PROJECT-BASED LEARNING MODEL: ITS IMPLICATION IN THE INITIAL TRAINING OF MATHEMATICS TEACHERS

Objective: The objective of the study was to analyze the benefits of a study curriculum under the competency-based approach and to analyze the implications of the implementation of the project-based learning model in the initial training of mathematics teachers in the context of university higher education. Theoretical Framework: The concepts related to the project-based learning model, its characteristics, its phases and educational competencies in the twenty-first century are systematically addressed. The relevance of the curriculum under the competency approach in university education is also analyzed from a critical perspective. Method: The research approach was mixed, non-experimental design, 30 teachers and 54 students in initial training for mathematics teachers participated in the study; the data were collected through a Likert questionnaire, the analysis and interpretation of the results was carried out with the Minitab software, complemented with the information from the semi-structured interview applied to the teachers. Results and Discussion: On average, more than 77% of teachers and more than 72% of students who attended the study have a favorable opinion about the implementation of project-based learning (PBL) in university education; therefore, it favors and strengthens the development of mathematical skills and competencies in the initial training of mathematics teachers based on the realization of mathematicsof tasks and problem solving through collaborative activities for efficient learning of mathematical concepts and procedures. Research Implications: This research highlights the importance of the project-based learning model from the theoretical and practical perspective of competency-based curricular implementation. Theoretically, the concepts of project-based learning are deepened and contextualized, while the practical implication translates into collaborative work between students and the teacher in problem solving. Originality/Value: The findings value the PBL model as a viable alternative for the efficient initial training of secondary school mathematics teachers, with a broad mastery of conceptual, procedural and digital technology knowledge in problem solving.


INTRODUCTION
The Project-Based Learning (PBL) model has been applied since the 60s of the twentieth century in the medical school of McMaster University in Canada, with the intention of instituting a teaching system that would correct the deficiencies of the medical care system; then, in the 70s, the idea of the project method was rediscovered and raised, together with the conception of open curriculum and community education; Later, its application has spread worldwide to the different disciplines taught in higher education, such as engineering, economics, architecture, mathematics and law.
In recent years, the project method has been connected to the constructivist model, making strong inroads into education at different educational levels, distinguishing itself as a teaching technique or even as a philosophy of changes in the entire learning environment, aimed at the development of competencies and capacities in students.As an alternative to the learning model envisaged in the traditional curriculum with methodologies based on memorization, accumulation and mechanical repetition of data.
The authors of this article participated in the redesign of the curriculum under the competency-based approach and the practice of project-based learning (PBL) or PBL, an acronym that responds to the English expression Project Based Learning, at the Faculty of Education Sciences of the National University Hermilio Valdizán de Huánuco.
The curricular redesign was directed by the directors, under the advice of an external consultant, a specialist in the application of the model in university teaching, making the teachers participating in the workshops able to design learning sessions and activities based on 4 the PBL model for the development of their teaching action.during the initial training process of teachers of the mathematics specialty.In this way, to verify the potentialities of this didactic model and also to reflect on its virtues and shortcomings of its implementation, and the educational achievements that can be obtained from its application in practice.
For the proposal of competencies, to be achieved through the PBL didactic model, the PISA 2012 report (OECD, 2012) is taken into account, which defines mathematical competence as an individual's ability to formulate, use and interpret mathematics in a variety of contexts; This includes mathematical reasoning and the use of mathematical concepts, procedures, data and tools to describe, explain and predict phenomena.In addition to the OECD proposal, the use of digital technologies is essential, which constitute essential means and resources in higher education in the digital age.These are linked to fundamental mathematical skills (communication, mathematization, representation, reasoning and argumentation, design of strategies to solve problems, use of operations and formal and technical language and the use of mathematical tools).
From what has been described, the study had as a guiding question: What are the benefits of the curriculum under the competency-based approach through the project-based learning model in the initial training of mathematics teachers?To answer the question, its application in the classroom is described; within a student-centered environment, to develop skills in information management, formulation and resolution of contextual problems from a mathematical perspective.

PROJECT-BASED LEARNING
Project-Based Learning (PBL) is circumscribed within the methodologies that have expanded in recent years for teaching at different educational levels and in different disciplines, standing as one of the alternatives for the competency training of students.In general terms, it is an approach that tries to combine theory and practice, based on the practice of educational activities contextualized to the student as the best way to consolidate valuable, significant and lasting learning, as an alternative to traditional teaching based on lectures and the transmission of knowledge.that has constraints and limitations.Maldonado (2008) points out that PBL is a learning experience that involves the student in a complex and significant project, which allows the integral development of their capacities, skills, attitudes and values; it involves forming groups of people with different profiles, who work together to carry out projects with the purpose of solving real-life problems and requires the teacher to be a creator, a guide, who stimulates students to learn.
According to the Buck Institute of Education (2019), PBL is a learning methodology that seeks to promote the development of twenty-first century skills, preparing students for academic, personal, and professional success, helping them to face the challenges of their lives and the world they will inherit.In this learning model, students work on an authentic project that seeks to solve a real problem or a complex question, crafting a public product or presentation for a real audience; unleashing a contagious and creative energy among students and teachers.

CHARACTERISTICS OF PROJECT-BASED LEARNING
The PBL constitutes a new paradigm in which the student is given the leading role, removing the passive role of receptionist of the various contents within the framework of the teaching-learning process and working from their participation and criticism that allows them to achieve the determining aspects defined in the project.This implies that the teacher's work must be aimed at enabling each student to achieve their cognitive development by taking control of their own learning and thus allowing everyone to learn to socially construct knowledge.
Fernández-Cabezas (2017) highlights conceptual comprehension in PBL instead of memorizing facts, thus preserving knowledge, increasing students' performance, motivation, and ability to work in groups.This learning is characterized by being interdisciplinary and responding to the demands of reality, it is focused on teachers developing competencies in students that prepare them for life, through the development of projects aimed at achieving competencies in a specific area, as well as in areas integrated with the predominant practice of collaborative work for the development of new skills and competencies for their academic and personal training.
PBL helps students to think, reason, work in a team, solve problems, and all the qualities that will help them in their social and work life.In this model, collaborative work is encouraged in the study of a case or problem, focusing on generating viable solutions, thus assuming greater responsibility for their learning and stimulating at the same time the value of teamwork by developing a feeling of belonging to it.To this end, the role of the teacher is to promote the development of skills such as complex and critical thinking, cooperation, leadership, communication, multidisciplinary work, problem solving, use of technology and decision- 6 making; skills to be developed in the twenty-first century (UNESCO, 2013), which are summarized in Figure 1.In this way, students can plan, implement, and evaluate activities for purposes that have real-world application beyond the classroom (Cobo, 2017).It also facilitates the development of transversal skills such as group work, cooperative learning or effective communication (Garrigós & Valero-García, 2012, p.126).

Figure 1
The Six Educational Competencies of the 21st Century, UNESCO-PISA, 2013 The PBL methodology presents an important change that, in addition to affecting the learning objectives, attends to modifications in the roles played by the members involved in this process.This methodology considers that problem solving is the basis for identifying the topics or learning objectives in relation to the subject; and then submit to a process of validation and interpretation of results, the scaffolding of the project, the demands of the final product and collaborative work (Benjumeda et al., 2015).
In addition, PBL is characterised by focusing learning on questions and concerns that students express in relation to their immediate environment, and which, when properly formulated and guided, can be used to address different aspects of the curriculum in an integrated way.It also offers: critical affinity, practical relevance and action, from the processoriented approach, holistic learning, collective realization, of an interdisciplinary nature.On the other hand, it promotes the development of students' skills and enables the incorporation of ICT 7 in the teaching-learning process (Badia & García, 2006).Enabling synergistic interactions between teachers and students in a positive learning environment in the classroom, which facilitates communication and allows the teaching and learning process to be optimised (Toledo & Sánchez, 2018).

PROJECT-BASED LEARNING PHASES
In the PBL model, students develop research based on a challenging problem or question that is related to their interests and their real world.In this search, they must generate questions and look for different sources to find solutions.Throughout this process, the school community reflects on what, how and why they are learning.Finally, students present their projects to a real audience in their community.
According to Disla (2013), the project-based learning strategy, using a technological tool, develops in students the competence of collaborative work in higher education students, the strategy allows the members of a group to acquire and strengthen their knowledge with the opinions of the other members through the exchange of ideas.Thus, this methodology stimulates collaborative learning and gives more meaning and connection with the real world to what they learn.As benefits of PBL, students explore their own interests, share their talents, build skills in critical thinking, planning, communication, and creativity and innovation.
For the implementation of PBL, the following steps are suggested: Selection of the topic and posing of the question; collaborative team building; definition of objectives and description of the product to be developed; organization and planning, assignment of responsibilities; incontext research; data analysis; application of new knowledge, implementation; presentation of the project; collective response to the initial question, reflection on the experience; and, evaluation and socialization (systematization of the experience).
The steps indicated for the project are summarized in four phases: Choice and motivation, planning, execution and evaluation (Carrizo, 2001).The fulfillment of these phases is based on collaborative work, it is based on an adequate organization of classroom activities, together with the organization of work groups and the distribution of roles among their members (Moraga & Soto, 2016).Under this premise, the inclusion of PBL in the area of mathematics requires the inclusion of explicit demands for procedures and processes by the teacher so that, while maintaining the student's autonomy, the desired mathematical aspects can be worked on, including an order and scheme in the problem-solving processes (Planas, 2011).The PBL is a didactic methodology that organizes the teaching and learning process by developing projects collaboratively in groups of students; understanding the project as the learning process that the group must follow as a result of its learning (Badia & García, 2006).
As such, it engages students in a long process of inquiry in response to a complex question, problem, or challenge, exercising in the student the possibility of participating with voice and vote.In this sense, the development of projects becomes a didactic strategy that is part of the so-called active methodologies, which is conceived as the search for an intelligent solution to the approach of a problem or a task related to the real world (Martí et al., 2010).
Among Role of the student: is at the center of the learning process and manages it; must be involved in a systematic process of research; work autonomously that promotes intrinsic motivation and collaborative learning; makes decisions regarding learning goals, makes continuous improvement in their products, presentations or performances; reflects with their classmates on the task or project they develop collaboratively, and assesses the evaluation of the process (Cobo, 2017).Thus, he becomes the protagonist of the training process, becoming the builder and controller of his own learning, assuming all the academic and ethical responsibilities that this implies.Thus, PBL offers students the opportunity to learn through research in a dynamic and active way in real situations, which end with the development of authentic and useful products for a social need (Del Valle-Ramón et al., 2020).

Role of the teacher:
Approach the teaching-learning process from an interdisciplinary perspective, through the design and implementation of projects, related to an experience and a specific topic, to collaboratively address the projected objectives (Medina & Tapia, 2017).9 According to Restrepo (2005), he considers that teachers should be specialists in the method at the time of planning, supervision, and evaluation of the process; motivate, reinforce, provide clues, be flexible in the face of students' critical thinking and have time to attend to students' concerns and needs.Teachers are called upon to encourage curiosity, creativity, the development of critical thinking, be the guide, provide resources and apply feedback at the time they consider appropriate (Erazo, 2020).Well, PBL requires teachers who are always vigilant and attentive to the resolution of doubts or solve problems that arise during the process.(Cesar et al., 2021) In summary, in the context of PBL, the teacher carries out activities of planning, observation, accompaniment, permanent stimulation, and evaluation of learning achievements, serving as a guide and tutor of the student, for the practice of activities oriented towards selflearning and motivates them to plan, both independently and collectively, their process, put it into practice.Then, it systematically evaluates, provides feedback to the process and evaluates its results; be flexible in the face of students' critical thinking; from the development of projects aimed at solving complex problems, through the performance of tasks through collaborative activities.

METHODOLOGY
The research approach used in the study is mixed, the quality of activities, relationships, means, materials or instruments is studied, in the process of designing and implementing the competency-based curriculum under the PBL didactic model.Based on the explanation of the activities to be carried out (quantitative approach) as an understanding of these activities (qualitative approach), through which the research process is strengthened (Barrantes, 2014).

PARTICIPANTS
The study involved 30 teachers working in the Faculty of Education Sciences and 54 students in the last cycles of studies in initial training for mathematics teachers, whose selection was made in a non-probabilistic way in the intentional modality.The participants were asked for their opinion on the relevance of the curricular design under the competency-based approach and whose execution is carried out with the PBL methodology, providing them with the necessary time to reflect on their answers.
The fieldwork objective was to collect self-perception about the relevance, factivibility and effectiveness of the implementation of PBL in the training of mathematics teachers, for academic and professional preparation according to the requirements of education in the regional, national and global context.

DATA COLLECTION TECHNIQUES AND INSTRUMENTS
As for the information collection techniques used to analyse the reality under study, it has been mainly based on the use of documentary analysis (institutional educational model, didactic models, etc.); these have been complemented with a Likert-type survey (with answers: strongly agree (TA), agree (DA), partially agree (PA) and disagree (ED)), for teachers and students; and a semi-structured interview for teachers, so that they can express their opinion on the virtues of the project-based teaching-learning model.Both data collection instruments have been validated by an expert; In addition, the interview questions are consistent with the research questions that are raised.

DATA ANALYSIS
The analysis of the study data translates into the presentation and processing of quantitative data in tables and figures using the Minitab 20 software, interpreting the results obtained in the research process as a percentage.Meanwhile, the data from the interviews conducted with the study participants are summarized taking the most salient points.The analysis of the results obtained from the questionnaire (quantitative) and the semi-structured interview (qualitative) led to the specification of the objectives proposed for the research.

RESULTS
The survey of the 30 teachers, through the survey on the applied Likert-type scale, regarding the relevance of the curriculum based on the competency-based approach through the PBL didactic model.On average, 39.0% of respondents expressed total agreement; 38.1% responded that they agreed; 15.7% expressed partial agreement, and only 7.2% disapproved of the statements made, Table 1.The results of Table 1 show a majority approval, more than 77% of the respondents, of the implementation of the curriculum under the competency-based approach with implementation under the project-based learning didactic model.
The survey applied to the students, who had a process of induction and information on the fundamental aspects of the curricular redesign of the mathematics career, and on the PBL didactic model, yielded the results summarized in table 2.Where it can be observed that, on the motivation and approval of carrying out their educational activities through the curriculum based on the competency approach and the didactic model of PBL, based on the six statements raised.The result being the percentage obtained, 32.1% consider themselves to be in total agreement, 40.12% express agreement, 17.9% express partial agreement, and only 9.88% express their disagreement with continuing their university studies through the pedagogical proposal.In the semi-structured interview, six questions were formulated, which were summarized taking into account the fundamental ideas expressed by the interviewees, which are transcribed below: Do you consider the use of PBL's didactic model essential to motivate the process of teaching and learning mathematics?"The innovation of methodological strategies at the higher education level is very important, because students are trained to be educators of children and young people, especially in the area of mathematics we have an enormous responsibility to assume from university education [...].To this end, working with projects is one of the active and highly motivating ways to carry out the process of learning mathematics." What strategies would be the most appropriate to promote among students, within the PBL model, for the development of students' academic competencies?"Any strategy that leads the student to develop their capacities and competencies is important, through group or team work, so that students interact with each other in the construction of significant learning, as well as through tasks and problem solving so that they put into practice the conceptual and procedural knowledge acquired." In PBL is it important that the student knows how to transfer the learning achieved to the resolution of mathematical problems and the context?"According to the literature on the model, the first thing that is required to undertake an activity or practical action is the knowledge of previous knowledge, the knowledge of concepts, properties of the mathematical content under study, since this becomes the main tool to solve mathematical problems, contextualizing the environment and the 13 needs of the students, only in this way will it be possible to achieve the proposed objectives. in the integral formation of students." Is it possible to implement PBL in other professional careers so that they are applied to all teachers regardless of their specialization?"The PBL strategy or model can be applied in all subjects in higher education, for example, in social sciences subjects, it facilitates activities that truly have to do with real problems in the student's context, through collaborative work and socialization of ideas; also in natural sciences in the realization of activities of exploration of the physical environment and experimental activities in the classroom and laboratory; Thus, this didactic model has an applicability, even more robust than in mathematics, in other disciplines.".Do you think that at PBL we really work on the competencies of future teachers?"In PBL, students are responsible for their own learning process, they must set hypotheses and work goals, look for information, resources and means to solve the problem and organize tasks to transfer what they have learned to all subjects; through collaboration between peers, under the advice of the teacher; a fact that motivates students to share their opinions provide feedback to their classmates, individually or in groups through the implementation of a series of actions, interactions and resources, for the strengthening of professional training; put it into practice and evaluate it." What resources do you consider to be the most recommended to efficiently carry out the PBL didactic model?"In the PBL model, it is very interesting to carry out activities that truly have to do with problems of reality or context; the use of resources from digital technology is essential for this, since students are digital natives and have as a means and tools to motivate themselves and dynamize their learning; but the resources of the student's social, cultural and physical environment cannot be left aside; thus, for PBL to have the expected results in the area of mathematics, the convergence of contextual and digital resources would be essential."14 educational process with explicit goals aimed at the comprehensive development of students in a given area (Martí et al., 2010).Therefore, in this study, PBL has an acceptance of more than 77% of the teaching staff, who are motivated and trained for its implementation, as it will contribute to the improvement of their performance as professionals in their teaching practice.
From the perspective of the students, more than 72% of the students who received an induction talk to the PBL didactic model, have a favorable opinion for its implementation, since it will contribute to their integral development in their professional training, but it does seem clear that they will learn skills such as teamwork, search for information, use of collaborative tools, time management, among others.Thus, teamwork, discipline and autonomy are strengthened, it is a methodology that allows solving or proposing a solution to real-life problems and in that work, develop competencies, capacities and acquire knowledge (Castro-Valle, 2022).At the same time, the development of cooperation, collaborative, and communicative skills is promoted, and teamwork is strengthened (Burgos et al., 2021).
The results of the study coincide with Ruikar & Demian (2013) that the PBL model in the field of university education has significant value in teaching, since it facilitates the student's autonomous learning, at their own pace, as well as encourages participation and learning.This contributes to the consolidation of PBL, which has an optimal impact on the initial training of teachers in the specialty of mathematics, through the achievement of processes and procedures conducive to the achievement of significant learning; to have a reflective and critical perception of the importance of innovation and adaptation of contextualized didactic strategies, guaranteeing the acquisition of learning and skills such as autonomous study, the search for information, the preparation of presentations, teamwork, time planning and the ability to express oneself appropriately.

CONCLUSION
The results obtained in the study show a majority acceptance of project-based learning in the initial training of mathematics teachers.From the perspective of the students, it induces greater intrinsic motivation and predisposition to study and to achieve the learning of thematic In short, the project-based learning model supported by normative and pedagogical foundations in the implementation process, is seen as an alternative for the efficient initial training of mathematics teachers in university classrooms conceived from the implementation of the curriculum under the competency-based approach in a practical, structured, systemic and contextualized way.according to the needs and challenges of today's education.
Project-Based Learning Model: Its Implication in The Initial Training of Mathematics Teachers ___________________________________________________________________________ Rev. Gest.Soc.Ambient.| Miami | v.18.n.7 | p.1-16 | e08136 | 2024.8 2.4 ROLE OF THE SUBJECTS OF EDUCATION IN PBL the essential elements of Project-Based Learning, the following stand out: the question as a starting point; use of ICT and Web tools; research as a pedagogical strategy; work based on the interests of the students; development of projects taking into account the reality and the immediate context; and to make the learning goals explicit.Through this work methodology, the aim is to develop in students the skills and knowledge in response to the most frequent problems of daily life, thus making them more competent, from the assumption of a leading role within their own learning process, with the teacher becoming a guide and facilitator of it.constituting an open work model, where evaluation is part of the learning process itself (López-Meneses, 2016).Thus, in the PBL didactic model, the roles of the teacher and the student are shared, and on other occasions complementary.
The implementation of PBL has been a novel initiative in the initial training of mathematics teachers; from the stage of planning, reinforcement and evaluation of the pedagogical process based on a problematic situation and the pantheatic of challenging situations.Here, teachers have content and objectives, use real assessment, guide and guide the Project-Based Learning Model: Its Implication in The Initial Training of Mathematics Teachers ___________________________________________________________________________ Rev. Gest.Soc.Ambient.| Miami | v.18.n.7 | p.1-16 | e08136 | 2024.
content through the generation of questions, search for answers, development of research capacity, teamwork, autonomy and responsibility in collaborative work with their peers and empathy with the teacher.for the elaboration of a final product or process.The implementation of the competency-based training curriculum through the projectbased learning model in an interdisciplinary way makes mathematics teachers in initial training ___________________________________________________________________________ Rev. Gest.Soc.Ambient.| Miami | v.18.n.7 | p.1-16 | e08136 | 2024.15 the protagonists of their own learning, putting their conceptual and procedural knowledge afloat for the resolution of problems in the context, with theoretical arguments and relevant use of technology.

Table 1
Teachers' opinion on the implementation of ABS in teacher training.

Table 2
Opinion of mathematics teachers in initial training on PBL.PBL generates interest and permanent interaction between the teacher and the students, who are connected in the resolution of contextualized mathematical problems.