Before the COVID-19 pandemic, the country’s education system was in a process of profound transformation and faced various challenges. Despite the important reforms undertaken, problems related to the quality, equity and financing of education persisted.

The educational system was characterized by its high degree of privatization. An OECD (2017) report highlighted that it had one of the highest percentages of students in private schools among member countries. This situation contributed to deepening socioeconomic inequalities, since higher-income families tended to opt for private schools, perceived as having better quality.

However, the implementation of these reforms faced several obstacles. The resistance of some sectors and limitations in terms of resources and training for teachers were notable challenges (Raczynski and Muñoz, 2017) . Furthermore, improving educational quality required a comprehensive approach that included teacher professional development, improved school infrastructure, and an updated and relevant curriculum.

In summary, the country’s pre-pandemic education system presented a scenario of ongoing reforms with persistent challenges. The need to address inequity and improve educational quality was clear priorities. The COVID-19 pandemic, which arrived in this context, not only disrupted these processes, but also posed new challenges and amplified existing deficiencies.

Mathematical learning in secondary education is not only an intellectual construction that culminates years of basic training, but also an essential tool for understanding the world and effective participation in society. Mathematics, with its universal language and logical principles, acts as a pillar in the development of critical thinking, problem solving and analytical skills of students.

In the context of secondary education, mathematics transcends the only acquisition of computational skills; They promote a way of reasoning and facing complex challenges, both in related academic disciplines and in everyday situations. Mathematics education prepares students for advanced courses in science, technology, engineering, and mathematics (STEM), fields that are fundamental to the economic and technological progress of any nation.

In this sense, the analysis of the current situation of the country is crucial to identify gaps and opportunities within the educational system. It is essential to recognize the different realities and needs of students, as well as the capabilities and limitations of teachers and institutions. This recognition must be translated into concrete actions and well-founded strategies that allow overcoming barriers to mathematical learning and promoting equitable and quality education (Donoso-Díaz & Reyes, 2021) .

Mathematics and the geometry that is part of it are a cornerstone of the educational curriculum due to their ability to develop critical thinking, problem solving and preparation for life in a modern society. Your teaching must be carefully designed to overcome obstacles and maximize its relevance and applicability.

The closure of schools due to the pandemic brought a sudden transition to online education, imposing significant challenges on the teaching of mathematics and geometry during the upper technical-vocational secondary education cycle.

Educational innovation is a dynamic field that encompasses various strategies to improve teaching and learning. Among the most relevant types of innovation are:

Pedagogical Innovations: Project-based learning is a methodology that fosters critical and problem-solving skills (Thomas, 2000) . Problem-based learning is a methodology that encourages collaborative work, critical thinking and decision making (Dolmans et al., 2005) . Gamification, for its part, uses game elements to increase student motivation and engagement (Kapp, 2012) .

Technological Innovations: Augmented reality and virtual reality offer immersive learning experiences that can improve understanding of complex concepts (Wu et al., 2013) .

Curricular Innovations: The integration of 21st century skills into curricula seeks to prepare students for future challenges, promoting critical thinking and creativity (Voogt & Roblin, 2012) .

Organizational Innovations: Changes in the structure and culture of educational institutions are crucial to support the implementation of pedagogical and technological innovations.

Currently, there are a wide variety of methodologies that can be chosen and adapted according to the nature of the content. Nonetheless, many of these methodologies require training for teachers.

To ensure a representative and relevant sample, inclusion and exclusion criteria were defined. The inclusion criteria covered an age range of 16 to 18 years of students who are enrolled in the third and fourth years of secondary education in technical high schools in different communes of the capital. The exclusion criteria were students over 18 years of age, students in courses below the third year of secondary education and students from non-technical high schools.

Thus, the selection of the sample for the research was limited to this target population and of the 340 students identified as the total eligible population, 307 students were included in the sample for the final analysis. The study was carried out in three technical high schools in three different districts of the country’s capital. At Lyceum “A”, a technical establishment, 150 students participated; in Lyceum technical “B”, the sample included 104 students; and the Lyceum technical “C” contributed with 53 students.

Ethically, the study was meticulous to ensure fair and respectful treatment of all participants. Special attention was paid to obtaining informed consent in a transparent manner, guaranteeing the privacy and confidentiality of personal data. Participants were assured the right to withdraw from the study at any time without any repercussions. The research was carried out with full transparency, ensuring that students were fully informed about what aspects of their performance were being assessed and when.

To know the effect that the pandemic caused on secondary technical high school students, it is necessary, first, to evaluate the level of mastery of geometry content that post-pandemic students had. Then, it was interesting to know their self-perception of both the subject and the usefulness of geometry content in daily life. Finally, it was relevant to measure the students’ ability to apply geometry in a real environment because students who graduated from technical high schools use this knowledge in their jobs and/or in their higher studies.

These arguments generated three variables:

To account for these variables, three instruments were implemented, the test development process involved several stages of validation and adjustment. Initially, a pilot test was carried out with a group of students, which led to modifying the instruments to focus on more relevant areas of the geometry curriculum. In each adjustment stage, mathematics experts, teachers with a master’s degree in education, were consulted to validate the content (commission of experts) and, subsequently, a validation commission with doctors in education to ensure reliability (commission of validation). At each stage all the requested adjustments and revisions were made.

The first instrument called “Level of Mastery and Knowledge in Geometry (NDCG)” consists of questions selected from the university entrance tests of the years 2016, 2017, 2018, 2019 and 2020; the contents of these tests are topics that everything third year student year of high school technical must dominate. The selected questions are focused on area and volume of the unit of geometry, where, first, the question is presented, and four alternative answers are given. The instrument indicates the objective itself and establishes the score and approval range. The objective of this test is to measure the level of mastery of geometry content that students have.

The second instrument “Self-Perception of Knowledge and Mastery of Geometry (ACDG)” seeks to determine the perception that students have in relation to the usefulness of geometry in the real environment. It aims to quantitatively determine the proximity that geometry has in their daily lives and the usefulness it has as a tool to solve problems in real environments. There are five questions that must be answered by marking one of the three options presented in the rubric.

The third instrument “Applied Geometry Evaluation in a Real Environment (EGAER)” aims to measure the level of analysis and mastery that students may have when using geometry in real environments. In this test, two cases are presented where geometry must be used to solve them. For this instrument there is a scoring scale for each question and the rubric indicating the form of development that is expected to be carried out and the way of presenting the result.

The different tests used in the diagnosis (form A tests) show students with a low mastery of mathematical tools for solving abstract problems and for contextualized problems, where none of the courses of the three high schools could obtain the minimum passing score of 43.2 points, In addition, it can be seen that they have ignorance of the contents corresponding to their study cycle and lack of knowledge about contents that they must handle as a basis for the second cycle. Additionally, students show a lack of motivation towards geometry caused, perhaps, by the way in which the contents are presented, far from previous knowledge, and by the little usefulness they see of geometry for solving problems in their daily lives. The above could be observed through the average obtained in the instrument called “Geometry evaluation applied in a real environment— Form A”, which is 5.8 points out of a total score of 70 points and a minimum passing score of 43.2 points.

It can also be observed that the geometry content presented in the instruments is totally unknown to most students, this is reflected in the mode of the different instruments. It should be considered that the contents of the instruments were related to the calculation of area and volume, which are knowledge that is taught in courses lower than the 3rd year of secondary technical high school.

The COVID-19 pandemic has significantly altered the educational landscape, impacting teaching and learning methods around the world. As institutions adapted to school closures and the transition to online learning, the need arose to evaluate the educational consequences of these interruptions, particularly in fundamental and complex areas such as geometry, relevant support for the development of logical thinking, and analytical in secondary technical education.

Regarding the first research question: How has the educational disruption caused by the COVID-19 pandemic affected the achievement and understanding of geometry for technical students?

The answer is that the decisions implemented during the pandemic were indeed catastrophic for the subject of geometry. The results of the three tests showed extremely low results, there were even courses in some high schools that had 95% of their students with 0 scores in some of the tests. Showing that they do not master the necessary content both for their future jobs and for entering higher education. It must be considered that geometry is the basis for the work they do as technicians later in everyday life.

Putting the average together in a waterfall graph, the three high schools show increases in their scores. However, they also show slight decreases, except for the test 3, which only increased.

Additionally, the results of the “Level of mastery and knowledge in geometry (NDCG) tests— Table 1 ” show a correlation between performance in geometry and the perception of its applicability in real life ( Table 2 : Self-perception of Knowledge and Geometry Domain); That is to think, there is statistical evidence that the students who had acceptable to good results conceived geometry as having high applicability in real environments and the students who did not have good results had an unfavorable perception and were not motivated by the knowledge and use of that subject.

The second research question: What measures are feasible to propose for academic recovery? It refers to the possibility of measures for academic recovery. The answer is that there is indeed the possibility of academic recovery because there is currently a wide range of innovative methodological strategies both with active methodologies and with ICTs. However, it is necessary to choose the appropriate strategy according to the content and train in technologies. Also, adequate infrastructure is needed.

This text provides a framework for the study objective and research questions, establishing the basis for an in-depth and structured analysis of the effects of the pandemic on mathematical learning in technical-vocational secondary education in the country.

Additionally, these tests offer a quantitative measure of the knowledge and skills in geometry and mathematics that students have managed to retain and apply after an extended period of online education. Additionally, they allow comparisons to be made with standards prior to the pandemic event, to establish significant differences in performance.

According to data provided by the Agencia de Calidad de la Educación (2019) , Chilean students have presented, from 2009 to 2018, one of the lowest and below average results in mathematics. Highlighting the low mastery of students in mathematics, a condition that has not been improved.

Comparing these results with the results obtained with test 1 of this work, the students presented results close to the minimum scores, a similarity and confirmation of the condition presented by the students in the mathematics domain can be observed.

Both investigations agree that there is a relationship between the perception of mathematics and the performance of students. Demonstrating that the more useful the student perceives about the use of geometry, the better performance they will present in the evaluations.

For their part, Cordero et al. (2018) indicate that student performance is directly related to the student’s attitude towards mathematics.

García et al. (2019) point out that this low performance occurs mainly in students from a low socioeconomic level. In comparison with the present study, it can be seen that the three high schools where the instruments related to the perception and mastery of geometry were applied were high schools where the highest percentage of students were from a low socioeconomic level.

Chaparro et al. (2016) sought to identify profiles of high school students, based on variables related to academic performance, socioeconomic status, cultural capital and family organization. A total of 21,724 high school students, from the five municipalities of the state of Baja California, took part. A K-means cluster analysis was performed to identify the profiles. The analyses identified two clearly defined clusters: Cluster 1 grouped together students with high academic performance and who achieved higher scores for socioeconomic status, cultural capital and family involvement, whereas Cluster 2 brought together students with low academic achievement, and who also obtained lower scores for socioeconomic status and cultural capital and had less family involvement. The result showed that factors as commitment to learning, socioeconomic status, and family support were important factors affecting students’ performance. It can be deduced that the importance of socioeconomic and family factors can affect mathematics even more given the level of abstraction of this subject.

Bernal et al. (2021) examine the impact of geometric shape perception on mathematical creativity. The authors explore the influence of characteristics of geometric shapes [simplicity, complexity, symmetry, etc.] on students’ mathematical thinking and creativity. The results show that the complexity and symmetry of geometric shape have a significant impact on the generation of creative ideas in mathematics. Both studies agree that the perspective that students have on geometry is directly related to the usefulness that students can give to geometry. In this context, the present study observed that there is a relationship in the highest scores of the evaluations applied to the students with the perception they have of geometry and with the mastery they have of geometry and with the mastery they had in using geometry in solving real problems.