This study investigates how five secondary students think mathematically and process information in a technology-supported environment while solving mathematics problems. In the study, students were given open-ended problems to explore in an online dynamic learning environment and to solve the problems in computer environments. Given that all the work was done in the computer environments, both online and offline, students’ work was recorded by using screen capturing software. A new method, the frame analysis method, was used to describe and analyze students’ thinking processes while they were interacting with mathematical objects in the dynamic learning environment and solving mathematics problems. The frame analysis method is a microgenetic method based on information processing theory and is developed to analyze students’ work done in computer environments. Two reasons make the analysis method used in this study unique: (a) collecting data with minimized disturbance of the students and (b) analyzing students’ artefacts through researcher’s (teacher) perspective, meaning that integrates teachers within the analysis process. The frame analysis method consists of multiple steps to observe, describe, interpret, and analyze students’ mathematical thinking processes when they are solving mathematics problems. I described each step in detail to explain how the frame analysis method was used to monitor students’ mathematical thinking and to track their use of technology while solving problems. The data emerged from this study illustrates the importance of using dynamic learning environments in mathematics and the potential for transformation of mathematical representational systems from symbolic to visual. Moreover, data suggest that visual representation systems and linked multi-representational systems encourage students to interact with mathematical concepts and advance their mathematical understanding. Rather than dealing with the grammar of algebra only, students may benefit from direct interaction with the visually represented mathematical concepts. It appears that recording students’ problem-solving processes may engage teachers and mathematics educators to seek opportunities for implementing process-oriented assessment into their curriculum activities. Furthermore, students may benefit from sharing their work through peer collaboration, either online or offline, and metacognition and self-assessment. Suggestions for further studies include using audio and video recording in the frame analysis method.

Thesis (Ph. D.)--University of Washington, 2006. / Vita. Includes bibliographical references (leaves 109-124).

Em 1870, na Itália, nasce Maria Montessori. De uma criança e adolescente curiosa, surge uma mulher corajosa e com ideias à frente de seu tempo. Ingressa na Faculdade de medicina, tornando-se a primeira mulher a concluí-la em toda a Itália. Estuda filosofia, psicologia experimental e pedagogia na Faculdade de Filosofia da Universidade de Roma, voltando todos os seus esforços para a Educação. Após escrever livros e ministrar palestras sobre Educação, suas ideias percorrem o planeta e, hoje, são utilizadas em várias escolas, em diferentes países. Percebe-se nessas escolas que, em algum momento da vida do escolar, ocorre uma ruptura das ideias montessorianas, que param de ser usadas, normalmente, nos anos iniciais do ensino fundamental. O objetivo deste trabalho é apresentar a pesquisa que teve como questão norteadora: é possível fazer uma releitura da perspectiva montessoriana de modo a abordar alguns conteúdos que são trabalhados nos anos finais do ensino fundamental na disciplina de matemática? Diante de tal questão, foram elencados os seguintes objetivos: contextualizar a vida e as obras de Maria Montessori, inseridas no contexto social, político e econômico da Itália no século XIX e início do século XX; fazer uma releitura dos princípios montessorianos de modo a potencializar tais processos de ensino e aprendizagem da matemática nos anos finais do ensino fundamental, no contexto contemporâneo; selecionar, aplicar e analisar uma proposta de atividades fazendo uso dos princípios montessorianos em uma turma dos anos finais do ensino fundamental de uma escola de Porto Alegre, abordando os conteúdos de produtos notáveis e funções, utilizando tecnologias. Para contextualizar a vida e as obras de Maria Montessori, foram feitas leituras de suas obras e de obras a seu respeito, além de leituras que traçam um panorama da Itália na época em que viveu. Também foram consultadas obras que se utilizem do tripé: matemática, didática e tecnologia. Com isso, foi adquirido conhecimento para selecionar, aplicar e analisar uma proposta de ensino, seguindo os princípios montessorianos e fazendo uso de tecnologias, que poderá ser utilizada por professores de matemática nos anos finais do ensino fundamental. Com a pesquisa, verificou-se que é possível, no contexto contemporâneo de tempo e espaço escolar, manter vivos os princípios montessorianos, de modo que a ação do professor de matemática crie situações de aprendizagem que respeitem a individualidade e o ritmo de cada aluno e que promovam sua autoeducação, sem que se perca a conexão com o outro e com o mundo, na perspectiva de uma educação para a paz. / In 1870, in Italy, Maria Montessori was born. A curious child became a brave woman with ideas ahead of her time. She joined the Medical School, becoming the first woman to complete it throughout Italy. She studied philosophy, experimental psychology and pedagogy at the Faculty of Philosophy of Rome University, returning all her efforts to Education. After writing books and giving lectures on Education, her ideas roam the planet and are now used in several schools in different countries. It can be seen in these schools that, at some point in the school life, there is a rupture of Montessori’s ideas, when they stop being used, usually in the early years of elementary school. The aim of this work is to present the research that had as its guiding question: is it possible to make a rereading of Montessori’s approach in order to address some contents that are worked in the final years of elementary school in mathematics? Faced with this question, the following objectives were listed: contextualize life and work of Maria Montessori, inserted in the social, political and economic development of Italy in the nineteenth and early twentieth century; make a rereading of Montessori’s principles in order to enhance these processes of teaching and learning of mathematics in the final years of elementary school, in the contemporary context; select, implement and analyze a proposal of activities making use of Montessori’s principles in a class of final years of elementary education at a school in Porto Alegre, addressing the contents of remarkable products and functions, using technology. To contextualize the life and the work of Maria Montessori, were made readings of her works and of works about her, and readings painting a panorama of Italy at the time in which she lived. Were also consulted works that use the tripod: mathematics, teaching and technology. With these, it was acquired knowledge to select, implement and analyze an educational proposal, following the Montessori’s principles and making use of technologies that can be applied by math teachers in the final years of elementary school. Through this research, it was verified that it is possible, in the contemporary context of time and school space, to keep alive the Montessori’s principles, so that the math teacher's action creates learning situations that respect the individuality and the rhythm of each student and promotes his self-education, without losing the connection with each other and with the world, in a perspective of education for peace.

Esta pesquisa foi regida pela seguinte questão: quais são as contribuições do uso de software de matemática dinâmica para a compreensão e solução de questões de geometria e contagem da OBMEP? Em termos pedagógicos, o objetivo foi trabalhar com questões desafiadoras com apelo ao dinamismo, assim o banco de questões da OBMEP1 foi escolhido por possuir questões bem elaboradas e com enunciados claros e desafiadores. Pretendeu-se, portanto, apresentar o software GeoGebra2 como recurso para resolver questões de geometria e de contagem e, analisar a produção dos alunos, avaliando como o GeoGebra contribuiu para a construção do conhecimento matemático. A metodologia utilizada foi a pesquisa qualitativa para produzir um Experimento de Ensino. As atividades foram desenvolvidas no contra turno com alunos do 7º e 8º ano do Ensino Fundamental em uma escola municipal de Gravataí, no ano de 2017. A sequência didática produzida nesta pesquisa é o produto didático da dissertação. A coleta de registros foi feita a partir de gravações de áudio e vídeo, diário de campo e arquivos de GeoGebra que contribuíram para o desenvolvimento do trabalho. A análise dos dados coletados demonstrou que o GeoGebra é interessante para desenvolver o raciocínio em questões em que a prova do arrastar (BORBA, DA SILVA, GADANIDIS, 2015, p.23) seja necessária. Assim, o software contribuiu para a construção de conceitos e compreensão de propriedades em figuras geométricas sendo possível verificar o comportamento dessas figuras conforme a utilização dos recursos do software. O GeoGebra também demonstrou ser útil para organização de ideias em problemas de contagem. No entanto, deve-se cuidar para que não haja a domesticação do software, ou seja, para não o utilizar no lugar de outras tecnologias que já são satisfatórias. / This research was conducted by the following question: What are the contributions of the use of dynamic mathematic software for understanding and solving OBMEP geometry and counting questions? In pedagogical terms, the objective was to work with challenging questions with a call for dynamism, so the OBMEP3 questions database was chosen because it has well elaborated math problems with clear and challenging statements. It was intended, therefore, to present GeoGebra4 software as a resource to solve geometry and counting questions and to analyze the students' production, evaluating how did GeoGebra contribute to the construction of mathematical knowledge. The methodology used was the qualitative research to produce a Teaching Experiment. The activities were developed in the inverse shift with students of the 7th and 8th years of elementary school in a municipal public school in Gravataí, in the year 2017. The didactic sequence produced in this research is the didactic product of the dissertation. The collection of records was made from audio and video recordings, field notes and GeoGebra files that contributed to the development of the work. The analysis of collected data showed that GeoGebra is interesting to develop the reasoning in questions where the drag test (BORBA, DA SILVA, GADANIDIS, 2015, p.23) is necessary. Thus, software contributes to the construction of concepts and understanding of properties in geometric figures and it is possible to verify their behavior according to the use of software resources. GeoGebra has also been shown to be useful for organizing ideas into counting problems. However, care must be taken that there is no domestication of the software, in other words, software should not be used in place of other technologies that are already satisfactory.

Esta pesquisa foi regida pela seguinte questão: quais são as contribuições do uso de software de matemática dinâmica para a compreensão e solução de questões de geometria e contagem da OBMEP? Em termos pedagógicos, o objetivo foi trabalhar com questões desafiadoras com apelo ao dinamismo, assim o banco de questões da OBMEP1 foi escolhido por possuir questões bem elaboradas e com enunciados claros e desafiadores. Pretendeu-se, portanto, apresentar o software GeoGebra2 como recurso para resolver questões de geometria e de contagem e, analisar a produção dos alunos, avaliando como o GeoGebra contribuiu para a construção do conhecimento matemático. A metodologia utilizada foi a pesquisa qualitativa para produzir um Experimento de Ensino. As atividades foram desenvolvidas no contra turno com alunos do 7º e 8º ano do Ensino Fundamental em uma escola municipal de Gravataí, no ano de 2017. A sequência didática produzida nesta pesquisa é o produto didático da dissertação. A coleta de registros foi feita a partir de gravações de áudio e vídeo, diário de campo e arquivos de GeoGebra que contribuíram para o desenvolvimento do trabalho. A análise dos dados coletados demonstrou que o GeoGebra é interessante para desenvolver o raciocínio em questões em que a prova do arrastar (BORBA, DA SILVA, GADANIDIS, 2015, p.23) seja necessária. Assim, o software contribuiu para a construção de conceitos e compreensão de propriedades em figuras geométricas sendo possível verificar o comportamento dessas figuras conforme a utilização dos recursos do software. O GeoGebra também demonstrou ser útil para organização de ideias em problemas de contagem. No entanto, deve-se cuidar para que não haja a domesticação do software, ou seja, para não o utilizar no lugar de outras tecnologias que já são satisfatórias. / This research was conducted by the following question: What are the contributions of the use of dynamic mathematic software for understanding and solving OBMEP geometry and counting questions? In pedagogical terms, the objective was to work with challenging questions with a call for dynamism, so the OBMEP3 questions database was chosen because it has well elaborated math problems with clear and challenging statements. It was intended, therefore, to present GeoGebra4 software as a resource to solve geometry and counting questions and to analyze the students' production, evaluating how did GeoGebra contribute to the construction of mathematical knowledge. The methodology used was the qualitative research to produce a Teaching Experiment. The activities were developed in the inverse shift with students of the 7th and 8th years of elementary school in a municipal public school in Gravataí, in the year 2017. The didactic sequence produced in this research is the didactic product of the dissertation. The collection of records was made from audio and video recordings, field notes and GeoGebra files that contributed to the development of the work. The analysis of collected data showed that GeoGebra is interesting to develop the reasoning in questions where the drag test (BORBA, DA SILVA, GADANIDIS, 2015, p.23) is necessary. Thus, software contributes to the construction of concepts and understanding of properties in geometric figures and it is possible to verify their behavior according to the use of software resources. GeoGebra has also been shown to be useful for organizing ideas into counting problems. However, care must be taken that there is no domestication of the software, in other words, software should not be used in place of other technologies that are already satisfactory.

Em 1870, na Itália, nasce Maria Montessori. De uma criança e adolescente curiosa, surge uma mulher corajosa e com ideias à frente de seu tempo. Ingressa na Faculdade de medicina, tornando-se a primeira mulher a concluí-la em toda a Itália. Estuda filosofia, psicologia experimental e pedagogia na Faculdade de Filosofia da Universidade de Roma, voltando todos os seus esforços para a Educação. Após escrever livros e ministrar palestras sobre Educação, suas ideias percorrem o planeta e, hoje, são utilizadas em várias escolas, em diferentes países. Percebe-se nessas escolas que, em algum momento da vida do escolar, ocorre uma ruptura das ideias montessorianas, que param de ser usadas, normalmente, nos anos iniciais do ensino fundamental. O objetivo deste trabalho é apresentar a pesquisa que teve como questão norteadora: é possível fazer uma releitura da perspectiva montessoriana de modo a abordar alguns conteúdos que são trabalhados nos anos finais do ensino fundamental na disciplina de matemática? Diante de tal questão, foram elencados os seguintes objetivos: contextualizar a vida e as obras de Maria Montessori, inseridas no contexto social, político e econômico da Itália no século XIX e início do século XX; fazer uma releitura dos princípios montessorianos de modo a potencializar tais processos de ensino e aprendizagem da matemática nos anos finais do ensino fundamental, no contexto contemporâneo; selecionar, aplicar e analisar uma proposta de atividades fazendo uso dos princípios montessorianos em uma turma dos anos finais do ensino fundamental de uma escola de Porto Alegre, abordando os conteúdos de produtos notáveis e funções, utilizando tecnologias. Para contextualizar a vida e as obras de Maria Montessori, foram feitas leituras de suas obras e de obras a seu respeito, além de leituras que traçam um panorama da Itália na época em que viveu. Também foram consultadas obras que se utilizem do tripé: matemática, didática e tecnologia. Com isso, foi adquirido conhecimento para selecionar, aplicar e analisar uma proposta de ensino, seguindo os princípios montessorianos e fazendo uso de tecnologias, que poderá ser utilizada por professores de matemática nos anos finais do ensino fundamental. Com a pesquisa, verificou-se que é possível, no contexto contemporâneo de tempo e espaço escolar, manter vivos os princípios montessorianos, de modo que a ação do professor de matemática crie situações de aprendizagem que respeitem a individualidade e o ritmo de cada aluno e que promovam sua autoeducação, sem que se perca a conexão com o outro e com o mundo, na perspectiva de uma educação para a paz. / In 1870, in Italy, Maria Montessori was born. A curious child became a brave woman with ideas ahead of her time. She joined the Medical School, becoming the first woman to complete it throughout Italy. She studied philosophy, experimental psychology and pedagogy at the Faculty of Philosophy of Rome University, returning all her efforts to Education. After writing books and giving lectures on Education, her ideas roam the planet and are now used in several schools in different countries. It can be seen in these schools that, at some point in the school life, there is a rupture of Montessori’s ideas, when they stop being used, usually in the early years of elementary school. The aim of this work is to present the research that had as its guiding question: is it possible to make a rereading of Montessori’s approach in order to address some contents that are worked in the final years of elementary school in mathematics? Faced with this question, the following objectives were listed: contextualize life and work of Maria Montessori, inserted in the social, political and economic development of Italy in the nineteenth and early twentieth century; make a rereading of Montessori’s principles in order to enhance these processes of teaching and learning of mathematics in the final years of elementary school, in the contemporary context; select, implement and analyze a proposal of activities making use of Montessori’s principles in a class of final years of elementary education at a school in Porto Alegre, addressing the contents of remarkable products and functions, using technology. To contextualize the life and the work of Maria Montessori, were made readings of her works and of works about her, and readings painting a panorama of Italy at the time in which she lived. Were also consulted works that use the tripod: mathematics, teaching and technology. With these, it was acquired knowledge to select, implement and analyze an educational proposal, following the Montessori’s principles and making use of technologies that can be applied by math teachers in the final years of elementary school. Through this research, it was verified that it is possible, in the contemporary context of time and school space, to keep alive the Montessori’s principles, so that the math teacher's action creates learning situations that respect the individuality and the rhythm of each student and promotes his self-education, without losing the connection with each other and with the world, in a perspective of education for peace.

Although the Technological Pedagogical Content Knowledge (TPACK) framework has shown a lot of promise as a theoretical perspective, researchers find it difficult to use it in particular environments because the requirements of the framework change in specific contexts. The purpose of this study was to explore and produce more flexible ways of using the TPACK for inservice mathematics secondary teachers. Three such teachers at an urban public school were observed in their classrooms and interviewed about their experiences of teaching mathematics and integrating computer technology in their day-to-day activities. Each participant had over 10 years experience in teaching mathematics in secondary schools in Ontario, and expertise in using computers in mathematics curriculum. The research questions were: 1) How do secondary school mathematics teachers describe their ways of integrating technology? 2) What difficulties do teachers have when they try to integrate technology into mathematics classrooms? The findings from the first research question show that teachers displayed a high degree of integration of technology. Their activities were very clearly designed, conferring clear roles to the use of integrating computer technology in mathematics classes. Teachers had specific approaches to integrate computer technology: a) to allow students opportunities to learn and experiment with their mathematical knowledge; b) to help them pass the content to the students in the process of teaching mathematics; and c) to assess and evaluate students’ work, and give them feedback. The findings from the second research question reveal that teachers had difficulties in purchasing and maintaining the computer equipment. They had some difficulties in trying to integrate new technologies as these required time, preparation, and dedication. In addition, teachers had some difficulties in making students use computers in a significant way. The implication for teacher education is that inservice teachers should have opportunities to update their computer and pedagogical skills, a long term perspective in integrating technology in mathematics education, and professional and technical support from teaching colleagues and administrators. Finally, the integration of computer technology in mathematics requires more intensive teamwork and collaboration between teachers, technical support staff, and administrators.

No description available.

ABSTRACT EFFECTS OF GRAPHING CALCULATORS ON EIGHTH GRADE STUDENTS&amp / #8217 / ACHIEVEMENT IN GRAPHS OF LINEAR EQUATIONS AND CONCEPT OF SLOPE &Ouml / n&uuml / r, Yurdag&uuml / l M.S., Department of Elementary Science and Mathematics Education Supervisor: Assist. Prof. Dr. Ayhan K&uuml / rSat ERBAS May 2008, 76 pages The purpose of this study was to investigate the effects of graphing calculators on eight grade students&amp / #8217 / achievement in graphing linear equations and concept of slope. Pretest-posttest experimental-control group design was utilized in the study. While the students in experimental group (EG) received instruction about graphs of linear equations and concept of slope with graphing calculators, the students in control group (CG) was taught the same topics without using graphing calculators. There were 27 students (13 girls and 14 boys) in each group. Students in both EG and CG was administered an achievement test (i.e., MAT) consisting of questions related to graphing linear equations and slope concept before and after the instruction. Additionally, the teacher and six students from the EG were interviewed. The data obtained from students&amp / #8217 / post test scores of MAT were analyzed by Analysis of Variance (ANOVA). A statistically significant difference was found between the achievements of students in experimental and control groups. However, gender had no statistically significant effect on students&amp / #8217 / post test scores of MAT. Additionally, students&amp / #8217 / pre-test scores of MAT and their mathematics grades of the second semester of the seventh grade (MGS) were analysed by independent samples t-test. The results showed no statistically significant difference. On the other hand, the analysis of interview data revealed that graphing calculators affected students&amp / #8217 / attitudes towards mathematics in a positive way. Students had no considerable difficulty while using graphing calculators and they found studying with graphing calculators enjoyable. In summary, the results of the study showed that when graphing calculators used at elementary school level, they had positive effects on students&amp / #8217 / achievement and in some respects to their attitude. Consequently, integration of graphing calculators to elementary mathematics curriculum may be beneficial for students and teachers.

Although the Technological Pedagogical Content Knowledge (TPACK) framework has shown a lot of promise as a theoretical perspective, researchers find it difficult to use it in particular environments because the requirements of the framework change in specific contexts. The purpose of this study was to explore and produce more flexible ways of using the TPACK for inservice mathematics secondary teachers. Three such teachers at an urban public school were observed in their classrooms and interviewed about their experiences of teaching mathematics and integrating computer technology in their day-to-day activities. Each participant had over 10 years experience in teaching mathematics in secondary schools in Ontario, and expertise in using computers in mathematics curriculum. The research questions were: 1) How do secondary school mathematics teachers describe their ways of integrating technology? 2) What difficulties do teachers have when they try to integrate technology into mathematics classrooms? The findings from the first research question show that teachers displayed a high degree of integration of technology. Their activities were very clearly designed, conferring clear roles to the use of integrating computer technology in mathematics classes. Teachers had specific approaches to integrate computer technology: a) to allow students opportunities to learn and experiment with their mathematical knowledge; b) to help them pass the content to the students in the process of teaching mathematics; and c) to assess and evaluate students’ work, and give them feedback. The findings from the second research question reveal that teachers had difficulties in purchasing and maintaining the computer equipment. They had some difficulties in trying to integrate new technologies as these required time, preparation, and dedication. In addition, teachers had some difficulties in making students use computers in a significant way. The implication for teacher education is that inservice teachers should have opportunities to update their computer and pedagogical skills, a long term perspective in integrating technology in mathematics education, and professional and technical support from teaching colleagues and administrators. Finally, the integration of computer technology in mathematics requires more intensive teamwork and collaboration between teachers, technical support staff, and administrators.