Paula Verdugo-Hernández

La perspectiva histórica y epistemológica ha demostrado su relevancia por su correspondencia con el desarrollo cognitivo del estudiante en el Análisis Real. Sin embargo, hay pocos estudios desde esta perspectiva centrados en sucesiones y series. Desde una perspectiva histórica-epistemológica se presenta un análisis sobre las nociones de sucesiones y series, para discutir sobre su enseñanza y alcances didácticos, con especial foco en una idea fundamental y transversal en el Análisis Real: la convergencia. A partir de un análisis documental, se aborda la evolución del conocimiento que dio lugar a la definición formal del límite de sucesión y el desarrollo de funciones en serie de potencias, lo cual produjo el desarrollo moderno del Análisis Real a través del estudio sistemático de la convergencia. Se relacionan los problemas e interrogantes con los conocimientos y métodos disponibles en periodos históricos específicos, enfatizando los obstáculos que debieron ser resueltos para alcanzar este desarrollo moderno.

This study explores how a pre-service mathematics teacher mobilizes specialized knowledge while teaching three geometric concepts: similarity, homothety, and Thales’ theorem. Drawing on the mathematics teachers’ specialized Knowledge model and Duval’s (1995) theory of registers of semiotic representation, the study examines how knowledge domains are enacted through multiple representations. Data were collected from three consecutive lessons during the teacher’s practicum in a socioeconomically disadvantaged and traditionally structured classroom. Findings indicate that the pre-service teacher evidenced representational fluency and procedural clarity, particularly in the use of diagrams and gestures to convey proportional reasoning. However, conceptual generalizations and formative engagement with students’ thinking remained limited. The study underscores the importance of teacher education programs in explicitly linking representational practices with epistemic goals and student reasoning, especially in socioeconomically disadvantaged contexts where systemic constraints often restrict opportunities. This research contributes to ongoing discussions on pre-service teacher development and the pedagogical demands of geometry instruction in authentic classroom settings.

The teaching of geometry in Chile faces several challenges, as evidenced by the low performance of students in international assessments. In particular, the concept of homothety is impacted by teaching methodologies that emphasize rote memorization and procedural repetition rather than conceptual understanding. This study explores how a task sequence fosters mathematical work with the concept of homothety among 15- to 16-year-old students in a public high school in the Maule Region of Chile. Grounded in the mathematical working space (MWS) theory, this research provides a framework for analyzing how students engage with mathematical tasks. Employing a qualitative approach, specifically a case study, the findings reveal that the task sequence contributes to progressive mathematical reasoning. The study concludes that the designed activities align with the criteria for “emblematic tasks”, evidencing their fundamental role in supporting students’ conceptual understanding of homothety.

Introduction: This study addresses the teaching task contextualized in the teaching of different topics of mathematic (functions, sequences, and Thales’ theorem). To do so, the characteristics of the mathematical work and specialized knowledge that the teacher shows during teaching are analyzed, considering as a frame of reference the complementarity between two theoretical perspectives (MWS and MTSK). Methodology: The study is framed in the qualitative approach, based on an integrated multiple case study design, which allows analysis of the proposed mathematical performance (oral and/or written) of three teachers. Results: The analyses allow a deep examination of the development of mathematical work and demonstration of the relationships between the inter-conceptual connections as well as the domain changes that are expressed in teachers’ work. In general, the results reveal the privileging of algebraic treatments and emphasis on the procedural. Conclusion: Contributions are identified in the perspective of the connection between theories in the context of the models involved.

In this article, we analyse a lesson on Thales’s theorem in a Chilean secondary school classroom through the combination of two theories: Mathematics Teachers’ Specialised Knowledge (ThMTSK) and Mathematical Working Spaces (ThMWS). Both theories, first separately and then in relation to one another, are used to analyse two tasks proposed by the teacher in the classroom following a cross-methodology for networking of theories. Through a single case study research design, a content analysis of the transcript of the video recording of the lesson was conducted. The joint analysis of this lesson allows us to better understand the mathematical work taking place in the classroom. In particular, the results show the scope of each model and their complementarity through the detection of meeting points and blind spots, through the role of proof, representations, and the change between geometrical and numerical work in teaching Thales’s theorem. This allows for a deeper understanding of a teacher’s practice and teaching. Ultimately, relationships between the theoretical elements of both theories are established to show their complementarity. We conclude that networking between theories can contribute to the development of these theories by raising questions that involve examining their foundations and assumptions in greater depth.

This article aims to identify the knowledge that enables a prospective secondary school teacher to mobilize number sense components during a pre-professional practicum. Using a qualitative case study approach, four classroom episodes focused on fractions, percentages, and algebraic language are analyzed. The study involves one prospective teacher and 23 students from a multi-grade school in central-southern Chile. The analysis integrates teacher’s specialized knowledge and mathematical working spaces to explore how these frameworks inform the teacher’s knowledge. The findings reveal that the teacher employs personal theories, real-life examples, and graphical aids to enhance students’ understanding, though there is a reliance on rules and algorithms. While specialized knowledge aids in teaching, there is less emphasis on fostering students’ conceptual understanding and judgement. The study recommends improving teacher training programs to better equip prospective teachers in grasping number sense components, highlighting the need for more focused training on its development.

Introducción: La literatura reporta que el conocimiento profesional docente y su preparación son puntos clave en la formación de profesores de matemática. Por ello, este estudio presenta el proceso de diseño y validación de un instrumento que aporta información sobre los formadores de profesores. Metodología: La investigación emplea un diseño metodológico descriptivo, pues el instrumento indaga en los perfiles del formador de profesores de matemáticas. El proceso de validación de contenido se realiza por juicio de expertos, lo cual permitió analizar la consistencia interna de dimensiones teóricas, de meta-categorías y de los ítems, utilizando el coeficiente V de Aiken. Resultados: Se presentan cinco etapas de la construcción, que abarcan desde la revisión de literatura para el diseño del instrumento y la validación hasta la versión final del instrumento. Discusión: La discusión considera las implicaciones del estudio para la formación de formadores de profesores de matemáticas. Conclusiones: El estudio concluye que el instrumento desarrollado es una herramienta útil para evaluar el perfil académico de formadores de profesores de matemática. Finalmente, ofrecemos el instrumento en su versión final para ser usado en futuras investigaciones o bien para ser adaptado por investigadores para su utilización en otros contextos.

Both international examination results and scientific evidence related to the teaching and learning of school algebra point to the necessity for analyzing mathematical practice in the classroom. In this context, we have formulated the research objective of characterizing the mathematical work that teachers favor based on the examples that they implement in the teaching of topics within the domain of secondary school algebra. To this end, this study is situated within the interpretive paradigm utilizing an instrumental collective case study approach. The study design entails the selection of two representative cases of mathematics teachers in Chilean schools, with the objective of understanding and describing their mathematical work in the secondary school classroom based on the examples they present. The results show that the first teacher utilizes varied examples that activate distinct forms of mathematical thinking, promoting the activation of both instrumental and discursive genesis. In contrast, the second teacher follows a more traditional approach, with the use of examples to illustrate and motivate, and a direct presentation of knowledge, activating primarily semiotic genesis, in which a transition between the numerical and the algebraic is lacking. Ultimately, the results highlight the importance of deeper consideration of the choice of examples in the teaching of algebra and how these examples can influence students’ learning. Furthermore, additional ideas are put forward for future work in this line of research associated with the use of examples in the classroom.

Objective: The study aims to characterize the teaching proposals of the future Mathematics teacher in the context of their practical training in the final year of their university studies. Theoretical Framework: To achieve this, we consider the theory of Mathematical Workspaces, which allows for the analysis of both the mathematical activity that an individual engages in while solving a mathematical task, and the activity that is promoted during teaching. Method: A qualitative methodology is adopted through the design of an instrumental case study. The case pertains to a future teacher conducting a class on constructing a box plot. This class was observed and transcribed for analysis in light of the proposed mathematical work. Results and Discussion: The mathematical work exhibited by the future teacher includes a strong semiotic component and the use of non-material artifacts for quartile calculations. Students' prior knowledge is utilized in this context, with procedural aspects taking precedence over statistical thinking. Implications of the Research: The study raises concerns about the statistical education of mathematics teachers and its impact on future teaching proposals. Originality/Value: This research contributes to the study of statistics and initial teacher training in their influence on the future practices of Mathematics teachers. It provides a characterization of the mathematical work promoted by a future teacher and offers insights into concerns regarding the development of statistical thinking.