Interpretações dos números racionais: uma análise no 7º ano do ensino fundamental
Autor(a) principal: | |
---|---|
Data de Publicação: | 2023 |
Tipo de documento: | Dissertação |
Idioma: | por |
Título da fonte: | Repositório Institucional Manancial UFSM |
Texto Completo: | http://repositorio.ufsm.br/handle/1/28556 |
Resumo: | This research follows a qualitative approach, aiming to investigate understandings about interpretations of rational numbers in fractional representation, when activities that emphasize figural records are proposed. From this point of view, the semiotic representation registers proposed by Raymond Duval and the theory of proportional reasoning, developed by Susan Lamon, are adopted as theoretical framework. The text follows a multipaper structure, composed of four manuscripts, with their respective specific objectives (i) analyzing mobilizations of figural registers, linked to rational numbers in fractional representation, with support of the manipulative material Frac-Soma; (ii) investigating understandings about the measure interpretation, through the compensatory principle and the recursive partition principle; iii) exploring concepts related to comparison, ordering and equivalence of rational numbers in fractional representation in approaches of continuous and discrete quantities, when associated with the part-whole interpretation; iv) analyzing understandings about sharing and comparison of quantities through the unitization process and its relations with the quotient and operator interpretations. To meet these objectives, the sources for triangulation of results considered: students' protocols, systematized during the meetings in auxiliary sheets; audio and video recordings that reveal dialogues and gestures that occurred during the process of solving the activities; photographs that reveal moments of manipulation of the Frac-Soma pieces; teacher's/researcher's with reflections on the development of the sequence. Among the results, it is evident that the Frac-Soma, as a manipulative material, contributed to unleash figural records that are associated with operational apprehension, showing mereological and positional changes. Also, the process of successive partitioning of the unit, used in the making of the Frac-Soma, enhanced the acquisition of concepts related to the main notion of rational number in fractional representation, combining evidence of the interpretations part-whole, quotient and measure. Moreover, in the activities related to the measure interpretation, we identified signs of the compensatory principle and the principle of recursive partitioning when we established relations based on the fact that the smaller the unit of measure, the greater the number of units needed, and that whole divisions should consider subunits in accordance with the measure requested. Regarding the part-whole interpretation, the understanding of equivalence relations through the unitization process stands out. On the other hand, in this same interpretation there are difficulties regarding the conservation of area in figures that are not subdivided into parts of the same size, as well as in the process of determining fractions from discrete quantities. Regarding the notions related to sharing, the concepts were understood in a satisfactory manner, involving the necessary partitioning to understand the quotient interpretation. Finally, it should be noted that generalizations were identified from the multiplicative concepts associated with the operator interpretation. |
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2023-04-04T19:01:09Z2023-04-04T19:01:09Z2023-02-01http://repositorio.ufsm.br/handle/1/28556This research follows a qualitative approach, aiming to investigate understandings about interpretations of rational numbers in fractional representation, when activities that emphasize figural records are proposed. From this point of view, the semiotic representation registers proposed by Raymond Duval and the theory of proportional reasoning, developed by Susan Lamon, are adopted as theoretical framework. The text follows a multipaper structure, composed of four manuscripts, with their respective specific objectives (i) analyzing mobilizations of figural registers, linked to rational numbers in fractional representation, with support of the manipulative material Frac-Soma; (ii) investigating understandings about the measure interpretation, through the compensatory principle and the recursive partition principle; iii) exploring concepts related to comparison, ordering and equivalence of rational numbers in fractional representation in approaches of continuous and discrete quantities, when associated with the part-whole interpretation; iv) analyzing understandings about sharing and comparison of quantities through the unitization process and its relations with the quotient and operator interpretations. To meet these objectives, the sources for triangulation of results considered: students' protocols, systematized during the meetings in auxiliary sheets; audio and video recordings that reveal dialogues and gestures that occurred during the process of solving the activities; photographs that reveal moments of manipulation of the Frac-Soma pieces; teacher's/researcher's with reflections on the development of the sequence. Among the results, it is evident that the Frac-Soma, as a manipulative material, contributed to unleash figural records that are associated with operational apprehension, showing mereological and positional changes. Also, the process of successive partitioning of the unit, used in the making of the Frac-Soma, enhanced the acquisition of concepts related to the main notion of rational number in fractional representation, combining evidence of the interpretations part-whole, quotient and measure. Moreover, in the activities related to the measure interpretation, we identified signs of the compensatory principle and the principle of recursive partitioning when we established relations based on the fact that the smaller the unit of measure, the greater the number of units needed, and that whole divisions should consider subunits in accordance with the measure requested. Regarding the part-whole interpretation, the understanding of equivalence relations through the unitization process stands out. On the other hand, in this same interpretation there are difficulties regarding the conservation of area in figures that are not subdivided into parts of the same size, as well as in the process of determining fractions from discrete quantities. Regarding the notions related to sharing, the concepts were understood in a satisfactory manner, involving the necessary partitioning to understand the quotient interpretation. Finally, it should be noted that generalizations were identified from the multiplicative concepts associated with the operator interpretation.A presente pesquisa segue uma abordagem qualitativa, com objetivo de investigar entendimentos sobre interpretações de números racionais na representação fracionária, quando são propostas atividades que enfatizam registros figurais. Por essa ótica, adota-se como referencial teórico os registros de representação semiótica, propostos por Raymond Duval e a teoria do raciocínio proporcional, elaborada por Susan Lamon. O texto segue a estrutura multipapper, composto por quatro manuscritos, com os respectivos objetivos específicos: i) analisar mobilizações de registros figurais, vinculados aos números racionais na representação fracionária, com apoio do material manipulável Frac-Soma; ii) investigar entendimentos sobre a interpretação medida, por meio do princípio compensatório e do princípio de partição recursiva; iii) explorar conceitos relacionados à comparação, ordenação e equivalência de números racionais na representação fracionária em abordagens de quantidades contínuas e discretas, quando associados à interpretação parte-todo; iv) analisar entendimentos sobre partilha e comparação de quantidades por meio do processo de unitização e suas relações com as interpretações quociente e operador. Para atender tais objetivos, as fontes para triangulação dos resultados consideraram: protocolos dos alunos, sistematizados durante os encontros em folhas auxiliares; gravações em áudio e vídeo que revelam diálogos e gestos ocorridos no processo de resolução das atividades; fotografias que revelam momentos de manipulação das peças do Frac-Soma; e diário de bordo da professora/pesquisadora com reflexões sobre o desenvolvimento da sequência. Dentre os resultados, evidencia-se que o Frac-Soma, como material manipulável, contribuiu para desencadear registros figurais que se associam à apreensão operatória, evidenciando modificações mereológicas e posicionais. Também, verifica-se que o processo de particionamento sucessivo da unidade, utilizado na confecção do Frac-Soma, potencializou a aquisição de conceitos relativos à noção principal de número racional na representação fracionária, aliando indícios das interpretações parte-todo, quociente e medida. Além disso, nas atividades relativas à intepretação medida, foram identificados indícios do princípio compensatório e do princípio da partição recursiva ao serem estabelecidas relações direcionadas ao fato de que quanto menor for a unidade de medida, maior será a quantidade de unidades necessárias, bem como que divisões do inteiro devem considerar subunidades em conformidade com a medida solicitada. Ao que refere à interpretação parte-todo, destaca-se a compreensão das relações de equivalência por meio do processo de unitização. Por outro lado, nessa mesma interpretação, verificam-se dificuldades ao que se refere à determinação de números racionais em quantidades discretas, bem como ao utilizar o processo de conservação de área. Ao que se refere às noções relativas à partilha, os conceitos foram compreendidos de maneira satisfatória, envolvendo particionamentos necessários à compreensão da interpretação quociente. Por fim, cabe destacar que foram realizadas generalizações a partir dos conceitos multiplicativos associados à interpretação operador.porUniversidade Federal de Santa MariaCentro de Ciências Naturais e ExatasPrograma de Pós-Graduação em Educação Matemática e Ensino de FísicaUFSMBrasilEducaçãoAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccessNúmeros racionaisRegistros de representação semióticaFrac-SomaEducação matemáticaRational numbersRegisters of semiotic representationMathematics educationCNPQ::CIENCIAS HUMANAS::EDUCACAOInterpretações dos números racionais: uma análise no 7º ano do ensino fundamentalInterpretations of rational numbers: an analysis of the 7th grade of elementary schoolinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesisMariani, Rita de Cássia Pistóiahttp://lattes.cnpq.br/8330933788557081Soares, Maria Arlita da SilveiraPozebon, SimoneSantarosa, Maria Cecília Pereirahttp://lattes.cnpq.br/9421436472136718Winkelmann, Claudia Aparecida7008000000066006006006006006001587a7ea-0877-496d-a629-7a9c40ad00dd2895211b-f553-457a-8af2-6c66f861670abca38a64-5821-4f86-a72c-e216bc189697a4bb041a-5261-4eba-b607-31ed174e4a1c45617d3a-e723-41d1-9bfd-44a170c7bb8areponame:Repositório Institucional Manancial UFSMinstname:Universidade Federal de Santa Maria (UFSM)instacron:UFSMCC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8805http://repositorio.ufsm.br/bitstream/1/28556/2/license_rdf4460e5956bc1d1639be9ae6146a50347MD52LICENSElicense.txtlicense.txttext/plain; charset=utf-81956http://repositorio.ufsm.br/bitstream/1/28556/3/license.txt2f0571ecee68693bd5cd3f17c1e075dfMD53ORIGINALDIS_PPGEMEF_2023_WINKELMANN_CLAUDIA.pdfDIS_PPGEMEF_2023_WINKELMANN_CLAUDIA.pdfDissertação de mestradoapplication/pdf5757546http://repositorio.ufsm.br/bitstream/1/28556/1/DIS_PPGEMEF_2023_WINKELMANN_CLAUDIA.pdf47588371f23822342a4621dca2753567MD511/285562023-04-04 16:01:09.565oai:repositorio.ufsm.br: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ório Institucionalhttp://repositorio.ufsm.br/PUBhttp://repositorio.ufsm.br/oai/requestouvidoria@ufsm.bropendoar:39132023-04-04T19:01:09Repositório Institucional Manancial UFSM - Universidade Federal de Santa Maria (UFSM)false |
dc.title.por.fl_str_mv |
Interpretações dos números racionais: uma análise no 7º ano do ensino fundamental |
dc.title.alternative.eng.fl_str_mv |
Interpretations of rational numbers: an analysis of the 7th grade of elementary school |
title |
Interpretações dos números racionais: uma análise no 7º ano do ensino fundamental |
spellingShingle |
Interpretações dos números racionais: uma análise no 7º ano do ensino fundamental Winkelmann, Claudia Aparecida Números racionais Registros de representação semiótica Frac-Soma Educação matemática Rational numbers Registers of semiotic representation Mathematics education CNPQ::CIENCIAS HUMANAS::EDUCACAO |
title_short |
Interpretações dos números racionais: uma análise no 7º ano do ensino fundamental |
title_full |
Interpretações dos números racionais: uma análise no 7º ano do ensino fundamental |
title_fullStr |
Interpretações dos números racionais: uma análise no 7º ano do ensino fundamental |
title_full_unstemmed |
Interpretações dos números racionais: uma análise no 7º ano do ensino fundamental |
title_sort |
Interpretações dos números racionais: uma análise no 7º ano do ensino fundamental |
author |
Winkelmann, Claudia Aparecida |
author_facet |
Winkelmann, Claudia Aparecida |
author_role |
author |
dc.contributor.advisor1.fl_str_mv |
Mariani, Rita de Cássia Pistóia |
dc.contributor.advisor1Lattes.fl_str_mv |
http://lattes.cnpq.br/8330933788557081 |
dc.contributor.referee1.fl_str_mv |
Soares, Maria Arlita da Silveira |
dc.contributor.referee2.fl_str_mv |
Pozebon, Simone |
dc.contributor.referee3.fl_str_mv |
Santarosa, Maria Cecília Pereira |
dc.contributor.authorLattes.fl_str_mv |
http://lattes.cnpq.br/9421436472136718 |
dc.contributor.author.fl_str_mv |
Winkelmann, Claudia Aparecida |
contributor_str_mv |
Mariani, Rita de Cássia Pistóia Soares, Maria Arlita da Silveira Pozebon, Simone Santarosa, Maria Cecília Pereira |
dc.subject.por.fl_str_mv |
Números racionais Registros de representação semiótica Frac-Soma Educação matemática |
topic |
Números racionais Registros de representação semiótica Frac-Soma Educação matemática Rational numbers Registers of semiotic representation Mathematics education CNPQ::CIENCIAS HUMANAS::EDUCACAO |
dc.subject.eng.fl_str_mv |
Rational numbers Registers of semiotic representation Mathematics education |
dc.subject.cnpq.fl_str_mv |
CNPQ::CIENCIAS HUMANAS::EDUCACAO |
description |
This research follows a qualitative approach, aiming to investigate understandings about interpretations of rational numbers in fractional representation, when activities that emphasize figural records are proposed. From this point of view, the semiotic representation registers proposed by Raymond Duval and the theory of proportional reasoning, developed by Susan Lamon, are adopted as theoretical framework. The text follows a multipaper structure, composed of four manuscripts, with their respective specific objectives (i) analyzing mobilizations of figural registers, linked to rational numbers in fractional representation, with support of the manipulative material Frac-Soma; (ii) investigating understandings about the measure interpretation, through the compensatory principle and the recursive partition principle; iii) exploring concepts related to comparison, ordering and equivalence of rational numbers in fractional representation in approaches of continuous and discrete quantities, when associated with the part-whole interpretation; iv) analyzing understandings about sharing and comparison of quantities through the unitization process and its relations with the quotient and operator interpretations. To meet these objectives, the sources for triangulation of results considered: students' protocols, systematized during the meetings in auxiliary sheets; audio and video recordings that reveal dialogues and gestures that occurred during the process of solving the activities; photographs that reveal moments of manipulation of the Frac-Soma pieces; teacher's/researcher's with reflections on the development of the sequence. Among the results, it is evident that the Frac-Soma, as a manipulative material, contributed to unleash figural records that are associated with operational apprehension, showing mereological and positional changes. Also, the process of successive partitioning of the unit, used in the making of the Frac-Soma, enhanced the acquisition of concepts related to the main notion of rational number in fractional representation, combining evidence of the interpretations part-whole, quotient and measure. Moreover, in the activities related to the measure interpretation, we identified signs of the compensatory principle and the principle of recursive partitioning when we established relations based on the fact that the smaller the unit of measure, the greater the number of units needed, and that whole divisions should consider subunits in accordance with the measure requested. Regarding the part-whole interpretation, the understanding of equivalence relations through the unitization process stands out. On the other hand, in this same interpretation there are difficulties regarding the conservation of area in figures that are not subdivided into parts of the same size, as well as in the process of determining fractions from discrete quantities. Regarding the notions related to sharing, the concepts were understood in a satisfactory manner, involving the necessary partitioning to understand the quotient interpretation. Finally, it should be noted that generalizations were identified from the multiplicative concepts associated with the operator interpretation. |
publishDate |
2023 |
dc.date.accessioned.fl_str_mv |
2023-04-04T19:01:09Z |
dc.date.available.fl_str_mv |
2023-04-04T19:01:09Z |
dc.date.issued.fl_str_mv |
2023-02-01 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/masterThesis |
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masterThesis |
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http://repositorio.ufsm.br/handle/1/28556 |
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http://repositorio.ufsm.br/handle/1/28556 |
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por |
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por |
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700800000006 |
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600 600 600 600 600 600 |
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Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/ |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
Universidade Federal de Santa Maria Centro de Ciências Naturais e Exatas |
dc.publisher.program.fl_str_mv |
Programa de Pós-Graduação em Educação Matemática e Ensino de Física |
dc.publisher.initials.fl_str_mv |
UFSM |
dc.publisher.country.fl_str_mv |
Brasil |
dc.publisher.department.fl_str_mv |
Educação |
publisher.none.fl_str_mv |
Universidade Federal de Santa Maria Centro de Ciências Naturais e Exatas |
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