INTRODUCTION

According to ^{Amado (2015}), although the integration of technologies in teaching and learning mathematics is widely recommended, including by the Organization for Economic Cooperation and Development (OECD-Organização para a Cooperação e Desenvolvimento Econômico), and with an increase in technological resources in Brazilian schools, there is not the use of such resources in the classroom in the same proportion. There is a need for continuous investment in the pedagogical training of teachers for this purpose.

Teacher training, both initial and continuing, demands an organization that provides educators with access to what is a priority to ensure the maintenance of the quality of the teaching process. According to Even and Ball (²⁰⁰⁹), all countries face the challenge of preparing and maintaining mathematics teaching that is effective and prepares students for life in society. However, teacher training systems are based on resources already incorporated into the culture, organization, and nature of education, hindering modifications, in particular, digital skills such as Information and Communication Technologies (ICT) like personal computers, smartphones, interactive whiteboards, tablets, and netbooks, among others.

A distinction must be made here between the use and acquisition of digital technologies. Using these technologies is much more demanding than purchasing technological equipment. According to ^{Amado (2015}), technologies can significantly expand teaching, but their use needs to be together with good planning. The introduction of technologies into teaching and learning mathematics is quite complex for teachers, and it is the teacher who has the power to transform technological resources into useful tools for student learning. There needs to be constant support and monitoring for teachers in this regard, both those who are just starting their professional careers and those who already have extensive experience.

Therefore, it is not enough to simply want to take a computer or tablet to use as a technological tool in the classroom. It is necessary, above all, to know what one intends to do with the resource. The importance of good lesson planning, scoring objectives, activities to be developed and questions about the proposal are fundamental for the teacher to make good practice with digital technologies, consider ^{Amado and Carreira (2015}).

In this context of the need for continued training, the use of digital technologies, and the importance of planning, this research was developed. Pedagogical training meets the need that teachers feel about constant updating, in line with new educational practices and the profile of students, who have indiscriminate access to digital technologies in the environment outside the school, and the teacher cannot be far away from this context, given the countless possibilities that technologies present in the pedagogical flow and new educational methodologies.

The study was carried out to assist teachers in planning mathematics classes to integrate digital technologies into their teaching because as already mentioned, planning is a very important stage in the teaching process. Given the above, the research problem is: how can the mentoring strategy influence mathematics teachers in planning their classes integrated with digital technologies?

Mentoring requires a relationship between at least two people: the mentor/trainer, characterized by the most experienced teacher, and the apprentice/trainee. The effectiveness of the strategy occurs if the mentor manages to create an environment in which the apprentice feels integrated and welcomed, leaving him/her in a position to express his/her questions and doubts. The mentor is required to be sensitive enough to perceive the dosage of his/her action towards the apprentice, sometimes just supporting, sometimes being more incisive, exemplifying a certain procedure or technique.

The teachers who took part in this study were two public elementary and high school teachers, and the professionals were chosen based on a relationship already established with both and the availability and acceptance of planning their mathematics classes with a focus on the integration of technologies. At the beginning, an informal conversation with the teachers was proposed to understand their needs and expectations of the meetings. From this, it was found that technologies are far from the school context, although the institution where they teach has technological resources because they do not feel able to use them, they end up carrying out their practices in a traditional way.

DIGITAL TECHNOLOGIES IN THE EDUCATIONAL CONTEXT

According to the Regional Center for Studies for the Development of the Information Society (CETIC-Centro Regional de Estudos para o Desenvolvimento da Sociedade da Informação), in research carried out in Brazil, technologies are increasingly closer and available to the population, even considering the disparities between social classes. The use of the Internet and mobile technologies increases annually among children and adolescents between nine and seventeen years old (^{BARBOSA; COSTA, 2022}).

These data are important, both for researchers and public policymakers due to the implications they can have on the social environment, cognition processes, and the educational context. For example, using a single portable tool, it becomes possible to access different media for study, personal activities, work, and entertainment, at any time, saving time and “bringing” distances closer together.

The idea of shortening distances can be understood as the relationship between the periods lived, the training of educators, and the current educational practice. The modernization of materials and methodologies and the level of access to information enjoyed by students today only reinforces the need for constant updating by the educator to access the “universe” of students, motivating them to build their knowledge through the mediation of the educator as someone closer to their context.

In the case of this study, the mathematics curricular component is addressed with its challenges and possibilities, according to the National Common Curricular Base (BNCC-Base Nacional Comum Curricular):

[...] the teaching of Mathematics aims at a comprehensive understanding of the world and social practices, highlighting that teaching must be contextualized and interdisciplinary, but at the same time, following the development of the ability to abstract, perceive, and use the imagination (^{BRASIL, 2016}, p. 132).

While technological advances make it possible to facilitate daily activities, different types of experiences lead to different structuring of the brain, as highlighted by ^{Prensky (2001}). “Digital natives” (^{PRENSKY, 2001}b) have online skills and competencies, they easily acquire basic knowledge that allows them to (un)install applications, explore them, search for music, photograph, film, and share information, whether through communication applications or social networks. The user can generate content and new sources of information, transforming culture based on his/her experiences.

According to ^{Wankel and Blessinger (2012}), digital technologies have shown what it means to learn in the contemporary world, improving the vision of learning collectively using immersive technologies so that learning is more enjoyable and interesting to the student. Based on its use, it is suggested to explore software, applications, and different media. This is important for the process of building student learning and provides opportunities for personalizing teaching. When arriving at school, the devices are put away, which means that the content is covered traditionally, producing only training and practice. On the other hand, solutions are beginning to be disseminated that facilitate the development of applications for mobile devices capable of offering virtual laboratory simulations on these platforms (^{ZERVAS et al., 2014}).

The use of technology goes far beyond simply being present in a digital era or being considered a current tool, but also for providing, when well designed and appropriate to the context, the construction of knowledge by students with the educator mediation. ^{Neide and Quartieri (2016}, p. 10) state that “if we consider the student's cognitive structure, visualization can promote mathematical learning in several ways”. From this, ^{Amado and Carreira (2015}) emphasize that:

As a first consideration, we argue that students learn from the results of their work on relevant and interesting tasks and, above all, from the possibility of sharing and discussing their mathematical ideas with their colleagues and the teacher. Technological resources play an important role during class when students are encouraged to work autonomously, seeking to solve problems and questions that are proposed to them, dealing with ideas and mathematical relationships, thinking, reasoning, applying, and developing concepts (^{AMADO; CAREER, 2015}, p. 14).

Therefore, the authors highlight what ^{Piaget's constructivist theory (1936}) defends, where the students are the one who constructs their concepts based on experiences with the environment and:

The success of students' learning, in this type of class, depends on the implementation of a teaching strategy that presupposes different moments, but in which students' work with mathematical tasks, supported by teaching resources, occupies a central position. This differs from another perspective in which the teacher explains the content and the student then exercises structured questions aimed at the assimilation of rules, procedures, or facts (^{AMADO; CARREIRA, 2015}, p.14).

The simple fact of introducing technology into the classroom does not mean that it will help in the process of building student knowledge, as each activity requires the teacher to plan his/her class well so that the use of the technological resource is effective and is not a superficial and empty class. ^{Dullius and Quartieri (2015}) reinforce that planning, setting objectives, choosing materials, selecting tasks, and anticipating questions gain a central dimension in the teacher's practice with technological resources. This is where fundamental questions arise, such as: how can the computer or tablet be used? Is it expected that some learning will result from this use?

MENTORING

Mentoring consists of a strategy in which theoretical training is supported by practice. It presupposes a relationship between at least two people, one being the mentor/trainer and the other apprentice/trainee. The mentor needs to have characteristics that allow him/her to create an environment conducive to the apprentice, in which he/she feels comfortable expressing his/her doubts and questions.

According to ^{Amado (2015}), during the mentoring process, a professional with a deep technical and experiential background (mentor) proposes to help another person who has less knowledge and practice time. The mentor is the professional who acts provoking reflections and insights for the growth of another person with less experience and technical arsenal. The idea is for the mentors to share their experiences, learning, overcoming, and technical knowledge to show possibilities for action, problem and conflict resolution, and management, among other objectives. The apprentice will be able to test in practice the suggestions proposed by their mentor to, from there, check if it is a path/strategy that makes sense for them, or if it is a case of adjusting and adapting to achieve the results they want.

Mentoring has many proven results in medical and nursing professionals, for example. There is evidence of the use of mentoring strategies in teacher training in the United States, England, Norway, and Portugal, as ^{Amado (2007}) describes. In Brazil, we can mention, for example, the work of ^{Alcântara (2015}), which used mentoring to help teachers in the early years of elementary school in using tablets in mathematics classes.

According to ^{Sundli (2007}), mentoring can be a way of helping to develop the apprentice's capabilities, increasing their professional and personal potential. Continuing teacher training appears as a fundamental strategy to improve the quality of these professionals, as it aims to help teachers manage the demands of the classroom through knowledge of techniques and skills essential for their work with the students.

^{Amado (2015}) warns of the need for knowledge of the context in which teaching practice is being carried out, and that the mentor must have a solid knowledge of curriculum and classroom management, as well as experience in using technologies with an educational bias. You must encourage the apprentice's creativity, and learn from them, not setting up as a model to be followed but giving them support and allowing them to explore their ideas freely.

Mentoring relationships arise from the interaction between mentor and mentee and, according to ^{Kram (1983}, 1985), they unfold into four phases: initiation, cultivation, separation, and redefinition of roles. The initiation phase is the moment of establishing the mentoring relationship based on the initiatives of the mentor or mentee when efforts are made to break down blocks so that identification, empathy, respect, and trust can occur. In the cultivation phase, the mentee learns from the mentor, develops skills, acquires support for their activities, and gains trust. The separation phase of the relationship occurs when the mentee gains independence and reaches a degree of autonomy, being able to carry out their activities without the intensive monitoring of the mentor. At this stage, interactions can happen more spaced out and with shorter orientations. Finally, there is a need to redefine the bond between mentor and mentee, resulting in a reconfiguration of the mentoring relationship, making it more egalitarian, and autonomous, and a bond of friendship may emerge. Mentoring, then, as ^{Amado (2007}) explains, favors the construction of an interpersonal approach based on trust and support, providing enrichment of pedagogical practice, and bringing innovation to the classroom environment.

METHODOLOGICAL APPROACH

Regarding the approach to the problem, the research can be classified as qualitative, as it aimed to understand the meaning that the participants attributed to the object of analysis. The focus was to evaluate how the mentoring relationship helped two basic education mathematics teachers in developing plans to integrate digital technologies into their classes. Qualitative research, according to ^{Gerhardt and Silveira (2009}), involves descriptive data obtained through the researcher's direct contact with the situation studied and is concerned with portraying the participants' perspectives, that is, qualitative research aims to ascertain the quality of what is being studied. In the present research, the mentor was the researcher, who was, therefore, in direct contact with the situation investigated, and the data were analyzed seeking to portray the perspectives of the two participating teachers.

To develop the research, meetings between the mentor and the participating teachers were held, and, from there, records were made to obtain data. The meetings were recorded, the researcher made notes in a field diary and, in addition, the teachers' productions during the mentoring were used as analysis materials. Therefore, this study was born from the methodological proposal involving the mentoring strategy with teachers and sought to monitor their professional development for planning to integrate technologies into their classes, identifying in practice the application of the suggested changes.

To achieve the objectives of this proposal, seven meetings were held with teachers who participated in activities based on a mentoring relationship, focusing on planning their mathematics classes with the integration of digital technologies. The meetings took place in person at the teachers' homes, as they were holding virtual classes due to the Covid-19 pandemic.

The teachers who participated in this study were two teachers who teach in the ninth year and tenth year of high school (ninth of middle school and first of high school in Brazil) at a state public school in the state of Rio Grande do Sul, in Vale do Taquari. The research was carried out at the end of the second half of 2020 and the beginning of the first half of 2021, in the middle of the pandemic period.

The meetings were held in pairs, as suggested, and requested by the teachers. They encouraged proactivity and the exchange of experiences. Furthermore, the mentor sought to mediate the meetings, valuing the teachers' ideas, promoting exchanges, and seeking to provide an environment in which the educators felt involved, in which everyone was learning, without a holder of knowledge, but rather people willing to evolve during the process.

Initially, an activity by ^{Jesus (2018}) on function analysis was used as an example, using the GeoGebra software so that they could realize that certain content can be presented in different ways to students, diversifying the methodologies used and encouraging student interest in the class. From this, plans were created together, according to the teachers' interests in terms of the content they cover in their classes.

The mentoring process aimed to make the teachers feel increasingly confident and independent to carry out these plans and, as a result, develop professionally. Afterward, the importance of good planning was discussed and some software such as GeoGebra, Graphmatica, and Winplot were explored with the teachers so that they could become familiar with them and be able to use them in their classes.

After that, the mentor helped plan classes with the integration of digital technologies according to the needs of each educator. At first, when the teachers did not feel comfortable using the tools without assistance, the mentor made use of the material, helping, and encouraging the teachers' interest in the tools, and demystifying some existing prejudices, until they felt comfortable to develop and apply activities using technologies more concisely.

As a data collection instrument, in addition to filming and recordings, notes were used in a field diary and materials produced from each intervention throughout the planning, which made it possible to analyze the teachers' professional development. To complement the data collection obtained during the mentoring, a semi-structured interview was carried out with each of them to obtain evidence, from their points of view, of how the proposed mentoring model may have contributed to their development during the activities. ^{Gil (1999}, p. 120) explains that “the interviewer allows the interviewee to speak freely about the subject, but, when he deviates from the original topic, he makes an effort to return to it”. Given this, in this technique, the researcher has to be attentive to the direction of the subject being addressed since the focus of the initial theme cannot be lost.

Data were analyzed based on conversations and information, according to records and exchanges of materials, as well as the final questionnaire. The analysis was descriptive and chronological, as the development of professionals was analyzed and monitored in planning their classes with the integration of technologies, as the descriptive analysis consists of monitoring and describing the step-by-step professional development while being researched.

DATA ANALYSIS

In this article, we chose to present the analysis focusing on the data collected and the plans developed in meetings 1, 2, 5, and 6. However, a total of seven meetings were held, all demonstrating improvements and advances. The choice to present data from the meetings is because they make clear, from the activities, the changes that occurred in the teachers' methodology through understanding the increased complexity of interpretation proposed to students with the use of technologies, and which, according to them, enhance the active construction of knowledge by the student.

Activity 3 reformulated - Behavior of the graph of the function in the plane, according to the variation of the coefficient “a” in the quadratic function.

The objective of this activity is to make students relate the concavity of a quadratic function to the coefficient “a”.

Plot the following functions in GeoGebra:

1. f(x) = x² - x - 20

2. f(x) = x² - 3x - 4

3. f(x) = -x² + x + 12

4. f(x) = -x² + 6x - 5

Questions:

1.Fill in the table below by observing the graphs created in GeoGebra:

2. What is the relationship between the concavity of the curve in the Cartesian plane and the coefficient “a” of each function?

The reformulated Activity 3 allows you to visualize the suggested changes in mentoring. In the activity titled “Behavior of the graph of the function in the plane, according to the variation of the coefficient ‘a’ in the quadratic function”, there are not many changes in the initial idea. It is clear how positive the activities that the teachers developed were, as their understanding of how to make good use of digital technology, in this case, GeoGebra, in the mathematics discipline is evident.

The activities were analyzed, and it was found that there was a need for structural improvements in the lesson plan, considering that the guidelines already presented led to the understanding that it would be possible and viable to add a few more functions. Based on the work proposal of this research, Activity 4 shows what the teachers developed following what had been worked on and discussed during the mentoring meetings.

FINAL CONSIDERATIONS

This research aimed to verify the possibility of integrating digital technologies into mathematics classes, based on the planning of activities developed through the mentoring strategy with two basic education teachers from the public school system. The teachers, at the time of the research, did not use digital technologies in their pedagogical practices. The use of technologies in education is much more than simply making use of machines and equipment: technologies should be seen as drivers of learning by allowing interactive and meaningful teaching.

This investigation was developed in seven meetings with two teachers, which were recorded to later assist in data analysis, considering the growth of teachers as digital professionals. In addition to the recordings, diaries were used, as well as some questions during the meetings and lesson plans produced.

The meetings were held in pairs, as suggested, and requested by the teachers. They consisted of moments of very significant exchanges, knowledge, and trust. At the beginning of the meetings, during the first activities developed, the teachers reported that as the pedagogical proposals were carried out, instigating the students' reasoning, and making them create their concepts about the content in question, they realized that in their stories as teachers they took the “mysteries” of learning already revealed to the students, without allowing them to be the protagonists of their learning. As a result, it was noticed that the mentees taught classes in a “traditional” way, which is strongly linked to the training path of most teachers - which is a barrier to be broken. This further reinforces the need for continued training or even monitoring, with the possibility of integrating technologies, as the professional is also a reflection of their training trajectory.

Given this, the teachers were satisfied with the activities, as they realized, and constantly emphasized, the importance of teaching in this way, enabling the autonomous construction of knowledge. It can be said, then, that this objective was achieved, because according to the reports and productions of the teachers, it was noticed that there was a significant development of the teachers in planning with the integration of technologies, accompanied in the mentoring process, each meeting held.

From the meetings onwards, the teachers began to allow themselves and feel comfortable with the technologies, until the moment came when they planned their classes alone. In this way, it was possible to identify that the mentoring strategy had a significant influence on the professional development of the teachers, as they did not feel alone when carrying out their planning, resolving their doubts about possible questions regarding the activities, and as reported by them, the mentor served as support and encouraged them more.

Given this, the process was very productive, as the activities were developed following the objective that mentoring requires: being incisive at the beginning of the process and becoming less present as soon as the mentee becomes comfortable. to produce alone. Finally, we reiterate the reflection on the importance of the mentoring strategy for the integration of technologies in mathematics classes, as the world is increasingly competitive and requires professionals who keep up with technological developments.