The Division Algorithm is often considered by students to be a complex subject. As a teacher, I met with students who were lacking in this content in all grades of Elementary and High School. In this work, we will deepen the theme, investigating how the division has been approached in textbooks available in public and private schools in Minas Gerais. We will also analyze the alternative suggestions present in PROFMAT’s theses and we will also present a diagnostic test created to understand the doubts and gaps of the students. Finally, we will study some theories about the construction of knowledge and the use of concrete material and games as a teaching methodology to propose ludic activities that can contribute to the approach of this content.

This thesis is a literature review that seeks to connect topics in the History of Mathematics with Spatial Geometry of geometric solids. Therefore, as part of the literature study, we begin by presenting a literature review on the productions of PROFMAT in Spatial Geometry and the History of Mathematics. We also conduct a study on Book XI of Euclid’s work The Elements, by Euclid of Alexandria (c. 300 BCE), from the perspective of two translations: the English version by Henry Billingsley in 1570 and the Brazilian translation by Irineu Bicudo in 2009. We briefly discuss the context of mathematics and sciences during the period in which Euclid possibly lived in the Ancient Age and Billingsley in the European Renaissance. We consider how the support of concrete material can aid in visualizing three-dimensional objects. We provide a brief history of pop-up books (paper elements that pop out of the book when turning the page) and propose a guide for constructing the ten main geometric solids using this resource, as defined in Book XI of Euclid in Billingsley’s version. The construction of these objects is done with ruler and compass, instruments widely used in the mathematics of Euclid’s time and revived during Billingsley’s period. At the end of the construction of each solid, we list some motivating questions to guide the discussion about the elements of each figure.

This thesis starts with the observation the challenge teachers face in attracting attention to their students. In Math, in whice there is normally greater resistence and difficulty by the students, this challenge is particularmente manifested in more abstract contents such as the study of polynomials. One solution to this stand-off is to create strategies that make classes more dynamic and engaging for students. Drawing on scholars such as Piaget, Vygotsky, Grando, and Lara, one strategy implemented was the use of games in the classroom, which generates greater interest from students and fosters learning that goes beyond the subject itself, developing logical thinking, teamwork, and strategy creation. We chose to work with the game of Checkers, which has simple rules and requires strategic thinking. A parallel was drawn between the game of Checkers and polynomials, emphasizing that both can be approached by to achieve goals through rules through rules. Taking these aspects into consideration, the product created was the game "Polidamas" (or PolyCheckers), which combines the joint of polynomials and the game of Checkers, reinforcing polynomial concepts and operations. Additionally, a Didactic Sequence was developed, including the study of polynomials and the application of the game "Polidamas" in the classroom. In conclusion, this study contributes to the teaching of polynomials and can be used by teachers as a foundation for instruction and classroom implementation.

Trigonometry and its functions are fundamental not only in Mathematics, but also play an essential role in the study of periodic physical characteristics, mechanics, astronomy, music, topography, engineering and other areas. In this context, this project seeks to support teachers in their approach to Trigonometry topics, taking into account the guidelines of the National Common Curriculum Base (BNCC) and the integration of Digital Information and Communication Technologies (DICTs) in Mathematics Teaching. The project offers teachers full support in practicing these concepts. Throughout this work, teachers will find support for the use of various Active Methodologies, as well as a detailed study of the possibilities offered by works already published on the ProfMat website. In addition, the project has a solid theoretical basis capable of fully supporting teachers who seek this support. The development of the project is exploratory in nature, with a direct approach to the object of research, involving a study of the trigonometry content presented in the BNCC and the construction of didactic sequences for each of these contents. As a result, the project presents an educational product accessible via a link available in the work. This product includes the practical application of one of the activity sequences, which is part of GeoGebrabook. In addition, the final considerations cover not only the entire development of the dissertation, but also highlight the possibilities for applying the material and the openness to continuing the project. The overall aim of this research is to propose an educational product, GeoGebrabook, in order to provide both the teacher and the student with complete material for the study of Trigonometry and Trigonometric Functions, based on Active Methodologies, Critical Mathematics Education and Digital Information and Communication Technologies (DICTs), through applets, which enable more dynamic learning, making the student the main part of the teaching and learning process, while the teacher plays the role of mediator in this educational context.

In this dissertation we present a research whose main objective was to develop an educational product that had as its prerogative the use of Digital Technologies as a resource for Teaching Mathematics. To base ourselves theoretically, we reviewed the literature on the use of technologies in the context of Education, from a historical, political and practical perspective, until we arrived at the use of technologies in the context of Teaching Mathematics, the focus of our research. Based on our reference, we decided to work with material that had already been produced, available and intended for Teaching High School Mathematics so that, based on this material, we could propose improvements based on the use of technologies. The material chosen was made available free of charge by the Working Group of Mathematics teachers of the Federal Center for Technological Education of Minas Gerais, GTMAT, through a website. After an analysis of this material, using as a reference some of the criteria for evaluating digital teaching material proposed by Godoi and Padovani, we developed interactive quizzes using the Kahoot application, in order to improve what was already available, from a perspective of student use. and the teacher. From this improvement, we developed an educational product for use by teachers who want to make use of quizzes in learning situations, either by reproducing or adapting the material for the context of their classes. In addition, we hope to be able to continue our work on improving the GTMAT material, contributing to the site dialoguing with technologies that are increasingly inserted in the context of teaching in general.

With RSA Cryptography as background, we present, generally speaking, the mathematical theory that underlies that encryption model, as well as its contextualization as part of the modern way of life and its security, and then proposes a product for Basic Education (BE). For this, we show the Brazilian National Common Core Curriculum (BNCC) view about the use of technologies in the mathematics teaching in addition to possibilities for the use of calculators in Elementary School (ES) classes. Before, however, basic elements of Modular Arithmetic are presented, building the mathematical model that supports RSA Cryptography - in addition to the totient function, as an additional topic. Thus, guided by the question “Is it possible to bring the RSA Cryptography theme to the Basic Education classroom?” our goal is to createactivities in which, from the use of technologies, modular arithmetic concepts are addressed, leading students to understand the RSA Encryption method. As a result, we present the final product, aimed at Basic Education, consisting of a proposal for didactic sequence for the 6th year of Elementary School, which aims to introduce the concept of primality using calculators, and two proposals for classes for the High School, approaching the calculation of residual powers through Modular Arithmetic, sometimes with manual calculations, sometimes with electronic spreadsheets. There are also suggestions on how can be approached the activities proposed in the High School. Finally, it is concluded that this work can be a guideline for future studies, research and practices, so that its elaboration and conclusion were successful in the sense of giving
continuity to the formation of a teacher who teaches mathematics in Basic Education.

The RPG, an acronym for Roleplaying Game, is a game that mixes roleplay and a set of probability-based luck tests. The objectives of this dissertation are to explore the application of RPG in mathematics education in order to develop a system that is easy to apply in a school environment and, based on this system, a Probability manual for
Elementary School. To develop the system, the following was explored: the trend of Games in Education, Roleplay in education and the existing works with similar objectives. For the Probability manual, we explored three Brazilian syllabuses and the mathematical definition of the topics to be discussed in class.
 

The concept of Probability is part of a society’s day-to-day, whether it is in simple situations, for instance event occurrence; or more complex circumstances, like decision-making processes that require logical thinking. Therefore, understanding probability is fundamental both in and outside the academic environment. That way, we have explored interesting and relevant ways to deal with this subject in a dynamic and playful approach, choosing to apply it on educational-biased games. Thus, the present study uses the Monty Hall Problem (MHP), greatly discussed by scholars, as a background to improve assimilation and comprehension of probability conceptions and properties. In doing so, we have developed a GeoGebra Book, made of theoretical materials involving diverse complexity levels of probability content. In addition, the digital book includes solved exercises in games, such as dice and cards, as well as applets that allow its use through adapted games and PMH expansion into other doors. The final product will serve as support material for high school students and teachers, or anyone enthusiastic about the topic. The proposed e-book connects applied probability axioms to game playing; encouraging users to foster and enhance concepts that will help them solve problems on a daily basis.

Several are the difficulties reported by teachers and students throughout the teaching-and- learning process in mathematics, particularly in what regards the content of Probability. The present dissertation has as its objective to analyze how a didactic sequence, developed through the use of Team-based learning (TBL) strategy may contribute to enhance not only teaching, but also student learning outcomes. The sequence in question, had initially been planned to be developed in remote (online) environment due to the COVID-19 pandemic. Nevertheless, in August 2021, students were allowed to partially return to school in smaller groups. Each of these groups would attend school, in person, every two weeks. After that, the nearly total return of students finally took place in September 2021. Hence, it was decided to adapt and change our initial planning and apply both the questionnaires and the didactic sequence in presential manner. In order to develop the project and the didactic sequence, an initial diagnostic questionnaire was applied, two different methodological paths were built and developed and students were subjected to a final questionnaire. The methodological path 1, involved the concepts of Sample Space, Event and Randomized Experimentation, the classic definition of Probability and the Probability of the union of two or more events. The methodological path 2, referred to the Probability of Independent Events and Conditional Probability. Both methodological paths were developed, based on TBL Teaching-and-Learning Strategy, in a class of 43 students of the 11th grade of high school, from a private school in Belo Horizonte. The data collected along the applications was registered and analyzed through a qualitative approach of interpretative base, being open-ended answers of students analyzed in a way that these would reveal relevant information regarding the students’ learning in what regards the content of Probability. These answers showed, for instance, that students apprehended the classic definition of Probability, but they also highlighted that the same students presented gaps related to contents from junior high school and from the 10th grade. It is important to emphasize that we were returning from an extended period of pandemic, what might have worsened this perception of possible learning gaps and in knowledge regarding the content of previous years. With the development of the didactic sequence, the participation and the answers from students,it was possible to notice that the teacher, as a mediator/tutor, has an important role in the construction of the students’ mathematical knowledge and that the teacher should always follow the signs of time, proposing different teaching-learning strategies in order foster the construction of learners’ mathematical knowledge, placing them as protagonists in the process, making mathematics interesting and meaningful for them.

This work is based on the studies of Markov Chains and Chapman-Kolmogorov Equations. To understand the text in Annexes A and B, there is a relevant review about Matrix and Probability, topics covered in the basic education level, usually in the second year of high school. After the review of those topics, it is possible to approach basic concepts of Markov Chains and their different applications. Markov Chains have numerous applications in economics, genetics, mathematical problems, and their main characteristic is that the determination of future probability does not depend on past probabilities, but it only depends on present probability. The objective of this work is to introduce the basic concepts of Markov Chains to second year high school students using an approach of Matrices and Probability that they are already familiar with, presenting them with interesting examples, instigating their curiosity, seeking to motivate them during the learning process and demonstrating how mathematics is useful, necessary and can be applied in several ways in the world around us. Finally, with the suggestion of a didactic sequence, we want teachers and students to have a rich experience and a significant learning of the basic concepts of Markov Chains and Chapman-Kolmogorov Equations after Matrices and Probability. The didactic sequence has five consecutive stages with an increasing degree of complexity, seeking to provide students with a specific, defined learning process and that they, through their abilities, reach new levels of learning.

Many young people, after graduating from high school, will deal with loans and financing to pay for goods and services, such as vehicles and houses, credit cards, graduation and various courses. In these circumstances, this public lacks strategies that help them to decide on the acquisition of these, and other items, in a conscious and responsible manner. This scenario justifies the presentation of this research whose objective is to promote learning about Amortization Systems for High School students, applied to situations of loans and/or financing, in the context of Financial Education, a mandatory transversal theme by the BNCC (2018). Such learning permeates through mathematical knowledge of this segment, namely, “Interest” and “Progressions”. In this dissertation, we present the development of the research, conducted in the light of Didactic Engineering, and the construction of an educational product (book) developed throughout the work, whose proposal is to assist teachers in teaching the most usual Amortization Systems: the PRICE Table and the SAC System. We present the organization of the book and the report of its applications in two private schools in Belo Horizonte – MG, carried out in classes of first and third years of High School, with the intention of adapting the proposed didactic situations. The validation of the material was based on analyzes supported by the evolution of the learning objectives of Bloom's Taxonomy and also in the development of mathematical competences at BNCC. Finally, the dissertation ends with the expectation of presenting to teachers and researchers in Mathematics, and related areas, an educational product of great value in promoting Financial Education for teachers and students, providing knowledge and learning that influence the behavior of a new generation of consumers.

This dissertation aims at contributing to the teacher's role, presenting activities on the study of the maximums and minimums of functions and on Newton's Binomial, involving a different approach from that presented in textbooks. For the development and elaboration of activities, we used active learning methodologies as a methodological reference, in particular the Flipped Classroom, collaborative work, the use of technologies and the application of the History of Mathematics as a structuring axis, in order to consolidate the content. The activities were applied in classes of the 2nd and 3rd grades of High School in two private schools in Belo Horizonte and the data collected during the application were analyzed using a qualitative approach of an interpretative nature and took into account the students' posture and responses, their behaviors and their discursive productions throughout the development of the activities. In addition to the activities, we present discussions and investigations that show Mathematics as an evolving science, in two specic moments of History. The rst, with Pierre de Fermat and the birth of Calculus, and the second with Sir Isaac Newton and the expansion of his binomial. At both times, we seek to apply methods, the activities developed enable a greater involvement of the students, encouraging the development of research and study skills, possibilities of interactions between students-Mathematics-teachers, in addition to presenting a more engaging and contextualized Mathematics.

The performance of brazilian students in the editions of the Programme for International Student Assessment (PISA) until 2018, with a focus on Mathematics, showed that they obtained a grade below the international average. Based on the result obtained in PISA of 2012, a text presented by the Brazilian Mathematical Society raised, among others, the following question to be answered by teachers: "Should my teaching emphasize mathematical concepts or how those concepts are applied in the real world?" As an answer, the teaching of mathematics in its applied form was encouraged, instead of just in its pure form. Among the areas of Mathematics education included in the basic education plan, it was decided, then, to present an application of Analytical Geometry and Linear Algebra. The purpose of this work is an application of matrix algebraic groups in Robotics, through a programming environment and a robot simulator, as an alternative and viable material for the teaching-learning of Mathematics. Among the possible activities, this work deals with the Matrix Groups Applied on Forward Kinematics of a robotic arm, which is the description of the positioning and orientation of the end-effector, that is last element of this equipment, with information on the location of its base, the size of its links and the position of its joints. The matrix groups was applied through the notation presented by Denavit and Hartenberg. As final product, some activities that can be applied in the school environment are presented. As an incentive to apply the methods suggested by this work, free programming and robotics simulation tools were used.

The mathematical content taught in Basic Education is greatly in uenced by external
evaluations that students perform at the end of High School, mainly the National High
School Exam. Thus, content that is not in National High School Exam has been relegated
by schools and teachers. One of the contents relegated, in the author's experience, is
Matrix Algebra, content that has assumed great importance today. The use of matrices
to perform geometric transformations and to solve linear systems are two examples of
this importance. Geometric transformations through matrices have been widely used in
the movement of robotic arms and in computer graphics, while linear systems are able
to model different situations, both in Mathematics itself and in other areas of knowledge.
On the other hand, Matrix Algebra is an abstract content and difficult to contextualize
due to the large number of operations that are necessary to represent a single problem
situation. Thus, this dissertation aims to present contextualized activities for teaching
Matrix Algebra with use of technological resources. For the construction of these activities,
we used the concept of computational thinking and investigative scenarios. Through a
bibliographic analysis, we found that computational thinking has many intersections with
Mathematics and, combined with investigative approach, it was possible to develop three
activities that have an empirical profile and put the student in a more active position
in their learning process. We believe that activities with this scope, promote more than
contextualized teaching, promote the creative and investigative spirit, as well as a re ection
on how mathematics teaching needs to connect with reality and promote meaningful
education.

The present work made a start on the restless concerns of the researcher professor with his activities as a Calculus teacher as well as a High School teacher. Bothered by the high failure and dropout rates in Calculus classes, he started to reflect what could be done in Basic Education and what could minimize these difficulties. The research was developed with High School freshmen and the objective was to analyse the development of teaching real functions of a real variable through a didactic sequence where the variational character of the functions stands out in contrast to the current static approach in the field of set theory. It is a qualitative research and the methodological procedures used were participant observation, questionnaires and documentary analysis. The development of the activities was supported by the GeoGebra software and it was found that this tool favored the visualization of the function graphs and helped the student develop an intuition of the variational character of the functions, enabling a dynamic approach to this content that is often overlooked. We must highlight here the importance of continuing research on this topic, given its relevance in Basic Education, in order to establish a dialogue with the Higher Education stage, minimizing the impacts of the transition between levels of education and promoting a better interpretation of the world.