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This study, entitled “Comparative analysis in mathematics between the second year of the Bachillerato General Unificado (BGU) and the first year of the International Baccalaureate (IB) at Hermano Miguel De La Salle High School, is an investigation that answers the following question: What common and different elements do these two modalities have in relation to the curriculum, methodology, and the learning level? To respond to this question, a mixed-methods approach research was carried out since an exploration was done to gather data, and thus contrast fundamental aspects of the curriculum of the Bachillerato General Unificado with that of the IB Diploma Programme (DP). Additionally, a group of mathematics teachers from the two modalities were interviewed to identify the methodologies and assessment techniques they use in their clases; students from the first IB and the second BGU took a mathematics general knowledge test to determine the learning level they had achieved; Student ́s T was applied for heterogeneous independent groups. Finally, the outcomes revealed differences concerning the elements of the BGU and the IB curricula. According to the teachers’ responses, they use the same methodology with both classes; nonetheless, the assessment differs due to each curriculum’s requirements. The test showed that the students from the BGU and the IB are highly familiar with structured-based questions, and the students from the IB demonstrated a higher learning level in exercises involving mathematical processes.

Mathematics is taught without relating it to reality, this causes disinterest and low learning results, consequences of an obsolete methodology that focuses teaching on a lecture and learning on the mere repetition of steps. That is why the present research aimed to analyze the application of didactical situations, based on Guy Brousseau’s theory, for the teaching of functions, while relating its content with real situations and Tracker and GeoGebra software are used; with the purpose of determining if the applied methodology generates better knowledge and skills than the traditional methodology. To achieve this goal, quasi-experimental research was developed by selecting two groups of students, one experimental and a control one, with a total of 56 students. 4 didactical situations were developed in which the experimental group was intervened while the control group worked with a traditional methodology; among the taught subtopics are: concept of function, graph, modeling, domain, range and increasing or decreasing. A pre-test and post-test were also applied to both groups and with the results two Student t tests were carried out. The results and discussion allowed to conclude that before the intervention, the groups did not have a significant difference in their knowledge, but after the intervention there was a significant difference between the two groups, which evidenced and confirmed that didactical situations generated better learning in the students of the experimental group and related the contents with the reality.

The didactic guide designed aims to provide teachers with didactic resources and technological tools to carry out a series of activities for teaching systems of linear equations, which are proposed by the Ecuadorian curriculum 2016, as well as to promote learning through the construction of knowledge according to constructivist methodologies. Through the development of the interviews conducted, it was possible to identify the problems that exist within the teaching process, such as the use of traditional methodologies and the scarce use of technological resources for this subject. Based on the pertinent analysis, this guide has been developed to suggest alternatives based on constructivist methodologies and innovative techniques.

The present didactic project analyzes the learning of sequences oriented in Precalculus with the use of Tics, therefore, it emphasizes the relation of necessary contents later in Calculus, since its objective is to support the learning of students who need to understand or reinforce their knowledge related to it, because, although the topic of sequences is presented within the curriculum, it is not related to the variational since many times it is not considered necessary. It is important within the subject that each knowledge is clearly specified and ready to be connected with future learning, as mentioned by constructivism, where the student builds his knowledge from previous knowledge. It is also worth mentioning that another significant part is the use of Tics, taking into account the target audience and how the influence of technology is part of the student's immediate environment, seeking to encourage its correct use and increase interest in education and especially in learning. For this, both a diagnostic evaluation and a survey directed to students of first and second cycle of the Faculty of Philosophy, of the Pedagogy of Experimental Sciences Career were applied, so that after completing the analysis of the diagnostic evaluation, the need of the didactic guide is confirmed and with the inspection of the survey, the acceptance of the Tics used is measured.

The present degree work, entitled 'Didactic proposal for the teaching of frequency distribution tables and statistical graphs through the Project-Based Learning (ABPt) approach', addresses the problem investigated in the private educational unit 'Las Pencas'. The research was carried out through participant observation and a survey addressed to a small group of students. The objective of this work is to develop a didactic proposal that incorporates methodological strategies based on the Project-Based Learning (ABPt) approach. This proposal seeks to promote the critical development of students in relation to statistics, specifically in the field of frequency distribution tables. Since, it has been identified that students usually have a superficial knowledge of its meaning and its application in daily life. Therefore, it is sought that the mathematics teacher uses various methodological strategies to promote meaningful learning in this topic.

The present graduation project entitled "Learning basic concepts of probability through the application of games in the first year of high school" proposes the use of games in student learning, addressing the skills outlined in the Ecuadorian national curriculum. The proposal gains strength due to the identification of knowledge deficits among students regarding basic probability concepts, as confirmed through surveys and diagnostic evaluations conducted with students at the Fiscomisional "San José de la Salle" Educational Unit. It is important to note that while these concepts are covered in the eighth grade in textbooks, survey analysis indicates otherwise. This is where the relevance of the proposal lies, as its objective is not only to promote the learning of basic concepts but also to provide the necessary leveling through engaging activities for adolescents, enabling them to understand three key themes of probability: its concepts, types of events, and probability rules. The aim is to foster meaningful learning that will be retained in their memory. Accordingly, a series of activities in the form of a teaching guide and games have been designed as didactic materials to aid students in their learning process based on constructivism, as this pedagogical approach is proposed by the Ministry of Education in the Ecuadorian national curriculum.

The use of technological resources is important in descriptive geometry classes, since it is difficult to explain three-dimensional figures on a blackboard that is a plane. In this sense, the objective of this study was to develop descriptive geometry classes with the use of a software simulator and to evaluate the results of the incorporation of this resource. For this purpose, two classes were designed using the GeoEnZo simulator software and plastic education as support for the class. Then, a survey was made to 39 students of a university degree in mathematics education about the use of the simulator software for descriptive geometry classes. The students indicated that their college teachers used the blackboard and the geometric game to develop the classes, and they worked on sheets of paper. Meanwhile, in the classes where the simulator software was used, the teacher’s explanation was much better in relation to understanding the topics and learning descriptive geometry. In addition, they indicated that the software simulator allowed them to solve and draw more quickly, and that it improved the presentation of their drawings. As part of the conclusions, it is determined that the use of technological resources in the classroom helps to improve the students’ spatial understanding, and therefore, to achieve quality learning.

This degree work called strategies for teaching basic operations with real numbers in eighth grade of the General Basic Education School Victor Alvarez Torres aims to develop strategies for teaching and learning that best suit the student contexts present, through interviews with teachers of the educational unit could be identified that there is still a classical education with mechanical and repetitive processes, where they do not take into account the learning styles of each student, In addition, in higher years students did not master the basic operations, which is why it is important for the teacher, who through the use of various teachinglearning strategies will change their classes, also take into account the learning styles using the VAK model, visual, auditory and kinesthetic, the same that contributes that the student has interest in the study of mathematics and can relate them to everyday life, thus meeting the school objectives set out in the educational curriculum of Ecuador. Therefore, a didactic guide for the teacher and a notebook for the student are proposed, a guide which is worked in a theoretical and practical way, in which the topics of basic operations are deepened in a creative and dynamic way, with the help of strategies focused on the mentioned learning styles, to achieve a significant learning in the basic operations with real numbers.

The present degree work named "Didactic guide for teaching-learning of the clearance and resolution of linear and quadratic algebraic equations" responds to the skills raised in the educational curriculum corresponding to the area of mathematics of Ecuador and aims to develop a didactic proposal containing methodological strategies with the use of digital and tangible materials to achieve a reflection in the student on the clearing of unknowns taking algebraic properties as a starting point. All this based on the analyzed information on the educational reality, obtained through semi-structured interviews aimed at teachers corresponding to the area of mathematics and physics of the "University of Cuenca" in the career of "Pedagogy of experimental sciences". This proposal aims to position algebraic properties such as the clearing of unknowns as essential in solving equations, because students usually don’t dig deep enough and aren’t able to reflect on their importance, which is why it is sought that the teacher of mathematics makes use of several methodological strategies with the use of concrete material that encourage a significant development.

In this research work, the problem of teaching linear functions in mathematics is analyzed and its main purpose is the understanding of this topic to help students achieve significant learning. Based on this approach, didactic guides are elaborated with constructivist pedagogy, gamification and technological applications for the teaching of linear functions as main points. Emphasis is placed on teaching through gamification, these technological resources are: Retomates, Symbaloo and Nearpod, which help the teacher to have control in work groups and organization in their tasks to generate active learning. For this, a brief investigation was made on these technological resources, in addition, what is the role that the teacher plays within constructivist pedagogy, and how gamification allows class activities to be more active. In fact, these activities are elaborated in a didactic proposal for the teaching of linear functions that consists of four classes that consider gamification and the use of technological resources, being these resources used to work on the anticipation, construction and consolidation of each class.

The present degree work refers to the elaboration of a methodological strategy and technological resources as an aid to the teacher in the planning and use of didactical material for the development of the corresponding topics in applications of derivatives. Several investigations indicate that there are shortcomings in teachers when it comes to executing constructivist methodologies in their teaching, since most are limited to performing exercises mechanically and the student becomes a repetitive algorithm machine. However, it is pertinent and possible to ensure that the learnings, techniques, knowledge and innovation of resources serve to create adequate tools that can improve the educational process for meaningful learning. For this, an investigation has been carried out through the technique of interviewing teachers in the area of mathematics of two unified general high school institutions in which it has been confirmed that teachers still use traditional methodologies, not very innovative and with resources that are not adequate , which makes difficult the development of a constructivist class. It should be noted that mathematics is one of the subjects that requires several strategies so that meaningful learning can be achieved, however, several inconsistencies are observed within it. Therefore, a methodological strategy has been developed that contains various didactic and technological resources that serve as pedagogical support and focused on the constructivist paradigm for an adequate use of material, directly benefiting the teacher in the advance of their classes since a well-structured sequence of topics to be developed with strong resources and elaborated in the virtual environment will provide better results in the classroom

The vertiginous development of information and communication technologies - ICT - has revolutionized society and education in particular, but not all that potential is exploited in the classroom because there are difficulties due to the teacher, the infrastructure, and the student. This work is a case study that aims to investigate how ICT is used in secondary education. The research has a quantitative, descriptive, and correlational approach; a survey was applied to recently graduated students to find out how ICT was used at school. It was found that educational institutions have deficiencies in their technological infrastructure, in addition, teachers and students use technological resources in a limited way, wasting their potential; Students spend a good part of their time on social networks and video games, despite acknowledging that it does not benefit them. This information is important because, based on these data; decisions can be made that optimize the use of ICT in Ecuadorian secondary education., El vertiginoso desarrollo de las tecnologías de la información y comunicación –TIC– ha revolucionado a la sociedad y a la educación en particular, pero en el aula no se aprovecha todo ese potencial, porque hay dificultades debidas al docente, la infraestructura y el estudiante. Este trabajo es un estudio de caso que tiene como objetivo investigar cómo se usan las TIC en la educación secundaria. La investigación tiene un enfoque cuantitativo, de tipo descriptivo y correlacional, se aplicó una encuesta a estudiantes que acaban de ingresar a la universidad y están recién graduados, para averiguar cómo se usaron las TIC en el colegio. Se encontró que las instituciones educativas tienen deficiencias en su infraestructura tecnológica, además, que los docentes y los estudiantes utilizan los recursos tecnológicos de forma limitada, desaprovechando el potencial que estos tienen; los estudiantes emplean buena parte de su tiempo en redes sociales y videojuegos, a pesar de reconocer que no les beneficia. Es importante esta información porque a partir de estos datos se pueden tomar decisiones que optimicen el uso de las TIC en la educación secundaria ecuatoriana.

The present degree work refers to the development of methodological strategies for teaching congruence and similarity of triangles with the support of didactic resources. Several researches point out that Geometry is focused on the memorization of formulas, concepts and the drawing of figures in plates, however, when the teacher poses problems of the context, students find it difficult to reason, reflect and visualize the geometric content. In addition, by applying a survey to students in the second cycle of the Pedagogy of Experimental Science Career: Mathematics and Physics, it is corroborated that the resources most frequently used are the blackboard, markers and slides, resources typical of traditional teaching. Therefore, a didactic guide for teachers with active methodological strategies that are useful and appropriate is elaborated, with the purpose of proposing innovative classes based on constructivism and at the same time, motivating students to identify, interpret and understand definitions, properties, theorems, etc. to apply them in the resolution of situations that arise in their lives. Likewise, the proposal is supported by didactic resources since they favor the teaching-learning process, since they are adapted to any content and promote the achievement of significant learning.

This degree work called Methodological Strategy and Resources for the teaching of Counting Techniques aims to develop a methodological strategy and concrete material based on constructivism for an adequate teaching process in the classes of counting techniques, this based on the fact that it was possible to verify problems within the teaching process, through semi-structured interviews with teachers in the area of mathematics in the Educational Unit "La Inmaculada" that prevent to meet the constructivist objectives that are within the Ecuadorian educational reform. The importance of establishing methodological strategies and resources is generated due to the fact that education is in a process of change where it is required to satisfy and cover all the needs of the students, as a consequence of this change it is sought that teachers transform their traditional classes into constructivist practices where the student is the protagonist in this process.the result of the above mentioned leads us to present a guide for the teacher composed of a practical and a theoretical guide that go hand in hand with concrete material, both linked to achieve dynamic classes that generate significant knowledge within the study of Counting Techniques.

This degree work presents the development of a didactic guide for the teaching of concepts of derivatives directed towards teachers of third year of high school. Some research shows that teachers are lacking in the application of constructivist strategies such as: didactic and technological resources for the development of classes on concepts of derivatives. It is essential to establish which resources can help teachers to improve their teaching process in the classroom, so that students can obtain significant knowledge. For this purpose, interviews were conducted with certain teachers of Mathematics in the third year of high school at the La Inmaculada Educational Unit, where they suggest that it would be appropriate to implement activities with technological resources, practical exercises applied to real life that stimulate the student's learning process and that at the same time are easy to use for the teacher. Therefore, a didactic guide is elaborated that contains as pedagogical and constructivist support the use of concrete material and easily accessible technological resources such as: videos, virtual games, demonstrations and simulations in geogebra, which encourage students to identify, demonstrate, understand and interpret when and how to apply the derivatives in events that occur in everyday life.

This research analyzes the problems that students face during the transition between school and the first semester of university studies, in mathematics. For this, at the end of the semester new students were surveyed, it was inquired about their previous formation in mathematics, the process of entrance to the university and the subjects taken in the area of ​​mathematics, in addition the academic records were reviewed once finalized the first semester. It was found that many students come from families where those responsible for the family have no or no primary education; also, that they don´t have good bases in mathematics, and that studies in the career are difficult. It is provided with information to reflect on the selection process, pre-university course and the curricular design of the career. It is very important to know the students who have just entered and their experience during the first semester, not only through statistical figures, but to understand the circumstances they go through., Esta investigación analiza los problemas que enfrentan los estudiantes durante la transición entre el colegio y el primer semestre de estudios universitarios, en las matemáticas. Para esto, al final del semestre se encuestó a estudiantes recién ingresados, se averiguó sobre su formación previa en matemáticas, el proceso de ingreso a la universidad y las asignaturas tomadas del área de matemáticas, además se revisó los registros académicos una vez finalizado el primer semestre. Se encontró que muchos estudiantes provienen de familias donde los responsables del hogar tienen formación primaria o ninguna; también, que no tienen buenas bases en matemáticas, y que los estudios en la carrera son difíciles. Se aporta con información para reflexionar sobre el proceso de selección, curso preuniversitario y el diseño curricular de la carrera. Es muy importante conocer a los estudiantes que recién ingresan y su vivencia durante el primer semestre, no solo a través cifras estadísticas, sino comprender las circunstancias que ellos atraviesan.

Studies show that in higher education institutions, students experience difficulties during the course of their studies, failing some subjects and in other cases, withdrawing from university. Knowing this reality can lead the authorities to make academic decisions to face this problem and improve terminal efficiency, which is an important indicator for institutional accreditation. The objective of this study was to determine the failure and dropout rates of the students of the different faculties, and at what point in the studies these rates are higher. The research was carried out on 1900 students from the University of Cuenca, Ecuador, a cohort that began their studies in March 2013, and a longitudinal follow-up was carried out until July 2018, using the documentary research technique, for which the academic databases owned by the university. It was found that both repetition and dropout rates are higher in the first semesters of study in all faculties, and that Medical Sciences and Architecture have low rates of loss of subjects and dropout of their students, while some faculties from the Social Sciences area they have high rates of failure and desertion. It is important to know these indicators for each faculty and at all times in the students' trajectory, so that the authorities make decisions that allow improving the academic offer.

This work refers to the development of methods and resources for teaching projections and solid views, several researches show that there is a shortfall in the development of spatial visualization skills in students, since a series of mental procedures are involved to operate or transform flat representations into three-dimensional objects. One of the causes of this problem is that the teacher uses the board for his explanations, but it is difficult to represent a three-dimensional object, on a flat surface like the board. The results obtained from tests and interviews reaffirm the difficulties that may arise when obtaining the projections and views of a solid or the identification of the correct positions of the faces of an object when it has been turned over. For this reason, a series of innovative activities have been designed that with the support of teaching resources achieve the skills of visualization and spatial orientation in students. Tangible teaching resources facilitate the manipulation and visualization of the various positions that solids can have, allowing students to observe the views and projections as they actually are. Therefore, a text was prepared for the teacher containing various active methodological strategies with guidelines for the proper use of the specific material, the objective is to propose several alternative options for the development of the subject, and at the same time, students enhance their visualization and spatial orientation capabilities. Also, ideas are proposed to design activities that allow the student to transform and internalize his learnings, so, these activities follow the most significant principles of David Ausubel's constructivist pedagogical model where the real protagonist of learning is the student.