This investigation concerns the ways in which young children (ages 5 to 8) compare quantities and how they work out the difference between them. The experiments involved children's understanding of mathematical problems and their ability to make use of graphical information in such problems. Each child was shown a series of illustrations, each representing two sets of quantities where the numerical difference was represented either discontinuously or continuously. The children were asked Equalize and Compare questions about each illustration and had to choose the correct answer from the set which represented the choice stimuli. Children's use of strategies was observed. In Experiment 1 (5-to-8-year-olds), only the younger children (5-to-6-year-olds) were observed to perform much more accurately on the Equalize-type question than on the Compare in both discontinuous and continuous conditions. The 7-to-8-year-olds reached a ceiling effect in performance, suggesting that by this age they can already deal with different types of arithmetic problems and with different types of graphical information. Experiment 2 (5-to-6-year-olds) repeated the first experiment presenting the graphical information on a microcomputer, but the discontinuous and continuous conditions were subdivided on the basis of the use of the comparative term "more" or "less". Children are helped significantly by the use of discontinuous material and by the use of "more" in Equalize-type questions only. These results did not support those of Experiment 1 where the Equalize and Compare difference was significant with both discontinuous and continuous material. Experiment 3 introduced part-whole manipulations in order to find out why Compare questions are more difficult to solve than Equalize questions. Five-to-6-year-olds' performance on Compare word problems was not affected by this type of manipulation. Experiment 4 explored the Equalize and Compare difference by presenting the material in a story-telling context. Again, the 5-to-6-year-olds' performance on Compare word problems was not affected by this type of manipulation. However, Equalize questions were helped by the use of the comparative term "more", as in Experiments 2 and 3, and by the presentation of discontinuous material, as in Experiment 2. Experiments 5 and 6 explored children's (5-to-8-year-olds) performance on Equalize- and Compare-type questions using spatial imagery manipulations. Experiment 5 involved manipulations of display in order to examine children's relative ease with Equalize word problems. Again, children's performance was not affected by this type of manipulation. In addition to the display manipulations, Experiment 6 introduced different level manipulations. However, in this experiment, the comparative pair was not represented in the choice stimuli. Children's performance on Compare word problems improved. There was no sign of the Equalize and Compare distinction which may be due to the fact that there was no representation of the comparative pair. The results show that the Equalize and Compare difference is due to a combination of their inherent structural and linguistic factors. Furthermore, the difficulty children have with Compare word problems is non-number-specific, but their relative ease with Equalize word problems is number-specific. Such type results indicate that children represent these two problems very differently.

In this thesis we study four groups of students in grade 8, 9 and 10 when they try to solve a classical mathematical problem: Which rectangle with given circumference has the largest area? The aim of the study was too see how the students did to solve a mathematichal problem? The survey shows that students have rather poor strategies to solve mathematical problems. The most common mistake is that students don’t put much energy to understand the problem before trying to solve it. They have no strategies. This was clearly obvious when you look at Balacheff’s theory in an article from 1988. His first, and lowest, level is called naive empiricism. Typical for that level was that the student’s efforts to solve the problem just consisted of social interaction without any direction and structure. One reason can be that the students don’t recognize mathematical laws and general concepts well enough. Another problem is that they don’t check their results. Why they don’t do this is hard to say. Earlier results indicating that one reason can be that the students don’t take tasks in school as an intellectual challenge. The just consider it like something the must do.

<p>In this thesis we study four groups of students in grade 8, 9 and 10 when they try to solve a classical mathematical problem: Which rectangle with given circumference has the largest area? The aim of the study was too see how the students did to solve a mathematichal problem?</p><p>The survey shows that students have rather poor strategies to solve mathematical problems. The most common mistake is that students don’t put much energy to understand the problem before trying to solve it. They have no strategies. This was clearly obvious when you look at Balacheff’s theory in an article from 1988. His first, and lowest, level is called naive empiricism. Typical for that level was that the student’s efforts to solve the problem just consisted of social interaction without any direction and structure. One reason can be that the students don’t recognize mathematical laws and general concepts well enough. Another problem is that they don’t check their results. Why they don’t do this is hard to say. Earlier results indicating that one reason can be that the students don’t take tasks in school as an intellectual challenge. The just consider it like something the must do.</p>

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This diploma work is focused on the introduction of environmental topics in teaching mathematics at the secondary schools including grammar schools. The thesis are a response to the lack of those topics in teaching at the secondary schools, which are not specified for physical sciences and timeliness of environmental problems in our community as a whole. The diploma work content are solved mathematical problems with methodological notes for math teachers. Logic interface of environmental problems with students knowledge of biology and geography for the relevant grade is the major effort of the thesis. Annexes contain worksheets for students. The aim of this diploma work is to inspire math teachers at the secondary schools to engage these problems or the environmental problems of their own in teaching.

This thesis provides some new insight into children’s strategies and behaviours relating to problem solving. Problem solving is one of the main aims in the renewed mathematics National Curriculum 2014 and has appeared in the Using and Applying strands of previous National Curriculums. A review of the literature provided some analysis of the types of published problem solving activities and attempted to construct a definition of problem solving activities. The literature review also demonstrated this study’s relevance. It is embedded in the fact that at the time of this study there was very little current research on problem solving and in particular practitioner research. This research was conducted through practitioner research in a focus institution. The motivation for this research was, centred round the curiosity as to whether the children (Year Six, aged 10 -11 years old) in the focus institution could apply their mathematics to problem solving activities. There was some concern that these children were learning mathematics in such a way as to pass examinations and were not appreciating the subject. A case study approach was adopted using in-depth observations in one focus institution. The observations of a sample of Year Six children engaged in mathematical problem solving activities generated rich data in the form of audio, video recordings, field notes and work samples. The data was analysed using the method of thematic analysis utilising Nvivo 10 to code the data. These codes were further condensed to final overarching themes. Further discussion of the data shows both mathematical and non-mathematical overarching themes. These themes are discussed in more depth within this study. It is hoped that this study provides some new insights into children’s strategies and behaviours relating to problem solving in mathematics.

Orientador: Nelson Antonio Pirola / Banca: Dione Lucchesi de Carvalho / Banca: Rita Melissa Lepre / Resumo: O objetivo do presente estudo foi investigar as contribuições que as classes de Correção de Fluxo proporcionaram aos alunos em termos de solução de problemas matemáticos, sob a visão de alunos, professores e ATP. Para tanto foram investigados: alunos egressos das classes de Correção de Fluxo, alunos que freqüentaram apenas classes regulares, professoras que lecionaram no projeto em questão e que hoje trabalham com os alunos egressos e o assistente técnico pedagógico que realizou as capacitações durante o desenvolvimento do projeto. Trata-se de uma amostra por conveniência. Foi possível constatar através de uma análise fenomenológica que o Projeto Correção de Fluxo não conseguiu atingir seus objetivos, ou seja, não promoveu a inclusão dos alunos que dele fizeram parte, pois os mesmos não resolvem problemas básicos envolvendo os números inteiros. Foi possível verificar que a exclusão destes alunos apenas foi adiada na instituição escolar investigada. Tais resultados apontam para a necessidade de melhorias em termos de políticas públicas educacionais, assim como de suas medidas operacionais. / Abstract: The objetive of the present study was investigate the contributions that the classes of Correction of Flow provided to the students in terns of solution of mathematical problems, under the student's vision, teacher and TPA. For so much whe investigated: regressor students of the classes of "Flow Correction", students that just frequented regular classes, teachers that taught in the project in subject and that today work with the egressor students and the pedagogia technical assistant that realized the trainings during the development of the project. It is a sample for convenience. It was possibible to verify through a fenomenologic analysis that the project "Flow correction" didn't get to reach its objectives, that is to say, it didn't promote the student's inclusion that took past of its, because the same ones don't solve basic problems, involving the whole numbers. It was possible to verify that these student's exclusion was just postponed in the investigated school institution. Such results point for the need of improvements in tenns of educational public politics, as wele as for their operational measures. / Mestre

Made available in DSpace on 2014-06-11T19:24:49Z (GMT). No. of bitstreams: 0 Previous issue date: 2007-09-10Bitstream added on 2014-06-13T19:11:34Z : No. of bitstreams: 1 buranello_lva_me_bauru.pdf: 1083474 bytes, checksum: 25ccdb14b48213f528ada08bfe779485 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Secretaria da Educação de SP / O objetivo do presente estudo foi investigar as contribuições que as classes de Correção de Fluxo proporcionaram aos alunos em termos de solução de problemas matemáticos, sob a visão de alunos, professores e ATP. Para tanto foram investigados: alunos egressos das classes de Correção de Fluxo, alunos que freqüentaram apenas classes regulares, professoras que lecionaram no projeto em questão e que hoje trabalham com os alunos egressos e o assistente técnico pedagógico que realizou as capacitações durante o desenvolvimento do projeto. Trata-se de uma amostra por conveniência. Foi possível constatar através de uma análise fenomenológica que o Projeto Correção de Fluxo não conseguiu atingir seus objetivos, ou seja, não promoveu a inclusão dos alunos que dele fizeram parte, pois os mesmos não resolvem problemas básicos envolvendo os números inteiros. Foi possível verificar que a exclusão destes alunos apenas foi adiada na instituição escolar investigada. Tais resultados apontam para a necessidade de melhorias em termos de políticas públicas educacionais, assim como de suas medidas operacionais. / The objetive of the present study was investigate the contributions that the classes of Correction of Flow provided to the students in terns of solution of mathematical problems, under the student`s vision, teacher and TPA. For so much whe investigated: regressor students of the classes of “Flow Correction”, students that just frequented regular classes, teachers that taught in the project in subject and that today work with the egressor students and the pedagogia technical assistant that realized the trainings during the development of the project. It is a sample for convenience. It was possibible to verify through a fenomenologic analysis that the project “Flow correction” didn’t get to reach its objectives, that is to say, it didn’t promote the student’s inclusion that took past of its, because the same ones don’t solve basic problems, involving the whole numbers. It was possible to verify that these student’s exclusion was just postponed in the investigated school institution. Such results point for the need of improvements in tenns of educational public politics, as wele as for their operational measures.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2016-01-29T10:28:13Z No. of bitstreams: 2 Dissertação - Cleuber Divino de Moraes - 2015.pdf: 1392384 bytes, checksum: 3331941fe4b509dad4a62fac6c0d8ea0 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2016-01-29T10:29:41Z (GMT) No. of bitstreams: 2 Dissertação - Cleuber Divino de Moraes - 2015.pdf: 1392384 bytes, checksum: 3331941fe4b509dad4a62fac6c0d8ea0 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2016-01-29T10:29:41Z (GMT). No. of bitstreams: 2 Dissertação - Cleuber Divino de Moraes - 2015.pdf: 1392384 bytes, checksum: 3331941fe4b509dad4a62fac6c0d8ea0 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-10-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work aims to evaluate how much the methodology of problem solving altogether with the use of classical music as ambient sound, can contribute to the learning of Mathematics. This work makes a contextualization about the methodology of solving problems, it describes some factors that can contribute to hamper Math teaching, leading to a reflection towards understanding the scene where the main object of this research is, the students. This research was developed at Escola Estadual Segismundo Pereira, in Uberlândia MG, with students of the 8th and 9th grades of the elementary school. Aiming to develop this evaluation we used interesting problems, selected according to the students’ knowledge level and that could make possible the involvement of the contents they study during the year also aiming to give the students more opportunity to review the contents which were not consolidated in previous years. The implemented activities followed the stages of the methodology, always directing the student to the solution of each problem and leading him to develop the learning by himself with the teacher’s orientations. With the application of this methodology and the use of music it was possible, with the research, to evaluate from the activities they developed and socialized through the presentation of the resolutions on the blackboard, the observations made during the classes, through their performance on the regular Math tests besides the data obtained from questionnairies that were applied at the beginning and at the final phase of the research, that the used procedure contributed to ease the side conversations, provided more freedom to expose their ideas, awaked on the teacher a reflection about the classroom teaching practice so that it could lead him to seek different methodologies to achieve satisfatory results in his classes. In the researcher’s point of view, one of the most important facts was to enable the students, more freedom to talk with the teacher about the approached content, it allowed them to learn Mathematics from their own questions, providing the students greater interaction with the teacher. / Este trabalho busca avaliar o quanto a metodologia de resolução de problemas juntamente com o uso da música clássica em som ambiente, pode contribuir para o aprendizado de matemática. O trabalho faz uma contextualização sobre a metodologia resolução de problemas, descreve sobre alguns fatores que podem contribuir para dificultar o ensino de matemática levando a uma reflexão no sentido de compreender o cenário onde se encontra o objeto principal dessa pesquisa, os alunos. A pesquisa foi desenvolvida na Escola Estadual Segismundo Pereira, em Uberlândia MG, com alunos de 8º e 9º ano do ensino fundamental. Com intuito de desenvolver esta pesquisa utilizamos problemas interessantes, selecionados de acordo com o nível de conhecimento dos alunos e que possibilitassem o envolvimento de conteúdos que estudam no decorrer do ano e que também possibilitasse oportunidade de rever conteúdos que não foram consolidados em séries anteriores. As atividades aplicadas seguiram as etapas da metodologia, sempre direcionando o aluno para a solução de cada problema e fazendo com que ele próprio desenvolvesse seu aprendizado com a orientação do professor. Diante da aplicação desta metodologia e o uso da música foi possível, com a pesquisa, avaliar através das atividades que desenvolveram e socializaram por meio de apresentações das soluções no quadro, das observações feitas durante as aulas, pelo desempenho nas avaliações regulares da disciplina de matemática, além dos dados obtidos em questionários que foram aplicados no início e na fase final da pesquisa, que o procedimento utilizado contribuiu para amenizar as conversas paralelas, proporcionar aos discentes maior liberdade para exporem suas ideias, despertar no docente uma reflexão sobre a prática de ensino utilizada em sala de aula, de modo que o levasse a buscar metodologias diferentes para alcançar resultados satisfatórios em suas aulas. No ponto de vista do pesquisador, um dos fatos mais importante foi possibilitar aos discentes maior liberdade para dialogar com o professor sobre os conteúdos abordados, fazendo com que eles aprendessem matemática a partir de seus questionamentos, proporcionando aos discentes uma maior interação com o professor.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-04-13T19:29:25Z No. of bitstreams: 1 carloshenriquedasilvacavaca.pdf: 13890089 bytes, checksum: 21d2ffa22eccbd951ccbff944bdab00e (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-04-24T03:23:29Z (GMT) No. of bitstreams: 1 carloshenriquedasilvacavaca.pdf: 13890089 bytes, checksum: 21d2ffa22eccbd951ccbff944bdab00e (MD5) / Made available in DSpace on 2016-04-24T03:23:29Z (GMT). No. of bitstreams: 1 carloshenriquedasilvacavaca.pdf: 13890089 bytes, checksum: 21d2ffa22eccbd951ccbff944bdab00e (MD5) Previous issue date: 2015-12-17 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Este trabalho é o relato de uma pesquisa realizada com um grupo de alunos do segundo ano do ensino médio da EPCAR (Escola Preparatória de Cadetes do Ar) sobre o ensino de Matemática Financeira com o uso de problemas do cotidiano como estratégia de estímulo para tal aprendizado. Durante oito encontros com um grupo de dez alunos, ocorreu a seguinte ordem na abordagem da Matemática Financeira: apresentação dos conceitos, resolução de problemas simples e comuns encontrados nos livros didáticos e, finalmente, resolução de problemas levados à sala de aula, tirados, dentre outras fontes, de folheto de loja, constatando os valores informados em produtos vendidos à prestação, e também analisando criticamente esses valores. São estes problemas a principal razão desta pesquisa. Consideramos que os livros didáticos não exploram de modo consistente toda riqueza e proximidade da realidade que a Matemática Financeira oferece. Entendemos que o uso de folhetos de lojas que anunciam seus produtos, análise de boletos, consulta sobre empréstimos e aplicações financeiras, análise de financiamentos, são exemplos que podem e devem ser usados no ensino da Matemática Financeira por trazerem a realidade que qualquer cidadão vai se deparar em algum momento na trajetória da sua vida. Acreditamos que ao levar para sala de aula este tipo de situação há significante estímulo para o aprendizado da Matemática Financeira. Além disso, descobrimos ao longo da pesquisa a necessidade de se estabelecer um currículo comum na abordagem deste conteúdo para alunos do ensino médio, pois observamos que não há um consenso nos livros didáticos sobre até onde deve ser levado esse assunto. / This study is an account of a research carried out with a group of 16-year-old high school students in EPCAR (Preparatory School of Air Cadets) on Financial Mathematics teaching by means of using everyday mathematical problems as a stimulus strategy for such learning. Throughout eight meetings with a group of ten students, the following scenario took place regarding Financial Mathematics approach: presenting concepts; solving simple and common mathematical problems found in textbooks; and finally problems from other sources such as store flyers brought to classroom in which it was possible not only to notice the installment payment prices of goods as well as critically analyze them. These problems are the main reason of this research. We believe that textbooks do not approach consistently all the richness and reality Financial Mathematics is able to provide. We understand that the use of store flyers that advertise their products, the analysis of bills together with the examination of loans and investments, and the financing analysis, are examples which can and shall be used in Financial Mathematics teaching for they are capable of bringing the reality any citizen will face some time in his/ her life. We believe that by presenting the classroom this kind of activity there is a significant stimulus for learning Financial Mathematics. In addition, we have discovered during the research the need to set a common high school curriculum to focus this subject for we have observed that there is no consensus in textbooks about how far this content should be dealt with.