The present research has the following question: What are the effects of applying  a didactic sequence based on the Theory of Semiotic Representations Records  for teaching Measures of Central Tendency on students in a 9th grade class of  Elementary School? To answer this, the general objective is to analyze the effects  of a didactic sequence in teaching Measures of Central Tendency and its  implications for the learning process of 9th grade students of Elementary School.  The specific objectives were established: to conduct a study of official documents  on the teaching and learning of measures of central tendency, to analyze the  results obtained by the didactic sequence and to construct a didactic sequence  from the perspective of the Theory of Semiotic Representation Records. The  research was based on Duval's Theory of Semiotic Representation Records  (2012) and the Didactic Engineering of Artigue (1996) is being adopted as the  methodological framework, based on its four phases. In the preliminary analyses,  the study of official documents, the literature review, the mathematical object and  the records of semiotic representation and the knowledge of the students are  being worked on. In the a priori analysis of the activities, we produced a Didactic  Sequence related to the content of Measures of Central Tendency (Mode,  Arithmetic Mean and Median), based on the assumptions of the Theory of  Records of Semiotic Representation, in which the concepts to be worked on  through their representations in natural, tabular and graphic language. In the last  stage of the study, which is the experimentation and a posteriori analysis of the  activities, we will present the results, analyses and conclusions obtained after the  application of the didactic sequence and the knowledge of the students.

This study aims to analyze how Artificial Intelligence (AI) tools can support and reshape pedagogical work in Mathematics teaching. It is grounded on the premise that AI, as discussed by Oliveira (2018), Santos et al. (2023), and Coelho (2024), has been impacting different dimensions of social and educational life. To investigate this transformation in the context of teachers’ work, a bibliographic study and a systematic literature review were conducted with the support of the Parsif.al platform, in addition to the application of a questionnaire (closed- and open-ended questions) to 135 Mathematics teachers in Basic Education, seeking to understand practices and the most time-demanding activities. The results indicated recognition of AI’s potential to support lesson planning, the production of instructional materials, and assessment, as well as challenges related to infrastructure, teacher education, and ethical issues, as highlighted by Bates and Sangrà (2011) and O’Neil (2016). Within a normative and formative framework, this study incorporated UNESCO’s AI Competency Framework for Teachers (2024/2025), which defines the knowledge, skills, and values required for AI use in teaching, outlining 15 competencies across five dimensions, organized into progression levels (acquire, deepen, and create), with emphasis, among other aspects, on a human-centered mindset and AI ethics. In light of this scenario and the needs expressed by the participants, the study proposed the development of a practical manual to support Mathematics teachers in the critical and responsible use of AI tools, aiming to contribute to the organization of pedagogical work. It is concluded that AI integration requires validation criteria, didactic intentionality, and continuous professional development in order to expand possibilities without replacing teachers’ professional judgment.

This research aimed to diagnose the learning difficulties of high school students in relation to problems involving quadratic equations. The study assumes that the teaching of this content often prioritizes the memorization of formulas and mechanical procedures, to the detriment of conceptual understanding and the ability to perform mathematical modeling. To this end, a quantitative and descriptive investigation was carried out, applying questionnaires and tests to 86 first-year high school students from public schools in the municipality of Ananindeua, Pará. The data collection instrument included questions about socioeconomic profile, study habits, recall of mathematical concepts, and a test with problems classified into easy, medium, and difficult levels. The data were analyzed using descriptive statistics and Fisher's exact test to verify associations between variables. The results indicated that, although most students recalled basic concepts such as Bhaskara's formula (94%), the perception of difficulty with these topics was predominantly classified as "regular" or "difficult". In the applied test, a high rate of blank answers (59.6%) and errors (25.9%) was observed, with only 14.5% correct answers. The greatest difficulties were concentrated in the interpretation and modeling of contextualized problems, especially in translating natural language into algebraic language. Statistical analysis did not reveal significant associations between performance and variables such as sex, age, or parental education, except in specific cases. It is concluded that the students' difficulties are more related to weaknesses in conceptual understanding and problem-solving than to isolated socioeconomic factors. The study recommends the adoption of pedagogical approaches that integrate multiple representations, historical contextualization, and an emphasis on problem-solving, aiming for more meaningful and less mechanical learning. As a practical contribution resulting from these conclusions, a complementary educational product was developed to help teachers and students overcome the identified modeling and interpretation difficulties.

This dissertation was developed within the Professional Master's Program in
Mathematics Teaching at the State University of Pará (UEPA). The research arises
from the need to investigate the importance of the Pythagorean Theorem in
Elementary School, as it is a central mathematical object related to various stages
and areas of knowledge. The guiding question seeks to understand the
characteristics of the teaching, learning, and assessment process of the Pythagorean
Theorem in the 9th grade of public schools in the State of Pará, from the teachers'
perspective. The general objective was to diagnose the difficulties presented by
students in the teaching-learning process of this theorem, based on the viewpoint of
mathematics teachers. The adopted methodology is characterized as a diagnostic
study with a mixed approach (qualitative and quantitative). The methodological path
was structured in seven stages, including a pilot test with 30 students for instrument
validation and the definitive application of an electronic questionnaire via Google
Forms to 32 public school teachers from municipalities such as Belém, Ananindeua,
and Abaetetuba. Data analysis included inferential statistical treatment using Fisher's
Exact Test in the JAMOVI software. The results reveal that student difficulties are
influenced by gaps in continuing education and a shortage of didactic resources. The
analysis via Fisher's Exact Test proved that teacher profile variables—such as
education level, length of service, and participation in courses—have no significant
association with their practices or perceptions, highlighting that the mapped
obstacles are of a structural and systemic nature within the Pará public school
system. Inability to perform basic operations was identified as the students' most
frequent failure. Based on the bibliographic surveys and the diagnostic results, this
work also presents an Educational Product consisting of a sequence of seven
experimental activities, structured to enable meaningful learning and reduce the
difficulties reported by teachers. In summary, the research demonstrates that gaps in
basic content and the limited use of diversified methodologies are the main obstacles
to mastering the Pythagorean Theorem.

This analytical dissertation addresses the following research question: What
difficulties do elementary school students encounter in the teaching-learning
process of Quadratic Equations? Its objective is to analyze the teaching of
quadratic equations to elementary school students. As a theoretical foundation, we
draw upon the works of Moran (2017) and Tahan (1961), and as methodological
procedures, we employ literature reviews, field research using a questionnaire to
profile the students, and a learning assessment test with questions related to
essential topics in the Quadratic equation curriculum, divided into difficulty levels,
targeting students at a public elementary school located in the city of Belém, Pará,
which included the participation of 86 students in the 8th and 9th grades, aged
between 12 and 16 years old. The main findings of this study show that the
teaching of Quadratic Equations is significantly affected by students’ difficulty in
coping with the level of abstraction required by Algebra, in understanding the
applicability of Mathematics to their daily lives, and in the need for methodologies
that stimulate mathematical reasoning through meaningful strategies. We conclude
that the difficulties in teaching Quadratic equations highlight the need to improve
the teaching-learning process through more dynamic and engaging approaches. In
this regard, methodologies that stimulate students’ interest and inquiry, combined
with teachers’ initiatives to supplement the guidelines in official documents, prove
to be fundamental. The development of teaching strategies applicable in the
classroom thus contributes to more meaningful and effective learning of
Mathematics.

The teaching of Geometry in lower secondary education (Grades 6–9) often relies on the memorization of formulas, distancing itself from an investigative and visual approach. Such practice results in learning difficulties and student disengagement. This study proposes a didactic intervention for the study of polygons in the 9th grade, using geometric drawing and manipulable materials as tools for active and meaningful learning. The research is grounded in Didactic Engineering (Artigue) and the Theory of Didactic Situations (Brousseau), which guide the design, implementation, and analysis of a didactic sequence. The general objective is to design, implement, and analyze the effects of this sequence, focusing on the property of the sum of the interior angles of polygons, in a 9th-grade class at a public school in Belém. The specific objectives include designing the sequence, implementing it in the classroom, analyzing the evolution of students’ conceptions regarding triangulation and the generalization of the formula, and evaluating the contribution of this approach to active participation and mathematical argumentation. The sequence is structured through progressive practical activities, from triangles to polygons with a greater number of sides, mediated by the teacher. It is expected that this research will contribute not only to students’ learning, but also offer a reflective model of a didactic sequence that encourages more investigative methodologies in the teaching of geometry.

ABSTRACT This dissertation had as its research problem understanding how the teaching of metric relations in right triangles, in the 9th grade of Elementary Education, has been occurring in a public school in the municipality of Magalhães Barata-PA. In light of this scenario, the general objective was defined as analyzing the teaching of this content with a view to identifying how it is being developed in the aforementioned school, unfolding into the specific objectives of identifying the profile of 9th grade students, ascertaining how they report having had access to the content, verifying the results of a learning assessment test, and developing an educational product aimed at improving this teaching. Methodologically, the research is characterized as applied, with a quali-quantitative approach, drawing on theoretical-methodological contributions from authors such as Gil (2008), Creswell (2014), and Lakatos and Marconi (1996), in addition to the pedagogical foundations of Shulman (2014) and the TPACK model by Mishra and Koehler (2006). Data were collected through a structured questionnaire and an assessment test applied to students who had already studied the content, complemented by a validation questionnaire answered by practicing Mathematics teachers. The results showed that students maintain a predominantly unenthusiastic relationship with Mathematics, associate assessments with feelings of tension, and presented unsatisfactory performance on the assessment test, with an error rate higher than the success rate on all questions, even the most elementary ones, indicating weaknesses in the consolidation of the content on metric relations in right triangles. Based on this diagnosis, the educational product Metricinha was developed, composed of a didactic guide and a mobile application grounded in the TPACK model, whose validation with Mathematics teachers indicated unanimous approval and a broadly positive evaluation, with a specific suggestion for improvement related to the algebraic handling of the application's calculators. It is concluded that, in the reality investigated, the teaching of metric relations in right triangles still privileges the procedural application of formulas over conceptual understanding, reinforcing the need for pedagogical practices that articulate content, pedagogy, and technology. In this sense, the educational product Metricinha is established as a viable and well-evaluated alternative for improving this teaching process, contributing both to teaching practice and to the field of research in Mathematics Education. As necessary follow-up steps, the application of the product directly with 9th grade students and the improvement of the algebraic handling of the application's calculators are indicated as promising paths for future research. 

This dissertation presents the results of research that addressed the following
question: What are the effects of a didactic sequence based on experimental
activities on the learning of First-Degree Equations by 7th-grade students in
Elementary Education? The overall objective was to analyze the pedagogical
potential of a didactic sequence grounded in experimental activities for teaching First-
Degree Equations in the 7th grade. The investigation, structured according to the
principles of Didactic Engineering by Michelle Artigue, encompassed the stages of
Prior Analysis, Didactic Sequence and A Priori Analysis, Experimentation, and A
Posteriori Analysis. The theoretical foundation rested on three pillars: the Teaching of
Mathematics through Experimental Activities proposed by Sá, Raymond Duval's
Theory of Registers of Semiotic Representation, and First-Degree Equations with
one unknown (Ribeiro, 2007; Barbeiro, 2012; Kieran, 1992). The Didactic Sequence
was applied to 31 students from a 7th-grade class in a public school in Belém over
seven teaching sessions, integrating lists of experimental activities, historical videos,
and collaborative discussions. The methodology prioritized progressive knowledge
construction through activities that explored the notion of equality, the additive and
multiplicative principles of equivalence, and formal resolution methods. The results
showed significant learning gains, substantial improvement in post-test performance
compared to pre-test performance validated by Student's t-test, and drastic reduction
in unanswered questions. Progress in conversion between natural language and
algebraic register was evidenced by the comprehension of equality principles and
correct application of the transposition method. Additional analyses revealed
significant association between participation frequency in activities and post-test
performance, as well as between video use and the ability to relate content to
everyday situations. As an educational product, a Didactic Sequence was developed
with detailed scripts and possibilities for adaptation to different school contexts.

This study aimed to investigate the teaching of the right circular cylinder in basic education and to analyze the potential of applying a Didactic Sequence (DS) designed to teach this topic to a third-year high school class at a state school in Belém. Specifically, the goal was to examine the potential of a DS involving the right circular cylinder for this student group. The methodological framework drew upon the concepts of Didactic Sequences proposed by Zabala (1998) and Cabral (2017) specifically the structure of Articulated Units of Conceptual Reconstruction (UARC) while utilizing manipulatives such as plastic bottles shaped like *Matapis* (traditional cylindrical fish traps) to support the DS implementation. To identify challenges in teaching and learning about cylinders, the study included a literature review on plane and spatial geometry education, research into textbooks and educational materials, and surveys involving students who had previously studied the topic and basic education teachers. To address identified difficulties and knowledge gaps, activities involving cylinders were developed as part of the DS, aiming to mitigate these issues. Data analysis focused on the workshops and revealed evidence of learning within the verbal interactions among students, teachers, and the subject matter. The results indicated significant learning, as students grasped the formulas and concepts through their own investigations supported by the plastic manipulatives and activity sheets used in the DS. This demonstrated the effectiveness of the developed sequence for teaching the right circular cylinder using the *Matapi* model, leading to the conclusion that the applied methodology offers a viable and effective alternative for teaching this topic to students

This dissertation investigates possibilities for teaching Plane Geometry in the context of Rural Education, with emphasis on the concepts of area and perimeter and on an approach grounded in Mathematical Modeling and Gamification. The research was conducted in the community of Jupuuba, located in the rural area of the municipality of Moju, Pará, with 18 first-year high school students from a public school that operates under the Modular Teaching Organization System (SOME). The study aimed to contribute to the teaching and learning process of Plane Geometry through a proposal designed to connect mathematical knowledge with the experiences and spaces of rural students. Methodologically, the study is characterized as qualitative and intervention-based research, developed through the design and implementation of a didactic sequence consisting of five activities carried out over five weeks. The activities addressed area, perimeter, and properties of plane geometric figures through the manipulation of the Tangram, the construction and investigation of circles, the use of the digital tools Plickers and Mathigon–Polypad, the representation and analysis of plots of land related to rural practices, the measurement and planning of real spaces, and the creation of geometric art and construction of kites. Data were produced through a field diary, activity sheets and students’ written records, photographs, audio recordings, digital productions, and responses collected using Plickers. Analysis of the evidence indicated that Mathematical Modeling was particularly relevant in situations in which students mobilized geometric knowledge to represent spaces, compare possibilities, interpret situations related to their context, and support decision-making. Gamification, in turn, contributed mainly to organizing situations involving challenges, participation, and feedback, although the results also showed that engagement and learning are not equivalent dimensions. Difficulties were identified in calculating area, interpreting problem situations, and distinguishing between area and perimeter, indicating the need for revisiting concepts and teacher mediation throughout the process. It is concluded that the connection between Mathematical Modeling, Gamification, contextualization, and pedagogical mediation can broaden the possibilities for approaching Plane Geometry in Rural Education by fostering situations in which students mobilize mathematical knowledge in an investigative manner and in connection with concrete situations, without implying that these approaches automatically guarantee learning.

This research aims to analyze and identify the potential of a didactic sequence,
structured according to the Articulated Units of Conceptual Reconstruction (UARC),
aimed at teaching the concept of rate of change. The investigation seeks to answer how
this intervention can favor the learning of 1st-year high school students in the state
public school system. The theoretical framework that supports the intervention is based
on the Theory of Didactical Situations (Brousseau), the concepts of Didactic Sequence
(Zabala), and UARC (Cabral), articulated with Microgenetic Analysis (Góes) and
Discourse Analysis (Mortimer; Scott) as investigative lenses. Methodologically, the
diagnostic stage was carried out at the Augusto Meira State High School, involving
questionnaires administered students (in person), as well as a knowledge test. The
diagnostic results showed that, although students recognize the teachers' notorious
content mastery, there is an affective detachment regarding the subject and significant
gaps in understanding the mathematical object in question. Given the lack of specific
literature on the topic, the study ratifies the essential need to explore the rate of change
through systematized pedagogical approaches, aiming to mitigate teaching and
learning difficulties in Basic Education.

This dissertation presents the methodological path and the results of a study conducted within a Professional Master’s Program in Mathematics Education on the teaching of regular polygons in Elementary Education, seeking to answer the following research question: What potentialities does a didactic sequence aimed at teaching the inscription and circumscription of regular polygons, developed based on the Articulated Units of Conceptual Reconstruction (UARCs), present for ninth-grade students in the state public school system? The general objective was to identify the potentialities of a didactic sequence, developed according to the UARCs, for teaching the inscription and circumscription of regular polygons. The study is characterized as applied research, with a qualitative, exploratory, and descriptive approach. Theoretically, it was grounded in Brousseau’s Theory of Didactical Situations (2008) and Cabral’s Articulated Units of Conceptual Reconstruction (2017); methodologically, it was supported by Góes’ Microgenetic Analysis (2000) and Mortimer and Scott’s Discourse Analysis (2002). The methodological path involved a literature review, an analysis of official educational documents, an analysis of three ninth-grade mathematics textbooks, an investigation of students’ perceptions of regular polygons, the development of a mathematical text on the object of study, the construction of the didactic sequence, its implementation with an Elementary Education class, and the analysis of the dialogues and written records produced by the students. The results indicated that the literature emphasizes investigative approaches to the teaching of Geometry, while the textbooks analyzed, although offering contributions regarding visual representations, contextualization, and construction activities, still reveal limitations concerning the development of mathematical argumentation, geometric deduction, and the relationships among side length, radius, and apothem. In addition, the analysis of students’ perceptions revealed conceptual difficulties related to the identification of properties and metric relationships of regular polygons. During the implementation, the didactic sequence showed evidence of conceptual reconstruction, manifested in the identification of inscribed and circumscribed polygons, the mobilization of relationships among side length, radius, and apothem, the retrieval of previously constructed knowledge across the UARCs, and the interactive patterns of initiation, response, feedback, reformulation, and appropriation. It is concluded that the didactic sequence presented potentialities for teaching the inscription and circumscription of regular polygons by fostering student participation, teacher mediation, the articulation between visual perception and mathematical formalization, and the progressive construction of geometric meanings. Finally, this dissertation resulted in the development of an Educational Product entitled Regular Polygons: Inscription and Circumscription, available at the link below.

This research investigates the potential of an integrated didactic sequence approach for teaching the calculation of areas of plane figures, mediated by the use of dynamic geometry, with emphasis on the GeoGebra software. The study starts from the observation that the traditional teaching of Plane Geometry, frequently based on the memorization of formulas, has contributed to persistent difficulties among Basic Education students in the conceptual understanding of areas and perimeters. Based on analyses of ENEM (Brazilian National High School Exam) results and observations in school contexts, it is evident that many students show low performance in questions that require mastery of the geometric properties of plane figures. In this context, a teaching and learning sequence based on investigative and exploratory practices, supported by GeoGebra, is proposed, which allows for the dynamic manipulation of figures, the visualization of properties, and mathematical experimentation. The research, of a qualitative nature, will be developed with 9th-grade students from a public school, analyzing student records and productions throughout the application of the didactic sequence. This approach is expected to contribute to overcoming mechanistic teaching, promoting meaningful learning, the development of logical-spatial reasoning, and a conceptual understanding of calculating the areas of plane figures, in accordance with the guidelines of the National Common Curriculum Base.

This research project addresses the teaching of round bodies with the aid of GeoGebra software, with the aim of developing and testing a didactic sequence for teaching round bodies using GeoGebra. The study aims to investigate the impact of using a didactic sequence with GeoGebra software on enhancing the learning of round bodies. This is relevant because Mathematics, especially in terms of spatial geometry, is often considered one of the most challenging subjects in the school curriculum, as reported by many high school students. These challenges arise from students' difficulty in identifying and understanding geometric objects, both planar and spatial, and performing calculations related to these objects. To achieve these objectives, the research adopts a qualitative approach, based on Rabardel's instrumental approach, Semiotics, and microgenetic analysis methods. The research has the potential to contribute to the improvement of pedagogical practices by enabling a more practical and contextualized approach to teaching round bodies. It aims not only to evaluate the effectiveness of using GeoGebra in combination with a structured didactic sequence but also to create an educational product that supports mathematics teachers

This master's dissertation presents the results of a research developed in the Professional Master's Program in Mathematics Teaching (PPGEM), of the State University of Pará, intending to answer the following problem: Does a didactic sequence using a Learning Object created in GeoGebra, enhance the learning of operations with integers? The research aimed to analyze what effects a didactic sequence with the use of a Learning Object created in GeoGebra has on the learning of operations with integers. The research methodology used was the assumptions of Artigue's Didactic Engineering, which has four phases: (I) Preliminary Analysis, we present a study on the state of the art, where research was carried out on works on the subject of operations with integers, in this phase we also presented historical factors about integers and their operations, and we presented the GeoGebras software that is the generator of the learning object of our research. (II) Conceptions and Analyses a Priori, in this phase, the Teaching of Mathematics by Activities was chosen as the theoretical foundation, which will be the teaching methodology that will support the sequence of activities produced.  (III) Experimentation. We applied a sequence of eight rediscovery activities, where students had to use the Learning Object, to apply the commands determined in the activity script. (IV) Validation and Posterior Analysis, in this phase we validated the results achieved in the experimentation, considering that the approach used made it possible to improve the effectiveness of the students' learning. The results obtained showed a positive qualitative performance of the subjects involved, as the activities contributed to the development of cognitive and practical skills.

The integration of Digital Information and Communication Technologies (DICT) in education has significantly changed pedagogical practices, especially in the teaching of mathematics. Among these tools, GeoGebra stands out as a widely used software due to its dynamic and interactive interface that allows for visual and investigative exploration of mathematical concepts. This study aims to analyze the applicability of an activity guide developed with GeoGebra for teaching metric relations in right triangles, based on Raymond Duval's Theory of Semiotic Representation Registers (TRRS) and Shulman's TPACK model. The research is characterized as applied, exploratory and qualitative-quantitative, structured through a theoretical review, the development and application of the activity guide, followed by the analysis of responses from teachers in public and private schools in the state of Pará, Brazil. The results show that the use of GeoGebra improves the visualization and understanding of metric relationships, promotes the integration of different registers of mathematical representation, and strengthens teaching through more dynamic and meaningful practices. The study also highlights the importance of educational materials, such as the proposed activity guide, and points to the feasibility of its application in different educational contexts, contributing to teacher training and student learning. Finally, it recommends the development of new guides for other mathematical topics, such as systems of equations, quadratic functions, triangle similarity, and analytic geometry.

This research was based on the importance of financial mathematics in everyday life and the need to understand how teachers deal with teaching and learning the concepts of simple interest in high school classes in public schools in the state of Pará, carried out during the Professional Master's Degree in Mathematics Teaching, in which the following research question was sought: What potential does a didactic sequence designed to teach simple interest to high school students in public schools, structured according to the Articulated Units of Conceptual Reconstruction, in line with microgenetics and discourse analysis, have? In order to obtain elements that would answer this question, the general objective was to analyze the potential of a didactic sequence, structured according to the Articulated Units of Conceptual Reconstruction, for high school students in public schools, with a view to identifying potential from the data resulting from its application and the analysis of discourse and microgenetics. The research included a literature review, field research with teachers, analysis of textbooks, and a historical and epistemological study of the mathematical subject of Teaching Simple Interest. The methodology used contributed to a better understanding of the teaching of this subject. The inclusion of simple interest in official documents that guide education reflects its importance, especially in relation to the skills for making financial decisions, managing personal finances, and preparing for success in their personal and professional lives. Different methodologies were found in textbooks that contributed to the development of this proposal. The expected results include the identification of challenges faced by teachers, effective pedagogical strategies, perceptions of student performance, and suggestions for improving the teaching of these concepts. This proposal showed the desired potential and can contribute to the process of teaching simple interest, encourage the use of other pedagogical practices, and contribute to the promotion and success of students in mastering financial mathematics. It is worth noting that the final research report generated an Educational Product entitled SIMPLE INTEREST – A TEACHING SEQUENCE, available at www.produtoeducacional.ppgemuepa.br.

This paper presents the results of a research that sought to answer the following question: What are the results of the application of a didactic sequence, based on the methodology of Teaching through Experimental Activities for teaching power, in the learning of 6th grade elementary school students, in terms of conceptual and procedural knowledge of this mathematical object? The objective was to investigate the results in the learning of 6th grade elementary school students, from the application of a didactic sequence designed for teaching and learning power, based on the methodology of Teaching through Experimental Activities. The research method used was Didactic Engineering, consisting of four phases: prior analysis; design and a priori analysis; experimentation; and a posteriori analysis and validation. The preliminary analyses present the results of the survey on the teaching of exponentiation, based on its historical, mathematical and curricular aspects, as well as some discussions of a didactic and epistemological nature. In addition, a survey was conducted using an online questionnaire with 70 mathematics teachers working in the public school system. The design and a priori analysis phase results in the development of tests and the didactic sequence, which culminated in the production of a didactic sequence based on the principles of Didactic Engineering and Teaching through Experimental Activities. In the experimental phase, a pre-test and a post-test were applied to two 6th grade classes of elementary school, which together had 60 students, and which took place at a public school in the municipality of Ananindeua, located in the metropolitan region of Belém. The information produced during the experimentation was systematized and analyzed both qualitatively and quantitatively. The analysis and validation of the results was carried out using data collected by checking the records of the students' conclusions in each activity carried out, by comparing the results of the post-test with the pre-test; analysis of the errors that occurred in the post-test, by applying Fisher's Exact Test, Cramer's V and the Hypothesis Test for paired samples. The results of the analysis of the application of the proposed Didactic Sequence indicate that it was successful. The results obtained by Fisher's Exact Test and Cramer's V proved that the socio-educational factors did not significantly interfere in the students' performance, since the correlation measured by Cramer's V between the socio-educational variables analyzed and the results of the post-test showed that it varied from weak to moderate. The Hypothesis Test statistically proved that the increase in the scores in the post-test occurred due to the application of the didactic sequence used, concluding that the methodologies used are directly related to the result obtained in the experiment. In addition, in relation to the students, the proposed activities were well evaluated.

This work presents a research proposal on the teaching of Combinatorial Analysis at secondary level that seeks to answer the following research question: “What are the possible effects that the application of a Didactic Sequence, based on problem solving and the use of activities experiments, can have on the understanding and performance in solving questions on the subject in high school students?”, The objective of the study is to analyze the possible effects that the application of a Didactic Sequence, based on problem solving and the use of experimental activities, may have on the understanding and performance in solving questions on the subject in high school students. The research methodology that will be adopted is Didactic Engineering. The experiment is expected to be carried out in at least one high school class in the state network in the urban region of the municipality of Parauapebas in Pará. The theoretical framework adopted in our research is based, among others, on the work of Almouloud and Silva (2012 ), Artigue (2018), Rosas (2018), Sá (2019), Sá (2022) and Sá, Mafra e Fossa (2022). The results of the Preliminary Analysis indicate that according to teachers in the city, the subjects that students have the most difficulty with involve solving problem situations about Simple Arrangement and solving problem situations about Simple Combination. The review of studies on the subject shows that many issues related to the content on screen could be approached in different ways in the classroom, using different resources and adopting diferente methodologies, with the possibility of presenting better results in the teaching- learning process. The results of the work to be developed are expected to produce an experimental study on the teaching of Combinatorial Analysis through Experimental Activities and the development of an educational product in the form of a Didactic Sequence for teaching the aforementioned subject.

This research is an extension and in-depth study of an undergraduate thesis in Mathematics, addressing the challenges related to consolidating students' understanding of exponential functions. Among the various methodological alternatives for teaching mathematics, the use of games has garnered interest from educators due to its motivating and playful nature. In this context, the Roleplaying Game (RPG) emerges as a potential tool for implementation in school settings. However, this modality is still not widely known among teachers. Envisioning the possibility of interrelating exponential functions and RPGs, we posed the following research question: What variation of a Roleplaying Game can be developed for high school students to consolidate their understanding of concepts related to exponential functions? To gather evidence and arguments to answer this question, the general objective was defined as: To develop a Gamebook, a variation of an RPG, for high school students, with the aim of consolidating their understanding of content related to exponential functions. This research is classified as Applied in nature, as it aims to generate knowledge for practical application toward solving specific problems. Its approach is Qualitative, focusing on developing a deeper understanding of a problem that cannot be measured solely by numbers. The objectives are Exploratory, as it investigates possibilities and scenarios to gain familiarity and new insights. The procedures are Bibliographical, based on a literature review of theoretical references published in physical or electronic media. The analysis of textbooks revealed that many questions are of the "manipulation" type, characterized by algorithmic calculations with an emphasis on routine dexterity, systematization, and the application of rules and formulas. In summary, the textbooks present a similar structure with few variations. Therefore, this proposal was developed to fill a gap identified in the analyzed teaching materials. The design was based on Brousseau's (1996, 2008) Theory of Didactic Situations and the use of games in mathematics education as described by Grando (1995, 2000). The articulation between these frameworks guided the model for creating the RPG proposal. After highlighting the potential of RPGs in education and selecting the Gamebook as the chosen variant—due to its ease of implementation by teachers unfamiliar with RPGs—the Gamebook titled “Contaminant Conspiracy” was developed. The game was implemented with graduating high school students, and the validation results demonstrated its effectiveness in promoting engagement, collaborative thinking, and the meaningful learning of an abstract concept through narrative. Furthermore, it was evident that the proposal contributed to the development of the participants' reading and textual analysis skills. Based on the results, adjustments were made to the material, leading to the creation of an Educational Product, available at the link provided2

This work presents the results of a research project developed within the Professional Master’s Program in Mathematics Education (PPGEM) at the State University of Pará, which seeks to answer the following question: how can Financial Education be introduced in high school in the riverside communities of Marajó in a contextualized, meaningful way that promotes autonomy and citizenship? Accordingly, this dissertation aims to investigate the potential of a didactic sequence in Financial Education related to mathematical concepts, designed from the riverside sociocultural reality of the Marajó archipelago, within the teaching and learning process of high school students. Grounded in the principles of Mathematics Education and in Freirean assumptions of a liberating education, the research proposes a contextualized approach capable of integrating mathematical, social, and cultural knowledge into the students’ civic formation. The investigation included a literature review that revealed a lack of studies addressing this topic in the Marajó region and underscored the need for academic research in riverside contexts. Methodologically, it is characterized as qualitative and exploratory research, whose main strategy was the design, implementation, and analysis of a didactic sequence structured according to the model of Articulable Units of Conceptual Reconstruction (UARC). The activities developed addressed Financial Education topics linked to students’ daily lives—such as consumption, financial planning, and conscious decision-making—aiming at the construction of critical mathematical thinking. The results showed that the use of UARC favored the articulation between theory and practice, fostering a dynamic, participatory, and meaningful learning environment. It was also observed that students’ interest in Mathematics increased, along with the perception that financial knowledge can serve as a tool for critical understanding and transformation of reality. The research therefore revealed that Financial Education teaching, when contextualized to the Riverside territory and integrated with Mathematics, contributes to students’ holistic development and reaffirms the school’s role as a space for emancipation and cultural appreciation. It is concluded that the methodological proposal presented constitutes a viable alternative for introducing Financial Education in high school, especially in Amazonian contexts, pointing to new pathways for pedagogical practices and future research in the field. As part of the study, an educational product was developed in the form of a booklet containing the didactic sequence and its main empirical aspects.

This work presents results of research that sought to answer the following question: how can teaching operations with integers through experimental activities impact the development of solving questions involving operations with integers? The objective was to analyze the possible impacts of applying a didactic sequence based on teaching mathematics through experimental activities on the performance of students in the 7th year of elementary school, in solving questions involving integers. The methodology adopted in the research was Didactic Engineering. The experiment was carried out in a class with 32 students from the 7th year of Elementary School at a municipal public school in Parauapebas-PA. The information produced during the experiment was systematized and analyzed both qualitatively and quantitatively. The results of the analysis of the application of the proposed Didactic Sequence indicate that it was successful. The analysis of pre- and post-test performances indicated that students performed satisfactorily, demonstrating the effectiveness of the Didactic Sequence. The analysis of the errors that occurred after the post-test indicates that the students' lack of attention when carrying out the activities caused the greatest number of errors. Fisher's Exact Test indicated that socio-educational factors had no impact on student performance. The only association identified in the study was in relation to students' frequency of studies outside of school. The results of the hypothesis test confirmed that the Didactic Sequence had a positive impact on the development of students, as after its application students achieved better performance. In the evaluation carried out by the students, the proposed activities were well evaluated. Based on the research results, a didactic product was created in the form of a didactic sequence entitled: Didactic Sequence for Teaching Operations with Integers through Experimental Activities, which was validated experimentally in the classroom and by the evaluation of the examining board of this work. and is available with free access at the address 

I present the research developed, through the Postgraduate Program in Mathematics Teaching – PPGEM, at the State University of Pará, regarding teaching and learning about Fundamental Trigonometric Equations. The guiding research question was: How will a Didactic Sequence, developed according to the structuring model – UARC, with the help of concrete material and software, contribute to improvements in the teaching and learning process of Trigonometric Equations? To answer this question, the general objective was defined to propose the development of a Didactic Sequence to enhance the teaching and learning of Fundamental Trigonometric Equations. The didactic sequence is structured according to the construct of Articulated Units of Conceptual Reconstruction (UARC), according to Cabral (2017). Dissertations and textbooks on the topic were analyzed, in addition to a questionnaire being sent via Google Forms to high school mathematics teachers and graduating students with questions relevant to the research, the data collected corroborated the methodological objectives outlined. The didactic sequence was applied to an experimental class and in the process control class the content was taught in a traditional way. After completing the classes and dynamics, a learning verification test was given to both classes. The analysis of the results is modeled according to the theories Microgenetic Analysis by Góes (2000) and Discourse Analysis by Mortimer and Scoot (2002). The transcribed evidence of learning, together with the average scores from the learning verification test of the experimental class, indicated that the use of the didactic sequence for teaching fundamental trigonometric equations with the aid of manipulative materials proved to be an efficient methodological alternative for the education.

The present research had as guiding question the following question: how can the use of manipulable materials contribute to the teaching of flat figures area for students with blind people? In order to answer this question we defined as objective: to investigate the process of teaching learning area of flat figures with blind students through the use of experimental activities with manipulable materials. The main purpose of this research was to develop a teaching proposal based on the Didactic Engineering of Michelle Artigue, the Teaching by Experimental Activities and Problem Solving of Sá.  The research was developed based on the four phases of Didactic Engineering: previous analyses, a priori analyses, experimentation and validation. We propose in this research a didactic sequence for the teaching of flat figures area using as a methodological resource the Teaching by Experimental Activities and Problem Solving, and as a didactic resource the use of manipulative materials.

This work presents results of a study that aims to analyze possible effects of applying a didactic sequence based on teaching through experimental activities by Sá and Mafra for teaching ratio and proportion on the conceptual and procedural learning of students in the 9th year of elementary education. We used Michele Artigue's Didactic Engineering as a research methodology, following its methodological steps: 1st phase, preliminary analysis; 2nd phase a priori analysis; 3rd phase experimentation; 4th phase a posteriori analysis and validation. The phase, preliminary analysis, will present theoretical references that support and conceptualize activity theory, teaching through experimental activities, the theory of semiotic representation records, problem solving, Bloom's taxonomy, the following aspects: historical; mathematical and curricular studies, studies on the teaching-learning of ratio and proportion. The a priori phase, conception and analysis, presents the sequence of experimental activities suggested for the experimentation phase, which was applied to 32 students in the ninth year of elementary school, in a school in the public education network of Novo Repartimento-Pará, still presents the pre-test and post-test, as well as a priori analyzes of all test questions and each activity in the sequence, in addition to expectations for a posteriori analyzes in relation to the tests and each activity.

This work aims to present a study on the sine and cosine trigonometric functions through experimental activities with the support of the GeoGebra Application on Smartphones. The choice of the topic investigated, trigonometric functions, was due to the fact that it is a content that most high school students, more specifically in the 2nd year, consider complex and excessively abstract. The guiding question of this research is: What are the contributions of applying a didactic sequence, prepared from the perspective of Experimental Activities, that can enhance the process of Teaching and Learning the Sine and Cosine Functions with students in the 2nd year of High School with the help from GeoGebra on smartphones? The general objective of this research is: Evaluate the application of a didactic sequence, with the aim of improving the Teaching and Learning of 2nd year high school students on the content of Trigonometric Functions sine and cosine, using the GeoGebra application on smartphones. The locus of this research will be the public school EEEM Walter Bezerra Falcão located in the municipality of Ananindeua/PA. The subjects will be 2nd year high school students. In order to achieve our objective described above. We use problem solving as a teaching methodology, based on Polya (1995) and Sá (2019), and we will rely on the ideas and assumptions of Didactic Engineering, a methodology from the Mathematical Didactics aspect of Artigue (1995), which presents four stages. The first stage, called preliminary analyses, contains bibliographic research such as dissertations, articles, theses, educational documents, analyzes of books that address relevant points on the topic and, also, a consultation with mathematics teachers about the content covered. In the second stage, called conceptions and a priori analysis, we present our didactic sequence, based on the experimental activities of Corrêa (2016), where we build applications in GeoGebra to be used in 9 activities on the sine and cosine functions. The third stage, experimentation, will be carried out in the following months, so that in the last stage we can carry out further analysis and validation of the activities applied in the didactic sequence. From this, an educational product will be created relating to our teaching sequence and also an online book on the GeoGebra website composed of multiple choice questions.

This research report is the result of research linked to the Postgraduate Program in Mathematics Teaching at the University of the State of Pará (UEPA), a professional master's course in Mathematics Teaching, which sought to identify to what extent a didactic sequence prepared according to Does the structuring model of Articulated Units of Conceptual Reconstruction (UARCs) enhance the teaching and learning process of Modular Function in high school? To obtain elements that answer this question, the general objective was to analyze the potential of a didactic sequence, structured according to the UARCs, for teaching modular function to high school students in public schools, with a view to identifying potential potential based on the data resulting from its application and analyzed according to discourse analysis and Microgenetics. For the development of the research, a theoretical foundation was carried out based on Brousseau's Theory of Didactic Situations, correlated with the Didactic Sequence according to Zabala's perceptions, as well as Cabral's Articulated Units of Conceptual Reconstruction. Regarding students' learning perceptions after applying the didactic sequence, the theories of Góes' Microgenetic Analysis and Discourse Analysis according to Mortimer and Scott were used. To support the elaboration of activities in the didactic sequence, a review of the literature on the topic was carried out, in addition, to identify the perceptions of students who graduated from the 1st year of high school and also of high school math teachers from public schools, about modular teaching and learning processes. Information obtained from textbooks approved by the PNLD supported a better understanding of the topic and its methodological approach in these books. (UNDER CONSTRUCTION) After combining this information, the activities that made up the didactic sequence were developed in line with the structures of the UARCs. After the application, the interactions between students and teacher/researcher were transcribed with the aim of identifying signs of learning during the process through discourse analysis and microgenetics.

This research focused on the following question: In what aspects can the application of a didactic sequence contribute to the teaching process of 1st Grade Polynomial Function in the 1st grade of high school? To answer it, the objective was to analyze signs of learning based on the application of a Didactic Sequence, supported by the Theory of Semiotic Representation Records, on 1st Degree Polynomial Function for 1st grade high school students. To develop the research, it was based on the Theory of Records of semiotic representation by Duval (2012) and the approach to the Theory of Didactic Situations from the perspectives of D’Amore (2007). Regarding the methodological framework, Artigue's Didactic Engineering (1996) was adopted, based on its four phases. In the preliminary analyses, the study of official curricular documents was carried out, as well as the literature review of works (theses and dissertations) that address the theme of this study. After that, a diagnostic activity of student knowledge was applied, based on the assumptions of Duval's Theory of Semiotic Representation Registers (2012) to verify the main difficulties of 1st grade high school students regarding the content of 1st Degree Polynomial Function and contribute to the production of the Didactic Sequence. In the construction of situations and a priori analysis, we produced a Didactic Sequence referring to the content of 1st Grade Polynomial Function, based on the assumptions of the Theory of Semiotic Representation Records, in which the concepts were worked on through their representations in natural language, tabular, graphical and algebraic. As a result, it was observed in the a posteriori analysis that the majority of students surveyed managed to obtain a satisfactory result regarding the concepts, domain and image, root/equation, growth and decrease and the sign of a 1st degree Polynomial Function, worked on in four mathematical representations: natural language, tabular, graphic and algebraic. Furthermore, during the process of applying the Didactic Sequence it was possible to observe the development of autonomy, logical reasoning and group work. It is considered that the didactic device has effective potential to work on mathematical objects in the classroom. It is expected that the Educational Product associated with this research will contribute to the initial and continuing training of teachers who work in Basic Education.

This paper presents the results of a research that sought to answer the following question: What are the possible effects of applying a didactic sequence, based on teaching mathematics through experimental activities, on the understanding of the concepts of investment and capitalization and on the performance of third-year high school students from a public school in the state of Pará, in the city of Belém, in solving questions on the subject? And it aimed to analyze possible effects of applying a didactic sequence for teaching capitalization. The teaching sequence used the financial products savings, direct treasury, capitalization bonds and consortia, and the research analyzed the effects of the teaching sequence on students' learning of the content by observing their performance in solving questions on the topic. The research methodology adopted was didactic engineering. We used a socio-educational questionnaire to obtain information about the profile of the participating students; a pre-test, in order to gather information about the level of prior knowledge of the students regarding the topic; application of the proposed teaching sequence; and the completion of a post-test, in order to verify the knowledge acquired on the topic after the intervention of the teaching sequence. From the data collected, comparing the students' performance in solving the pre-test and post-test, it was possible to verify that the proposed didactic sequence had positive effects on the students' performance and learning in this content, since the data analysis showed an improvement in the students' performance in the post-test in relation to the pre-test. When performing the Fisher test to verify the association or not between some variables, the test showed an association between the variables study habit and performance in the post-test, help to study mathematics and performance in the post-test, who helps in studying mathematics and performance in the post-test and liking for mathematics and performance in the post-test.

The present work presents the results of a research that aimed to analyze the effects of applying a didactic sequence for teaching Function Bijection through experimental activities on class participation, concept construction and performance in problem solving and questions that address the subject and its application. The experimental part of the research was developed in a state public school in Imperatriz in the state of Maranhão, with 10 high school students as subjects, in order to investigate the following scientific problem: how teaching through experimental activities influences teaching and learning of Function Bijection and its application in comparing the cardinality of numerical sets? Didactic Engineering was adopted as a research methodology. The analysis of the results took place through the notes of the students in the activities, confrontation of a priori and a posteriori analyzes, comparing the results of the pre-test with the post-test, applying Pearson's Linear Correlation Coefficient and the Test of Hypothesis. The results of the comparison showed a considerable increase in the post-test scores; the hypothesis test proved that the post-test scores had a significant improvement in relation to the pre-test and the correlation analysis showed that none of the socioeconomic factors raised had direct interference in the results obtained, showing that there were improvements in learning and that good The result of the experiment is due, above all, to the methodology used. 

The final results of a Professional Master's Dissertation in Mathematics teaching are presented, whose objective was to identify the didactic potential of a Didactic Sequence built specifically for the teaching of Algebraic Expressions. Curriculum guidelines for teaching mathematics in elementary school and a preliminary bibliographic study on the teaching of Algebraic Expressions through dissertations, articles and textbooks on the subject were considered. Among the issues raised were the difficulty in giving meaning to algebraic symbolism, as well as the development of algebraic thinking and symbolic algebraic language. We also sought to investigate the perceptions of mathematics teachers about learning Algebraic Expressions and a historical and epistemological deepening of this mathematical object. In these preliminary researches, the relevance of an interactive and collaborative teaching with gradual construction of knowledge was verified. For that, the research was guided by the question “What contributions can a Didactic Sequence structured according to the Articulated Units of Conceptual Reconstruction (UARC) bring to reduce learning difficulties in teaching algebraic expressions directed to the 8th grade classes of elementary school?” To answer this question and achieve the objective of the research, the methodological path was guided by the Theory of Didactic Situations, and a structured didactic sequence was elaborated according to Cabral (2017). Considering the reality of the return to face-to-face classes after the COVID-19 pandemic, we opted for concrete, manipulative didactic material with educational and playful intentions to create an attractive and collaborative learning environment. Through the geometric properties of the pieces of the Tangram puzzle, six sequenced, planned and articulated activities on Algebraic Expressions were developed.

28 students from public schools in the city of Cametá-PA participated in the Experimentation. The data collected from the experimentation in audio and written protocol were analyzed according to Microgenetic Analysis and Discourse Analysis. The results showed that the constructed didactic sequence presented didactic potential in the pedagogical, epistemological and learning dimensions and that the dialogical quality of the didactic episodes helped to minimize difficulties diagnosed in preliminary research. In this way, the research and its construct presented developments related to affectivity in relation to mathematics and form the source and basis for future research.

This research suggests verifying the use of a didactic sequence, in Guy Brousseau's Theory of Didactic Situations (TSD), elaborated according to the stages of Didactic Engineering, with problems that focus on real situations, encouraging research and activities related to the emergence of logarithms, contributing to the construction and understanding of the logarithmic function by the students. The study is based on the didactics of mathematics that is concerned with the results of experiences in the classroom. The methodology used follows the principles of Didactic Engineering, which is justified by the fact that it is a concept that gives importance to both the theoretical and experimental dimensions of the research and whose purpose is to analyze didactic situations, the object of study in mathematics education. The sequence activities were elaborated with the 1st year graduation students, from the State University of the State of Pará, in the resolution of problems through the logarithmic function. In this work, the productions of 19 (nineteen) students were analyzed, organized in pairs and trio. It can be inferred from the a posteriori analyses, that the activities of the sequence contributed to the understanding of the logarithmic function being the inverse of the exponential function. From the analysis carried out, it was possible to verify that the students participating in the research obtained a greater understanding about the teaching of the logarithmic function by the teacher-researcher, than in traditional classes. Therefore, the use of trends in Mathematics Education, applied to a certain content, facilitated even more the teaching and learning process of the same, and made the learning of the concept of logarithm and logarithmic function more significant, allowing the students to use this concept to understanding of real-world phenomena.

This research aims to present the results of a research, whose objective is to investigate the potentiality of a didactic sequence in the learning of trigonometric relations in the right triangle, sine, cosine and tangent functions, using a free software called GeoGebra, whose focus is to apply a didactic experiment, as an alternative for   teaching trigonometry. The focus of the research will be the students of a public high school, of the Federal Institute of Education of Maranhão, in São Luís, on Campus Monte Castelo. The subjects were 35 students (who were divided into two groups) of the second year of high school. As a research instrument, a didactic sequence with 12 activities was used. The research methodology adopted was Rabardel's Instrumentation theory, Duval's theory of Semiotic Representations and Microgenetic Analysis. of Mathematics, served as a subsidy for the construction of the 12 activities of the didactic sequence, for learning the basic concepts, properties, and definitions of trigonometric ratios in the right triangle, of the sine, cosine, and tangent functions. The research results from the written records and recorded voices, show us that the learning of the students/participants was gradual and that the subjects had satisfactory   results, since they point out that the activities in the GeoGebra application is a didactic resource that makes the activities more dynamic, favoring the construction of learning, in relation to the proposed trigonometric contents. Thus, we conclude that the learning of the subjects involved with our object of study, demonstrated at least two registers of representation, that is, in two different languages, feeling motivated to create possibilities for problem solutions.

The Interest in the subject emerged during the development of the subject Topics in the History of Mathematics. Research indicates that teachers should rely on diverse teaching methods, and the History of Mathematics emerges with the Methodological Tendencies of Mathematics Education. In light of this, the question arose: what historical clippings can be elaborated from a history of probability, according to Chaquiam (2022), for use in the classroom? The general objective of this study was to develop historical excerpts based on a history of probability, for use in the classroom. The research had a qualitative nature, drawing on bibliographic and documentary sources, anchored by the Methodological Diagram of Chaquiam (2022), considered a guide in the bibliographic review for the survey of historical and literary facts; in construct a diagram about a history of probability; produce the historical clippings and in the elaboration of activities. The research was based on official curriculum documents, surveys teachers' perceptions, analysis of textbooks and other documents and authors that guide the History of Mathematics in Education and the History of Probability. Data collection had to be conducted through a virtual questionnaire due to the pandemic caused by COVID-19, limiting the validation of the material to mathematics teachers only. The results indicated a 90% or higher acceptance of the Historical Text and the set of activities, stipulating an accessible language and activities that expand the students' vocabulary and promote their autonomy in the teaching process. The result of the complementary evaluation revealed that 97.2% of teachers agreed with the use of the Educational Product, corroborating its validation to be used in the classroom. Teacher’s suggestions were incorporated to enhance the product. Finally, it is recommended to validate the material with students to analyze its effectiveness, production and new historical clippings, elaboration of additional activities and adaptation of texts and activities for different levels of education, according to the specific needs of students in each locality or region.

This work presents the results of a research that aimed to evaluate the effects of applying a didactic sequence based on the teaching of mathematics through experimental activities on the performance of problem solving and the understanding of affine function concepts by 9th grade students. fundamental year. The research sought to answer what effects does the application of a didactic sequence based on experimental activities have on the performance of question resolution and the understanding of affine function concepts of students in the 9th grade of elementary school?? The research methodology used was Didactic Engineering. And as research resources: bibliographical analysis of other works that directly or indirectly addressed the theme addressed here, application of questionnaires and tests. The didactic sequence elaborated in this work took place in six meetings, and was applied to 44 students of the 9th grade of the final years of elementary school in two public schools in the municipality of Altamira-PA. The analysis of the collected data occurred by verifying the record of the students' conclusions in each activity carried out, by comparing the results of the post-test in relation to the pre-test, by applying Pearson's Linear Correlation Coefficient and the Hypothesis Test and some observations in loco, these strategies of statistical inferences confirmed that the didactic sequences by applied activities produced better results for the understanding of the concept of affine function, as well as, in the interpretations and solutions of problems involving this theme. We also verified that the students obtained progressive advances within the process of institutionalization of the approached mathematical concepts and definitions; in addition, we noticed a more expressive participation of the students in the solutions of the questions, as perceived in the Post-test. In view of this and the results in the comparison of performance in the Pre and Post-test, we can say that the objective of this work was achieved and the problem raised initially was answered. The result is similar to others from researchers and authors who applied Didactic Engineering as a line of research in the application of didactic sequence by activities involving the affine function.

The History of Mathematics is highlighted by literature and official documents as an
efficient teaching resource for improving the student's teaching and learning
process, however, despite the efficiency it can provide, there is a scarcity of
theoretical references that make use of the History of Mathematics. Mathematics,
which led me to become interested in this study, more specifically for the teaching
of trigonometry, a fact that led me to the following research question: “What
historical insights can be made from a history of trigonometry in order to integrate
history and mathematics in a didactic-pedagogical context for use in the teaching
process in Basic Education?”. To answer this question, the aim was to prepare
texts with excerpts from the history of trigonometry, using the methodological
diagram proposed by Chaquiam (2017, 2020, 2022), so that they can be used as a
teaching resource during the teaching process of the contents involved. . To this
end, this work reports a bibliographical research that falls within the area of
Mathematics Education, in the line of History of Mathematics, in which we review
the studies carried out by Costa (2003), Chaquiam (2017, 2020, 2022) Gomes
(2009 ), Mendes (1997), Miguel (1993), Oliveira (2009), Prado (1990), Viana e
Silva (2007) and among other studies that point to the History of Mathematics and
warn the importance of using this didactic resource in teaching of trigonometry.
Furthermore, a questionnaire was administered via Google Forms to teachers from
different education networks with the aim of carrying out a diagnostic survey of the
teaching process of trigonometry content. And another questionnaire, also via
Google Forms, was applied with the aim of analyzing the historical text, its excerpts
and their respective proposed activities. Given this context, from the literature, it
was observed that the History of Mathematics has been seen as relevant to the
teaching and learning process and that it can be considered as a teaching
resource, capable of encouraging students during math classes. mathematics.
Furthermore, based on the results of the questionnaires, we can see some
difficulties experienced by teachers during the process of teaching trigonometric
content and the need for innovative teaching resources that can improve this
teaching, which allows us to conclude that the History of Mathematics in teaching
mathematics, particularly trigonometry, can be efficient in improving the
mathematics teaching and learning process. The results of this research generated
the educational product entitled: “Teaching trigonometry through historical
clippings2

”, available at the link indicated in the footer.

The partial results of a professional master's research in Mathematics teaching are presented, whose objective was to investigate the didactic potential of a didactic sequence built specifically for teaching Congruence of Triangles. The problem raised in preliminary research indicated that the mathematical object Congruence of Triangles has been treated as a tool for the development of other mathematical objects, such as the study of quadrilaterals, so that some characteristics and properties of this object are not treated in research and teaching textbooks of Teaching Fundamental and, consequently, is not studied in all its complexity. This complexity was revealed throughout the research through a historical-epistemological study on the mathematical object, according to Chaquiam (2020; 2022). In the review of studies, methodological indications and didactic materials that have potential for teaching geometry were raised, the isometric mesh was chosen as a resource for the construction of knowledge that involves cases of congruence of triangles. The didactic sequence structured as Articulated Units of Conceptual Reconstruction (UARC) was adopted as a teaching proposal, according to Cabral (2017). As a guide for the methodological path of the research, the Theory of Didactic Situations (TSD) developed by Brousseau and the methodology developed by Silva, Chaquiam, Cabral (2022) were adopted. The construct developed presents three activities planned and articulated with each other for the development of knowledge and skills involving Congruence of Triangles. After carrying out the instructions in the qualification exam, the intention is to apply the sequence in a regular class of the 8th year of Elementary School, in the public school system. The data collected in the experiment will be audio transcriptions and written records of the research subjects, analyzed according to Microgenetic Analysis and Discourse Analysis. In the elaboration of this didactic material, the dialogical quality of the teaching and learning process was considered, as well as collaborative learning. The initial research presented indicates new perspectives on the teaching of geometry, both as a field of knowledge and as an essential element for the development of mathematical arguments.

This investigation was developed from the restlessness as a high school teacher in teaching the content of Sine Function and the difficulty on the part of students in learning this subject. In this context, this investigation seeks to answer the following research question: How is it possible to teach sine function, through activities that are associated with periodic phenomena of nature, with the help of pedagogical and technological resources? Thus, this research aimed to: The elaboration and analysis of a didactic sequence, mediated by geogebra, for the teaching and learning of sine function content. In order to achieve the proposed objective, we chose to carry out a quantitative-qualitative research, in which we initially made a bibliographic study based on articles, dissertations, theses and books. Then, we conducted field research through questionnaires to 35 teachers of Mathematics in the teaching profession who teach the content of the sine function in their classroom practice. The theoretical framework of the research was based on the conception of Brousseau's Theory of Didactic Situations (1996) and the teaching by activities in the view of Sá and Jucá (2017). These conceptions served to guide the analysis of the research and the production process of the educational product. From this biographical review and the methodology described, a sequence of activities was built, in which the elaboration of the educational product was preceded by a field research, in which we consulted a group of fifty high school students, to know their perspectives on the teaching of the Sino function, through activities mediated by the Geogebra software as a practical guide defined by "Sino Function: A didactic sequence for high school, inspired by natural phenomena". Thus, it was concluded that from the proposed activities, it was demonstrated by the students an excellent mastery and understanding of the proposed theme, consolidating the educational product as a great possibility of activity for the teaching of the sine function.

This work presents the results of a study on first grade problems with one unknown. The guiding question is: What are the effects of a didactic sequence based on experimental activities on student performance in solving problems with an unknown? The research methodology adopted was Didactic Engineering. And with the aim of investigating the effects caused by this sequence in the teaching and learning process of grade one problems with one unknown. And as methodological teaching resources: teaching through activities, with the use of some teaching resources to enhance and speed up problem solving; and for analysis and proof of learning we will use Duval's Theory of Semiotic Representations. The application of the didactic sequence developed in this work took place in a 7th year class of elementary school II at a public school located in the city of Óbidos-PA. And to achieve our objective, we will rely on David Paul Ausubel's theory of meaningful learning, characterized by the interaction between prior knowledge and new knowledge that contribute to giving meaning to the subject about the subject being worked on.

This work shows the progress of the stages of an investigation that aims to analyze the impacts of developing experimental activities on learning 1st grade problems and equations with an unknown with students from a 7th year elementary school class at a public school in city of Parauapebas/PA. The methodological approach used to conduct this study was Didactic Engineering. The information began to be produced from the month of May 2022, with theoretical studies on Didactic Engineering, then the preliminary analysis stage was experienced, which included historical data on the subject, with mathematical aspects, with review of current literature that deals with the mathematical object addressed and with information from 52 public school teachers who gave their opinion on teaching 1st grade problems with an unknown. The data were obtained through the application of questionnaires and their treatment was carried out through tables and graphs with their appropriate quantitative analyses. The text exposes the set of activities that make up a sequence of experimental activities proposed for the experimentation phase. The a priori analysis stage was also experienced, revealing expectations about each activity in the experimental sequence. We are currently in the experimentation phase, to later carry out post-mortem analyzes and validation of the activities developed during the application of the didactic sequence. The systematization of the information produced occurred through the construction of statistical tables and graphs. The results and validation of the didactic sequence will be considered based on the comparison between a priori analysis and a posteriori analysis. Based on the results of the study, an educational product will be created in the format of a didactic sequence.

This work presents the partial results of research developed in the Professional Master's Program in Mathematics 

Teaching (PPGEM) at the State University of Pará, with the purpose of answering the following problem: How to 

apply a didactic sequence, based on the teaching of does mathematics through experimental activities influence 

the performance of solving additive problems with natural numbers? With this purpose, the objective behind this

 investigation is: to analyze the effects of applying a didactic sequence, based on teaching mathematics through 

experimental activities, on the performance of solving additive questions with natural numbers of students in the 

6th year of elementary school . The research methodology that will be adopted is Didactic Engineering of the 1st 

Generation (ED1G), consisting of four stages: initially we present the previous analyses, consisting of a review of

 studies on additive problems and a consultation with students in the 6th year of Education Fundamental about the

 teaching-learning process of this content. In the second stage, called a priori conception and analysis, descriptions

 of Teaching by Activities, Problem Solving, the use of Games as a methodology for teaching mathematics and

 a didactic sequence for teaching additive problems are presented, consisting of 15 activities and organized 

according to the methodologies mentioned. The third stage of the study is called experimentation, according 

to ED1G. In the fourth stage, a posteriori analysis and validation, analyzes of activities and evaluative tests 

will be carried out using a qualitative and quantitative approach. At the end of the research, an educational 

product will be obtained based on the studies and results achieved, organized according to a didactic sequence

 for teaching additive problem solving, enabling the teacher to use didactic sequences through experimental

 activities and problem solving. , with the intention of contributing significantly to the teaching and learning 

of this object of study.

This work presents the partial results of a research carried out during the Professional Master's course in Mathematics Teaching, as well as the methodological path and the planning of the stages to be completed. In studies of the guidelines of Brazilian education, literature review regarding teaching and learning of tangent function and mathematics textbooks, it was possible to observe the potential didactic-pedagogical methodologies currently researched and adopted. On the other hand, from the students' perceptions, I verified the difficulties in teaching and learning the tangent function. Thus, we establish the following guiding question: To what extent does a Didactic Sequence elaborated according to the structuring model of the Articulated Units of Conceptual Reconstruction enhance the teaching and learning process of tangent function? The general objective is to study the didactic potential of a Didactic Sequence elaborated specifically for the teaching and learning of tangent function in the high school of a public school in Abaetetuba-Pará, and structured according to Articulated Units of Conceptual Reconstruction. From the literature review on teaching and learning of tangent function and theoretical and methodological studies, I adopted the Theory of Didactic Situations as the central axis for the development of a didactic sequence structured according to Cabral's Conceptual Reconstruction Units, (2017). With this information and background, it was possible to develop a didactic sequence for teaching the Tangent Function, improved in discussions in a research group. For the analysis of results resulting from the application of the didactic sequence, it is intended to adopt the Microgenetic Analysis, by Góes (2000), and Discourse Analysis, by Mortimer and Scott (2002) to validate the educational product built. This text also includes the planning of the next research steps involving the application of the didactic sequence with students from a public school in the state of Pará and identifying the evidence of learning in the students' speeches. Finally, the preparation of the final research report and the educational product can contribute to the training of mathematics teachers, as well as providing alternative material for teaching tangent functions.

We present a research that investigates the potential of a didactic sequence favorable to the learning of trigonometric inequalities in high school, from their algebraic and graphical representations in the trigonometric cycle. It is a research characterized by a qualitative and quantitative approach in which activities with gradual and increasing level of requirements were used. Each stage of this research is based on the theoretical assumptions of Didactic Engineering, a methodology adopted in research involving an experimental part. The research was guided by the following guiding question: What are the effects of applying a didactic sequence, different from the traditional model, to the resolution of trigonometric inequalities? As a research resource: Bibliographic analysis of other works such as articles, dissertations that presented their direct or indirect contributions on the theme of research, application of questionnaires and tests. The didactic sequence elaborated in this work will be applied in two group meetings in a room composed of 35 to 45 students from a 2nd year high school class in a Public School in Imperatriz - MA. The arguments arising from the data analysis will be highlighted in the a priori and posteriori analysis of Didactic Engineering. The research-related educational product will be presented with all the activities in the didactic sequence with an orientation for their use in the classroom.

This work presents the results of a research that aimed to evaluate the effects of a didactic sequence for the teaching of Cone Trunk by Activities on the conceptual aspects and performance of the resolution of questions involving the subject. The experimental part of the research was developed in a state public school in Belém/PA with 25 students from the 3rd year of High School, using Didactic Engineering as a research methodology. The analysis of the results was carried out by registering the students in the activities, comparing the a priori and a posteriori analyses, by comparing the results of the pre-test with the post-test, as well as by applying Pearson's Linear Correlation Coefficient and Hypothesis Test. The comparison results showed an increase in post-test scores; the analysis of correlations showed that none of the socioeconomic factors raised had a direct influence on the results obtained. The use of Geogebra as an educational tool was fundamental for the post-test scores to have a statistical improvement in relation to the pre-test, noting that the good result of the experiment was mainly due to the methodology used.

This paper presents parts of a research proposal that aims to analyze the difficulties of students when studying permutation and the effects that a didactic sequence, built through problem solving, provides for the construction of knowledge, with a view to improving teaching and permutation learning in a 2nd high school class. As theoretical and methodological references, we used Didactic Engineering in our research, which, according to Artigue (1996), consists of four stages (previous analysis, a priori analysis, experimentation and validation), we adopted the Microgenetic Analysis proposed by Góes (2000), Resolution of Problems in the conception of Polya (1945) and the Didactic Sequence, in the vision of Zabala (1998). The experimentation will take place in the first semester of 2021. In this stage, we will present the proposed didactic sequence, in a class of the 2nd year of high school in a public school in the city of Codó-MA. During the application of the didactic sequence, we will make video recording and filming for data description and analysis through Microgenetic Analysis. Finally, in the a posteriori analysis and validation, we will write step by step the analysis of the results obtained with the application of the didactic sequence. Our results will be presented in the final version of the dissertation, together with an educational product so that we can contribute to teaching and learning in the pedagogical practice of mathematics teachers when teaching the Permutation content.

Keywords: Initial and continuing education. Information and Communication Technology. Maths teacher.

The teaching and learning process in Mathematics has been treated as one of the research fields in the area of Mathematics Education. In view of this process, the object studied in this research was the content of remarkable products, with which, we proposed to present the potential of a didactic sequence with the use of manipulative materials for the teaching of remarkable products with students of the 8th year of Elementary School of a public school of the State Network of Belém do Pará. For this, it was necessary to base its construction on Brousseau's Theory of Didactic Situations, associated with the Didactic Sequence based on Zabala's perceptions, as well as on Cabral's Articulated Conceptual Reconstruction Units and on didactic materials that can be manipulated with Lorenzato. For the analysis of evidence of student learning in the application of the Didactic Sequence, Microgenetic Analysis with Góes and Discourse Analysis with Mortimen and Scott were used. With this theoretical background, it was essential to carry out analyzes of the object from different approaches, including carrying out a review of studies, understanding the perception of students and teachers from the completion of questionnaires, in addition to identifying the depth of approaches to books didactics of Basic Education and to elaborate a solid mathematical text to subsidize the construction of the Didactic Sequence. Which, was structured in 5 activities with the identities: square of the sum of two terms; square of the sum of three terms; product of the sum by the difference of two terms; square of the difference of two terms and cube of the sum of two terms 

The process of apprehension of mathematical knowledge by students is marked by several difficulties. One of the fields of mathematics that is configured as one of these obstacles is algebra. In this field, the literature points out, among other disciplines, difficulties in learning the 2nd degree polynomial equation. In this sense, the main motivation that guided the construction of this work, emerged from the following research question:What is the effect of applying a Didactic Sequence in the light of Vergnaud's Conceptual Fields for learning 2nd degree Equation from Problems? Teaching research focuses on seeking to improve the teaching and learning process of this mathematical object. This work presents the partial considerations research in development that aims to study as didactic potentialities and the evidence of learning in the application of a didactic sequence elaborated based on problem solving and on the Conceptual Fields theory focused on the teaching of polynomial equation of the 2nd degree. For this, we take as a theoretical reference the considerations of Vergnaud and Vygotski about conceptual fields and social interactions in the learning process and considerations about the use of problem solving in the teaching of mathematics. We study the polynomial equation of the 2nd degree in the scientific scope from the research and the scientific community in which we approach historical aspects and methods of resolution and in the didactic scope from the official documents and the review of studies. Thus, we elaborated a didactic sequence composed of 3 activities preceded and closed with pre-test and post-test composed of 8 questions, respectively. Our partial results are composed of the previous analyzes of each activity, and the comparison of the process based on the tests. We expect that a didactic proposal corroborates for an alternative approach to teaching high school equations so that they favor students' ability to solve problems and an understanding of this object at a conceptual level.

This work presents partial results of a research that has as objective to analyze the effects that a didactic sequence by activities has on the performance in solving multiplicative problems that involve the idea of a rectangular configuration in a 6th grade elementary school class. As methodological foundation of the work we used the four stages of didactic engineering. Firstly, we performed the previous analyzes, composed by a review of studies on multiplicative problems and a diagnostic consultation with students of the 7th year of primary education about the teaching and learning process of this content. In the second stage, a priori conception and analysis, we present the theoretical basis with the presentation of Teaching by Activities as an alternative methodology for teaching mathematics, and a didactic sequence for the teaching of multiplicative problems involving the idea of a composite, rectangular configuration by 06 activities elaborated according to the aforementioned methodology. The third stage of research, experimentation, will be developed in the first half of 2019. In it we will carry out the application of the socioeducational questionnaire, the pre-test, the proposed didactic sequence and the post-test, in a 6th grade elementary school of a state public school in the municipality of Ananindeua in the metropolitan region of Belém-PA. In the fourth and last step of the research, a posteriori analysis and validation, we describe how the analysis of the results obtained during the previous phase will be performed. The statistical analysis of the information collected will occur through the percentage comparison of the results, using the Pearson linear correlation coefficient and the Hypothesis Test. The validation of the results collected will be effected by crossing the data obtained in the a priori and a posteriori analyzes. The final results and conclusions will be presented in the dissertation from this research, which will culminate in the construction of an educational product that can collaborate to improve the pedagogical practice of mathematics teachers by teaching multiplicative problems involving the idea of a rectangular configuration.

This work presents a research, aiming to analyze the learning generated from a Didactic Sequence on Irrational Numbers in high school students. The experimental part of the research was developed with a sample of 6 high school students from Jaderlândia (Ananindeua / PA) and Marambaia (Belém / PA). The research methodology was the Didactic Engineering of Michèle Artigue (1988). The analysis of the results found in the students' records, during an activity, was carried out by comparing the results of the pre-test with the post-test and was carried out using the Hypothesis Test. In the results, the students performed well in the activities; the comparison of pre-test and post-test data known that the Didactic Sequence developed learning about Irrational Numbers; the hypothesis test proved that the post-test scores had a statistically significant improvement in relation to the pre-test, it proved that the application achieved the adequate results.

This paper presents partial results of a research which aims at analyzing the effects of a didactic sequence through activities on the performance of the resolution of additive problems in a group of 6th year of elementary school students. For the methodological approach, the Didactic Engineering one was chosen, which consists of four stages. Initially, it is presented the previous analyzes, composed by a review of studies on additive problems and a consultation with the 7th year of elementary school students about the teaching-learning process of this content. The second stage of the work, conception and analysis a priori, presents the description of the Teaching by Activities and Games as methodologies for the teaching of mathematics, and a didactic sequence for the teaching of additive problems, composed by 13 activities in which it will be used the methodologies mentioned. The third stage of the research, experimentation, will take place in the first semester of 2019. In this process, it will be described how the questionnaire, the pre-test, the development of the proposed didactic sequence and the post-test will be carried out in a group of 6th year of elementary public school students in the municipality of Ananindeua-PA. In the last step of the research, the posteriori analysis and validation, it will be described how the analysis of the results obtained will be conducted during the experimentation. The statistical techniques of analysis of the produced information will occur by means of the percentage comparison of the results, using Pearson's linear correlation coefficient and Hypothesis Test. The validation of the results will be accomplished by confrontation between the data obtained in a priori and a posteriori analyzes. The results and conclusions will be presented in the final version of the dissertation, culminating in an educational product that will contribute to the pedagogical practice of mathematics teachers when teaching additives problems in basic education.

This paper presents the results of a research on the teaching of trigonometric equations, whose scientific problem was: what effects does a didactic sequence of experimental activities have on the performance of students in solving trigonometric equations in a 3rd year high school class? And its general objective was to analyze the effects of applying a didactic sequence through experimental activities on the performance of students in solving questions about trigonometric equations. The experimental part of the research was developed in a virtual format with a group of 10 students from the 3rd year of High School of a public school in the municipality of Ananindeua in Belém do Pará and Didactic Engineering was adopted as a research methodology. The analysis of the results was carried out by registering the students in the activities, comparing the a priori and a posteriori analyses, comparing the results of the pre-test with the post-test, analyzing the errors that occurred in the post-test, as well as the application Pearson's Linear Correlation Coefficient and Hypothesis Test. The comparison results showed an increase in the post-test scores; the hypothesis test proved that the post-test scores had a statistically significant improvement in relation to the pre-test and the analysis of the correlations showed that none of the socioeconomic factors raised had direct interference in the results obtained, noting that the good result of the experiment was due to above all to the methodology used.

This research is a dissertation of the Professional Master's Program in Teaching Mathematics (PMPEM) of the State University of Pará (UEPA) and consists of action research of an applied nature. For its realization we start from the present problematic: "the use of a geogebra application can improve the teaching of exponential equation"? To answer this question, we have developed a didactic sequence (SD) aimed at teaching exponential equation from the use of a geogebra application, which can be used on both computers and mobile devices. This didactic sequence will be applied in a public high school, located in Belém do Pará, targeting students in the 3rd year of high school. We will test the following hypotheses: a) the use of the geogebra application applied to the students of this school promoted a more concrete and more objective learning; b) students felt more motivated when they learned math through technology. The general objective is to test the effectiveness of a didactic sequence aided by a geogebra application to teach exponential function in a classroom environment and to establish specific objectives: Define and detail the theories and methodologies to be adopted; To do a study on the teaching of exponential equations in literature review and textbook analysis; Conceptualize and describe the mathematical object: exponential equations; Construct a didactic sequence on the teaching of exponential equations; Present the procedures and planning for application of the didactic sequence and analysis of results. We hope that at the end of this research our objectives will be achieved and that there has been success in the aspect of the conversion of languages and the relations between subject-object, subject-instrument, and instrument-object in instrumental activities.

This dissertation addresses the teaching of decimal numbers through experimental activities in elementary school where we aim to answer what are the effects of a didactic sequence through experimental activities on learning decimal numbers with students in the 6th year of elementary school? And to answer this question, we aim to analyze the effects of a didactic sequence by experimental activity on the learning of decimal numbers with students in the 6th grade of elementary school. Therefore, we searched the literature for authors who deal with this theme such as Jucá, Sá and Lopes. The experimental part of the research was developed virtually with a sample of eight students, three from the municipality of Abaetetuba and five from Ananindeua, for this we adopted didactic engineering as a research methodology, addressing its four phases and methodology teaching, teaching by activities. To analyze the results, we used the records in the observations and conclusions of the students in the activities, the confrontation of a priori and a posteriori analysis, the comparison between the results of the pre-test and the post-test, we also analyzed the errors that occurred after the post-test, we applied Pearson's Linear Correlation Coefficient and finally we applied these results to the hypothesis test to validate the experiment. The results showed an increase in the post-test scores; the hypothesis test proved that the post-test scores improved statistically compared to the pre-test scores and the analysis of the correlation with socio-educational factors showed no interference in the results, thus noting that the satisfactory result was given by the methodology. used which proves to be an excellent instrument to be used by teachers to apply the contents of mathematics

This research was developed in order to analyze the learning signs resulting from the application of a didactic sequence involving the four most elementary arithmetic operations with integers, in a class of the 3rd stage of youth and adult education (EJA) of elementary school. a state public school in Belém do Pará. In this sense we use the model of Articulated Units of Conceptual Reconstruction (UARC), according to Cabral (2017), for the elaboration of the proposed didactic sequence. For the analysis of learning signs resulting from verbal interactions during the application of the didactic sequence we used the notions of Microgenetic Analysis, according to Góes (2000), and Discourse Analysis, according to Mortimer and Scott (2002). The results of the analysis on the communicative approaches and the interactive patterns showed strong learning signs, showing that the students actively participated in the (re) structuring of knowledge, seeking the assimilation of the algorithms to add, subtract, multiply and divide integers. Finally, these analyzes were corroborated by the positive quantitative results of the Final Restrictive Evaluative Intervention.

This paper portrays research conducted during the Professional Master's Degree in Mathematics Teaching and presents the partial results and the planning of the stages to be finalized, whose main objective is to study the didactic potential of a Didactic Sequence designed specifically for the teaching and learning of the concept. of function in the high school of a public school in Belém do Pará, and structured according to Articulated Units of Conceptual Reconstruction. Achieving this goal means getting answers to the following guiding question To what extent does a Didactic Sequence elaborated according to the structuring model of the Articulated Units of Conceptual Reconstruction enhance the process of teaching and learning of concept of function? From the literature review on the teaching and learning of the concept of function and theoretical and methodological studies, I adopted the Theory of Didactic Situations as the central axis for the development of a didactic sequence structured according to the Articulated Units of Conceptual Reconstruction, Cabral (2017), with a view to teaching the concept of function. For the analysis of results resulting from the application of the didactic sequence, I adopted Góes's (2000) Microgenetic Analysis and Mortimer and Scott's (2002) Discourse Analysis. In the studies of the Brazilian education guidelines, literature review regarding the teaching and learning of the concept of function and mathematics textbooks, it was possible to observe the potential didactic-pedagogical methodologies currently researched and adopted. On the other hand, from the students' perceptions, I verified the difficulties in teaching and learning the concept of function. With this information and reasons, it was possible to elaborate a didactic sequence for teaching the concept of function, improved in discussions in a research group. This text also includes the planning of the next steps of the research involving the application of the didactic sequence with students from a public school in the state of Pará, to later make use of Discourse Analysis and Mycogenetics to identify the learning signs in the students. student speeches. Finally, the elaboration of the final research report and the educational product can contribute to the formation of mathematics teachers, as well as providing alternative material for teaching the concept of function.

This dissertation addresses the teaching of polygons by activities, since it is a theme in the field of geometry very present in everyday life. The research problem took place on what are the effects, the application of a didactic sequence, built on the foundations of Teaching by Activities, addressing the theme of polygons, causes, in terms of teaching student learning? And its general objective was to analyze the effects of the application of a didactic sequence for teaching polygons to students in the 8th year of elementary school, built based on the teaching methodology called activity-based teaching. The research methodology adopted was Didactic Engineering, covering all its phases (previous analysis, a priori analysis, experimentation and validation) and as teaching methodology we adopt teaching by activities. The experiment took place in a public school in the municipality of Uruçuí-PI, with 34 students from the 8th grade of regular elementary school. The analysis of the results occurred through the verification of the registration of the observations and conclusions of the students described in the applied activities, confrontation between a priori and a posteriori analyzes, in addition to the analysis of the performance in solving questions in the pre-test (before the experiments) and post-test (post-experiment). Among the main results verified after analysis are: acceptance of students to the new teaching methodology, commitment to perform the tasks described in the activities and reasonable participation through speech in the institutionalization of content. The conclusion was that the didactic sequence tested with the students had a positive impact on their learning, which shows to be a viable path for the mathematical content addressed and can be adopted by schools and teachers, in the teaching process.

This research was developed with the purpose of studying the potentialities of a didactic sequence created on the optics of the Conceptual Reconstruction Articulated Units (UARC) of Cabral (2017) for the teaching-learning process of Afim Function. It was applied in a first year high school class of a state public school in the city of Bujaru, in the state of Pará. As main theoretical contributions for the elaboration of our research, we adopted Didactic Engineering according to Artigue (1996), the Didactic Sequences in (1998), Curriculum Parameters for High School (2002), Articulated Units for Reconstruction proposed by Cabral (2017), Microgenetic Analysis proposed by Goes (2000) and Mortimer and Scott's Discourse Analysis (2002) ). The didactic sequence was composed of five UARC's, where we define the representation of the related function, the rate of variation, the graphical representation, the growth and decrement of the function and the zero of the related function. In all the stages we did the video recording of the application of the didactic sequence. Through Microgenetics and Discourse Analysis I made the signs of learning and the validation of my didactic sequence. Among the results obtained by the application of the didactic sequence on the UARC's optics for the teaching-learning process, as for the teacher: better organization by the teacher; the teacher becomes the mediator; improves the teacher's capacity for argument.As for the student: stimulation the perception by the students of the standards and regularities in each topic; autonomy on the part of the student; students justify their procedures. Knowledge is constructed, it ceases to be conceptual only and becomes experimental.

This thesis presents the results of a research that had as objective to evaluate the effects of a didactic sequence for the teaching of Polyhedra by Activities on the conceptual aspects and performance of the resolution of questions involving the subject. The experimental part of the research was developed in a federal public school in Tucuruí / PA with 26 students of the 3rd year of high school, was adopted as a methodology of research Didactic Engineering. The analysis of the results was based on the students' registration in the activities, comparing the a priori and a
posteriori analyses, comparing the results of the pre-test with the post-test, analysing the errors occurred in the post-test, as well as the application the Pearson Linear Correlation Coefficient and the Hypothesis Test. The results of the comparison indicated an increase in the post-test scores; the hypothesis test showed that the post-test scores improved statistically in relation to the pre-test and the correlation analysis showed that none of the socioeconomic factors raised had direct interference in the results obtained, noting that the good result of the experiment was due to especially the methodology used.

This study aims to investigate the learning of high school students about difficulties in Matrix subjects. The research carried out here is classified as exploratory and descriptive in terms of purposes, and bibliographical and field research on means. A bibliographical research was carried out initially, considering that from the researches
and already existent documents on the difficulties of learning in the Mathematics classes, more specifically in the teaching of Matrices it was possible to contribute in the study proposed here. Field research was conducted in a high school classroom of a state school in the municipality of Abaetetuba. For data collection, a questionnaire composed of 07 questions focused on learning difficulties in Mathematics classes was applied, focusing on Matrix teaching. To analyze the data, a quantitative approach was used, using descriptive statistics. In the course of the study, we intend to highlight the complexity of the content about Matrices and the difficulties faced by many students. It is verified that the students believe that the didactic sequence can improve their performance in learning Matrices, highlighting the need to contextualize the their reality.

This work presents the results of a research that had as objective to evaluate the effects of a sequence for the teaching of divisibility of natural numbers through activities on the conceptual aspects, properties and performance in the resolution of questions related to the theme. The experimentation took place in a municipal public school in São João de Pirabas / PA with 13 students from the 6th year of elementary school, where didactic engineering was adopted as methodology. The analysis of the results was verified through the verification of the students' observations and conclusions in the activities, confrontation of the a priori and a posteriori Analyzes, as well as the analysis of the performance in the resolution of questions. The results showed a good behavior of the students regarding the observations and conclusions that presented for each activity, where they were able to solve questions related to the subject, leading us to conclude that the experiment had a positive effect, and that this fact is mainly due to the methodology used.

The aim of this research is to note how third year high school students appropriated conics when interacting with a “Geogebra” dynamic geometry software, as well as what mathematical knowledge mobilizes when developing a sequence of activities that understand this subject. Thus, the research intended to answer the following question: Does the use of a sequence of activities using Geogebra enhance the learning of conics? The locus chosen for the research was a state public high school located in Curuçá-Pa. The research subjects were students of the 3rd year of high school. The theories behind the research were Rabardel's Theory to understand how students interact with Geogebra and what mathematical contents move in the development of the sequence of activities and the theory of semiotic representation records to analyze all data collected from written annotations. and audio and video recordings. An evaluation and subsequent validation of the sequence of activities with teachers of the state public school system was made, before being applied with the students in the classroom. The sequence of activities applied has a total of 16 activities, one focused on the generation of conics from a double cone, five activities focused on Parable teaching, five activities on Elipse teaching, and five activities focused on Hyperbole teaching. The results led to the materialization of the different semiotic records of our mathematical object, which is an indispensable quality for its apprehension. The activities presented contributed to the knowledge and graphic understanding, algebraic and in its natural language of the mathematical object approached. Overall, there was a very significant student interaction with conics mediated by the use of Geogebra throughout its application, which resulted in more meaningful and engaging learning for students. We can also highlight the willingness and commitment of students to carry out the activities of our teaching sequence.

This work aimed at researching the teaching of the straight circular cylinder in basic education and analyzing the potential of applying a Didactic Sequence designed to teach this content in a second grade of a high school in a state school in the municipality of Abaetetuba, soon it aimed to analyze the potential of applying a Didactic Sequence -(SD) involving the teaching of a straight circular cylinder for a 2nd grade class in high school. For that, the theoretical contributions used in the didactic methodology of the experiment were the Theories of Didactic Situations of Brousseau (1986), linked to the concepts of Didactic Sequences of Zabala (1998), and structures of UARC's of Cabral (2017), were chosen as instruments support for the application of SD, concrete manipulable materials and activity sheets composing 5 UARC's. To probe the difficulties encountered in teaching and learning about cylinder, a literature review on the teaching of spatial geometry was carried out, research on formal and didactic books on the content, research through questionnaires involving 100 students graduated from the content and 41 teachers from the basic education. After verifying the difficulties and gaps in the content, five activities were developed involving cylinders that composed the SD in order to minimize such difficulties and gaps. Data analysis was performed through Discourse Analysis and Microgenetic Analysis, which validated the experiment carried out, as they pointed out as a result the evidence of learning present in verbal interactions between students and teachers participating in the experiment, in addition to the evidence of learning pointed out, verified in a Restricted Evaluative Intervention -(IAr) that the students learned the content significantly, because they learned the formula and concepts through the researches made by themselves with the support of the concrete manipulable material and activity sheets used in the UARC's, thus showing that the sequence produced was useful for teaching straight cylinder with circular base, concluding that the methodology applied in the experiment is a good alternative to teach straight circular cylinder.

This present paper will present the results of a research which has the goal for using the software geogebra as a tool for the position spatial geometry teaching and learning aid, emphasizing the plane and line teaching process, from the construction of a didactic sequence. As a theoretical basis of this paper it was used mathematical didactic based on teaching by activities and use of information and communication technologies. For the production of the necessary information and research formation, it was executing previous analyzes through a study review that it was looked for what it had been already researched and produced about the subject approached, it was done a research with 25 teachers from the public regular schools to identify how the plane and line have been teaching at the school environment and there was also a research with a sample of 100 students of a state school in the city of Belém, in the state of Pará. The results and the analyze indicated that the position geometry teaching, relating to the plane and line teaching has been taught in an unsatisfactory way, which the main difficulties presented by the students consisted in the conceptual area, perspective and identification of the geometricentities and the relation of the technical terms with the properties. The methodology applied for this research was the use of information and communication technologies, based on the
Didactic Engineering, in which activities were created using the geogebra for 3rd grade students of high school of public school. The experiment accomplishment occurred in three moments, in a 19 students classroom of a public school from the metropolitan region of Belém, first, a mini-course was applied explaining the main geogebra’s functionalities, second, it was developed a didactic sequence with 15 proposed activities and third, it was applied a post-test for validating and verifying the obtained data; The obtained results during and after the experimentation indicated positive effects which led an improvement in students’ performance in the study of the position geometry and activities resolution, it was also observed that the students learned the relations, concepts, postulates and properties with motivation and cooperation.

This work presents reflections on the teaching and learning process of the solid
geometric sphere, with students of the third year of high school in a public school in
the state of Pará, through a didactic sequence using applets from Geogebra 5.0
software. For the construction of the didactic sequence, the Articulated Units of
Conceptual Reconstruction (UARC) was applied as a model just as proposed by
Cabral (2017), considering previous studies composed of a review of studies, an
analysis of textbooks used by public schools and the results of a teaching and learning
research made with teachers and students. In order to analyze the results from the
application of the didactic sequence, it was applied the Microgenetic Analysis of the
verbal interactions of teacher / student and student/student according to Góes (2000),
the Discourse Analysis according to Scott and Mortiner (2002), besides the student’s
development in solving questions in comparison with a control group. From the results
obtained, students are expected to improve the visualization process, recognize
elements and solve questions that make use of the area and volume ideas of the solid
sphere and its parts.

Our research aims to present results of studies on teaching complex numbers, in order to verify a sequence of activities involving use records, using technological tools, makes their learning meaningful. For this, we propose to perform document analysis with algebraic, trigonometric, geometric (graphical representation by the Geogébra platform) and through the programming programming language in the construction of algorithms, which are performed as functions of a calculator. which does not operate a set of complex numbers, the students participating in the didactic experiment through Raymond Duval's (1993) semiotic representation record theory. The research group will be a state public school in the city of Abaetetuba / PA and the subjects will be 25 (twenty five) students of the 2nd year of high school. A research technique will be created in Didactical Art Engineering, where the phases will be initiated by a sequence of activities or the historical moment of the appearance of complex numbers, followed by activities that aim at constructing concepts and manipulating selection, subtraction operations. , multiplication, division, enhancement and radiation in the set of complex numbers.

This research answered the following question: what is the potential of a didactic sequence, based on Didactic Engineering theories, in teaching high school problems with one variable? and with the objective of investigating the effects caused by this sequence in the process of teaching and learning of second degree problems with one variable. We adopted as research methodology the Didactic Engineering. And as methodological teaching resources: teaching activities and problem solving method in Poland, using some resources of Information and Communication Technology to enhance and expedite problem solving; for analysis and proof of learning that were used as theories of Duval Semiotics. The didactic sequence elaborated in this work occurred in four meetings and was applied to a 9th grade elementary school class from a public school located in the city of Barcarena - PA. After further analysis of the written records, the sequence of elaborated and applied activities, students 'interest, enthusiasm, and ability to solve problems that occur in grade two polynomial equations were evidenced by improving students' written records. Given this, as well as the qualitative results observed, we can conclude what the objective of this work was achieved.

Important studies critically reveal the absence and abandonment of geometry teaching, Lorenzato (1995) and Pavanello (1993) present the various factors that place geometry apart from the school environment. In this context of abandonment, this work represents, in a broad way, a contribution to the study and teaching of geometry by putting into discussion the importance of teaching this knowledge for the education of the student. In a specific way, this study proposed to evaluate the potential of a didactic sequence, structured according to the Articulated Units of Conceptual Reconstruction (UARC) produced by Cabral (2017), developed for teaching the cone to 2nd year of high school at a public school in the state of Pará (Brazil). The choice we made for the theory of didactic situations reveals the learning environment conceived by this work and justifies the choice of microgenetic analysis and discourse analysis as methodologies for analyzing the data produced in the application of the didactic sequence. This work also comprises a study review that allowed us to learn about significant researches that address experiences and bring diagnoses of geometry teaching, as well as analysis of textbooks, a diagnostic study of the teaching and learning of the cone that approached 100 (one hundred) students and 25 (twenty five) teachers from the public school. In addition, the study presents the construction, application and analysis of the didactic sequence. The results of the analysis of the discursive interactions showed learning situations showing significant assimilations about the cone. In this way, the students explored knowledge of the mathematical field in which, they carried out analyzes on the elements that characterize the cone, reflected on the aspects that classify the cone in straight and oblique, made important abstractions proposed by the planning and experiment activities to study of cone volume. The didactic sequence produced a learning environment permeated with communicative interactions that revealed significant evidence of learning. Finally, the analyzes reaffirm that the didactic sequence is potentially favorable to the learning of the cone