3.1 Profile of Research Subjects

Aiming to identify the profile of teachers participating in the training course, a questionnaire was administered, via Google forms, consisting of thirteen questions. Seven Mathematics teachers who work in public and/or private education networks, from different locations, participated in the investigation (Table 1).

It was found that the teachers participating in the investigation formed a very varied group in terms of age; all participants in the Continuing Education course had a degree in Mathematics and at least one postgraduate course. Regarding weekly workload, one participant works up to 20 hours per week, one works from 21 to 30 hours per week, three work from 31 to 40 hours per week and one works over 40 hours per week.

When asked whether they had already worked with Digital Technologies in their classes before the pandemic period in which the course meetings were held, the seven respondents reported that they had previously been exposed to projects that dealt with the integration of Digital Technologies into pedagogical planning. Among them, four teachers indicated that their teaching practices underwent changes after the experience in question, either through the use of the technological resources studied or through improvements when using them. Furthermore, two teachers highlighted that they only had contact with this approach during their academic Training.

It was asked whether the teachers involved in the research had previous experience with Distance Learning (EaD) before the pandemic period, considering that they were all working with remote teaching during the quarantine. It was found that three teachers stated that they had never had contact with distance learning, while three teachers reported having previously been students in distance learning courses. Furthermore, one teacher highlighted having offered help informally through a digital environment.

Participants were asked whether they felt prepared to use Digital Technologies in the classroom. Six teachers stated that they felt prepared and of these, 1 teacher highlighted that despite feeling prepared he was not successful due to the technological impasses faced in the school environment. One teacher highlighted that, sometimes, he does not feel prepared due to the students' lack of equipment.

Regarding being prepared to work with distance learning, two teachers stated that they did not feel prepared, highlighting the student's difficulty in learning in this environment. A third teacher responded "more or less", mentioning the lack of appreciation and feedback from students. On the other hand, the other four teachers stated that they were prepared, with three of them highlighting the positive experience they had with their students during the quarantine period, which favored their adaptation to this teaching modality. The fourth teacher attributed his preparation to his search for new Knowledge through courses and Training.

They were also asked if their students used cell phones in the classroom and in which subject, and 6 teachers responded that cell phones are used by students, of which 4 stated that the use occurs in the Mathematics subject, 1 teacher stated that students only use the resource to attend classes and 1 teacher highlighted only the English language subject. Another teacher highlighted that because he changed schools, he did not have the opportunity to work on the resource with his students, although he used it at his previous school.

When asked what they think about the use of cell phones in the classroom, 3 teachers highlighted the importance of guided use of this resource, 1 teacher highlighted that this is an important resource and that it should not be "left aside", 1 teacher stated that the cell phone is a great ally, 1 teacher highlighted that "if used well, it helps a lot in classes" and 1 teacher highlighted that the cell phone must be linked to a pedagogical proposal and also shared a concern with the use of the tool, since, according to him, the tool is susceptible to distracting students when carrying out the proposed activities.

Finally, teachers were asked to suggest a Digital Technology resource to be used in Elementary School Mathematics classes in the final years (6th to 9th year). The resources mentioned included the use of cell phones, software GeoGebra, games and calculators. Additionally, one professor suggested an interactive course, although he did not specify any in particular.

A relevant point is that, although the majority of teachers have declared that they feel prepared to use Digital Technologies in their classes, many still show caution regarding the use of cell phones during teaching activities. They highlighted the importance of a well-structured pedagogical proposal for the use of these resources, highlighting a reflective process on the integration of Digital Technologies in teaching.

3.2 Structure of the Continuing Training courseIntegrating mobile devices in Mathematics Education

The Continuing Training course was offered to Mathematics teachers from public and private schools who worked in the final years of Elementary School. There were six meetings that took place weekly via the Meet platform from 7pm to 10pm and 12 hours of complementary activities, totaling 30 hours. The material presented at each meeting was available to participants in a class created on Google Classroom, along with recommendations for additional reading and/or questionnaires relating to the topic discussed at each meeting.

The meetings of the continuing education course offered (Figure 2) were scheduled with the aim of enabling teachers to take ownership of the concepts covered through discussions and reflections focused on teaching practice.

At the first meeting, the participants and the researcher were presented, as well as an overview of the proposed course. Also, the aim was to initially discuss the insertion and integration of Digital Technologies in Mathematics Education, in addition to encouraging dialogue between participants regarding the previous experiences of participating teachers, linked to the theme of the Continuing Training course offered.

Source: The survey.

Figure 2 Summary of the organization of the Continuing Training course meetings offered 

At the second meeting, the different possibilities of free applications available for mobile devices were discussed, involving mathematical content aimed at the final years of Elementary School, with examples of applications being presented.

In the third meeting, Technological Pedagogical Content Knowledge – TPACK was discussed, discussing the different knowledge bases necessary for an effective integration of Digital Technologies in the classroom and the relationships between them, according to the work of ^{Mishra and Koehler (2006)}.

The fourth meeting was scheduled for discussions about the Didactic Sequence as an effective tool in the process of integrating mobile devices into teaching practice and two Didactic Sequences were presented, prepared by the researcher responsible for the investigation, that integrated mobile devices into their planning as a model for participants. Furthermore, Activity 1 was proposed, which aimed to create a Didactic Sequence by the participating teachers, with the content developed and the application(s) integrated into it being free to choose.

An extra moment was established between the fourth and fifth meetings, in which the researcher would be available if participants had doubts and/or issues to be discussed during their planning, with their participation not being mandatory. Of the seven participants, five SDs were considered for data analysis due to the criterion of integrating at least one application in the proposal.

The DS prepared by the participants were presented at the fifth meeting, providing a rich moment of reflection and discussion among the participants, with the aim of generating contributions for each Didactic Sequence presented. Frame Framepresents an overview of the SD presented at the aforementioned meeting.

At the sixth meeting, it was possible for each participant to clarify their perceptions regarding possible contributions generated from the Continuing Training course. Furthermore, a questionnaire was made available for teachers to evaluate the Continuing Training course.

At the end of each meeting, an online form developed to validate the subjects' participation was made available. In the analyses, the answers to the forms and the observations made by the researcher were taken into account.

4.1 Insert versus Integrate Digital Technologies in Mathematics Education

This theme was addressed in the first meeting of the Continuing Training course with the intention of highlighting that, for a technological resource to actually contribute to the teaching and learning process, it is necessary to guarantee its effective integration into teaching planning, which implies in a conscious and intentional use that is related to the other proposed activities, aiming to achieve the previously defined educational objectives, which differs from the insertion of technological resources, since this uses such resources in a way disconnected from the work in the classroom, which ends up determining technology "as an extra instrument, something more that is not in fact in line with the teacher's actions" (^{Bittar, 2011}, p. 159).

It was highlighted that the course sought paths and possibilities through collaboration between participating teachers for the integration of Digital Technologies, specifically, mobile devices such as smartphones and tablets.

After the researcher's discussion on the topic, the teachers were asked, "What are the paths taken so far, and what are the possibilities?", where the participants had the opportunity to make considerations, pointing out their points of view regarding the use of Digital Technologies in classroom based on their experiences, which yielded the following topics for analysis: i) the use of Digital Technologies by teachers is inevitable and ii ) the difficulties encountered in integrating Digital Technologies into teaching practice.

In frame 2 one can observe the comments from teachers P1 and P6 that highlight the first topic.

From discussions among participants, there was a consensus regarding the need to use Digital Technologies in teaching practice, considering that cell phones, tablets, among other technological resources, are integral elements of students' daily activities. In this sense, the participants considered it essential that such technologies be integrated into the Training of students in different areas of Knowledge.

Next, frame 3 presents the participants' statements that highlighted the second topic of analysis of this discussion.

The main points highlighted as difficulties faced by the teachers subject to the investigation are also highlighted, namely: the maturity of students when working with cell phones in the classroom, as students sometimes do not understand this resource as a tool pedagogical; the ease of distraction for students during the proposed activities, given the many possibilities that cell phones offer, especially if the student has access to the internet; the time available for the syllabus, given that there is a curriculum that teachers need to deal with and, given this, they find it difficult to allocate time for the use of Digital Technologies; the lack of resources on the part of students, especially considering remote teaching, as according to teachers, some students share the same tool with other family members, which makes it difficult to access and carry out the proposed activities, and can even generate disinterest of the student in carrying out these; and, the lack of support that some teachers still find in the schools where they teach, when superiors and colleagues do not consider Digital Technology as a tool that enhances the construction of Knowledge.

It is understood from these discussions that there are many challenges and even barriers that still need to be addressed. Be faced and overcome by teachers for an effective integration of Digital Technologies into their teaching practices. From the infrastructure, the social context of students, prejudices and even the adaptation of both students and teachers, consider the use of Digital Technologies, especially cell phones and other mobile devices, as an important tool in the teaching and learning process.

However, it is noteworthy that it is through Knowledge that these difficulties can be faced, Knowledge can be developed through continuing education courses such as the one proposed in this investigation, which make it possible to expand existing Knowledge through the exchange of experiences between peers and a deeper theoretical approach, which provides a basis for teachers' decision-making, aiming for the effective integration of the resources in question. Furthermore, it can be inferred from statements made by participants in the course offered that the discussion enabled teachers to critically reflect on their practices, as they state that programmatic contents are a recurring concern that can interfere with the use of Digital Technologies. To ^{Alarcão (1996)}, this reflection is an essential action that contributes to professional development and the construction of teaching identity, enabling teachers to be able to transform their practice and constitute themselves as autonomous subjects in search of changes in the educational context. It is considered even more important when this reflection makes it possible to seek paths that enable the integration of Digital Technologies into teaching practice, especially at a time when, often, these are the only resources that have ensured the continuity of fundamental activities in Education, due to the pandemic scenario that has been experienced in recent years.

4.2 Applications for Mathematics Education

The second meeting of the continuing education course offered was aimed at the presentation of applications focused on Mathematics Education. 10 applications were selected (Frame 4) available for the Android operating system, previously cataloged from the search carried out on the Google Play tool (https://play.google.com/store?hl = pt - BR&tab =r8), which were presented considering the application's potential, difficulty levels, content covered (when specified in the application), available tools and application configurations.

Frame 4 Applications presented at the second meeting of the course 

Name	Google Play search preview
2248
Testes de Matematicando
math Pieces
math Challenge
Mestre doTangram
Math Games
math duel
Mestre da Matemática Jogo Educativo e Exercícios
Matemática para a escola
Integer Calculation

Source: The survey.

After the presentation of the applications by the researcher, teachers were asked to give their considerations about the applications presented and their expectations regarding their use. Based on the comments that the teachers made during the aforementioned meeting (Frame 5), it is observed that they showed great interest in the applications covered and stated that they would provide a more attractive and dynamic class.

It is noteworthy that teachers P4 and P6 presented in their speeches points discussed in the first meeting of this continuing education course, as the integration of a technological resource includes the need to have mastery over the selected resource and to consider students with that you intend to use the resource, mainly considering your prior Knowledge (^{Mishra and Koehler, 2008}).

It should also be noted that in this meeting, P2 and P6 raised questions regarding the elaboration of a Didactic Sequence designed to integrate Digital Technologies and applications as a tool that contributes to the construction of students' Knowledge, as they do not envisage possibilities, since that the concept of SD had not yet been addressed, as detailed in Frame 6.

It is understood that such questions highlight what ^{Oliveira (2015, p. 62)} states that "the presence of technologies in the classroom does not ensure improvement in the teaching and learning process" (^{Oliveira, 2015}, p. 62) and that there is a need to seek paths and possibilities that support teachers in breaking down barriers related to the integration of Digital Technologies, especially mobile devices, considering that the presentation of applications and the clear interest of teachers in taking them to the classroom, was not enough to provide the research subjects with subsidies for the effective integration of such resources into teaching planning.

Furthermore, the importance of developing Knowledge related to TPACK stands out, especially Technological Knowledge, which allows the teacher to recognize when technology can help or impede the achievement of an objective (^{Mishra and Koehler, 2008}). It is understood that it is from this Knowledge that it is possible to determine whether a technology can be effectively integrated into teaching practice, contributing to the construction of Knowledge.

4.3 Technological Pedagogical Content Knowledge

The third meeting of the continuing education course was directed towards the discussion of Technological Pedagogical Content Knowledge (TPACK), in which the researcher presented the knowledge bases necessary for an effective integration of technologies in teaching practice, namely Pedagogical Knowledge, Knowledge Technological and Content Knowledge, in addition to the relationships between these bases (^{Mishra and Koehler, 2008}).

From the approach of this reference with the teachers participating in the investigation, it was possible to provide a reflection on the issues that involve the integration of a technological resource, as it is understood that "the presence of technologies in the classroom does not ensure improvement in the process of teaching and learning" (^{Oliveira, 2015}, p. 62) and goes beyond choosing a resource, as one must consider the content covered, the students' prior Knowledge, the potentialities and limitations of the technological resource that is intended to be used and the methodology that intended to be applied (^{Mishra and Koehler, 2008}).

After the researcher's argument, some teachers commented on the subject, as shown in Frame 7.

It is observed that the theme was well received by teachers, enabling a critical reflection on the participants' teaching practice, proving to be an important discussion in the professional development process.

It is understood that sometimes, in-service teachers lack theoretical support to guide their teaching practice. As ^{Zabala (1998)} states, there is a lot of talk about "updating" and that teachers must be open to new methodologies and resources that enhance the construction of Knowledge, however, it is highlighted that academic research often does not reach the teacher in practice.

Participants highlighted the importance of the knowledge bases presented and the relationships between them, especially with regard to Pedagogical Knowledge, as they agreed that it is from this knowledge base that the teacher will observe whether students have the necessary concepts to the construction of new Knowledge and maturity to develop certain activities.

Another point highlighted was in relation to the National Common Curricular Base (BNCC) which, according to teachers, favors the maturation of students in relation to the abstract contents of mathematics, since based on the normative document they begin to be developed in stages in different years, enabling greater student ownership, due to their maturity.

That said, it is understood that discussions like this contribute to the teacher feeling theoretically grounded in the exercise of their functions, seeking to integrate new resources, not because they are new or in evidence, but rather by appropriating them in order to explore all the potential they offer in the teaching and learning process.

4.4 Perceptions of teachers participating in the research regarding the Continuing Training course

Seeking to present to the reader the perception of teachers participating in the Continuing Training course developed, entitled “Integrating mobile devices into Mathematics Education”, an excerpt of the teachers' speech is presented at the last meeting held, as detailed in Frame 8.

From the answers to the aforementioned questionnaire, it is inferred that teachers have been seeking new Knowledge in relation to Digital Technologies, as they understand the need to integrate such resources into their teaching practices, especially since the pandemic period experienced, not only in Brazil but throughout the world, causing social distancing that aimed to reduce the spread of the SARS-CoV-2 virus that causes the disease called COVID-19, which put Digital Technologies in the spotlight, especially with regard to Education.

It is understood that the teachers participating in the investigation understand that it is not valid to just insert TD, but rather that they must be integrated, aiming to make greater use of these resources, seeking to enhance the construction of students' Knowledge, and for this reason they are looking for courses with this theme.

Regarding the continuing education course Integrating mobile devices in Mathematics Education, teachers highlighted that it enabled a rich exchange of experiences between peers, duly mediated by the research teacher, a fact that stands out as it provided a favorable environment for learning, according to the statement of teachers P2, P3, P6 and P7.

Understanding that "learning is more than receiving or obtaining information and knowing or understanding it, it is making learning part of the being, implying developing with it", as stated by ^{Alvarado-Prado, Freitas and Freitas (2010, p. 369)}, it is highlighted that, based on the answers in the "Final Assessment" questionnaire and the teachers' comments during the meetings, that the teachers felt part of the process of building their own Knowledge, as they felt at ease during the discussions and for being able to share their experiences and learn from the experiences of their colleagues. Factors that are in line with what ^{Alvarado-Prado, Freitas and Freitas (2010)} state when they highlight the importance of considering the trajectory of teachers, their conceptions of life, society, school, Education, their interests, needs, skills, as well as their fears, difficulties and limitations during a Continuing Training process with a view to the full professional and personal development of teachers in Training.

Furthermore, the statements of teachers P1, P3, P4 and P6 highlighted what ^{Figueiredo, Lobo da Costa and Llinares (2021)} state, that the sharing of experiences between teachers and trainers drives reflection on teaching practice, enabling that the teachers involved “prepare to face and live with changes and uncertainties” (^{Figueiredo, Lobo da Costa and Llinares, 2021}, p. 6).