this work was the development and application of a Virtual Learning Environment (VLE), based on the Moodle platform, for the study of Mathematic topics at the elementary education level. Furthermore, another purpose is to observe the student's work within this environment and, subsequently, the analysis of the results of this proposed teaching method. The VLE concept was developed with the ISMART foundation students in mind, which are studying at the Bandeirantes School, in Sao Paulo. This foundation is a nonprofit organization that grants scholarships to academically well performing students as well as those of low income families, which are selected through exams and interviews. The selected themes within VLE are quantities and also measurements, chosen for their importance and relevance in every-day life, divided into four units: Measurements of Length, Surface, Volume and Percentage. Emphasis is given to the concept of measurement, by studying the various quantities and diverse approaches, such as: historic, the use of nontraditional unit standards and interactive applications. In relation to the percentage theme, it is important that the students familiarize themselves with different representation of rational numbers (fraction, decimal and percentage). Besides that, this theme is part of the everyday information, where we are dealing with spreadsheets, graphics and statistic data. As part of the VLE, it was the intention of this study to seek real life problems that would involve the use of calculators for more complicated calculations. Furthermore, mental percentage calculations were practiced by applying a specially developed game. Resources were utilized in form of texts, images, audio, video, flash animations and Geogebra applications, besides PowerPoint exercises that were solved in a step by step method as well as other proposed exercises. The VLE was executed with ISMART students of 2011 and 2012, whereby some activities were developed at the School and others at home. / Este trabalho teve como objetivo o desenvolvimento e aplicação de um Ambiente Virtual de Aprendizagem (AVA) na plataforma Moodle, para o estudo de tópicos de Matemática do Ensino Fundamental. Teve ainda como proposta a observação do trabalho dos alunos nesse ambiente e a análise dos resultados. O ambiente foi desenvolvido tendo como público alvo os alunos da fundação ISMART que estudam no Colégio Bandeirantes, em São Paulo. Essa fundação é uma instituição sem fins lucrativos que concede bolsas de estudo a alunos estudiosos e de baixa renda, selecionados através de provas e entrevistas. Optou-se pelos temas Grandezas e Medidas e também Porcentagem, para serem trabalhados no ambiente, escolhidos pelas necessidades dos alunos e pela relevância no cotidiano das pessoas. No trabalho com Grandezas e Medidas procurou-se dar ênfase ao conceito de medida, através do estudo de diferentes grandezas e de diversas abordagens: histórica, emprego de padrões não usuais de unidades e uso de atividades interativas. Quanto ao tema Porcentagem, é importante que os alunos se familiarizem com as diferentes representações dos números racionais (fracionária, decimal e percentual). Além disso, esse é um tema ligado ao tratamento da informação do dia a dia, ao lidarmos com tabelas, gráficos e dados estatísticos. Ao longo de todo o ambiente, procurou-se propor problemas do mundo real, envolvendo o uso da calculadora para cálculos mais elaborados. Trabalhou-se também com o cálculo mental de porcentagens através de um jogo que foi desenvolvido. Como recursos foram utilizados textos, imagens, áudio, vídeo, animações em flash e aplicativos Geogebra, além de exercícios resolvidos passo a passo no PowerPoint, e também exercícios propostos. O ambiente foi dividido em quatro unidades: Medidas de comprimento, Medidas de Superfície, Medidas de Volume e Porcentagem. O ambiente foi aplicado aos alunos do ISMART de 2011 e de 2012, sendo que algumas atividades foram desenvolvidas no Colégio e outras em casa.

Made available in DSpace on 2016-06-02T20:29:25Z (GMT). No. of bitstreams: 1 5860.pdf: 2193810 bytes, checksum: 8f6aaa00065dd7db0a219c6a2f7e41a9 (MD5) Previous issue date: 2014-03-21 / Financiadora de Estudos e Projetos / This paper is related to the application of GeoGebra software in the following topics: Ratio, Proportion, Thales Theorem, Similarities, Similarity of Triangles, Pythagorean Theorem, Trigonometric Ratios in the Triangle Rectangle, Number and circumference. Initially all the contents were thought by traditional method, board and chalk. Therefore the GeoGebra software was applied. It was developed to teachers and students (9th grade) of an elementary school involved in these activities. Positive and negative results were showed but there is a perspective of improvement. All the stages are separately related to help the understanding. The object is the analyses about the importance and relevancy of using computerized mathematical tools during the learning process. / Este trabalho trata da aplicação do software GeoGebra nos seguintes temas: Razão, Proporção, Teorema de Tales, Semelhança, Semelhança de Triângulos, Teorema de Pitágoras, Razões Trigonométricas no Triângulo Retângulo, o número e a Circunferência. Cada conteúdo foi trabalhado inicialmente do modo tradicional, giz e lousa, e posteriormente com o uso do GeoGebra. Desenvolvido de modo inédito para o professor e para os alunos de 8ª série (9° ano) envolvidos nas atividades, mostra resultados positivos e negativos com perspectiva de melhoras. Todas as etapas estão relatadas separadamente e o desenvolvimento foi feito de modo a favorecer uma compreensão adequada sobre o trabalho com o objetivo de propiciar a análise sobre a importância e relevância do uso de ferramentas matemáticas informatizadas no auxílio da aprendizagem.

Submitted by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-05-19T14:38:35Z No. of bitstreams: 2 license_rdf: 22190 bytes, checksum: 19e8a2b57ef43c09f4d7071d2153c97d (MD5) arquivototal.pdf: 4151214 bytes, checksum: 332991af15a25d2ea7775c9d71ef3216 (MD5) / Approved for entry into archive by Leonardo Americo (leonardo@sti.ufpb.br) on 2015-05-19T15:13:52Z (GMT) No. of bitstreams: 2 license_rdf: 22190 bytes, checksum: 19e8a2b57ef43c09f4d7071d2153c97d (MD5) arquivototal.pdf: 4151214 bytes, checksum: 332991af15a25d2ea7775c9d71ef3216 (MD5) / Made available in DSpace on 2015-05-19T15:13:52Z (GMT). No. of bitstreams: 2 license_rdf: 22190 bytes, checksum: 19e8a2b57ef43c09f4d7071d2153c97d (MD5) arquivototal.pdf: 4151214 bytes, checksum: 332991af15a25d2ea7775c9d71ef3216 (MD5) Previous issue date: 2013-08-12 / This work is the result of a research on the Euler line and the circumference of the nine points. The software geogebra was used to illustrate geometric constructions and present some practical activities for the study of notable points of the triangle, the Euler line and the circumference of the nine points to high school students. However, the work was based on the proof, with the use of Modern Algebra and Linear Algebra, the existence and properties of the object of this research, especially the universal property of points in the plane, critical in these demonstrations. / Este trabalho é o resultado de uma pesquisa sobre a reta de Euler e a circunferência dos nove pontos. Foi utilizado o software geogebra para ilustrar as construções geométricas e apresentar algumas atividades práticas para o estudo dos pontos notá- veis do triângulo, da reta de Euler e da circunferência dos nove pontos aos estudantes do Ensino Médio. Todavia, o trabalho se baseou nas demonstrações, com o uso da Álgebra Moderna e da Álgebra Linear, da existência e das propriedades do objeto desta pesquisa, sobretudo da propriedade universal dos pontos no plano, fundamental nestas demonstrações.

Submitted by Alysson Azevedo (roberto.alysson@yahoo.com.br) on 2018-10-25T15:46:55Z No. of bitstreams: 4 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertação de Mestrado Acadêmico - PPGECIM - Alysson R. G. 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Azevedo..pdf: 3636150 bytes, checksum: 4882d45771c5bbce4d8b48aa3dd73ded (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2018-10-29T19:45:22Z (GMT) No. of bitstreams: 3 Carta Encaminhamento Autodepósito.pdf: 236115 bytes, checksum: b7925c5a8a96b94e07f7af193785c687 (MD5) Dissertação de Mestrado Acadêmico - PPGECIM - Alysson R. G. Azevedo..pdf: 3636150 bytes, checksum: 4882d45771c5bbce4d8b48aa3dd73ded (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-10-29T19:45:22Z (GMT). No. of bitstreams: 3 Carta Encaminhamento Autodepósito.pdf: 236115 bytes, checksum: b7925c5a8a96b94e07f7af193785c687 (MD5) Dissertação de Mestrado Acadêmico - PPGECIM - Alysson R. G. Azevedo..pdf: 3636150 bytes, checksum: 4882d45771c5bbce4d8b48aa3dd73ded (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-06-26 / This work presents as the subject of the research guide: "How does the coordination of different registries contribute to the process of teaching and learning of topics of Analytical Geometry, especially Point, Line and Circumference in High School, from a sequence didactics in which constructions are proposed with the aid of software GeoGebra? ". For this, a sequence of activities with the use of GeoGebra software. However, it should be noted that this work does not summarize to a didactic sequence only. The main objective of this research is to analyze how the coordination of different records of semiotic celebration are made by the clients during the process of teaching and learning of concepts of Analytical Geometry in Teaching Middle, starting from a didactic sequence with the use of GeoGebra. The research by a differentiated and dynamic practice with the intention of improving the quality of the teaching-learning process was the main motivation the difficulties presented by the students. The interest in dealing with this subject is based on the low incomes of students found in large-scale assessments and in school assessments applied, in addition to finding the difficulties presented by these students in the process of converting records related to the topic addressed. Such activities were planned, implemented and evaluated. Based on the Theory of Semiotic Representation Registers by Raymond Duval. The didactic experience was carried out with 18 students from the 3rd year of the High School of a school of the state network of the municipality of Manaus, AM. The methodological course of this work was developed based on the stages of the didactic engineering methodology idealized by Michele Artigue. The analysis of the results that were obtained after the application of the Following the activities, it was verified that when using GeoGebra software, made possible a better understanding and interpretation of the Analytical Geometry in High School representing a methodology important in the teaching-learning process of Mathematics. / Este trabalho apresenta como questão norteadora de pesquisa: “De que maneira a coordenação de diferentes registros contribui para o processo de ensino e aprendizagem de tópicos da Geometria Analítica, especialmente Ponto, Reta e Circunferência no Ensino Médio, a partir de uma sequência didática em que são propostas construções com o auxílio do software GeoGebra?”. Para esta constatação foi desenvolvida uma sequência de atividades com o uso do software GeoGebra. No entanto, convém destacar que este trabalho não se resume a uma sequência didática somente. O principal objetivo desta pesquisa é analisar como a coordenação de diferentes registros de representação semiótica são utilizados pelos estudantes durante o processo de ensino e aprendizagem de conceitos de Geometria Analítica no Ensino Médio, a partir de uma sequência didática com o uso do GeoGebra. A investigação por uma prática diferenciada e dinâmica com o intuito de melhorar a qualidade do processo de ensino-aprendizagem, foi a principal motivação deste estudo diante das inúmeras dificuldades apresentadas pelos estudantes. O interesse em tratar deste assunto, baseia-se nos baixos rendimentos dos alunos constatados em avaliações em larga escala e nas avaliações escolares aplicadas, além da constatação das dificuldades apresentadas pelos alunos no processo de conversão de registros relacionados ao tema abordado. Tais atividades foram planejadas, implementadas e avaliadas a partir dos subsídios teóricos fundamentados pela Teoria dos Registros de Representação Semiótica de Raymond Duval. A experiência didática foi realizada com 18 alunos do 3º ano do Ensino Médio de uma escola da rede estadual do município de Manaus, AM. O percurso metodológico deste trabalho foi desenvolvido com base nas etapas da metodologia da engenharia didática idealizada por Michèle Artigue. A partir da análise dos resultados que foram obtidos após a aplicação da sequência das atividades, constatou-se que ao utilizar o software GeoGebra, possibilitou uma melhor compreensão e interpretação dos conceitos de Geometria Analítica no Ensino Médio a partir dos registros de representação semiótica envolvidos, demonstrando-se uma proposta eficiente no processo de ensino-aprendizagem de Matemática.

After the students in high school have learned to integrate a function, the calculation of the volume of a solid of revolution, like a rotated parabola, is taken as a good applied example. The next step is to calculate the volume of an object of reality which is interpreted as a solid of revolution of a given function f(x). The students do all these calculations in the same way and get the same result. Consequently the teachers can easily decide if a result is right or wrong. If the students have learned to work with a graphical or CAS calculator, they can calculate the volume of solids of revolution in reality by modelling a possible fitted function f(x). Every student has to decide which points of the curve that generates the solid of revolution can be taken and which function will suitably fit the curve. In Austrian high schools teachers use GeoGebra as a software which allows you to insert photographs or scanned material in the geometric window as a background picture. In this case the student and the teacher can control if the graph of the calculated function will fit the generating curve in a useful way.

<p>У докторској дисертацији презентовано је педагошко истраживање које се односи на увођење<br />математичког моделирања као савременог методичког приступа учењу математике, његову примену у<br />наставној пракси, као и на могућност да се кроз математичко моделирање проблема, појава и процеса који<br />се обрађују у неком другом наставном предмету оствари модернији, интердисциплинарни приступ<br />средњошколској настави. Све то у циљу афирмације иновативног, креативног и напредног математичког<br />мишљења, квалитетног, структуираног и функционалногзнања, које се у овом истраживању односи на<br />одређене наставне садржаје математичке анализе. У наставни процес интегрисан је осмишљен и израђен<br />пројекат чија је реализација омогућила успостављањемеђупредметне корелације, изучавање функција и<br />њихових примена у научним проблемима кроз процесе математичког моделирања. Нарочита пажња<br />посвећена је реализацији когнитивних активности ученика у свакој етапи процеса математичког<br />моделирања и остваривању њиховог вертикално-кумулативног поретка. Као посебна активност усмерена на<br />потребе и циљеве одабраних процеса математичког моделирања и усклађена са наставним садржајима<br />који су изучавани, уведен је и примењен нови методички приступ у обради функција датих помоћу<br />одређеног интеграла, са посебним акцентом на анализу логаритамске функције дате помоћу одређеног<br />интеграла, а затим и анализу експоненцијалне функције, као инверзне логаритамској. Један део<br />истраживања фокусиран је на увођење и реализацију назначене нове методичке концепције у обради<br />функција датих помоћу одређеног интеграла и на високошколском нивоу учења математичке анализе.<br />Испитивање ефикасности примењених нових, савременихметодичких приступа у раду са ученицима,<br />односно студентима, уз планску и систематску употребу рачунара са одговарајућом софтверском<br />подршком, обрађено је компаративном анализом резултата педагошких експеримената. На основу<br />резултата истраживања утврђен је позитиван утицај предложених методичких приступа на квалитет<br />математичког знања ученика, односно студената, и оствареност оптималних резултата у учењу и изучавању<br />наставних садржаја из области функције.</p> / <p>U doktorskoj disertaciji prezentovano je pedagoško istraživanje koje se odnosi na uvođenje<br />matematičkog modeliranja kao savremenog metodičkog pristupa učenju matematike, njegovu primenu u<br />nastavnoj praksi, kao i na mogućnost da se kroz matematičko modeliranje problema, pojava i procesa koji<br />se obrađuju u nekom drugom nastavnom predmetu ostvari moderniji, interdisciplinarni pristup<br />srednjoškolskoj nastavi. Sve to u cilju afirmacije inovativnog, kreativnog i naprednog matematičkog<br />mišljenja, kvalitetnog, struktuiranog i funkcionalnogznanja, koje se u ovom istraživanju odnosi na<br />određene nastavne sadržaje matematičke analize. U nastavni proces integrisan je osmišljen i izrađen<br />projekat čija je realizacija omogućila uspostavljanjemeđupredmetne korelacije, izučavanje funkcija i<br />njihovih primena u naučnim problemima kroz procese matematičkog modeliranja. Naročita pažnja<br />posvećena je realizaciji kognitivnih aktivnosti učenika u svakoj etapi procesa matematičkog<br />modeliranja i ostvarivanju njihovog vertikalno-kumulativnog poretka. Kao posebna aktivnost usmerena na<br />potrebe i ciljeve odabranih procesa matematičkog modeliranja i usklađena sa nastavnim sadržajima<br />koji su izučavani, uveden je i primenjen novi metodički pristup u obradi funkcija datih pomoću<br />određenog integrala, sa posebnim akcentom na analizu logaritamske funkcije date pomoću određenog<br />integrala, a zatim i analizu eksponencijalne funkcije, kao inverzne logaritamskoj. Jedan deo<br />istraživanja fokusiran je na uvođenje i realizaciju naznačene nove metodičke koncepcije u obradi<br />funkcija datih pomoću određenog integrala i na visokoškolskom nivou učenja matematičke analize.<br />Ispitivanje efikasnosti primenjenih novih, savremenihmetodičkih pristupa u radu sa učenicima,<br />odnosno studentima, uz plansku i sistematsku upotrebu računara sa odgovarajućom softverskom<br />podrškom, obrađeno je komparativnom analizom rezultata pedagoških eksperimenata. Na osnovu<br />rezultata istraživanja utvrđen je pozitivan uticaj predloženih metodičkih pristupa na kvalitet<br />matematičkog znanja učenika, odnosno studenata, i ostvarenost optimalnih rezultata u učenju i izučavanju<br />nastavnih sadržaja iz oblasti funkcije.</p> / <p>In this PhD dissertation pedagogical research related to introduction of mathematical modeling as a<br />modern methodical approach to learning mathematics is presented, its application in teaching practice,as well<br />as possibilities to achieve modern, interdisciplinary approach to high school teaching, using mathematical<br />modeling of problems, phenomenons and processes which are being looked at in another ciruculum subject. The<br />intention is to affirm innovative, creative and advanced mathematical thinking, high quality, structured and<br />functional knowledge, which in this study refers tospecific areas of mathematical analysis. A well thought<br />project was designed and constructed in such a way&nbsp; to integrate within the teaching process, and its<br />implementation has enabled the establishment of interdisciplinary correlation and the study of functions and<br />their use in scientific problems through the process of mathematical modelling. Special attention is paid to the<br />realization of cognitive activities of pupils in each stage of the process of mathematical modelling, and acheiving<br />their vertically-cumulative order. As a separate activity focused on the goals and needs of selected processes of<br />mathematical modeling, harmonized with teaching content being studied, a new methodical approach was<br />introduced and applied to treat functions given by&nbsp; a definite integral, with special emphasis on treatment of<br />logarithmic function given by a definite integral,&nbsp; and followed by treatment of the exponential function as the<br />inverse logarithmic. A part of the research is focused on the introduction and implementation of new methodical<br />concept in treatment of functions given by a definite integral and on university level of studying mathematical<br />analysis. Efficiency of these new and modern methodical approaches was tested on pupils and students,&nbsp; with<br />planned and systematic usage of computers and appropriate software, and processed by comparative analysis of<br />results obtained by pedagogical experiments. Results have shown positive impact of proposed methodical<br />approaches to the quality of pupils&rsquo; and students&rsquo;&nbsp; mathematical knowledge, and the achievement of optimal<br />results in learning and studying of teaching content on mathematical functions.</p>