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)

Abstract

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.