National Qualification Exam 2025.2

This dissertation, developed within the scope of PROFMAT/UFOB, proposes an innovative approach to the teaching of functions, starting from the premise that meaningful learning must transcend mere algebraic manipulation and abstract formalism. Motivated by the concerning educational reality in Bahia, where data from IEDE (2023) indicate that only a tiny fraction of students achieve an adequate level of proficiency in Mathematics, this research investigates how graphical visualization can act as a catalyst for understanding complex concepts. Grounded in the epistemology of Bento de Jesus Caraça, who advocates for not confusing a function with its defining instrument, and on Sergio Lorenzato’s assumptions regarding "seeing with one's hands", this work seeks to promote the transition from the concrete to the abstract through direct experimentation. Methodologically, a qualitative approach of an interpretive and descriptive nature was adopted, structured in a multipaper format, analyzing the engagement and autonomous concept construction of public school students. Article 1 discusses the relevance of using manipulatives in Mathematics teaching, grounded in a literature review and a state-of-the-art mapping conducted across the Brazilian Digital Library of Theses and Dissertations (BDTD), Google Scholar, and the PROFMAT repository, alongside a critical analysis of contemporary textbooks, highlighting the gaps that justify the need for new methodological approaches. Article 2 presents the report and analysis of the application of an innovative manipulative resource with high school students from the public school system of the state of Bahia. The essence of the activity consisted of student agency through the creation and handling of a low-cost material, composed of graph paper and acetate sheets, used for the visual and dynamic exploration of properties such as injectivity, surjectivity, and the construction of inverse functions. The study distinguishes itself from previous research by prioritizing geometric perception as the foundation for function inversion, aiming to prepare students not only for the rigor of Higher Education but also for the challenges of interpreting real-world phenomena demanded by evaluations such as PISA. It is concluded that student agency and the stimulation of visual perception result in a more robust and intuitive appropriation of mathematical knowledge, transforming students into true co-authors of their learning process.

The growing incorporation of digital technologies into educational contexts, reinforced by the Brazilian National Digital Education Policy (PNED) and the National Common Core Curriculum (BNCC), has demanded new pedagogical practices in Mathematics education. At the same time, Trigonometry remains one of the most challenging contents in basic education and the integration between Educational Robotics (ER) and block-based programming is still scarcely explored in the Brazilian context. In this scenario, this dissertation aims to analyze how the integration of ER and block-based programming can contribute to Mathematics teaching in upper secondary education, in the light of Papert’s Constructionism, seeking to understand both the possibilities of integrating these technologies for the development of logical reasoning and their contributions to Trigonometry learning. The research adopts a qualitative approach and is organized into two articles and an educational product. The first article, of an exploratory nature, is based on documentary analysis of PNED, BNCC, and theoretical references, investigating possibilities of integration between Mathematics, block-based programming, and ER for the development of logical reasoning. The second article adopts an empirical and participant design and consisted of the design, development, implementation, and evaluation of a workshop composed of three experimental sessions focused on the Pythagorean Theorem, the Law of Sines, and the Law of Cosines, carried out with eight third-year upper secondary students from a public school in Luís Eduardo Magalhães, Bahia, using field notes, audiovisual records, and task resolutions as data collection instruments. The results indicated that the integration of these technologies fosters engagement, motivation, and the construction of trigonometric concepts, promoting a shift from an exclusively algebraic approach to a relational understanding of mathematical objects, mobilizing the four pillars of Computational Thinking and converting errors into opportunities for active reconstruction of knowledge under teacher mediation. Progress in accuracy rates was also observed across the three experiments. It is concluded that the integration of ER and block-based programming, when intentionally and reflectively planned, constitutes a promising pedagogical strategy for teaching Trigonometry and results in a replicable didactic guide aimed at Mathematics teachers in basic education.

This dissertation addresses the challenges associated with teaching Arithmetic Progressions (AP) in high school—a subject often characterized by predominantly procedural and decontextualized approaches—and proposes a didactic sequence grounded in the Mathematical Modeling frameworks of Bassanezi and Burak, while aligning with the guidelines of the National Common Curricular Base (BNCC). Adopting a qualitative development-oriented approach, the research aims to analyze high school mathematics textbook series published in 2024 through the lens of Mathematical Modeling and to propose a didactic sequence for teaching AP that adheres to BNCC guidelines. The study involves documentary research to identify BNCC directives for teaching AP and establishes a theoretical framework on Mathematical Modeling to support both the textbook analysis and the development of the educational product. The proposed didactic sequence seeks to move beyond purely traditional methodologies centered on formula memorization; instead, it aims to guide students through a complete modeling cycle—ranging from the initial problem formulation and data collection based on real-world phenomena (such as training schedules or the emptying of PET bottles) to the construction and critical validation of algebraic models and the communication of results. The proposal aims to develop students' ability to identify patterns of constant variation, derive the general term formula for AP, and relate it to discrete affine functions, utilizing simple, low-cost materials. Although the didactic sequence has not been implemented in an actual classroom setting, it is anticipated that its application could foster a more natural, engaging, and critical approach to learning AP, thereby promoting student agency and a deeper understanding of mathematics as a tool for interpreting and intervening in reality.

This study explores the use of Euler's Relation as a pedagogical tool in investigative activities in Mathematics Education. Our objective is to highlight the importance of methodologies that promote critical thinking and student autonomy, commonly considered essential elements. Euler's Relation, valid for convex polyhedra, can be expressed through the formula V - A + F= 2 and emerges as a central concept that can be explored beyond mere memorization, encouraging a deeper investigation into its applications and implications. An extensive analysis of textbooks reveals that this relation is often presented superficially, without stimulating questioning or exploring its validity in different geometric contexts. This dissertation proposes a pedagogical intervention that employs investigative problems as a facilitating tool in this process. In this approach, students are encouraged to discover and validate Euler's equation through practical and theoretical experimentation, contributing to the development of stronger analytical and mathematical skills. As a final outcome, this approach can serve as an alternative resource for modernizing school curricula.

Youth and Adult Education (EJA) presents specific challenges in mathematics teaching, such
as the decontextualization of content and the lack of recognition of students' sociocultural
knowledge. In view of this, this dissertation investigates the intersections between the Theory
of Didactic Transposition (TTD) and Ethnomathematics, seeking to understand how these
approaches can be integrated to make mathematics teaching in EJA more meaningful and
inclusive.
Organized in a multipaper format, this research is composed of three interconnected articles.
The first article, "Intersections between Didactic Transposition and Ethnomathematics: a look
at Brazilian research", maps the academic production on TTD and Ethnomathematics in EJA,
highlighting the predominance of studies on Ethnomathematics and the scarcity of research that
articulates both approaches. The second article, "An epistemological reference model for
analyzing mathematics textbooks in EJA", proposes a methodology to evaluate the adequacy
of textbooks in this modality, based on the references of TTD and Ethnomathematics. The third
article, "Analysis of a mathematics textbook for EJA: connections between Didactic
Transposition and Ethnomathematics", applies this epistemological model to the analysis of a
textbook used in EJA, demonstrating how the transposition of content and the lack of
contextualization can impact student learning.
The results show that the integration between TTD and Ethnomathematics can enrich
mathematics teaching in EJA, enabling more accessible learning that is connected to the
students' reality. Furthermore, as a practical contribution, this research presents an educational
product, a didactic sequence based on these approaches, with the objective of offering support
to teachers in the implementation of more contextualized pedagogical practices. Thus, this study
reinforces the importance of a Mathematics Education that values the cultural diversity of
students, promoting a more equitable and meaningful teaching in EJA.

This dissertation presents the problem of developing mathematical skills in high school, with the theme being the integration of the Python programming language into interdisciplinary teaching sequences. The general objective is to investigate how the Python programming language, integrated into interdisciplinary didactic sequences, can contribute to the development of mathematical skills in high school, in line with the guidelines of the National Common Curricular Base (BNCC), the National Curricular Parameters (PCNs) and the National Digital Education Policy (PNED). The methodology used comprises a literature review, documentary analysis of normative documents for basic education and the proposition of an interdisciplinary didactic sequence, aimed at the 1st year of high school, which articulates the content of 2nd grade polynomial function with the fundamentals of programming in Python. The results indicate that the integration between programming and Mathematics, even in the theoretical-methodological scope, has the potential to favor the development of skills such as abstraction, decomposition, pattern recognition, modeling and logical reasoning, in line with the general and specific skills provided for in the BNCC and the PNED guidelines. The documentary analysis showed convergences between the legal frameworks, indicating openness and encouragement for the inclusion of computational thinking in the basic education curriculum, with the possibility of articulation with mathematical content to promote more meaningful and contextualized learning. The elaborate didactic sequence provides a model of interdisciplinary pedagogical practice that values student protagonism and the critical use of digital technologies. It is concluded that the proposal fulfills the objective of offering a structured framework to integrate Python and Mathematics in High School, serving as a basis for future implementations and research that wishes to explore the potential of this approach to expand students' engagement, conceptual understanding and intellectual autonomy.

The present work addresses the methodological implications of the notion of didactic accessibility in the permanence of mathematical praxeologies for deaf students, analyzing the high school Mathematics textbook. Initially, we will discuss the definition of didactic accessibility according to the conjuncture of conditions that allow the addition of aluminum to the combination, depending on the study format, classroom situations, resources and instructions. Next, mathematical praxeology was detailed as an approach to understanding educational practices and access to mathematical knowledge, highlighting the need for specific adaptations for deaf students. This also includes the use of interactive and gastronomic resources, such as GeoGebra software, to promote visual understanding of mathematical concepts. By adopting pedagogical practices and using tools like GeoGebra, it is possible to create a more inclusive and equitable learning environment. Finally, the research contributes to a discussion about the need to train qualified teachers to meet the demands of accessibility and inclusion, promoting Mathematics Education that considers the specificities of deaf students.

The present research aimed to understand how the use of investigative tasks can enhance the processes of teaching and learning statistics in high school. To achieve this understanding, our research was developed through four specific objectives, namely: to investigate how high school students plan, organize, collect, analyze, and produce a report on sample research; to analyze how students use statistical concepts in producing sample research with contextual reference to reality; to analyze students' performance before and after an investigative task involving sample research with contextual reference to reality; to produce a guidebook for teachers on working with investigative tasks for teaching statistics. We observed that, in their inquiries, students initially ask direct questions related to the research question. Without an interest (or knowledge of the importance of understanding a sample), they do not include delimiting questions for understanding the sample. However, the investigative task proposed to the students allowed for a diverse range of responses and patterns analyzed by them. In their processes of report writing, these students apply some transnumerative techniques such as constructing tables and graphs, calculating summary measures, and classifying variables, in addition to integrating statistics with the context in which they live. Despite the diagnostic tasks not showing significant advancement, we noticed a percentage improvement in performance when comparing the diagnostic tasks before and after the investigative task. The investigative practice facilitated the development of the investigative cycle, aiding in these students' statistical literacy. Practices are necessary for students to attribute meaning to the content worked on, as we propose in our study.

The approval of the National Common Curricular Base - BNCC for Secondary Education, in 2018, brings challenges to the school community, especially to the math teacher, who is faced with the need to break paradigms and reinvent their pedagogical practices. Because, in addition to knowing and understanding their proposals, totally reframed in search of teaching that promotes youth protagonism, it is necessary to develop and execute plans that enable the effective teaching of mathematics useful for the student's life, in order to improve the current scenario that indicates low levels of learning in the area and considering, also, the reduction of weekly workload destined to the discipline. The concerns generated by this context prompted investigations in the fields of Anthropological Theory of the Didactics - ATD and Cognitive Neuroscience - CN, with a special focus on the keys to learning by Jo Boaler, which, in turn, led us to translate the central problematic of this work as : the phenomenon of fragmentation of mathematical knowledge in curricula and the superficiality of mathematical tasks used in school institutions. From this delimitation, the objective was to identify the approximations and distances between the competences and skills proposed by BNCC for the area of Mathematics and its Technologies and teaching based on the six learning keys, as well as investigating the role of mathematical tasks in this scenario , in order to understand how ATD and recent studies on NC can influence the restructuring of teaching practice. In search of answers to these questions, a bibliographical and documentary research was carried out regarding the theories involved, in addition to a commented analysis of the BNCC, directing the eyes to the area of Mathematics and its Technologies. The close relationship between the proposals of Base, ATD and the keys to learning, identified as a result of this study, inspired the creation of a didactic mathematical (alternative) praxeological model that links the aforementioned theories and relates them to the BNCC principles. This model has infographics and diagrams that compile the main ideas defended, as well as a suggestion of curricular organization for high school mathematics, based on central axes. This model was designed to facilitate the teaching work in this period of adaptations, and shows the strong influence of the choice of tasks in the teaching of mathematics, highlighting the urgent need for changes in the way of conducting classes, with the teaching of mathematics as its guide. multiple, flexible and deep, at all praxeological levels (logos and praxis).

This research aimed to investigate the approach to teaching exponential functions in high school. For this, the methodological path of the study comprised: the understanding of how the approaches and discussions on exponential functions, associated with the GeoGebra tool, have been in the most recent Brazilian scientific literature; the analysis of the approach of the tasks on this concept in Mathematics textbooks focused on the aforementioned Basic Education stage and, finally, the proposition of an approach involving tasks with exponential functions through the didactic use of the GeoGebra software for the aforementioned school level. We found that, in the scientific world, the teaching of exponential functions is very much associated with contexts of reality, so that its application as a model for the study of physical (thermodynamic), financial (interest) and biological (bacterial reproduction) phenomena is recurrent in filtered searches. In addition, among the studies analyzed, there is a common concern in allowing students to establish relationships between the semiotic representations of this type of mathematical function, mainly algebraic and graphical. In this regard, the main didactic contribution, derived from the association between exponential functions and GeoGebra, is the experimental character that this software prints in the classroom, allowing students to develop their autonomy, starting to perceive relationships between different representations. We verified that the textbooks present, exclusively, tasks in the form of exercises and problems. Thus, they lack tasks with an open structure, which allow students to experience, during classes of exponential functions, investigative processes. In this way, they could exercise their capacity for conjecture, reflection and discovery through experimentation, instead of receiving and applying ready-made definitions and properties in an objective manner. We understand that such an experimental character can be achieved with the help of GeoGebra, as pointed out by recent research and the tasks proposed in this study.

The Chinese Remainder Theorem was a tool used, historically, to solve a problem about counting soldiers. It is basically an equivalence relationship between remainders. Nowadays, in addition to being part of the study of Arithmetic, it is used in encryption and password sharing, for example. Considering the current context, in which more and more technologies are used

in the teaching process, the purpose of this work is to make the didactic about the Chinese Theorem of Remains more playful, for this, an algorithm is developed that lists the step by step to solve problems of linear congruence systems and, then, a program capable of executing this script and generating a file with the registration of the calculations is implemented. To compose the theoretical framework, the concepts of divisibility, prime numbers and congruence are addressed, presenting the related definitions and solving examples. Then, we present the Chinese Remainder Theorem, the proposed algorithm, and demonstrate its effectiveness by solving problems with the computational implementation.

This research is a master’s thesis with the general objective of proposing andapplying an approach to the study of the Fundamental Theorem of Algebra (TFA) forhigh school teachers and students in an accessible and intuitive way. As problematic: How to approach the fundamental theorem of algebra for teachers and high school students in
an accessible and intuitive way? This is a qualitative, exploratory research in the form of a case study (GIL, 2002; YIN, 2001). As specific objectives we list: Investigate ways of working with TFA with high school students; establish a relationship between algebraic and graphical representations using TFA. We verified that difficulties may arise in working the proofs of this theorem with high school students due to the theoretical load required, especially with regard to topics on topology, outside the basic education syllabus. The results of this theorem, on the other hand, can be explored with these students, enhanced by the use of tools such as GeoGebra that allow a clear association between algebraic and graphical constructions.

The Teaching of Mathematics is marked by the learning difficulty and the rejection of students who do not find motivation to study it, reflecting on the students’ academic performance, as can be seen in the results of the main external evaluations, such as PISA. Thus, in an attempt to contribute to changing this situation, the present work seeks alternatives and proposals for teaching methodologies through the support of digital resources. The popularization of Mobile Digital Technologies - TDM (smartphones and tablets) is responsible for several transformations in contemporary society that are reflected in the school. Thus, it is proposed to evaluate, analyzing the quality, and relate apps with Mathematics worked in the first year of High School, in addition to suggesting pedagogical activities with the aim of using technology in favor of mathematical learning. In doing so, it is clear that TDM is an opportunity to favor an integration between the teaching-learning process and digital technologies, thus contributing to progress in the teaching of mathematics.

The Set of Irrational Numbers, although little exposed in textbooks, is of fundamental impor-
tance for mathematics, set this set has characteristics such as: density in relation to the set of
reals, infinite cardinality and non-enumerability. The irrationals have a greater measure than
the set of rationals, in a probabilistic analysis the choice of an irrational in an interval is almost
a event certain, in addition to the great presence in the real line, the irrationals are also present
in various objects and everyday situations , for example, on the A4 sheet of paper, in interest
capitalized continuously, in the ratio of the length of a circumference to its diameter, which can
be explained through the Geogebra software, but it should be noted that the set of irrational
numbers is not closed for the operations of addition and multiplication, hence, it is necessary a
thorough analysis of its arithmetic, substantiated with examples and applications. The BNCC
emphasizes the importance of locating irrational numbers on the real line, for which one of the
ways is the approximation by rationals that can be given through manipulations of continuous
fractions that can be obtained more quickly by Geogebra. The study of irrational numbers can
also be carried out through the theory of algebraic and transcendent, a theory that has led to
the solution of several problems, for example, the three classic problems of Euclidean geometry,
namely, duplication of the cube, general trisection of angles and square the circle. In general,
the study of irrational numbers leads to incursions in several topics of mathematics, which
makes its application in high school something essential, therefore, this work aims to build a
didactic material in which teachers can from the same to realize the importance of irrational
numbers and carry out didactic transposition for the classroom use of the highlighted numbers.

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This research aimed to investigate how the scientific literature, published in journals with scope in Mathematics Education, and textbooks, approved by the National Book and Didactic Material Program (PNLD), approach the topic of probability in Basic Education. In a textual structure in a multipaper format, this study was developed in three stages, each described by means of an article, with individual characteristics, which, its objectives are consequences of the general objective. The first article aims to analyze how the literature published in national journals, with scope in Mathematics Education, discusses the teaching and learning of probability in Basic Education. 29 articles were analyzed, which resulted in a synthesis on the teaching of probability based on the relationship between the categories discussed and the different conceptions on the theme. The objective of the second article is to analyze how the topic of probability is addressed in three high school textbooks approved by the National Book and Didactic Material Program (PNLD) 2018-2020. Among the results, it was highlighted that some approaches are more emphasized than others, in a perspective that encompasses the conceptions of probability. With regard to the third article, its objective is to analyze three tasks, two extracted from the textbook and a game of its own elaboration. This analysis was based on task markers described by Barbosa (2013) and expanded by Costa, Oliveira and Silva (2017), these being: the context of reference of the task, the use of language, the structure, the teaching objective, the pedagogical relationship and teaching focus. The aspects that make up the theoretical model called Mathematics Teacher’s Specialized Knowledge (MTSK) were also considered for analysis under the perspective of Carrilo et al. (2013) and Moriel Júnior and Carrillo, 2014). As a result of this third study, it was highlighted that the implementation of probability tasks when well planned and articulated can bring good results, and promote learning situations. In addition, it assists teachers in the mobilization of concepts and techniques in order to ensure that the student develops his learning capacity with autonomy. As a result of all the research, presented individually in each article, and thinking about the contribution to the pedagogical practice, a booklet was elaborated with the purpose of guiding the teacher on the ways of seeing the teaching and learning processes of the probability theme and reflecting on the their professional practice.

This work has as object of study the National High School Exam (ENEM). Due to the relevance that this exam has been conquering in the face of educational policies and educational institutions, the work aimed to understand how it has produced effects in the teaching and learning process, notably in the area of Mathematics and its Technologies. It has exploratory character, to analyze the Exam in terms of historical, theoretical and methodological aspects, as well as questions contained in the test in this area, bibliographical and documentary research was carried out. In order to identify the possible effects of the exam on the teaching and learning process in Mathematics area, field research was developed, with an approximation of the qualitative approach, having as locus two high schools located in Barreiras-BA city, one public and another one private. The research participants were teachers, pedagogical coordinators and students, who answered an online questionnaire, hosted on google. The information and data have pointed out, among other things, that the Exam: (i) it is a national and local reference exam; (ii) the Mathematics questions contained in the test are contextualized and consistent with MEC guidelines; (iii) it takes effect on the educational process, making it focus on the conduction of student learning.