El infinito potencial y actual: descripción de caminos cognitivos para su construcción en un cont... more El infinito potencial y actual: descripción de caminos cognitivos para su construcción en un contexto de paradojas
Revista Latinoamericana De Investigacion En Matematica Educativa, 2010
Resumen es: En este trabajo se presentan los resultados de un proyecto de largo alcance en Mexico... more Resumen es: En este trabajo se presentan los resultados de un proyecto de largo alcance en Mexico cuyo proposito consiste en profundizar en la forma en que los estud...
En este trabajo presentamos algunas dificultades que tienen los estudiantes en la representacion ... more En este trabajo presentamos algunas dificultades que tienen los estudiantes en la representacion grafica de sistemas de ecuaciones lineales con dos variables. Para ello, se diseno una entrevista basandonos en el acercamiento teorico de los modos de pensamiento geometrico y analitico en algebra lineal. Reportamos acerca del desempeno de los estudiantes ante situaciones presentadas en un modo geometrico.
para explicar el desarrollo de un esquema de espacio vectorial. Caracterizamos los niveles de des... more para explicar el desarrollo de un esquema de espacio vectorial. Caracterizamos los niveles de desarrollo de dicho esquema, a partir de una descomposición genética, como resultado de nuestro análisis teórico, que señala un camino por el cual los estudiantes pueden construir el concepto espacio vectorial. Con base en dicho análisis teórico diseñamos una entrevista semiestructurada para averiguar la viabilidad de las construcciones mentales que habíamos considerado en la descomposición genética preliminar, eligiendo las preguntas de tal manera que permitirían obtener información profunda respecto al desarrollo del esquema espacio vectorial. Este instrumento fue aplicado a los estudiantes matriculados en un programa de Licenciatura en Matemáticas. El análisis de los resultados obtenidos con estos datos empíricos permite refinar el análisis teórico y presentar una caracterización mejorada de los niveles de esquema del concepto espacio vectorial. Este análisis teórico, además de ser un modelo de aprendizaje, representa una herramienta didáctica que señala estrategias de enseñanza del concepto espacio vectorial.
Xiv Conferencia Interamericana De Educacion Matematica, Mar 31, 2015
Se presentan algunas observaciones y ejemplos de las ventajas brindadas por las representaciones ... more Se presentan algunas observaciones y ejemplos de las ventajas brindadas por las representaciones dinamicas para interiorizar concepciones Accion referentes a transformaciones lineales, en el sentido de la teoria APOE (Arnon et al., 2014). El analisis presentado es parte de una investigacion de doctorado sobre la construccion mental del concepto de transformacion lineal apoyada en la coordinacion de registros de representacion (Duval, 1999) estaticos y dinamicos. Se identifican a las representaciones dinamicas de vectores y transformaciones del plano como facilitadores del mecanismo de interiorizacion para lograr concepciones de linealidad; tras la aplicacion de actividades de ensenanza apoyadas fuertemente en representaciones graficas dinamicas.
Relime Revista Latinoamericana De Investigacion En Matematica Educativa, 2004
Llevar al estudiante a un conflicto cognitivo puede ser una manera de hacerle ver que los concept... more Llevar al estudiante a un conflicto cognitivo puede ser una manera de hacerle ver que los conceptos o métodos que maneja no son los adecuados para llegar a una conclusión satisfactoria en la resolución de un problema. Aparte de las discusiones en un ambiente de aprendizaje colectivo y el uso de juegos matemáticos, las situaciones matemáticas deben portar elementos que favorezcan un conflicto y faciliten el enfrentamiento de los conocimientos anteriores con nuevos conocimientos a adquirir. Asimismo, los maestros necesitan tener experiencias directas con este tipo de actividades en su nivel antes de aplicarlas en sus clases. En este artículo reportamos los resultados de una investigación que involucró la aplicación de una actividad de criptografía a grupos de maestros donde los conceptos de conjuntos y operaciones binarias entraron en juego. PALABRAS CLAVE: conflicto cognitivo, operaciones binarias, diseño de actividades, criptografía, formación de maestros Generating cognitive conflict by means of a cryptography activity that involves binary operations ABSTRACT Causing a cognitive conflict can be one way to make students aware of the inadequacy of the concepts that they possess and the methods that they use in solving a problem. Apart from discussions in a cooperative group and the use of mathematical games, the mathematical situations themselves have to possess certain characteristics to be able to provoke a conflict and furthermore facilitate the confrontation between the previous knowledge and new knowledge to be acquired. Moreover teachers need to have direct experiences at their level with this kind of activities, before they can employ them in their classes. In this paper we report the results of a study in which a cryptography activity was applied with a group of university and high school teachers, where the concepts of sets and binary operations came into play.
Revista Latinoamericana De Investigacion En Matematica Educativa, 2010
Proyecto académico sin fines de lucro, desarrollado bajo la iniciativa de acceso abierto construc... more Proyecto académico sin fines de lucro, desarrollado bajo la iniciativa de acceso abierto construcción de una descomposición genética 89
Revista Latinoamericana De Investigacion En Matematica Educativa, Mar 1, 2007
Proyecto académico sin fines de lucro, desarrollado bajo la iniciativa de acceso abierto • PALABR... more Proyecto académico sin fines de lucro, desarrollado bajo la iniciativa de acceso abierto • PALABRAS CLAVE: Contexto de ingeniería, matemáticas en uso, herramienta metodológica, pensamiento teórico, pensamiento práctico.
This paper reports the presence or absence of a systemic thinking in students, when they solve th... more This paper reports the presence or absence of a systemic thinking in students, when they solve the linear extension problem, which consists of determining a linear transformation through the images of the vectors of a basis. The problem is formulated geometrically, making use of the tools of the Cabri-géomètre II software. The difficulties that students present when face this problem,
The focus of this chapter is a discussion of the emergence of a possible new stage or structure a... more The focus of this chapter is a discussion of the emergence of a possible new stage or structure and the use of levels in APOS Theory. The potential new stage, Totality, would lie between Process and Object. At this point, the status of Totality and the use of levels described in this chapter are no more than tentative because evidence for a separate stage and/or the need for levels arose out of just two studies: fractions (Arnon 1998) and an extended study of the infinite repeating decimal \( 0.\bar{9} \) and its relation to 1 (Weller et al. 2009, 2011; Dubinsky et al. in press). It remains for future research to determine if Totality exists as a separate stage, if levels are really needed in these contexts, and to explore what the mental mechanism(s) for constructing them might be. Research is also needed to determine the role of Totality and levels for other contexts, both those involving infinite processes and those involving finite processes. It seems clear that explicit pedagogical strategies are needed to help most students construct each of the stages in APOS Theory and that levels which describe the progressions from one stage to another may point to such strategies. Moreover, observation of levels may serve to help evaluate students’ progress in making those constructions.
El infinito potencial y actual: descripción de caminos cognitivos para su construcción en un cont... more El infinito potencial y actual: descripción de caminos cognitivos para su construcción en un contexto de paradojas
Revista Latinoamericana De Investigacion En Matematica Educativa, 2010
Resumen es: En este trabajo se presentan los resultados de un proyecto de largo alcance en Mexico... more Resumen es: En este trabajo se presentan los resultados de un proyecto de largo alcance en Mexico cuyo proposito consiste en profundizar en la forma en que los estud...
En este trabajo presentamos algunas dificultades que tienen los estudiantes en la representacion ... more En este trabajo presentamos algunas dificultades que tienen los estudiantes en la representacion grafica de sistemas de ecuaciones lineales con dos variables. Para ello, se diseno una entrevista basandonos en el acercamiento teorico de los modos de pensamiento geometrico y analitico en algebra lineal. Reportamos acerca del desempeno de los estudiantes ante situaciones presentadas en un modo geometrico.
para explicar el desarrollo de un esquema de espacio vectorial. Caracterizamos los niveles de des... more para explicar el desarrollo de un esquema de espacio vectorial. Caracterizamos los niveles de desarrollo de dicho esquema, a partir de una descomposición genética, como resultado de nuestro análisis teórico, que señala un camino por el cual los estudiantes pueden construir el concepto espacio vectorial. Con base en dicho análisis teórico diseñamos una entrevista semiestructurada para averiguar la viabilidad de las construcciones mentales que habíamos considerado en la descomposición genética preliminar, eligiendo las preguntas de tal manera que permitirían obtener información profunda respecto al desarrollo del esquema espacio vectorial. Este instrumento fue aplicado a los estudiantes matriculados en un programa de Licenciatura en Matemáticas. El análisis de los resultados obtenidos con estos datos empíricos permite refinar el análisis teórico y presentar una caracterización mejorada de los niveles de esquema del concepto espacio vectorial. Este análisis teórico, además de ser un modelo de aprendizaje, representa una herramienta didáctica que señala estrategias de enseñanza del concepto espacio vectorial.
Xiv Conferencia Interamericana De Educacion Matematica, Mar 31, 2015
Se presentan algunas observaciones y ejemplos de las ventajas brindadas por las representaciones ... more Se presentan algunas observaciones y ejemplos de las ventajas brindadas por las representaciones dinamicas para interiorizar concepciones Accion referentes a transformaciones lineales, en el sentido de la teoria APOE (Arnon et al., 2014). El analisis presentado es parte de una investigacion de doctorado sobre la construccion mental del concepto de transformacion lineal apoyada en la coordinacion de registros de representacion (Duval, 1999) estaticos y dinamicos. Se identifican a las representaciones dinamicas de vectores y transformaciones del plano como facilitadores del mecanismo de interiorizacion para lograr concepciones de linealidad; tras la aplicacion de actividades de ensenanza apoyadas fuertemente en representaciones graficas dinamicas.
Relime Revista Latinoamericana De Investigacion En Matematica Educativa, 2004
Llevar al estudiante a un conflicto cognitivo puede ser una manera de hacerle ver que los concept... more Llevar al estudiante a un conflicto cognitivo puede ser una manera de hacerle ver que los conceptos o métodos que maneja no son los adecuados para llegar a una conclusión satisfactoria en la resolución de un problema. Aparte de las discusiones en un ambiente de aprendizaje colectivo y el uso de juegos matemáticos, las situaciones matemáticas deben portar elementos que favorezcan un conflicto y faciliten el enfrentamiento de los conocimientos anteriores con nuevos conocimientos a adquirir. Asimismo, los maestros necesitan tener experiencias directas con este tipo de actividades en su nivel antes de aplicarlas en sus clases. En este artículo reportamos los resultados de una investigación que involucró la aplicación de una actividad de criptografía a grupos de maestros donde los conceptos de conjuntos y operaciones binarias entraron en juego. PALABRAS CLAVE: conflicto cognitivo, operaciones binarias, diseño de actividades, criptografía, formación de maestros Generating cognitive conflict by means of a cryptography activity that involves binary operations ABSTRACT Causing a cognitive conflict can be one way to make students aware of the inadequacy of the concepts that they possess and the methods that they use in solving a problem. Apart from discussions in a cooperative group and the use of mathematical games, the mathematical situations themselves have to possess certain characteristics to be able to provoke a conflict and furthermore facilitate the confrontation between the previous knowledge and new knowledge to be acquired. Moreover teachers need to have direct experiences at their level with this kind of activities, before they can employ them in their classes. In this paper we report the results of a study in which a cryptography activity was applied with a group of university and high school teachers, where the concepts of sets and binary operations came into play.
Revista Latinoamericana De Investigacion En Matematica Educativa, 2010
Proyecto académico sin fines de lucro, desarrollado bajo la iniciativa de acceso abierto construc... more Proyecto académico sin fines de lucro, desarrollado bajo la iniciativa de acceso abierto construcción de una descomposición genética 89
Revista Latinoamericana De Investigacion En Matematica Educativa, Mar 1, 2007
Proyecto académico sin fines de lucro, desarrollado bajo la iniciativa de acceso abierto • PALABR... more Proyecto académico sin fines de lucro, desarrollado bajo la iniciativa de acceso abierto • PALABRAS CLAVE: Contexto de ingeniería, matemáticas en uso, herramienta metodológica, pensamiento teórico, pensamiento práctico.
This paper reports the presence or absence of a systemic thinking in students, when they solve th... more This paper reports the presence or absence of a systemic thinking in students, when they solve the linear extension problem, which consists of determining a linear transformation through the images of the vectors of a basis. The problem is formulated geometrically, making use of the tools of the Cabri-géomètre II software. The difficulties that students present when face this problem,
The focus of this chapter is a discussion of the emergence of a possible new stage or structure a... more The focus of this chapter is a discussion of the emergence of a possible new stage or structure and the use of levels in APOS Theory. The potential new stage, Totality, would lie between Process and Object. At this point, the status of Totality and the use of levels described in this chapter are no more than tentative because evidence for a separate stage and/or the need for levels arose out of just two studies: fractions (Arnon 1998) and an extended study of the infinite repeating decimal \( 0.\bar{9} \) and its relation to 1 (Weller et al. 2009, 2011; Dubinsky et al. in press). It remains for future research to determine if Totality exists as a separate stage, if levels are really needed in these contexts, and to explore what the mental mechanism(s) for constructing them might be. Research is also needed to determine the role of Totality and levels for other contexts, both those involving infinite processes and those involving finite processes. It seems clear that explicit pedagogical strategies are needed to help most students construct each of the stages in APOS Theory and that levels which describe the progressions from one stage to another may point to such strategies. Moreover, observation of levels may serve to help evaluate students’ progress in making those constructions.
Few would argue against the statement that theoretical thinking is, in principle, necessary for t... more Few would argue against the statement that theoretical thinking is, in principle, necessary for the learning of linear algebra at the university level. But is it also necessary de facto, i.e. for successfully completing a linear algebra course? Our study started with this question. In our research we encountered many theoretical and methodological problems. We needed to make it explicit what we meant by theoretical thinking in order to explain why we considered this mode of thinking necessary for the learning of linear algebra. This led us to proposing a model of theoretical thinking, which then needed to be operationalized in terms of concrete mathematical behaviors. Moreover, we needed to define some measures of a student's or a group of students' disposition to theoretical thinking in order to compare it with the students' grades in the courses. We developed our theory and methodology along with our study of data from interviews with 14 students who achieved high grades in a first linear algebra and some of whom obtained high grades also in a second linear algebra course. We also studied the final examination questions in these courses from the point of view of the level of theoretical thinking that was, in principle, needed to obtain full marks. The report has five chapters and is preceded by an introduction, where the research question is formulated and put in context. In Chapter I we describe our model of theoretical thinking. We argue that practical and theoretical thinking are deeply intertwined in the development of scientific knowledge, and we posit that theoretical thinking develops against practical thinking, which thereby also constitutes a condition for the existence and significance of theoretical thinking. We then justify why we think that theoretical thinking is necessary for understanding linear algebra. The chapter concludes with a summary of the postulated features of theoretical thinking. We name the main categories of these features "reflective", "systemic" and "analytic" thinking. In Chapter II we present our interviews with a group of 14 students who were selected for having achieved high grades in a section of a first linear algebra course (they were all taught by the same instructor). Six of these students achieved high grades also in a second linear algebra course. We explain our design of the seven interview questions and our method of coding students' responses: the code [1, 0] is used to represent theoretical behavior, [0, 1] represents practical behavior and [1, 1] represents a mixture of theoretical and practical behavior. Six questions in the interview had mathematical content and the seventh question was meant to incite the students to reveal their beliefs about, among others, truth, mathematical knowledge, and reasons for taking mathematics courses. In describing students' behavior in each question we explain how we related this behavior with our model of theoretical thinking. This leads us to an operationalization of the model in terms of features of students' behavior; we identify 18 "theoretical behavior" features In Chapter III we present the contents of the linear algebra courses taken by the interviewed students and we analyze in detail the final examinations given in these courses. We look at the examination questions from the point of view of the relevance of theoretical thinking in their solution. We point to ways of reformulating the questions so that theoretical thinking becomes more relevant in their solution. Chapter IV contains a quantitative evaluation of the students' theoretical behavior. We treat the numerical coding introduced in Chapter II as scores and we define three categories of statistical indices as functions of these scores: expectation, tendency and capability, which we apply to the group as a whole (group indices) and to individual students (individual indices). We use these indices to examine the relations between the students' disposition in the sense of expectation, tendency and capability, and their grades in the courses. We also apply the group indices to dress a profile of theoretical thinking strengths and weaknesses in the whole group of students, compared with the profile of the subgroup of students who achieved high grades in both courses. These profiles are formulated in terms of rankings of the 18 theoretical behavior features according to the indices of expectation, tendency and capability. We conclude with a comparison of the students' disposition to theoretical thinking and the relevance of this kind of thinking in solving the examination questions. In Chapter V we formulate some more general conclusions from the study and, looking back at our research methodology, we reflect on the empirical value of our findings. The report closes with some remarks on the theme of the idea of a university. We propose that the development of theoretical thinking is an important part of the mission of the university and that there is much room for improvement in mathematics courses in contributing to fulfill this mission.
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Papers by Asuman Oktac
We developed our theory and methodology along with our study of data from interviews with 14 students who achieved high grades in a first linear algebra and some of whom obtained high grades also in a second linear algebra course. We also studied the final examination questions in these courses from the point of view of the level of theoretical thinking that was, in principle, needed to obtain full marks.
The report has five chapters and is preceded by an introduction, where the research question is formulated and put in context.
In Chapter I we describe our model of theoretical thinking. We argue that practical and theoretical thinking are deeply intertwined in the development of scientific knowledge, and we posit that theoretical thinking develops against practical thinking, which thereby also constitutes a condition for the existence and significance of theoretical thinking. We then justify why we think that theoretical thinking is necessary for understanding linear algebra. The chapter concludes with a summary of the postulated features of theoretical thinking. We name the main categories of these features "reflective", "systemic" and "analytic" thinking.
In Chapter II we present our interviews with a group of 14 students who were selected for having achieved high grades in a section of a first linear algebra course (they were all taught by the same instructor). Six of these students achieved high grades also in a second linear algebra course. We explain our design of the seven interview questions and our method of coding students' responses: the code [1, 0] is used to represent theoretical behavior, [0, 1] represents practical behavior and [1, 1] represents a mixture of theoretical and practical behavior. Six questions in the interview had mathematical content and the seventh question was meant to incite the students to reveal their beliefs about, among others, truth, mathematical knowledge, and reasons for taking mathematics courses.
In describing students' behavior in each question we explain how we related this behavior with our model of theoretical thinking. This leads us to an operationalization of the model in terms of features of students' behavior; we identify 18 "theoretical behavior" features
In Chapter III we present the contents of the linear algebra courses taken by the interviewed students and we analyze in detail the final examinations given in these courses. We look at the examination questions from the point of view of the relevance of theoretical thinking in their solution. We point to ways of reformulating the questions so that theoretical thinking becomes more relevant in their solution.
Chapter IV contains a quantitative evaluation of the students' theoretical behavior. We treat the numerical coding introduced in Chapter II as scores and we define three categories of statistical indices as functions of these scores: expectation, tendency and capability, which we apply to the group as a whole (group indices) and to individual students (individual indices). We use these indices to examine the relations between the students' disposition in the sense of expectation, tendency and capability, and their grades in the courses. We also apply the group indices to dress a profile of theoretical thinking strengths and weaknesses in the whole group of students, compared with the profile of the subgroup of students who achieved high grades in both courses. These profiles are formulated in terms of rankings of the 18 theoretical behavior features according to the indices of expectation, tendency and capability. We conclude with a comparison of the students' disposition to theoretical thinking and the relevance of this kind of thinking in solving the examination questions.
In Chapter V we formulate some more general conclusions from the study and, looking back at our research methodology, we reflect on the empirical value of our findings.
The report closes with some remarks on the theme of the idea of a university. We propose that the development of theoretical thinking is an important part of the mission of the university and that there is much room for improvement in mathematics courses in contributing to fulfill this mission.