This research addresses the study, replication and analysis of a set of classes with the aim of promoting understanding of the curricular topic of fractions in the sixth year of elementary school with the support of cuisenaire bars. The set of classes was developed with students in the 6th year of elementary school, from a private school in the Federal District, according to the Exploratory Teaching approach (Canavarro, 2011). It was evident that the manipulative material promoted increased understanding of the meaning of fractional numbers, as well as problem situations in which operations with fractions were required. Furthermore, the value of playful and exploratory activities in the learning process was highlighted, as they contribute to the mastery of the mathematical object studied and allow students different ways of exploring and recording their mathematical productions.

The main goal of this work was to investigate the Mathematics Teaching-Learning-Assessment Methodology through Problem Solving relevance and contributions, applied to the content of flat paving, with a focus on regular and semi-regular tiles. To this end, a case study was carried out with students from a 2nd grade high school class at a public school located in São Sebastião/DF. With an emphasis on manipulation, exploration and investigation in the classroom, the didactic sequence applied started from a generating problem, thought and conceived with the intention of promoting the development of geometric/mathematical thinking through the Euclidean plane's tiling. The interest in problem-solving methodology in mathematics classes arises from the fact that it has proven effectiveness in improving student performance when teachers use it. This research had a qualitative approach and data collection took place during the development of activities, carried out in two meetings. The results obtained in this investigation shed light on the attitude of the students and the teacher-researcher, regarding working in groups and facing unusual situations through a new teaching approach. Despite the various challenges inherent to the work proposal of this research, this teaching strategy is seen as an alternative path to so-called traditional approaches. In this way, it is hoped that this research can serve for consultations and reflections on Mathematics teaching focused on problem solving.

This dissertation demonstrates that the incenter and ex-incenter of a triangle are solutions to a problem of maximizing and minimizing a real function, and proposes a mathematics activity aimed at high school level, to work on this topic in the classroom. Recognizing the relevance of technology in education, GeoGebra was used as an auxiliary tool to develop the activity. The proposal seeks a practical and visually stimulating approach to review concepts derived from algebra and geometry, encouraging active student participation. By integrating Geometry and Differential Calculus, the work promotes the integration of these two areas, helping students to tackle mathematical challenges more effectively and creatively, providing meaningful learning.

This study aims to map free digital tools for teaching and learning trigonometry. One of the main challenges in learning trigonometry is the difficulty students face in understanding mathematical concepts that are often presented abstractly. This challenge can be overcome by using various technologies that facilitate the connection between mathematical concepts and the practical world, as well as improve the visualization and understanding of concepts and properties. Ten studies from 2019 to 2023 were analyzed, focusing on the use of different digital tools in teaching trigonometry, such as GeoGebra, the PhET platform, Nearpod, Tour Trigonométrico, and Geocam Free, through classes, mini-courses, and workshops for students of different ages and academic levels. These studies show that the use of digital technologies and manipulable materials, combined with good lesson planning, has many positive effects on the teaching and learning of trigonometry. To analyze digital tools, this study selected 103 applications using different keywords on the Playstore and Apple Store platforms, which were filtered using inclusion criteria, exclusion criteria, and quality assessment. Seven applications were selected and analyzed based on research questions that investigate whether the tools are educational games or other types, their strengths, limitations, the content covered, and user interaction. The results of this analysis showed the existence of few free digital tools focused on the teaching and learning of trigonometry, that these applications do not cover all trigonometry content, and that there are no specific digital games for trigonometry. Additionally, different digital tools from those present in the analyzed studies were revealed, which can be used by teachers, students, and even parents and tutors to support students' development.

This work presents a detailed educational itinerary for High School that integrates educational robotics in a structured way into the school curriculum. The implementation of this educational itinerary will be discussed, ranging from the selection of the most suitable robotics kit for the target audience to the development of challenges and projects for competitions and tournaments. The objective is to explore the practical application of robotics as a teaching-learning tool, focusing on participation in competitions and tournaments that serve as catalysts for engagement and deepening mathematical knowledge among students. Additionally, a training course for teachers will be developed, essential for empowering educators to effectively use educational robotics, preparing them to guide their students through national competitions such as the Brazilian Robotics Olympiad.

The objective of this dissertation is to present the results of a research about how students apprehend and understand the several interpretations of the primitive term "staight line" in the context of finite geometry. To this end, we criated a matemathical task centered on the game Dobble which is a model for the finite projective geometry of order 7. In this task, the student is guided to explore the structure of the game and to create his own mini-Dobble. The analysis of several interpretations of the primitive term "straight line", through conversion of registers and change of context, is proposed to the student. During the development of the task, the student is asked to use several representations of his mini-Dobble, to reflect and to write down his ideas, which allow for the analysis of how he copes with the primitive term "straight line". The task was applied in several classes of highs chool students in the Federal District and its surrounding region. Based on the results of these applications and their analysis, suggestions of how to use and mediate the task in classroom were also appended.

In this work, we develop a brief introduction to the concepts of numerical sets,

functions, limits and derivatives. We present statistical data that synthesize the result of applying

Limits and Derivatives in a short course given to high school students. While reading, the reader

will come across a successful pedagogical proposal for teaching these concepts that, commonly,

high school students contemplate in higher education, specifically in the field of Exact Sciences.

The main objective of this dissertation is to make educators have a new vision about the improvement achieved in student learning when introducing technological devices in their classes. Initially, the idea of writing this work was to work only with mathematical software in the development of Mathematics. Although in the course of the study, it was seen that technology does not only revolve around electronic and modern equipment, but tools created a long time ago such as the ruler, the compass, the square and the Soroban. These tools, which were innovative in their time, are now underused. When collecting the opinion of Mathematics teachers, some believe that a lot of time is lost using such resources, but others believe in the advantages that such materials can offer in the development of different types of mathematical content. Initially, the importance and contributions that Mathematics provided to technology were addressed, both currently and in times when there was no electronic equipment, that is, through analog or manual equipment. It was also detailed about the importance of using such tools in the classroom, in addition to the challenges faced by the educator when leaving the traditional classroom and adopting a differentiated methodology using these resources. To defend this theory, a survey was carried out with teachers and elementary school students who gave their opinion on the use of these tools. Based on this research, work was carried out in the classroom, emphasizing the application of the contents using the materials presented in the body of this dissertation, proving the thesis that there are more advantages than disadvantages when introducing technology to Mathematics classes.

This paper aims to present reflections arising from the use of the test in phases as an assessment tool focused on learning in mathematics classes. Fourteen students from a third grade class of a public high school in the Federal District participated in this investigation. To carry out this investigation, we chose a qualitative and interpretive methodology and the use of a test composed of four discursive questions that was applied in two phases. After completing the test in phases, the students answered a questionnaire to identify their perceptions about this experience. At the end, we observed some positive points in the use of this test, such as: identification of gaps in student learning and opportunities to provide more efficient feedback.

Mathematics Education has gained space in the educational scenario, since the urgent need to improve the quality of teaching and learning in the various modalities of Basic Education has driven research in the area. In this context and in the search for methods that can positively 

influence the process of teaching and learning mathematics, Exploratory Teaching stands out as a didactic approach. Exploratory Teaching aims to contribute, from the critical and reflective dialogue between teachers and students, to the conceptual and mediational development in mathematics. In this research, we assume Exploratory Teaching having as conceptual cut the integers and rational numbers in Youth and Adult Education (EJA). The present study aims to make use of the analysis of students' records and participation in mathematical tasks - of an exploratory nature, and in diagnostic assessments, as a basis for identifying the main strategies presented by them when dealing with problems involving integers and rational numbers. In addition, it is intended to reflect on the teacher's posture in the pedagogical process to verify actions that influence students' decisions. The tasks were constructed and inspired by research on other works developed in relation to the contents involving integers and rational numbers. Two EJA classes of the second segment, 7th and 8th stages, with the presence of 36 students from a public school in the Federal District participated in this study. The stages of the research were: diagnostic evaluation, to assess the level of prior knowledge of the students, to give basis to the interventions and the elaboration of the mathematical tasks; applications of the tasks; and, finally, post-diagnostic evaluation, to analyze the evolution in the students' learning. The results were verified through the observation of critical dialog, participation and records in the evaluations and tasks. These results indicate that when students are placed in an environment of sharing and interaction, they produce creative answers to the problems presented and that learning rates grow significantly with the methodology adopted.

Teaching strategies that enable argumentation, communication and the construction of mathematical reasoning

provide students with ways to exercise their own citizenship.

In this sense, exploratory teaching provides a possibility to develop these skills, through mathematical tasks for the

Youth and Adult Education – YAE – classes, so that students can build their own knowledge. This master's thesis

has the general objective of analyzing the development of mathematical tasks, by EJA students, from public schools

of the Federal District, to verify aspects that indicate learning referent to the combinatorial reasoning. The research

presents the development of three mathematical tasks, the first one refers to a diagnostic assessment, with the

purpose of verifying skills considered necessary, under the eyes of the teacher-researcher, to develop

understanding about Combinatorial Analysis. The other two refers to tasks developed in groups, from the

perspective of exploratory teaching, for 71 students from two schools in the 3rd stage of the 3rd segment of YAE.

The theoretical foundation was based on authors such as Canavarro (2011) and Stein (2008) on exploratory

teaching and Ponte (2014) on mathematical tasks. The results showed that group work in mathematical tasks

favored mathematical argumentation and discussion. Thus, it was possible to conclude that the development of

mathematical tasks of Combinatorial Analysis developed from the perspective of Exploratory Teaching was able to

promote the learning of these students.

This research presents the study, observation, development, and analysis of a mathematics task (MT) in Financial Education (FE), applied to students in order to analyze the theme about conscious spending. The task was developed in a class of ninth grade, according to an Exploratory Teaching approach (Canavarro, 2011). Thus, essential information was produced and supported the creation of the meaningful tasks and allied to the educational objectives. Furthermore, tasks that provide an enriching experience of learning. The aim was to develop a MT of Financial Education to secondary school due to its applicability on daily routine of the student, that make them relate the content of Financial Mathematics to their daily lives, understanding debts, purchases paid in installments and to calculate discounts in products. The MT: “Decide with wisdom, which cellphone would you buy?” simulated the acquisition of a cellphone that encouraged the interest of the students, because it was related to their interests, motivating them to answer it. After developing the MT according to the principles of exploratory teaching and analyze the data, it became clear that the students were able to reflect on the available options of purchase in the market and ways of saving money. They also calculated percentages and found interest rate. It was concluded that the implementation of the MT provided a series of benefits to the students. Besides revisiting the content of Financial Mathematics, it gave autonomy to the students, encouraging the critical thinking and reflection, allowing them to relate the content studied in class to their everyday lives. The study also highlighted that the Financial Education was essential to give the students basic knowledge to act and make a decision in the face of a buying scenario. It is expected that this experience can help them to buy with more responsibility, recognizing their wishes and being able to save their money in the future.

Circular transformation is a subject little known to many teachers, as it is largely focused on solving Olympic problems, the vast majority of which are difficult. In general, it is an advanced technique that is used on a large scale in Mathematics Olympiads, in which it seeks new talents with the characteristic of being an intellectual competition, using challenging problems that require the student to have their creative capacity in solving them, and the Inversion is an important tool that assists in this task. In general, if there is no specific preparation, we are faced with several barriers that are hidden in theories. This fact may be common to both students and teachers and, especially, in the final stages of the Olympics. Thus, this work proposes an attempt to reduce the gap that occurs with mathematics in Basic Schools and in the Mathematics Olympiad. The dynamics of this proposal were based on experiences in preparing students for the Olympics and in some way trying to show how we can grow theoretically through the problems involved. Therefore, the problems here have detailed resolution, some copyright and others with due reference. Through the resolution of some chosen problems, the aim is to outline some interesting strategies for resolving them using little theory. In such resolutions, comments are made on these strategies and possible difficulties that the student and/or teacher may encounter. This experience indicated that there are several problems that encourage young people to enjoy mathematics and become interested in Olympic Problems, simply by developing their reasoning through subjects that are easy to learn.

The main goal of this work is to introduce the concept of a numerical semigroup, highlighting those generated by two elements and the relations between these generators and the invariants of the numerical semigroup. We use a geometric identification to associate each gap of a numerical semigroup with a point with non-negative integer coordinates on the cartesian plane through a bijection, which allows us to compute some invariants of the numerical semigroup, such as the genus, the conductor, the Frobenius number, the depth and the Kunz coordinates. Finally, we present a didactic sequence of activities for high school schools based on the geometric identification.

This work addresses the Machine Learning from theory to hands-on with an emphasis on Support Vector Machine classification algorithm associated with high school mathematics in order to improve knowledge and skills. This dissertation is based on new strategies and pedagogical guidelines that direct to the use and implementation of Machine Learn- ing as a didactic and technological resource in Mathematics. Our integrated approach includes many concepts, challenges and performance metrics in Machine Learning, also builds data augmentation and dimensionality reduction. This work describes how work linear, soft margin and kernel trick models, including two theoretically-based activities for the students. This dissertation contains a short Python tutorial in order to help the implementation. Six project steps are proposed with the Support Vector Machine, each one having performance evaluations by metrics, graphics and matrices.

Aiming to disseminate the culture of mathematical epidemiology, this work will provide an analysis of the basic Kermack-Mackendrick SIS and SIR compartmental models. Demographic effects and effects of pulse vaccination on the epidemic will be investigated using this last model. Due to its importance for epidemiological studies, emphasis will be placed on the basic reproduction number of the disease and its meaning as the number of secondary infections caused by an infectious individual. Furthermore, two proposals for introducing mathematical epidemiology in junior school and high school will be presented, one with the specific presentation of epidemiological models discretized as coupled recursive sequences, and the other as an interdisciplinary project involving computer programming, mathematics and related disciplines, with the purpose to build epidemiological models, implement them computationally and study the dynamics they describe.

This research paper aims at providing an overview of the teaching/learning process of quadratic function and second-degree equation content by a group of students in a public high school in the Federal District, Brazil, during the time of hybrid teaching imposed by the COVID-19 pandemic. This academic piece of work, which was conducted during the transition from hybrid teaching to school-based teaching, intends to analyze and diagnose difficulties that might have arisen or increased due to the remote teaching and social isolation between 2020 and 2021. In addition, it also approaches math-problem solving as a teaching methodology. It will also explain the TEACHING- LEARNING-ASSESSMENT methodology for Math teaching. At the end, this work will recommend some ludic and motivating exercises in order to present parts of the quadratic function content, and some math problems to be discussed after introducing specific content.

This dissertation presents the results of a mapping of research that discuss Lesson Study (LS) in Elementary School - Final Years published in the Annals of the I International Seminar on Lesson Study in Mathematics Teaching, held in May 2021. In a special way , we seek to characterize how problem solving is adopted and its results in terms of basic education students' mathematical learning. The analysis of the texts showed the increase in the number of studies at this level of schooling, at the same time that it showed the prevalence of studies in contexts of continuing education. Furthermore, the results reveal aspects of the adaptation carried out according to the specific contexts of Brazilian classrooms. It was evidenced that the studies that were more successful in relation to the students' learning instituted the resolution of problems throughout the stages of planning-execution-reflection of the SL developed.

Mathematics has always been the subject of big discussions, reflections and emotions. Its impact on society spans educational, social and even economic spheres. Focusing on the first two, this work aims to analyze and disseminate the experiences of former medalists in scientific mathematics olympiads, participants in scientific initiation programs who followed an academic career, in order to question and promote the positive consequences experienced by them. For this, seven graduate students from the University of Brasília were interviewed, verifying their main challenges, achievements, opinions and questions regarding their academic trajectory. Relating the experiences of each one with some theories of learning, as well as collecting graphic data, it was possible to discuss and bring analyzes and considerations about scientific research in Brazil, the role of the teacher, the impacts on future generations and the benefits of the incentive and promotion participation in Olympics and scientific initiation programs.