# Doctorado Interinstitucional en Educación con Énfasis en Educación Matemática

URI permanente para esta colecciónhttp://hdl.handle.net/11349/2102

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Ítem Articulación de argumentos del teorema fundamental del cálculo de Newton y de Leibniz para su enseñanza en la formación de ingenieros con el uso de recursos tecnológicosMuñoz Villate, Weimar; Leon Corredor, Olga Lucia; Leon Corredor Olga Lucia [0000-0003-4373-8630]Among the mathematical objects that make up mathematical analysis, the Fundamental Theorem of Calculus (FTC) stands out. However, the teaching and learning process of the FTC has difficulties. For example, some teachers present limitations when teaching it, because they have a low conceptual knowledge, sometimes even procedural, of the definite integral; they do not know how to improve their teaching environments; nor do they know how to create didactical sequences in order to improve the understanding of the theorem; or because they do not consider the complexity of the mathematical objects that compose it. For students the obstacles with the FTC range from having problems understanding previous mathematical notions (continuity, differentiability, ratio of change, etc.) to understanding that∫_a^x▒f(t)dt is a function that depends on x. This doctoral thesis shows that the history of mathematics is still a source of resources, which set with suitable educational software and framed in an appropriate didactical approach, allow the design of tasks for university students.Ítem La agrimensura en el diseño de trayectorias de enseñanza que promueven el aprendizaje de la geometría en la escuela ruralBarbosa Meléndez, Fredy Alejandro; León Corredor, Olga LucíaThe purpose of this doctoral thesis is to strengthen the training of rural mathematics teachers by promoting the incorporation of surveying in the design of teaching trajectories that improve the learning of school geometry, in particular, of angular amplitude magnitude. This thesis emerges from the literature review of nearly 500 publications that account for surveying and mathematics education in rural contexts. The research was developed during the COVID 19 health emergency, with three rural mathematics teachers who work in Wayúu ethno-educational schools, who through the platforms of: Meet, Zoom, Skype and WhatsApp formed a community of practice with the purpose and incorporate surveying in curricular design, respecting the ancestral knowledge of the Wayúu indigenous communities. The doctoral thesis shows that surveying promotes the reincorporation of the study of visual angles to the school curriculum, in addition, this practice can be a complement for the construction of Wayúu houses and corrals, favoring the transition between an empirical geometry to a theoretical geometry.Ítem Formación continuada de profesores de Preescolar y Básica Primaria en una educación Matemática escolar accesibleCastro Miguez, Luis Alexander; León Corredor, Olga LucíaThe research provides elements for the characterization of the effects that the incorporation of reflective processes of teachers has on the teaching practices of mathematics in learning environments accessible from a system for the continuous training of teachers. It is situated in a mixed approach that uses systematic, empirical and critical processes for its study from the collection and analysis of qualitative and quantitative data. According to its scope, it is exploratory in nature, where its research objects are systems. The methodological design has two components: a construction structure that is nourished by the science of design applied to education and a validation structure that is nourished by the techniques and elements of qualitative and quantitative research. From which the central research question is addressed: Are the continuing education strategies for teachers that incorporate reflective processes on the teaching practices of mathematics, are they devices for solving problems posed by the inclusion of vulnerable populations in basic education?Ítem Formas de expresión de modelos mentales [cronotópicos] de alumnos y profesor en clase de geometría analítica de grado 10°Aroca Araújo, Armando Alex; Calderón, Dora Inés; Vasco Uribe, Carlos EduardoThe research problem consisted of studying the mental models and their forms of expression of students and teachers when together they solve Analytical Geometry activities. The general objective was to characterize, determine and establish and compare some features of the mental models [chronotopic] and their forms of expression of students and a teacher with the mathematical activities proposed in the classroom, in the development of Analytical Geometry activities of the 10th grade. Methodologies derived from the Chronotopy Program were used, whose emphasis was placed on the subdisciplines of Topia or Topo*, that is, Topography, Topology, Topometry and Toponomy. For the collection of information, the audiovisual record of 14 classes was made during the 2017-2 semester, non-participant observation was made. The methodology combined ethnographic moments with other types of research. The ethnographic phase allowed the collection of data that allowed studying manifestations of mental models. Due to the above, it became necessary to distinguish ethnographic moments from others such as the study from the grounded theory. For the analysis of the information, the methodology of the grounded theory was used, under the categories of the Chronotopy Program. For the systematization of the data, ATLAS.ti8 was used. The theoretical framework emphasized three theoretical components, which is why it was called the Theoretical Ψrider: Mental Models, Semiotics and the Chronotopy Program. The elements that allowed identifying mental models [chronotopic] stand out in this theoretical framework. Among the results of the analysis, some of them can be highlighted: 1. A characterization and an identification methodology of the forms of expression of the mental models [chronotopic] of students and teacher of grade 10 when they develop Analytical Geometry activities in the plane . 2. An interpretation of the forms of expression of the mental models [chronotopic] of students and their teacher in the described context. 3. The generation of a set of elements related to mental models [chronotopic] that need to be considered in the didactics of analytical geometry. Among the recommendations, the development of future research on the teaching and learning of Analytical Geometry and in particular on the recognition of other forms of mathematical expression that coexist in the classroom is proposed.Ítem Formas de pensamiento aditivo en estudiantes de tercero de primaria (8-9 años): una aproximación desde la teoría de la objetivaciónPantano Mogollón, Óscar Leonardo; Vergel Causado, Rodolfo; Radford Hernández, LuisThe research entitled Forms of additive thinking in third grade students (8-9 years old): an approach from the Theory of Objectification characterizes forms of additive thinking that appear, are produced, through the encounter with historical-cultural arithmetic knowledge in the joint labor that emerges between third grade students of Primary Basic Education and the teacher in the process of solving additive tasks in the naturals. These forms of thinking are produced through sensitive and material forms of perception, gestures, corporeality, symbolization, discursivity and use of artifacts.Ítem Representaciones semióticas de objetos matemáticos y articulación de sentidos en situaciones de tratamiento. El caso de los profesores de matemáticasMejía Osorio, Gladys; Rojas Garzón, Pedro JavierThis research focuses on the cognitive transformation of treatment oriented: on the one hand, to document the phenomenon related to the difficulties encountered by some teachers to articulate the senses assigned to semiotic representations of the same mathematical object, obtained through treatment; and on the other, to establish similarities and differences of these difficulties with those that students encounter when solving mathematical tasks that require treatment reported in the literature. A collective case study is carried out with 11 mathematics teachers [5 primary and 6 secondary], selected as a result of the solutions made by 64 teachers [32 primary and 32 secondary], who do not carry out a minimum semiotic articulation in three Tasks, that is, those teachers who from the syntactic point of view admit the equivalence between expressions or representations, but from the semantic aspect these expressions or representations are associated with different mathematical objects or situations, thus preventing the relationship between the assigned meanings. these. The processes of assigning meanings of 11 mathematics teachers are described and analyzed in relation to four specific tasks, which investigates the meaning assigned to certain semiotic representations that requires the performance of treatment transformations, as well as their necessary articulation. A qualitative research approach of a descriptive-interpretive type is assumed, from two theoretical perspectives: the theory of semiotic transformations proposed by Raymond Duval (1993, 2017) that allowed to describe, consolidate and locate the research problem in the field of research of the transformations; and the theory of the ontosemiotic approach proposed by, Juan D. Godino and his collaborators (1994, 2019) in as much, provides tools to analyze and explain the solutions given by teachers that highlight the personal meanings given to semiotic representations obtained by treatment. The present study provided elements that made it possible to identify the difficulties that mathematics teachers encounter in articulating the senses assigned to semiotic representations obtained through treatment: the difficulties that mathematics teachers encounter were contrasted with the difficulties encountered by students, aspects that allow us to conclude that both teachers and students adequately perform the required treatments that allow them to admit the equivalence between expressions from the syntactic plane, but giving meaning and meaning to the expressions related in the tasks prevents them from making the same recognitionÍtem Narrativas de profesores de matemáticas sobre su experiencia profesional y de formación: Aproximación a las subjetividades emergentesSalazar Amaya, Claudia; Rojas Garzón, Pedro Javier; Calderon, Dora InésThis research’s purpose was unveiling the mathematics teachers’ subjectivities from a narrative-autobiographic study with a hermeneutic nature. This research assumes the linguistic and narrative turn for evidencing the subjectivation processes, the subjectivities constitution, and the teachers’ subjunctivize possibility. The problem that it confronts is the lack of comprehension about the ways how the mathematics teachers’ subjectivities are built through the professional experience and formation trajectories, which leads to the intentional actions’ absence in the initial and continued training -of these teachers- that contribute to the constitution of these subjectivities. This lack of comprehension obeys to the way as we have configured the reasoning (or pedagogical reasoning) features in Mathematics Education, and we have ignored other thinking ways required in teaching for constructing sense and interpreting sense for others. Therefore, the questions that guided the inquiry are related with the types of professional and formation experiences which are reported in the teachers’ narrative wefts and the senses that are unveiled about their subjectivities in the hermeneutic cycle developed by them. This inquiry is framed in the interpretative paradigm and it is developed by means of qualitative methods of inductive nature. It commits with the cognitive communion between the teachers, the researcher and the double hermeneutic that it implies. For the research development, it proposed a learning environment where nine mathematics teachers experienced the creation and interpretation of autobiographic narrative wefts about their professional experience and formation trajectory in three phases: preconfiguration, configuration and reconfiguration. In reconfiguration phase, the teachers and researcher made paradigmatic and narrative analysis of extracted data of their wefts, analysis that helped teachers to take distance of their experiences, they lived identification processes and they entered in subjunctive mode. As results, the teachers identified their narrating, saying and making skills in all the modalizations (to know, must, to want and can); they recognized the transition between the own identification and the own ipseity in the own comprehension, in other words, between a morally neutral posture and a responsible subject posture; they identified the others relevance in the own professional identity configuration, specially the academic peers who play the adaptative or routine experts role in their early years of professional experience; the artisanal knowledge that is from teachers community which they belong; the cultural practices who let they the early links with the mathematics practices; and, the interactions with students in different levels and contexts.Ítem Constitución dialéctica de los procesos de objetivación y subjetivación con una ética comunitaria en actividades desarrolladas al abordar tareas de generalización de patronesLasprilla Herrera, Adriana; León Corredor, Olga Lucía; Radford, LuisResearch on pedagogical practice in mathematics education requires the study of relationships, which is in itself a relational practice that is based on and with the connections that occur between the intentions for education, the scenarios and actors of the educational process, the contents and mediations of the educational process, among other aspects. In general, the studies that have been conducted about this practice have focused their interest on the development of teaching-learning processes that are characterized by having a strong inclination to attend to the development of mathematical concepts, so that important aspects such as the formation in values are left aside or these end up being invisibilized or ignored (Radford, 2014a; León and Lasprilla, 2019). However, in mathematics education research, the interest should be placed both in the development of mathematical concepts and in the formation of values and it is extremely important that both are given equal primacy, because giving greater value to the contents and downplaying the importance of values formation allows students, teachers, school knowledge and the structures of educational institutions to be thought of in an incomplete way and, therefore, limits are established to rethink education in a more holistic way. That is why it is necessary to propose interventions in the classroom that start from a simultaneous interest between ethical education and mathematics, where ethics goes far beyond promoting collaborative work and truly generate practices that allow solidarity, respect and care for the other. The proposal of the present study consists of finding empirical evidence to characterize the dialectical relationship between the processes of subjectivation and objectivation with community ethics during the development of joint work in mathematics classes.Ítem Estudio exploratorio sobre el proceso de autovalidación de respuestas a preguntas capciosas sobre probabilidad en un ambiente virtual con estudiantes de ingenieríaLeón Parada, Fernando; León Corredor, Olga Lucía; Vasco Uribe, Carlos EduardoThis document is a doctoral thesis whose objective is to characterize aspects in the developmental of self-validation processes in the answers to probability questions. The exploratory type of research, with a mixed methodology that considers as qualitative and independent variables the combinations between two forms of questions, tricky versus transparent, with two types of metacognitive strategies, concept maps versus online text consultation. The fieldwork was carried out with two groups of engineering students who were studying probability and statistics in two Colombian public universities during 2019. The quantitative variables, defined on the frequencies of the factual learning trajectories in the tests in a virtual self-regulated learning environment, showed that the majority did not realize the probability fallacies that linked the problems and chose inappropriate responses. They found reasons to give up on them with a concept map, which rarely happened when the strategy was to consult a text on the web. The tricky question inciding in results on the phenomenon of cognitive biases by making him doubt and decrease his retrospective confidence in the initial answer, and the conceptual map that feeds his consciousness in the error and makes him give up his self-validation process to solve your problem.Ítem El proceso de autovalidación de respuestas a preguntas capciosas de probabilidad en un ambiente virtual; estudio exploratorio con estudiantes de ingenieríaLeón Parada, Fernando; León Corredor, Olga Lucía; Vasco Uribe, Carlos EduardoThis document is a doctoral thesis whose objective is to characterize aspects in the developmental of self-validation processes in the answers to probability questions. The exploratory type of research, with a mixed methodology that considers as qualitative and independent variables the combinations between two forms of questions, tricky versus transparent, with two types of metacognitive strategies, concept maps versus online text consultation. The fieldwork was carried out with two groups of engineering students who were studying probability and statistics in two Colombian public universities during 2019. The quantitative variables, defined on the frequencies of the factual learning trajectories in the tests in a virtual self-regulated learning environment, showed that the majority did not realize the probability fallacies that linked the problems and chose inappropriate responses. They found reasons to give up on them with a concept map, which rarely happened when the strategy was to consult a text on the web. The tricky question inciding in results on the phenomenon of cognitive biases by making him doubt and decrease his retrospective confidence in the initial answer, and the conceptual map that feeds his consciousness in the error and makes him give up his self-validation process to solve your problem.Ítem Un laboratorio de prácticas docentes para la formación de profesores de matemáticasMalagón Patiño, María Rocío; D'Amore, BrunoThe teaching practices of mathematic teachers have been the subject of particular interest in the last decades for the national and international community of researchers in the field, given the consensus around its influence in the analysis of learning achievements by students. What do these practices consist in?, what factor haves affected them?, moreover, how to get teachers to reflect on them and turn them into objects of study? These are still open questions from which studies have been generated in national and international debates.Ítem Las problemáticas semióticas en las representaciones de los conjuntos infinitos en la práctica docenteBecerra Galindo, Héctor Mauricio; D'Amore, BrunoThe present doctoral research on semiotic problems in the representations of infinite sets in teaching practice, arises from the teaching and learning processes of infinite sets of numbers, where difficulties are evident in students regarding its cognitive construction. These are associated with the objective difficulty of the students in the face of the infinity theme, the general theme of the formation of a noetic versus the semiotic representations (Duval's paradox) and the development of a “semiotic consciousness” by the teachers; with "semiotic consciousness" we mean the conscious knowledge about the systems of representations, which are mobilized in mathematical activity and which is specific to semiotics. This research is especially focus on the characterization of the teachers' “semiotic awareness” manifestations of the representations of the infinite sets, based on their reflection on the productions and comments of their students.Ítem Articulación y cambios de sentido en situaciones de tratamiento de representaciones simbólicas de objetos matemáticosRojas Garzón, Pedro Javier; D' Amore, BrunoIn the process of teaching and learning of mathematics the use of representations of objects in a variety of semiotic representation systems, more specifically in diversity of semiotic records (Duval, 1999), but especially it becomes necessary to appropriate possibilities to transform a semiotic representation of one mathematical object in another. Such transformations between semiotic representations they occur both within the same semiotic representation register as between differentiated records, transformations that Duval calls treatments and conversions, respectively. Duval recognizes conversion as one of the fundamental cognitive operations for the subject's access to a true understanding, and focuses on the difficulties of mathematics learning in this process. However, in mathematics, the treatment transformations between semiotic representations - within the variety of records used - not only are they fundamental, but they could be source of difficulties in the processes of understanding mathematics by students. It is usually stated that cognitive problems are related to conversion, while what is related to treatment is not usually considered as a relevant problem for the construction of the mathematical object. That is, this author stands out explicitly the complexity involved in recognizing the same object through of completely different representations, as produced in semiotic systems heterogeneous (Conversion), but does not highlight the complexity associated with transformations made within the same semiotic system of representation (treatment). The present investigation is oriented to document the phenomenon related to the difficulties encountered by some students to articulate the senses assigned to semiotic representations of the same mathematical object, obtained by treatment. A description and analysis of the allocation processes of senses of nine students, six of 9th grade and three of 11th grade, based on work performed by them in three small groups in relation to specific tasks, in which inquire about the meaning assigned to certain semiotic representations and it is required to perform treatment transformations A qualitative research approach is assumed, performing a descriptive-interpretative analysis, from different perspectives theoretical, taking as reference works by Bruno D’Amore, Raymond Duval, Juan D. Godino and Luis Radford. Thus, this work is situated in a semiotic context, and studies in a general way the relationship semiosis-noesis in the construction of mathematical knowledge by students of 9th and 11th grades of basic and secondary education, respectively; study that without being exhaustive, includes aspects about mathematical activity, communication about objects emerging mathematicians and the cognitive construction of mathematical objects.Ítem Una perspectiva sistémica para el estudio de los programas de formación de profesores de matemáticasGil Chaves, Diana; León Corredor, Olga LucíaThe present research creates a conceptual and methodological system for the study of mathematics teacher training programs (PFPM), in order to transcend product-focused views (high quality accreditation of programs) or in mass evaluations ( Saber Pro tests), and provide important perspectives and reflections on research, theory, practice and policy in teacher training, both locally and globally. This involved taking two conceptual and methodological orientations for the development of the research, one of them was the Theory of processes and systems (TGPS) of Vasco (2014) and the other the notion of field and its internal fields (Bourdieu and Wacquant, 1995 ; Díaz, 1995; Zuluaga & Herrera, 2009). The system was constructed from three fields of education, the field of mathematics teacher training, this field explores in the programs the approaches on the meaning that the training of mathematics teachers have for the PFPM; the field of the curriculum, this field seeks to understand the approaches on the curricular organization of the PFPM and the field of mathematics didactics that interrogates the program about the approaches that the PFPMs express about the didactics of mathematics and its place in the construction of the identity of the students for math teacher. The system was applied to three PFPM in Colombia.Ítem Dificultades, conflictos y obstáculos en las rácticas educativas universitarias de iniciación al cálculo diferencial —PEUC— en estudiantes de ingenieríaNeira Sanabria, Gloria Ines; Vergel Causado, Rodolfo; Vasco Uribe, Carlos EduardoThis research focuses on identifying, describing, characterizing and explaining the difficulties, conflicts and obstacles that can be inferred from the study of some university educational practices of the initial work in the differential calculus in first semester students of Engineering, how they emerge, and what possible causal relationships could be established for its explanation, initially configuring a comprehension framework for the conceptualization of some theoretical constructs related to the notion of obstacle. The theories of Brousseau, Sierpinska, Artigue, Godino, Radford and D'Amore, as well as the studies of Sfard on reification, and of Tall on conceptual image ("concept image") and procept ("procept") were used. . The research is qualitative, descriptive-interpretative; The type of study is exploratory, with the case study method, in which non-participant observation and structured task-based interviewing were used.Ítem Análisis didáctico a un proceso de instrucción del método de integración por partesMateus Nieves, Enrique; Vergel Causado, Rodolfo; Vasco Uribe, Carlos EduardoThis is a systematic didactic analysis that analyzes a process of instruction of the method of integration by parts. It is a qualitative research based on the case study and focused on a particular educational context. The observation methodology used was the description of class sessions. We apply the ontosemiotic approach of mathematical cognition as a theoretical framework and develop the proposed analysis categories. Specifically, the notion of didactic suitability (description, explanation and evaluation) of the mathematical instruction processes. This analysis shows an X-ray of what happened in the classroom and why. It is important to recognize that the relationship between the relationship of teaching and the meanings assigned to mathematical objects that institutionalize are not disconnected from the context in which they develop given the implication that it has in the evolution and construction of meaning implemented ; for that reason we propose to do less formal teaching processes where other techniques are included, such as the incorporation of Tics among others. This research in which we tried to look at the role of context in the processes followed by a teacher chosen to institutionalize the construction of meaning of the method of integration by parts (MIP) can be a guide for other teachers interested in improving their pedagogical practices and be replicated in other objects of basic, medium or higher level, typical of the teaching of mathematics.Ítem Cambios de concepciones de estudiantes para profesor sobre su gestión del proceso de enseñanza-aprendizaje en ambientes de aprendizaje fundamentados en la resolución de problemasBohórquez Arenas, Luis Ángel; D'Amore, BrunoIn the last decades, several studies have been carried out, from which the change of beliefs and conceptions of teachers has been investigated, as stated by Pehkonen, Ahtee, Tikkanen and Laine (2011); However, from the ones reported by Bobis, Way, Anderson and Martin (2016), the question: under what conditions do changes occur in the beliefs and conceptions of the teacher, formulated a decade ago by Pehkonen (2006), still remains valid . For this reason, it is considered pertinent from this research to establish a characterization of belief, conception and management, so that a characterization of the changes achieved by students for teachers can be presented and a description of the factors that support or limit such changes.Ítem Formas de pensamiento algebraico temprano en alumnos de cuarto y quinto grados de educación básica primaria (9-10 años)Vergel Causado, Rodolfo; Vasco Uribe, Carlos EduardoThe possibility of promoting the development of algebraic thinking in the early years of schooling is an issue that increasingly generates more interest for research in mathematics education. In particular, generalization of patterns is considered one of the most important ways of introducing algebra in school. However this necessarily demand develop an enlarged perspective on the nature of school algebra, consider a dialectical relationship between forms of algebraic thinking and ways of solving the problems of generalization pattern, which introduces a problem in terms of the constitution of algebraic thinking in young students. In this process of generalization of patterns we must consider that acts of knowledge by students include different sensory modalities, such as tactile, perceptual, the kinesthetic, etc., which become integral parts of cognitive processes. This is what has been called in the international context (Arzarello, 2006) the multimodal nature of human cognition. We are therefore faced with the need to recognize all those discursive situations (oral and written), gestural and procedural evidencing in student attempts to build explanations and arguments on general structures and mindsets, and their arguments and explanations are supported by individuals or situations into concrete actions. In epistemological terms, we are suggesting ways of conceptualizing, knowing and thinking can not be adequately described only in terms of discursive practices. It is important to consider the cognitive, physical and perceptual students mobilize resources when working with mathematical ideas. These resources or modalities include symbolic and oral communication as well as drawings, gestures, manipulation of artifacts and body movement (Arzarello, 2006; Radford, Edwards & Arzarello, 2009).