COMPREENSÃO DOS CONCEITOS DE DERIVADA CLÁSSICA E DERIVADA FRACA: ANÁLISE SEGUNDO O MODELO COGNITIVO APOS
Autor(a) principal: | |
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Data de Publicação: | 2017 |
Tipo de documento: | Tese |
Idioma: | por |
Título da fonte: | Repositório Institucional Universidade Franciscana |
Texto Completo: | http://www.tede.universidadefranciscana.edu.br:8080/handle/UFN-BDTD/601 |
Resumo: | The present study is on the field of Mathematics Education in higher education and is focused on the teaching and learning of Calculus concepts, specifically related to the concepts of classical derivative and weak derivative. The work, developed in the context of a qualitative research, aims to investigate how some students of the Master degree course in Teaching Mathematics of a community institution in Rio Grande do Sul comprehend the concepts of classical derivative and weak derivative. The APOS theory serves as a theoretical and methodological reference for the elaboration of the genetic decomposition in which the possible mental constructions used by the students were described in what the understanding of the concepts of classical derivative and weak derivative concern. We proposed some activities on the historical constructions of the classical concept, the derivative in the present times, and the passage from the classical derivative to the weak derivative. The teaching situations were developed in the classroom in the second semester of 2016, within the subject of Fundamentals of Differential and Integral Calculus. The basis for these activities was the ACE teaching cycle. The results obtained by analyzing the students' records in the proposed activities and the observations recorded in the field diary indicate that the students were able to coordinate actions and processes in order to obtain the derivative and verify if a function is differentiable. They could also coordinate the interpretations of the derivative, such as slope of the tangent line, instantaneous velocity and rate of variation, besides using mechanisms of generalization and reversibility in the analysis of the graphs of functions and their derivatives and encapsulating the processes necessary for a satisfactory understanding of the concept of the classical derivative. By means of the integral equation, the integration formula by parts, and the fundamental theorem of Calculus, the students were able to coordinate the function and intervals, the functions with compact support by means of internalizing actions and processes for the encapsulation of the mathematical object and weak derivative. Although there are some errors in these processes, there is evidence that the concepts of classical derivative and weak derivative have been understood by the students. These evidences developed mental mechanisms of reflective abstraction that allowed the construction of the mental structures of action, process, object and scheme present in the genetic decomposition that allowed them to understand the concepts. Moreover, the treatment with the historical context of the derivative and the collaborative work of the students were significant factors to obtain the results of the research. |
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Bisognin, VanildeIsaia, Silvia Maria de AguiarCiscar, Salvador LlinaresFerreira, Marcio ViolanteSant'Ana, Marilanine de FragaCury, Helena NoronhaAlves, Marcos AlexandreRachelli, Janice2018-08-20T17:53:01Z2017-10-03Rachelli, Janice. COMPREENSÃO DOS CONCEITOS DE DERIVADA CLÁSSICA E DERIVADA FRACA: ANÁLISE SEGUNDO O MODELO COGNITIVO APOS. 2017. 294f. Tese( Programa de Pós-Graduação em Ensino de Ciências e Matemática) - Centro Universitário Franciscano, Santa Maria - RS .http://www.tede.universidadefranciscana.edu.br:8080/handle/UFN-BDTD/601The present study is on the field of Mathematics Education in higher education and is focused on the teaching and learning of Calculus concepts, specifically related to the concepts of classical derivative and weak derivative. The work, developed in the context of a qualitative research, aims to investigate how some students of the Master degree course in Teaching Mathematics of a community institution in Rio Grande do Sul comprehend the concepts of classical derivative and weak derivative. The APOS theory serves as a theoretical and methodological reference for the elaboration of the genetic decomposition in which the possible mental constructions used by the students were described in what the understanding of the concepts of classical derivative and weak derivative concern. We proposed some activities on the historical constructions of the classical concept, the derivative in the present times, and the passage from the classical derivative to the weak derivative. The teaching situations were developed in the classroom in the second semester of 2016, within the subject of Fundamentals of Differential and Integral Calculus. The basis for these activities was the ACE teaching cycle. The results obtained by analyzing the students' records in the proposed activities and the observations recorded in the field diary indicate that the students were able to coordinate actions and processes in order to obtain the derivative and verify if a function is differentiable. They could also coordinate the interpretations of the derivative, such as slope of the tangent line, instantaneous velocity and rate of variation, besides using mechanisms of generalization and reversibility in the analysis of the graphs of functions and their derivatives and encapsulating the processes necessary for a satisfactory understanding of the concept of the classical derivative. By means of the integral equation, the integration formula by parts, and the fundamental theorem of Calculus, the students were able to coordinate the function and intervals, the functions with compact support by means of internalizing actions and processes for the encapsulation of the mathematical object and weak derivative. Although there are some errors in these processes, there is evidence that the concepts of classical derivative and weak derivative have been understood by the students. These evidences developed mental mechanisms of reflective abstraction that allowed the construction of the mental structures of action, process, object and scheme present in the genetic decomposition that allowed them to understand the concepts. Moreover, the treatment with the historical context of the derivative and the collaborative work of the students were significant factors to obtain the results of the research.O presente estudo se situa no campo da Educação Matemática no ensino superior e se insere na linha de investigação voltada ao ensino e aprendizagem de conceitos do Cálculo, especificamente ligados aos conceitos de derivada clássica e derivada fraca. O trabalho, desenvolvido no contexto de uma pesquisa qualitativa, teve como objetivo investigar como se dá a compreensão dos conceitos de derivada clássica e derivada fraca por estudantes de um curso de mestrado em Ensino de Matemática de uma instituição comunitária do Rio Grande do Sul. Tendo a teoria APOS como referencial teórico e metodológico, elaborou-se a decomposição genética, em que foram descritas as possíveis construções mentais utilizadas pelos estudantes para a compreensão dos conceitos de derivada clássica e derivada fraca. Foram organizadas situações de ensino compostas por atividades sobre as construções históricas do conceito clássico, a derivada nos tempos atuais e a passagem da derivada clássica para a derivada fraca. As situações de ensino foram desenvolvidas em sala de aula, no segundo semestre de 2016, na disciplina de Fundamentos de Cálculo Diferencial e Integral, tendo como base o ciclo de ensino ACE. Os resultados obtidos, por meio da análise dos registros dos alunos nas atividades propostas e das observações anotadas no diário de campo, indicam que os estudantes foram capazes de coordenar ações e processos para obter a derivada e verificar se uma função é diferenciável, coordenar as interpretações da derivada como inclinação da reta tangente, velocidade instantânea e taxa de variação, além de, utilizar mecanismos de generalização e reversibilidade na análise dos gráficos das funções e suas derivadas e de encapsular os processos necessários para a compreensão, de forma satisfatória, do conceito da derivada clássica. Por meio da equação integral, da fórmula de integração por partes e do teorema fundamental do Cálculo, os alunos coordenaram a função e os intervalos, funções com suporte compacto, interiorizando ações e processos para a encapsulação do objeto matemático, derivada fraca. Embora com alguns erros cometidos nesses processos, há evidências de que houve compreensão dos conceitos de derivada clássica e derivada fraca pelos estudantes. Estes evidenciaram desenvolver mecanismos mentais de abstração reflexionante que possibilitaram a construção das estruturas mentais de ação, processo, objeto e esquema presentes na decomposição genética que lhes permitiu compreender os conceitos. Além do mais, o trato com o contexto histórico da derivada e o trabalho colaborativo dos alunos foram fatores significativos para a obtenção dos resultados da pesquisa.Submitted by MARCIA ROVADOSCHI (marciar@unifra.br) on 2018-08-20T17:53:01Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Tese_JaniceRachelli.pdf: 4818944 bytes, checksum: 0c02a81d2b4c04364b21e1ddddc2fe58 (MD5)Made available in DSpace on 2018-08-20T17:53:01Z (GMT). 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Weak derivative.Ciências e MatemáticaCOMPREENSÃO DOS CONCEITOS DE DERIVADA CLÁSSICA E DERIVADA FRACA: ANÁLISE SEGUNDO O MODELO COGNITIVO APOSinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisreponame:Repositório Institucional Universidade Franciscanainstname:Universidade Franciscana (UFN)instacron:UFNTEXTTese_JaniceRachelli.pdf.txtTese_JaniceRachelli.pdf.txttext/plain542971http://tede.universidadefranciscana.edu.br:8080/bitstream/UFN-BDTD/601/6/Tese_JaniceRachelli.pdf.txt328878a3d8a7a4294e8ce96e763b64efMD56THUMBNAILTese_JaniceRachelli.pdf.jpgTese_JaniceRachelli.pdf.jpgimage/jpeg3404http://tede.universidadefranciscana.edu.br:8080/bitstream/UFN-BDTD/601/7/Tese_JaniceRachelli.pdf.jpg6728fc9c9aef1f6fddac8fcf14d36638MD57LICENSElicense.txtlicense.txttext/plain; charset=utf-8309http://tede.universidadefranciscana.edu.br:8080/bitstream/UFN-BDTD/601/1/license.txte4ae80c7384074d77a55dabcdffdf13aMD51CC-LICENSElicense_urllicense_urltext/plain; charset=utf-849http://tede.universidadefranciscana.edu.br:8080/bitstream/UFN-BDTD/601/2/license_url4afdbb8c545fd630ea7db775da747b2fMD52license_textlicense_texttext/html; charset=utf-80http://tede.universidadefranciscana.edu.br:8080/bitstream/UFN-BDTD/601/3/license_textd41d8cd98f00b204e9800998ecf8427eMD53license_rdflicense_rdfapplication/rdf+xml; charset=utf-80http://tede.universidadefranciscana.edu.br:8080/bitstream/UFN-BDTD/601/4/license_rdfd41d8cd98f00b204e9800998ecf8427eMD54ORIGINALTese_JaniceRachelli.pdfTese_JaniceRachelli.pdfapplication/pdf4818944http://tede.universidadefranciscana.edu.br:8080/bitstream/UFN-BDTD/601/5/Tese_JaniceRachelli.pdf0c02a81d2b4c04364b21e1ddddc2fe58MD55UFN-BDTD/6012018-08-22 01:00:54.816oai:tede.universidadefranciscana.edu.br:UFN-BDTD/601RXN0ZSB0cmFiYWxobyBzZXLDoSBsaWNlbmNpYWRvIHNvYiBhIExpY2Vuw6dhIEF0cmlidWnDp8Ojby1Ow6NvQ29tZXJjaWFsLVNlbURlcml2YcOnw7VlcyA0LjAgSW50ZXJuYWNpb25hbCBDcmVhdGl2ZSBDb21tb25zLiBQYXJhIHZpc3VhbGl6YXIgdW1hIGPDs3BpYSBkZXN0YSBsaWNlbsOnYSwgdmlzaXRlIGh0dHA6Ly9jcmVhdGl2ZWNvbW1vbnMub3JnL2xpY2Vuc2VzL2J5LW5jLW5kLzQuMC8gb3UgbWFuZGUgdW1hIGNhcnRhIHBhcmEgQ3JlYXRpdmUgQ29tbW9ucywgUE8gQm94IDE4NjYsIE1vdW50YWluIFZpZXcsIENBIDk0MDQyLCBVU0EuRepositório de Publicaçõeshttp://www.tede.universidadefranciscana.edu.br:8080/http://www.tede.universidadefranciscana.edu.br:8080/oai/requestopendoar:2018-08-22T04:00:54Repositório Institucional Universidade Franciscana - Universidade Franciscana (UFN)false |
dc.title.por.fl_str_mv |
COMPREENSÃO DOS CONCEITOS DE DERIVADA CLÁSSICA E DERIVADA FRACA: ANÁLISE SEGUNDO O MODELO COGNITIVO APOS |
title |
COMPREENSÃO DOS CONCEITOS DE DERIVADA CLÁSSICA E DERIVADA FRACA: ANÁLISE SEGUNDO O MODELO COGNITIVO APOS |
spellingShingle |
COMPREENSÃO DOS CONCEITOS DE DERIVADA CLÁSSICA E DERIVADA FRACA: ANÁLISE SEGUNDO O MODELO COGNITIVO APOS Rachelli, Janice Educação Matemática. Teoria APOS. Cálculo. Derivada clássica. Derivada fraca. Mathematics Education. APOS Theory. Calculus. Classical derivative. Weak derivative. Ciências e Matemática |
title_short |
COMPREENSÃO DOS CONCEITOS DE DERIVADA CLÁSSICA E DERIVADA FRACA: ANÁLISE SEGUNDO O MODELO COGNITIVO APOS |
title_full |
COMPREENSÃO DOS CONCEITOS DE DERIVADA CLÁSSICA E DERIVADA FRACA: ANÁLISE SEGUNDO O MODELO COGNITIVO APOS |
title_fullStr |
COMPREENSÃO DOS CONCEITOS DE DERIVADA CLÁSSICA E DERIVADA FRACA: ANÁLISE SEGUNDO O MODELO COGNITIVO APOS |
title_full_unstemmed |
COMPREENSÃO DOS CONCEITOS DE DERIVADA CLÁSSICA E DERIVADA FRACA: ANÁLISE SEGUNDO O MODELO COGNITIVO APOS |
title_sort |
COMPREENSÃO DOS CONCEITOS DE DERIVADA CLÁSSICA E DERIVADA FRACA: ANÁLISE SEGUNDO O MODELO COGNITIVO APOS |
author |
Rachelli, Janice |
author_facet |
Rachelli, Janice |
author_role |
author |
dc.contributor.advisor1.fl_str_mv |
Bisognin, Vanilde |
dc.contributor.advisor-co1.fl_str_mv |
Isaia, Silvia Maria de Aguiar |
dc.contributor.referee1.fl_str_mv |
Ciscar, Salvador Llinares |
dc.contributor.referee2.fl_str_mv |
Ferreira, Marcio Violante |
dc.contributor.referee3.fl_str_mv |
Sant'Ana, Marilanine de Fraga |
dc.contributor.referee4.fl_str_mv |
Cury, Helena Noronha |
dc.contributor.referee5.fl_str_mv |
Alves, Marcos Alexandre |
dc.contributor.author.fl_str_mv |
Rachelli, Janice |
contributor_str_mv |
Bisognin, Vanilde Isaia, Silvia Maria de Aguiar Ciscar, Salvador Llinares Ferreira, Marcio Violante Sant'Ana, Marilanine de Fraga Cury, Helena Noronha Alves, Marcos Alexandre |
dc.subject.por.fl_str_mv |
Educação Matemática. Teoria APOS. Cálculo. Derivada clássica. Derivada fraca. |
topic |
Educação Matemática. Teoria APOS. Cálculo. Derivada clássica. Derivada fraca. Mathematics Education. APOS Theory. Calculus. Classical derivative. Weak derivative. Ciências e Matemática |
dc.subject.eng.fl_str_mv |
Mathematics Education. APOS Theory. Calculus. Classical derivative. Weak derivative. |
dc.subject.cnpq.fl_str_mv |
Ciências e Matemática |
description |
The present study is on the field of Mathematics Education in higher education and is focused on the teaching and learning of Calculus concepts, specifically related to the concepts of classical derivative and weak derivative. The work, developed in the context of a qualitative research, aims to investigate how some students of the Master degree course in Teaching Mathematics of a community institution in Rio Grande do Sul comprehend the concepts of classical derivative and weak derivative. The APOS theory serves as a theoretical and methodological reference for the elaboration of the genetic decomposition in which the possible mental constructions used by the students were described in what the understanding of the concepts of classical derivative and weak derivative concern. We proposed some activities on the historical constructions of the classical concept, the derivative in the present times, and the passage from the classical derivative to the weak derivative. The teaching situations were developed in the classroom in the second semester of 2016, within the subject of Fundamentals of Differential and Integral Calculus. The basis for these activities was the ACE teaching cycle. The results obtained by analyzing the students' records in the proposed activities and the observations recorded in the field diary indicate that the students were able to coordinate actions and processes in order to obtain the derivative and verify if a function is differentiable. They could also coordinate the interpretations of the derivative, such as slope of the tangent line, instantaneous velocity and rate of variation, besides using mechanisms of generalization and reversibility in the analysis of the graphs of functions and their derivatives and encapsulating the processes necessary for a satisfactory understanding of the concept of the classical derivative. By means of the integral equation, the integration formula by parts, and the fundamental theorem of Calculus, the students were able to coordinate the function and intervals, the functions with compact support by means of internalizing actions and processes for the encapsulation of the mathematical object and weak derivative. Although there are some errors in these processes, there is evidence that the concepts of classical derivative and weak derivative have been understood by the students. These evidences developed mental mechanisms of reflective abstraction that allowed the construction of the mental structures of action, process, object and scheme present in the genetic decomposition that allowed them to understand the concepts. Moreover, the treatment with the historical context of the derivative and the collaborative work of the students were significant factors to obtain the results of the research. |
publishDate |
2017 |
dc.date.issued.fl_str_mv |
2017-10-03 |
dc.date.accessioned.fl_str_mv |
2018-08-20T17:53:01Z |
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info:eu-repo/semantics/publishedVersion |
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info:eu-repo/semantics/doctoralThesis |
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doctoralThesis |
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publishedVersion |
dc.identifier.citation.fl_str_mv |
Rachelli, Janice. COMPREENSÃO DOS CONCEITOS DE DERIVADA CLÁSSICA E DERIVADA FRACA: ANÁLISE SEGUNDO O MODELO COGNITIVO APOS. 2017. 294f. Tese( Programa de Pós-Graduação em Ensino de Ciências e Matemática) - Centro Universitário Franciscano, Santa Maria - RS . |
dc.identifier.uri.fl_str_mv |
http://www.tede.universidadefranciscana.edu.br:8080/handle/UFN-BDTD/601 |
identifier_str_mv |
Rachelli, Janice. COMPREENSÃO DOS CONCEITOS DE DERIVADA CLÁSSICA E DERIVADA FRACA: ANÁLISE SEGUNDO O MODELO COGNITIVO APOS. 2017. 294f. Tese( Programa de Pós-Graduação em Ensino de Ciências e Matemática) - Centro Universitário Franciscano, Santa Maria - RS . |
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http://www.tede.universidadefranciscana.edu.br:8080/handle/UFN-BDTD/601 |
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por |
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por |
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http://creativecommons.org/licenses/by-nc-nd/4.0/ info:eu-repo/semantics/openAccess |
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Centro Universitário Franciscano |
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UNIFRA |
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Brasil |
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Ensino de Ciências e Matemática |
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Centro Universitário Franciscano |
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