Doctorado Interinstitucional en Educación con Énfasis en Educación Matemática
URI permanente para esta colecciónhttp://hdl.handle.net/11349/2102
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Ítem Constitución de los vectores de una ética de orientación comunitaria y su repercusión en el grado de sofisticación de las formas de pensamiento variacional emergentes en estudiantes de tercero de primaria (8-9 años)(Universidad Distrital Francisco José de Caldas) Diaz Fernández, Lina Marcela; Vergel Causado, Rodolfo; Vergel Causado, Rodolfo [0000-0002-0925-3982]In this research, we propose to characterize more relational and dialogic forms of interaction in the classroom, in addition to identifying the elements that foster the emergence of vectors of such ethics in this group of students and their impact on the forms of variational thinking that students encounter and become objects of awareness. Studying only what happens in the state of knowledge implies leaving aside a source of information about how the subject, the way in which they relate and their culture, enter into interaction with what they learn, how they learn it and the very disposition to develop tasks proposed by the teacher. For this reason, we consider it necessary to study the manifestations of community ethics that emerge and how these, in fact, can directly influence the teaching-learning process of variational thinking and the sophistication of the forms in which this thinking can appear.Ítem Implicaciones de las subjetividades éticas de los profesores en las prácticas matemáticas escolares: éticas imperantes en la clase de matemáticas(Universidad Distrital Francisco José de Caldas) Clavijo Riveros, Martha Cecilia; D'Amore, BrunoThis theoretical-practical research focuses on an analysis of the implications that teachers' ethical subjectivities may have in the educational context. It shows that the term “prevailing ethics” is a relevant component for problematizing mathematics education, both in theory and in school practices. At the theoretical level, it is evident that this has gone unnoticed by several theoretical approaches and can be positioned as a focus of cross-cutting analysis. At the level of school practices, its spontaneous nature in teachers' practices and its potential to enable teachers to understand and problematize school mathematics practices were also perceived. The research was conducted with primary school mathematics teachers in Bogotá. This analysis is based on the recognition that mathematics education extends to the training of individuals in interaction with others, in situations determined by social and power relations. To this end, this research is based on the analysis of studies that articulate mathematics teaching and ethics. One finding of this research is that, although teachers' ethical subjectivities influence the ethics prevailing in mathematical practices, this usually occurs mainly in an unconscious manner. In addition, teachers initially show a lack of awareness of the implications of ethical subjectivities in mathematics teaching. An additional finding is that the repercussions of prevailing ethics on mathematical practices and society in general have been largely overlooked. Thus, considering ethical subjectivities and prevailing ethics enables a critical analysis of mathematics teaching. To carry out this research, a teaching laboratory was implemented, understood as a structured space for collaborative learning and critical reflection, designed specifically to explore and analyze the implications of ethical subjectivities in mathematical practices. The main objective of this laboratory is to scrutinize ethical subjectivities, prevailing ethics, and their implications. Four teachers participated in a process of critical self-reflection on their practices. During the process, a transition was evident from subjectivities centered on obedience and knowledge to more cooperative and caring ones. A relevant aspect of this research is that teachers, through the laboratory, begin to question their own ethical subjectivities and identify their influences on teaching decisions. They recognized that these subjectivities came from their training and teaching experience. This process of self-reflection and analysis allows teachers to find features of their own spontaneous ethical subjectivities and understand how they influence their mathematical practices. This study highlights the importance of integrating critical reflection on the ethical subjectivities that influence mathematics teaching, recognizing their role in shaping mathematical practices, as these subjectivities can shape certain prevailing ethics in mathematical practices. By incorporating these ethical considerations, the aim is to enrich mathematics teaching, as awareness of ethical subjectivities and their implications for mathematical practices could promote pedagogical practices that favor the development of critical, reflective, and socially engaged mathematical thinking.Ítem Desarrollo de un análisis didáctico-matemático conjunto. Un estudio de los factores de cambio en las creencias y concepciones de profesores universitarios sobre la enseñanza y aprendizaje del espacio vectorial(Universidad Distrital Francisco José de Caldas) Fúneme Mateus, Cristian Camilo; D'Amore, BrunoLinear Algebra occupies a foundational position in the education of mathematicians, engineers, and scientists, serving as both a core theoretical framework and a gateway to abstract reasoning in the mathematical sciences. Yet, despite its centrality, its teaching and learning continue to pose significant challenges. The concept of the vector space represents a particularly critical juncture, demanding that learners shift from computational reasoning toward structural and relational thinking. Extensive research indicates that these difficulties are not merely epistemic or cognitive, but are deeply embedded in faculty beliefs, conceptions, and didactic–mathematical knowledge, which collectively shape instructional practices and influence how students construct meaning. This dissertation investigates university professors’ beliefs about the teaching and learning of the vector space, and the factors that contribute to or constrain their transformation. It is grounded in the premise that teachers’ beliefs are dynamic, evolving constructs that emerge from the interplay between mathematical understanding, pedagogical experience, and institutional context. Recognizing this complexity is essential to fostering a more reflective, research-informed approach to mathematics teacher education at the university level. Employing a qualitative, descriptive–interpretive research design, the study explores how professors conceptualize the mathematical objects associated with the vector space, how these conceptions are enacted in their pedagogical practices, and how reflective engagement may facilitate belief transformation. A research–training laboratory was implemented to document participants’ initial beliefs, observe the evolution of their thinking throughout the process, and identify key factors underlying change. Analytical categories were derived from a theoretical framework integrating the constructs of belief, conception, and didactic–mathematical knowledge, providing a coherent lens for interpretation. The findings reveal that meaningful transformation in teachers’ beliefs does not arise from content exposure alone but from the nature of the formative experience—specifically, the dialogic, reflective, and analytical quality of professional interactions that enable educators to reexamine the epistemological and pedagogical foundations of their practice. The study corroborates international evidence highlighting the role of didactic–mathematical analysis in cultivating teachers’ professional vision, deepening their understanding of mathematical structures, and enhancing their capacity for pedagogical reasoning.Ítem Diseño de actos de institucionalización en el marco de la Ingeniería didáctica: reflexión sobre el proceso de construcción del saber en la Teoría de situaciones didácticas(Universidad Distrital Francisco José de Caldas) Santos Torres, Julián Humberto; Acosta Gempeler, Martín Eduardo; Acosta Gempeler, Martín Eduardo [0000-0003-1002-069X]After implementing a Didactical Engineering (DE) on the parabola as a geometric locus (Santos, 2016) within the framework of the Theory of Didactical Situations (TDS), we began to question how to support the teacher during the moments in which continuity between knowledge and knowing is constructed. Following a substantial theoretical review, we recognized that DE designs describe in detail the moments of knowledge construction within adidactical situations, but they do not address the moments in which knowledge becomes decontextualized into knowing — in other words, they do not describe in detail the process of institutionalization. We propose the hypothesis that institutionalization acts can be designed in the a priori analyses of DE, enabling the teacher to manage their interventions appropriately during whole-class discussions and thus effectively guide the construction of continuity between knowledge and knowing. To test this hypothesis, we propose analyzing, through certain categories of the TDS (validation, devolvement, and institutionalization), the data collected after implementing two DEs: one on the parabola as a geometric locus and another on the Pythagorean theorem. In both, the milieu is interwoven with Dynamic Geometry Software (DGS), and a priori designed acts of devolvement and institutionalization are included. We expect this project to contribute to problematizing the process of institutionalization within the framework of the TDS, fostering an evolution in the conception of institutionalization itself. Furthermore, we aim to propose an evolution of DE as an instrument for the production of knowing, in which acts of institutionalization can be described in greater detail, thus promoting the construction of continuity between knowledge and knowing.Ítem Relaciones entre el proceso de configuración de una actividad en tanto que una labor conjunta y la evolución en las formas de pensar algebraicamente sobre generalización de patrones con estudiantes de grado quinto de primaria (9-11 años)(Universidad Distrital Francisco José de Caldas) Joya Cruz, Sindy Paola; Vergel Causado, Rodolfo; Vergel Causado, Rodolfo [0000-0002-0925-3982]This doctoral thesis stems from a series of needs that have been recognized from my role as a classroom teacher, in which we identify that students arrive in class with particular interests, tastes, needs, and motivations that are often opposed to the approaches established by the educational institution, ideas that are not even in line with what I, as a teacher, would like to teach my students in relation to mathematics, life, and the world. In particular, about the teaching of school algebra, we identified that it has focused on the manipulation of symbolic expressions and the solution of fictitious problems (Valenzuela & Gutiérrez, 2018); However, it should be noted that algebraic thinking is not the manipulation of alphanumeric signs, but rather thinking in certain distinctive ways (Radford, 2008b) in which indeterminate quantities are recognized, idiosyncratic ways of representing and operating with those quantities, and above all, the way of approaching these indeterminate quantities analytically (Radford, 2018b). The emergence of algebraic thinking and how it is understood has been the subject of recent observation and study. Some authors, such as Kaput (2008), Radford (2018d), Pincheira and Alsina (2021b), and Vergel (2016b) suggest that elementary school students can perform algebraic generalization processes in which they establish rules that lead them to indicate both the concrete and the abstract in numerical and figural situations without the need to resort to the use of alphanumeric signs. In fact, one of the difficulties students have in performing algebraic generalizations is subject to the transition from the concrete perceptual to the non-existent in perception (Radford, 2013a), which requires an awareness of the spatial structure of sequences, as well as the recognition of at least one commonality, which implies a mobilization of semiotic resources (Vergel, 2015b). Considering the approaches of Vergel (2016c, p. 25), it is noted that “there is a gap between students' ability to recognize and verbally express a certain degree of generality and their ability to use algebraic notation with ease.” Therefore, it is appropriate to determine the semiotic-cultural practices present in mathematical activity that allow us to observe how mathematical ideas emerge that are expressed to characterize a commonality. We rely on the multimodal theoretical conception of human thought, under which the inclusion of the body in the act of knowing is considered important (Vergel, 2015a) to analyze the way mathematics is done. We also recognize that learning involves elements other than knowledge, such as the formation of subjects, classroom relationships, ethics, and the development of critical positions, among others. In this regard, Radford (2021a, p. 45) points out that "[...] learning mathematics involves emotions and affections in ways that deeply touch, affect, and shape us. Therefore, classrooms not only produce knowledge, but also subjectivities (that is, unique human beings)." Classrooms are filled with individuals who are in constant interaction, immersed in a culture of which they are a part, constantly transforming themselves and, in turn, generating culture. This leads us to recognize the need to identify how interaction occurs in the classroom and how, through joint work, social relationships are established around knowledge. Another of our observations refers to the fact that the types of interaction established in the classroom preserve traditional vertical structures, in which it is assumed that the teacher is above the student and only their voice has meaning and resonance in the classroom. Our efforts aim to promote a horizontal structure in which students feel recognized and can actively participate in class tasks and proposals, without fear of sharing their experiences, observations, and questions. This reflection shows that the type of relationship between teachers and students and among students themselves is decisive for the encounter with knowledge and for the constitution of subjects. Our perspective also calls us to be aware that learning does not only take place during class time and, therefore, there is an urgent need to seek to create a mathematics classroom with a critical perspective, with a human touch, an encounter with the world and a presence of ethics, where respect, commitment, and teamwork prevail, where there is the possibility of sharing, experimenting, and learning mathematics together. We will investigate the generalization processes and strategies used by fifth-grade students and their teacher, taking into account Objectification Theory (Radford, 2023a), strata of generality (Radford, 2010a), and the semiotic means of objectification (Radford, 2010a) through the inclusion of the body, recognizing cognitive, physical, and perceptual resources (Vergel, 2015c); as well as the impact of community ethics vectors: responsibility, commitment to collective work, and care for others (Lasprilla et al., 2021; Radford, 2023). Recognizing that there is a distinction between Activity and Joint Labor that is determined by the presence or absence of community ethics (Lasprilla, 2021b), we hope to analyze the relationships between the process of configuring an activity as joint labor and the evolution of algebraic thinking in fifth-grade students (9-11 years old) at Colegio Isabel II (Bogotá, Colombia) when performing tasks related to pattern generalization, in which the production of algebraic knowledge is linked to the establishment of non-alienating forms of human collaboration that demonstrate community ethics. We will identify both the emergence and evolution of algebraic thinking (Radford, 2008b, 2021f, 2022c; Vergel, 2015c; Vergel & Rojas, 2018) and the constitution of subjectivities (Leóntiev, 1984; Radford, 2017c; Radford & Lasprilla, 2020; Vergel, 2024) through a multimodal approach involving perception, gestures, mathematical symbols, and natural language. Qualitative data are collected to analyze activity, collective consciousness, classroom organization, community ethics, teacher management, algebraic thinking, attunement, and social relationships. From the cultural semiotics proposed by Objectification Theory (Radford, 2023a), we recognize that the most significant contributions of this work are related to building evidence that allows us to reflect on the materialization of the activity that is taking shape as joint labor between teachers and students as they perform different tasks to promote the emergence and evolution of algebraic thinking. presenting an idea of a reimagined mathematics classroom in which emancipatory practice (Radford, 2021e) prevails, enabling encounters with knowledge and beings, as well as recognizing that as community ethics take shape, more sophisticated ways of doing and thinking will emerge. The expected results allow us to reflect on non-alienating forms of human collaboration, as a characteristic of joint labor, which are based on the vectors of community ethics: commitment to joint labor, care for others, and responsibility (Radford, 2017d, 2020c); and on ways of producing knowledge related to the development of algebraic thinking and the strata of generality: factual, contextual, and symbolic (Radford, 2010b). Similarly, it allows us to reflect on new ways of perceiving the mathematics classroom in search of emancipatory practices. The work is divided into six chapters. Chapter 1 presents the state of the art, emphasizing activity and algebraic thinking. Chapter 2 outlines the research problem based on what has been discussed previously. Chapter 3 provides a theoretical description based on the principles of Objectification Theory, and we describe elements related to activity such as Joint Labor, community ethics, algebraic thinking, and multimodal analysis. Chapter 4 describes the research configuration based on the tasks designed and the methodological elements for data collection. Chapter 5 discusses the research results based on the constitution of the data and its analysis. Finally, Chapter 6 is devoted to the generation of theory, where the propositional nature of theoretical construction that underpins this study is articulated with solidity and coherence. Through the elements of reflection and the recommendations presented, the academic community is invited to look beyond traditional forms of teaching and analysis, exploring the possibilities that emerge when the activity is constituted as joint labor. This chapter not only culminates the research journey, but also opens horizons for new questions, dialogues, and practices that promote a more meaningful, critical, and collaborative mathematics education.Ítem Cambio de concepciones de profesores en ejercicio sobre la enseñanza inclusiva de las matemáticas. Una apuesta por la articulación entre la ideología, la práctica y la posibilidad de desarrollo(Universidad Distrital Francisco José de Caldas) Garzón Muñoz, Angélica Lorena; Bohórquez Arenas, Luis Ángel; Bohórquez Arenas Luis Ángel [0000-0002-1340-9214]This thesis aimed to determine the characteristics of the changes in practicing teachers' conceptions about inclusive teaching in mathematics when participating in a continuing education program and to identify the aspects associated with the training course that led to changes. Five teacher conceptions about inclusive mathematics teaching were identified, three types of changes characterized by the articulation between ideology, practice, and the possibility of development were described, and the aspects of the course that influenced the changes in teachers' concepts were identified, such as the types of activities implemented, the texts designed, and interactions between colleges.Ítem La emergencia del pensamiento proporcional durante la actividad matemática de estudiantes de quinto grado en un aula mediada por una ética de orientación comunitariaMoreno León, Rafael; Vergel Causado, Rodolfo; Camelo Bustos, Francisco Javier; Vergel Causado, Rodolfo [0000-0002-0925-3982]; Camelo Bustos, Francisco Javier [0000-0002-8627-4816]For more than sixty years, research has been carried out on proportional thinking in the field of Mathematics Education. During this time, Piaget's research on the subject has influenced many authors, who consider mathematical thinking with a certain orientation towards the individual who learns (Kantian subject) and arises from a certain vision of the human being inherited in education from other social areas. This individual construct or recreates the knowledge associated with proportional thinking, which leads, according to a dialectical materialist vision, to two kinds of alienation: by the product of the activity (knowledge) and by the activity of learning itself. Furthermore, the study of proportional thinking has forgotten the issue of subjectivity in the classroom. Therefore, from a multimodal research perspective on human cognition, we ask ourselves: What relationships link the forms of social interaction that promote a community-oriented ethic and the emergence of proportional thinking in a fifth-grade classroom of Primary Education? The analysis carried out is based on the dialectical method. A central axis of this method is to conceive that all phenomena must be studied as processes in constant movement and change, articulated as an organic whole. The results of our research offer a broader view of proportional thinking in fifth grade, identifying classroom conditions that favor human collaboration and the impact of non-alienating social interactions on the emergence of forms of proportional thinking.Ítem Posibles cambios de concepciones de profesores universitarios sobre las causas de los errores (de sus estudiantes) en el aprendizaje de la matemáticaRamírez Bernal, Henry Alexander; D'Amore, BrunoThe doctoral thesis sought to deepen the description, characterization and understanding of the possible changes in the conceptions of a group of mathematics teachers (in practice) in the first semesters of University about the causes of their students' mathematics errors. Through reflection and discussion exercises in focus groups, and using theoretical frameworks from Mathematics education such as Brousseau's theory of obstacles and Duval's theory of semiotic representations, the study sought to obtain information that would allow for a descriptive qualitative analysis to determine and characterize: What changes occur in teachers' conceptions about the causes of their students' mathematical errors as a result of these reflections and analyses? What factors influence these changes? If there are no changes, what is the reason? How does peer discussion impact changes in teachers' conceptions on this topic? The study revealed some gradual and differentiated changes in the conceptions of participating teachers regarding the causes of students' errors when learning mathematics.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.