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Construcción de los espacios vectoriales R2 y R3 desde la teoría Apoe

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dc.contributor Parraguez González, Marcela Cecilia
dc.creator Rodríguez Jara, Miguel Alejandro
Fecha Ingreso dc.date.accessioned 2021-10-13T19:20:39Z
Fecha Disponible dc.date.available 2021-10-13T19:20:39Z
Fecha en Repositorio dc.date.issued 2021-10-13
Resumen dc.description <p>Esta investigaci&oacute;n, ya concluida, se enfoc&oacute; en documentar la construcci&oacute;n de los conceptos espacio vectorial R2 y espacio vectorial R3 desde la teor&iacute;a APOE. Para ello se dise&ntilde;aron y documentaron descomposiciones gen&eacute;ticas a la luz del ciclo de investigaci&oacute;n que dicha teor&iacute;a provee. Los aspectos hist&oacute;ricos epistemol&oacute;gicos que se consideraron y los diversos antecedentes que se recopilaron procuran sustentar la pertinencia de reconstruir los conceptos en cuesti&oacute;n desde aquellas ideas matem&aacute;ticas fundamentales, a saber ecuaci&oacute;n lineal homog&eacute;nea (en su sentido amplio), ecuaci&oacute;n lineal no homog&eacute;nea y sus respectivos conjuntos soluci&oacute;n. En ese escenario los conceptos previos conjunto, operaci&oacute;n binaria as&iacute; como las geometr&iacute;as asociadas a R2 y R3, desde ideas elementales como la dilataci&oacute;n y contracci&oacute;n de un segmento dirigido, inciden fuertemente en dicha reconstrucci&oacute;n.Abordamos la investigaci&oacute;n como un estudio de caso m&uacute;ltiple, desde una perspectiva hermen&eacute;utica centrada en un objeto matem&aacute;tico, lo que puso de relieve la importancia de elaborar cuestionarios en sinton&iacute;a con un modelo cognitivo para indagar en aquellas construcciones y mecanismos mentales que son necesarios para la reconstrucci&oacute;n de los conceptos espacio vectorial R2 y R3. All&iacute; las entrevistas semiestructuradas permitieron ahondar en aquellos aspectos que los cuestionarios no consideraron o no quedaron expl&iacute;citos desde los argumentos que manifestaron los estudiantes. Los antecedentes recopilados evidenciaron la importancia de las construcciones objeto conjunto soluci&oacute;n de una ecuaci&oacute;n lineal homog&eacute;nea y conjunto soluci&oacute;n de una ecuaci&oacute;n lineal no homog&eacute;nea en la reconstrucci&oacute;n de los espacios vectoriales en cuesti&oacute;n.Para finalizar, dentro de los principales resultados destaca la importancia de comenzar la reconstrucci&oacute;n de los conceptos ya aludidos atendiendo a la construcci&oacute;n acci&oacute;n asociar n&uacute;meros a los t&eacute;rminos de una ecuaci&oacute;n lineal homog&eacute;nea y la construcci&oacute;n acci&oacute;n asignar un n&uacute;mero a una de las inc&oacute;gnitas de una ecuaci&oacute;n lineal homog&eacute;nea, as&iacute; como las construcciones proceso m&uacute;ltiplo escalar de una soluci&oacute;n, todas las soluciones, &aacute;lgebra de soluciones, cartesiano R2 y cartesiano R3</p>
Resumen dc.description <p></p><p>This research, already concluded, is focused on documenting the construction of the concepts of vectorial space R2 and vectorial space R3 based on APOS theory. For the above mentioned, genetic decompositions were designed and documented in the light of the research cycle provided by such theory. The epistemologic historical aspects that were considered and the several precedents compiled try to support the approppriateness of reconstructing the concepts in question from those fundamental mathematical ideas, that is to say: homogenous linear equation (in a broad sense), nonhomogenous linear equation and the respective solution sets. In that scenario, the previous concepts: set, binary operation as well as geometries associated to R2 and R3, starting from basic ideas like the expansion and contraction of a directed line segment, strongly affect this reconstruction.<br />We faced the research as a multiple case study, from a hermeneutics perspective focused on a mathematical object, which highlighted the importance of creating questionnaires accordingly with the cognitive models to investigate in those constructions and mental mechanisms that are necessary for the reconstruction of the vectorial space concepts R2 and R3. There, the semistructured interviews allowed to go deep is those aspects that the questionnaires did not consider or that were not explicit from the arguments that the students showed. The compiled precedents showed the relevance of the constructing mental objects, solution set of a homogenous linear equation and solution set of nonhomogenous linear equation in the reconstruction of the vectorial spaces in question.</p><p>Finally, among the main results, it is stressed the importance to begin the reconstruction of the concepts already mentioned paying attention to constructing mental action to associate numbers to the terms of an homogenous linear equation and constructing mental action to assign a number to one of the variables of a homogenous linear equation, as well as constructing mental process: scalar multiplier of a solution, all the solutions, algebra of solutions, cartesian R2 and cartesian R3</p><p></p>
Resumen dc.description last modification
Resumen dc.description Doctor en Didáctica de la Matemática
Resumen dc.description DOCTORADO EN DIDACTICA DE LA MATEMATICA
Resumen dc.description <p>Esta investigaci&oacute;n, ya concluida, se enfoc&oacute; en documentar la construcci&oacute;n de los conceptos espacio vectorial R2 y espacio vectorial R3 desde la teor&iacute;a APOE. Para ello se dise&ntilde;aron y documentaron descomposiciones gen&eacute;ticas a la luz del ciclo de investigaci&oacute;n que dicha teor&iacute;a provee. Los aspectos hist&oacute;ricos epistemol&oacute;gicos que se consideraron y los diversos antecedentes que se recopilaron procuran sustentar la pertinencia de reconstruir los conceptos en cuesti&oacute;n desde aquellas ideas matem&aacute;ticas fundamentales, a saber ecuaci&oacute;n lineal homog&eacute;nea (en su sentido amplio), ecuaci&oacute;n lineal no homog&eacute;nea y sus respectivos conjuntos soluci&oacute;n. En ese escenario los conceptos previos conjunto, operaci&oacute;n binaria as&iacute; como las geometr&iacute;as asociadas a R2 y R3, desde ideas elementales como la dilataci&oacute;n y contracci&oacute;n de un segmento dirigido, inciden fuertemente en dicha reconstrucci&oacute;n.Abordamos la investigaci&oacute;n como un estudio de caso m&uacute;ltiple, desde una perspectiva hermen&eacute;utica centrada en un objeto matem&aacute;tico, lo que puso de relieve la importancia de elaborar cuestionarios en sinton&iacute;a con un modelo cognitivo para indagar en aquellas construcciones y mecanismos mentales que son necesarios para la reconstrucci&oacute;n de los conceptos espacio vectorial R2 y R3. All&iacute; las entrevistas semiestructuradas permitieron ahondar en aquellos aspectos que los cuestionarios no consideraron o no quedaron expl&iacute;citos desde los argumentos que manifestaron los estudiantes. Los antecedentes recopilados evidenciaron la importancia de las construcciones objeto conjunto soluci&oacute;n de una ecuaci&oacute;n lineal homog&eacute;nea y conjunto soluci&oacute;n de una ecuaci&oacute;n lineal no homog&eacute;nea en la reconstrucci&oacute;n de los espacios vectoriales en cuesti&oacute;n.Para finalizar, dentro de los principales resultados destaca la importancia de comenzar la reconstrucci&oacute;n de los conceptos ya aludidos atendiendo a la construcci&oacute;n acci&oacute;n asociar n&uacute;meros a los t&eacute;rminos de una ecuaci&oacute;n lineal homog&eacute;nea y la construcci&oacute;n acci&oacute;n asignar un n&uacute;mero a una de las inc&oacute;gnitas de una ecuaci&oacute;n lineal homog&eacute;nea, as&iacute; como las construcciones proceso m&uacute;ltiplo escalar de una soluci&oacute;n, todas las soluciones, &aacute;lgebra de soluciones, cartesiano R2 y cartesiano R3</p>
Resumen dc.description <p></p><p>This research, already concluded, is focused on documenting the construction of the concepts of vectorial space R2 and vectorial space R3 based on APOS theory. For the above mentioned, genetic decompositions were designed and documented in the light of the research cycle provided by such theory. The epistemologic historical aspects that were considered and the several precedents compiled try to support the approppriateness of reconstructing the concepts in question from those fundamental mathematical ideas, that is to say: homogenous linear equation (in a broad sense), nonhomogenous linear equation and the respective solution sets. In that scenario, the previous concepts: set, binary operation as well as geometries associated to R2 and R3, starting from basic ideas like the expansion and contraction of a directed line segment, strongly affect this reconstruction.<br />We faced the research as a multiple case study, from a hermeneutics perspective focused on a mathematical object, which highlighted the importance of creating questionnaires accordingly with the cognitive models to investigate in those constructions and mental mechanisms that are necessary for the reconstruction of the vectorial space concepts R2 and R3. There, the semistructured interviews allowed to go deep is those aspects that the questionnaires did not consider or that were not explicit from the arguments that the students showed. The compiled precedents showed the relevance of the constructing mental objects, solution set of a homogenous linear equation and solution set of nonhomogenous linear equation in the reconstruction of the vectorial spaces in question.</p><p>Finally, among the main results, it is stressed the importance to begin the reconstruction of the concepts already mentioned paying attention to constructing mental action to associate numbers to the terms of an homogenous linear equation and constructing mental action to assign a number to one of the variables of a homogenous linear equation, as well as constructing mental process: scalar multiplier of a solution, all the solutions, algebra of solutions, cartesian R2 and cartesian R3</p><p></p>
Formato dc.format PDF
Lenguaje dc.language spa
dc.rights sin documento
dc.source http://opac.pucv.cl/pucv_txt/txt-8500/UCE8579_01.pdf
Materia dc.subject TRANSFORMACION LINEAL
Materia dc.subject Espacios vectorales
Materia dc.subject ENSEÑANZA DE LAS MATEMATICAS
Materia dc.subject Algebra lineal
Title dc.title Construcción de los espacios vectoriales R2 y R3 desde la teoría Apoe
Tipo dc.type texto


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