https://repositorio.ufu.br/handle/123456789/19678 2026-09-24T18:11:15Z https://repositorio.ufu.br/handle/123456789/50379 Title: Redes neurais artifciais: uma abordagem didática e aplicações em problemas de matemática aplicada Abstract: This work presents an introductory study on arti cial neural networks, with emphasis on the mathematical foundations that support their structure and learning process. The main objective was to understand how these models operate from a mathematical perspective, especially highlighting the role of linear combination and gradient in the construction and training of neural networks. To this end, concepts from linear algebra, di erential calculus, and analytic geometry were revisited, establishing an appropriate theoretical basis for the interpretation of the arti cial neuron, activation functions, the cost function, and the backpropagation algorithm. Initially, the modeling of the arti cial neuron was discussed based on the linear combination of inputs, weights, and bias, emphasizing its relationship with lines, hyperplanes, and decision boundaries. Next, some of the main activation functions used in neural networks were presented, highlighting the importance of nonlinearity for the representational capacity of the model. Subsequently, the cost function, gradient, and chain rule were addressed, showing how these elements are articulated in the backpropagation process and in the updating of parameters through the gradient descent method. As a way of concretely illustrating the concepts studied, two computational applications were developed in Python. The rst consisted of a linear tting problem involving temperatures in degrees Celsius and Fahrenheit, in which the network successfully learned the expected relationship between input and output. The second involved a nonlinear classi cation problem in the plane, in which the network had to distinguish points located inside and outside a circle. In this case, the importance of the hidden layer and nonlinear activation functions for learning a decision boundary more compatible with the geometry of the problem was observed. In both applications, the matrix formulation proved particularly useful for organizing the calculations and supporting the computational implementation. Thus, it is concluded that the study of neural networks may constitute a relevant opportunity to connect Mathematics with contemporary topics in computing, contributing to a more grounded understanding of these models and opening possibilities for future developments in more general architectures and new applications. 2026-03-20T00:00:00Z https://repositorio.ufu.br/handle/123456789/49996 Title: “E agora, cinco?”: uma história virtual do conceito para o ensino da tangente no ensino médio Abstract: This study aims to analyze the contributions of an intentional organization of teaching, grounded in the Logical-Historical Movement and the Guiding Teaching Activity, to the appropriation of the concept of tangent by high school students. It is based on the understanding that Mathematics teaching should provide students with access to the historical conditions that gave rise to concepts, fostering the development of theoretical thinking. The research is characterized as qualitative and was developed through the elaboration and implementation of a Learning Triggering Situation (LTS), which addressed historical elements related to the emergence of the concept of tangent. The activity was organized to engage students through a Virtual History of the Concept, designed to highlight the need for the concept of tangent and enable the investigation of its conceptual nexuses. For the analysis of the empirical material, two analytical axes were established: the understanding of the historical necessity of the concept of tangent and the identification of the conceptual nexuses and students' appropriation of this concept. The results of this research indicate that the articulation between the study of the Logical-Historical Movement of concepts and the organization of teaching through the Guiding Teaching Activity constitutes a powerful approach to the teaching of trigonometry in the school context, insofar as it enables students to approach the conditions that gave rise to mathematical concepts and fosters an understanding of the conceptual relationships that structure them. 2026-03-20T00:00:00Z https://repositorio.ufu.br/handle/123456789/49994 Title: Construção de bases de Gröbner para ideais polinomiais sobre um corpo K Abstract: This work presents an introduction to the theory of Gröbner bases for polynomial ideals in the ring K[x1,...,xn] over an arbitrary field K. The aim of this work is to develop the main foundations of this theory, establishing the results necessary for the construction and characterization of Gröbner bases. To this end, it adopts a theoretical approach, in which the concepts, definitions, and proofs are developed progressively. Initially, it presents fundamental results on ring theory, monomial orderings, and the division algorithm in several indeterminates, extending to the multivariate case tools previously available only for polynomials in one indeterminate. Next, it characterizes monomial ideals and proves Dickson's Lemma and Hilbert's Basis Theorem, from which the existence of Gröbner bases for every polynomial ideal follows. Subsequently, it establishes Buchberger's Criterion and Buchberger's Algorithm, culminating in the characterization of minimal and reduced Gröbner bases. The results developed make it evident that Gröbner bases provide effective methods for solving problems involving polynomial ideals and can serve as a foundation for the future study of applications of the theory. 2026-08-04T00:00:00Z https://repositorio.ufu.br/handle/123456789/49264 Title: Tecendo histórias: narrativas e marcos legais no cenário da educação matemática inclusiva mineira Abstract: This work seeks to understand how the movement of implementation of Inclusive Education in Brazil has taken place. The research is guided by the following question: what can be said about the movement to implement Inclusive Education in schools, especially in the context of Mathematics Education? To achieve the objective, a study of national and international legal frameworks that underlie Inclusive Education was initially carried out, as well as theoretical references in the area of Mathematics Education that discuss inclusion and diversity in school. The research adopts a qualitative approach, based on the methodology of Oral History, as practiced by the Oral History and Mathematics Education Group (Ghoem). As methodological procedures, interviews were conducted with educators who work or research inclusive practices in the school context. The participants were invited virtually and, after signing the Informed Consent Form, they previously received a set of guiding questions. The interviews took place in person or virtually, were recorded in audio and later submitted to the processes of transcription and textualization, according to the assumptions of Oral History. The analytical movement was developed from the narrative analysis of narratives, articulating the narratives of the participants to the theoretical references and documents analyzed throughout the research. The results indicate that, despite important advances in the legislative field, the effectiveness of Inclusive Education in schools still faces significant challenges. Among the main obstacles identified are the insufficiency of initial and continuing training of teachers to deal with the diversity present in the classrooms, the permanence of structural prejudices in the school environment and the lack of investment in public policies that sustain inclusive pedagogical practices. The narratives also show that the construction of a more inclusive education involves the recognition of the singularities of students, the appreciation of differences and the need for a more sensitive and humanized teaching attitude. Thus, it is concluded that the implementation of Inclusive Education does not occur in a linear way, but constitutes a process in constant construction, marked by tensions, advances and challenges. In this scenario, Mathematics Education can play an important role in promoting pedagogical practices committed to social justice, contributing to the construction of a more equitable and welcoming school. 2026-03-18T00:00:00Z