Teacher training can be understood as a pathway for promoting moments of lesson planning, teaching practice, and knowledge production, guided by discussions linked to mathematical concepts regarding their pedagogical, didactic, and specific aspects. In this context, opportunities to reflect on diverse interpretations and experiences of mathematicians and mathematics educators, and to create a variety of specific communicative configurations that address mathematical concepts—including the concept of function—occur within the context of teacher training (Santos, 2017). The topic of functions is configured as one of the four dimensions of Algebra, along with equations, structural dimension, and generalized arithmetic (Ponte; Branco; Matos, 2009; Usiskin, 1994). Its concept plays an important role in describing, interpreting, and constructing graphs to represent natural phenomena, as well as in establishing connections with other mathematical concepts (Gonçalves, 2015). Conversely, Tabach and Nachlieli (2015) argue that teaching the concept of function can present didactic setbacks, since the correct use of this concept by students does not imply that they possess the knowledge to use it in identifying other mathematical objects, even if these have already been explained in the classroom. From this perspective, Ribeiro and Cury (2015) present research addressing the algebraic knowledge of teachers and students, regarding the learning difficulties of the concept of function. In their research, classroom practices are considered the foundations for modifications in initial and continuing education courses for mathematics teachers regarding the teaching of Algebra. Pedagogical practice puts into action the professional knowledge of the mathematics teacher, and the understanding of the concept of function is mobilized in both initial and continuing education (Pazuch; Ribeiro, 2017). Furthermore, Pazuch and Ribeiro (2017) point out that, given the learning difficulties and potential obstacles associated with the context of pedagogical practice, the concepts can be re-signified. In this aspect, different and new meanings are attributed to the concept of function and to the teachers' knowledge for teaching in the continuous process of training. Therefore, this article aims to describe theoretical and methodological aspects of research on the concept of function in continuing education. This literature review aims to systematize the theoretical and methodological aspects present in the selected research articles and synthesize information related to the teaching of the concept of function in the continuing education of the mathematics teacher. Thus, the defined methodological procedures aim to identify, select, and critically evaluate relevant research in the area of Mathematics Education to potentially use them in the production and analysis of data in future studies.

Knopf (2006) notes that reviews in the field of education commonly start from a non-systematic sample with non-explicit criteria, failing to specify the methods used to identify, select, and analyze the studies reviewed. While this model allows for general interpretations and critical discussions of the literature review's works, these reviews do not discriminate the inclusion or exclusion of specific works from the literature. Considering these limitations, this literature review uses the principles established by the Preferred Reporting Items for Systematic reviews and Meta-Analyses (PRISMA) because of its clear instructions and the model's recognition and utility in different research areas (Moher et al., 2009). The review incorporated stages of: (i) identification, screening, and selection of publications; (ii) reading and critical evaluation of the selected articles; and (iii) presentation of the results.

The search and selection of articles were carried out in the Scielo, Math Educ Database, Web of Science (WoS), and Scopus databases. The choice of these databases is due to their reach and dissemination of works, based on the availability of access to article indexing databases provided by the Coordination for the Improvement of Higher Education Personnel (CAPES), through the CAPES Periodicals Portal. This provides the research with unique breadth. Due to this guarantee of representativeness, the use of the databases aims to facilitate the organization of the review's article bank and the quantification of specific information. It is important to emphasize that a limitation of the research is the non-exhaustive exploration of all existing databases. The Scopus and WoS databases have 309 and 114 indexed Brazilian journal titles, respectively, proving that Scopus has greater coverage than WoS, as it contains more than double the number of journals incorporated internationally and nationally (Rodrigues; Quartiero; Neubert, 2015). However, Packer (2011) expresses that there are obstacles to the integration of Brazilian and other developing country scientific productions into international scientific information databases. For these reasons, we chose to use the Scielo database, which has strong representation in Brazil (Vicente et al., 2017), due to the number of publications in Portuguese and Spanish in the fields of Teaching and Education. The Math Educ Database specializes in the field of Mathematics Education, as well as in publications dedicated to communicating mathematical skills in schools and universities (Ruffer-Henn; Wegner, 2010). The determination of the four databases for this literature review covers relevant articles nationally and internationally classified in the field of Mathematics Education. Thus, we sought to identify the search terms that would be appropriate to the study's objectives, using the terms "concept of function" and "teacher training" present in the text's body, organized in Table 1. All searches were carried out on July 31, 2019, considering all types of publications in the period from January 2009 to the end of July 2019. Thus, we established an initial publication year for the inclusion of works in our database, in order to present results in Mathematics Education research on this topic—teaching the concept of function from the perspective of teacher training—in the aforementioned period. The research organized in each database considered all fields (the entire structure and body of the works) in which the search expressions presented in Table 1 could be identified. Table 2 shows the search strategies, along with the number of results computed for each database, including the initial database for the literature review. Initially, publications identified in the screening process and characterized as letters, editorials, reviews, comments, chapters, or complete books were excluded. Subsequently, the selection process grouped them into these categories: a) continuing education of mathematics teachers; b) initial training of mathematics teachers; c) teaching and learning of basic education students; d) theoretical studies and literature reviews; and e) the term "concept of function" present only in the publication's references—that is, there is no discussion of this topic; the reference is presented in the article to justify another mathematical topic and does not fit within the scope of this research. The systematic review included four stages (Figure 1): identification (preliminary survey of publications), screening (exclusion of repeated, unavailable publications, or those not classified as articles, i.e., book chapters, reports, among others), selection (detailed analysis of articles: after reading the abstracts, objectives, and methodologies of the texts, those whose conceptions and discussions on the concept of function were not the main focus of the review were excluded), and inclusion (database closure), totaling 16 articles that make up the analysis corpus of this text. After the article selection stage, articles that addressed, regarding the term function, other conceptions not related to the field of Mathematics were excluded—for example, the meaning of the term as a natural activity or characteristic of something, or as an obligation to fulfill, a role to play.

This section presents interpretive syntheses of each article. According to Fiorentini and Crecci (2017), an interpretive synthesis is not a summary of the analyzed article, but an analytical interpretive production made by the researchers, highlighting its objective or research question and how the question was answered. The analytical and interpretive syntheses elaborated include information from the 16 selected articles, which are the final analysis corpus. Categories were established a priori to classify the articles according to the predominant methodological approach in each continuing education context chosen by the authors of each article (Table 3). The articles were categorized into four groups based on their predominant methodological approach: (1) Those that organized continuing education programs/courses using only data produced by participating teachers. (2) Those that organized continuing education programs/courses, using data from both teachers and their students. (3) Those that analyzed teaching actions/practices in the classroom, using data from both teachers and students. (4) Those that analyzed teaching actions/practices in the classroom, focusing solely on teacher actions. A final fifth group encompassing studies that used evaluative forms or questionnaires, along with teacher interviews. The findings from each group reveal key aspects of teaching the concept of function in continuing education. Group 1 highlights the use of tasks in continuing education programs/courses, particularly Task-Based Activities (TBAs), as instruments to provide professional learning opportunities. These tasks serve as indicators of teachers' mastery of the concept of function, proficiency in algebraic reasoning, and overcoming resistance to studying the concept of function through self-diagnostic practices and analyzing student understanding. The topic of the concept of function is linked to discussions about teachers' work in continuing education processes, including other types of functions and related content, expressing teachers' knowledge of the content and curriculum. Group 2, incorporating the analysis of student results, adds professional knowledge related to students' understanding of the concept of function. Formative moments for teachers show changes in student performance on tasks aimed at the transition of representations of the concept of function. They also offer plausible (cognitive and epistemological) reasons for students' actions related to the concept of function. Group 3 focuses on the interactions between teachers' knowledge, the concept of function, and students' understanding. Studies in this group highlight didactic variables that aid in planning and building tasks using technology, emphasizing questions that address the meanings of covariation and correspondence of the concept of function, as well as representations, their connections, and the transition between them during concept review with students. Conversely, other studies reveal the generation of distinct understandings of the concept of function by students due to the teachers' methodological approach choices, through the notions of action and process of the concept of function. The findings reinforce the conclusion that Specialized Content Knowledge is a necessary condition for students to establish mathematical connections with the concept of function. Group 4 reflects on the knowledge mobilized by teachers in teaching the concept of function, specifically Common Content Knowledge. This group's research examines how teachers interact with components related to the concept of function, such as its representation, the concept of function associated with covariations, the calculation of images and pre-images, among others. It shows how the way teachers interact with components related to the concept of function allows for the construction of the understanding of the concept of function in flow. This promotes significant experiences that stimulate intellectual need, making the descriptive part of research on teaching the concept of function procedural. The final group, using evaluative forms and questionnaires along with interviews, reveals strategies for analyzing and classifying teachers' knowledge used after verifying the answers given in tasks focusing on the concept of function. The development of generic evaluation tasks is associated with considerations of heterogeneous teaching profiles, which add a broad promotion of the relationships between teachers' knowledge and actions for teaching functions. In summary, the findings from all groups reveal convergent aspects that configure theoretical and methodological aspects of research on the concept of function in continuing education, whether through participation in continuing education courses/programs or through classroom observation along with individual and/or collective discussions. The interaction of teachers in continuing education to study the concept of function showed methodological strategies and theoretical considerations regarding a mathematical concept and its interference in students' understanding of the content. The construction of the concept of function by students involved establishing relationships and transitions between arithmetic and algebraic knowledge through different approaches. Participation in formative processes showed, through the construction, application, and diagnostic analysis of TBAs and mathematical tasks, possibilities and difficulties for both teachers and their students regarding the concept of function and classroom dynamics.

The findings of this systematic literature review highlight several theoretical and methodological directions for research on teaching the concept of function in continuing teacher education. The integration of teachers' knowledge, the concept of function, and students' understanding emerges as a crucial aspect. The use of Task-Based Activities (TBAs) and other mathematical tasks as diagnostic tools, accompanied by collaborative discussions about pedagogical practice, is shown to be a valuable approach for evaluating the results of continuing education. The importance of coordinating the transition between arithmetic and algebraic knowledge through the development of functional thinking, the analysis and convergence of curricula, and the investigation of patterns and different representations of the concept are emphasized. The use of technology as a didactic resource to visualize the transition between representations of the concept of function and the knowledge mobilized by teachers is also highlighted. The study underscores the need for teachers to master didactic-pedagogical resources based on didactic variables for teaching the different representations of the concept, and to manage the continuing education process and student assessment strategies. This review generates implications for future empirical studies involving continuing education with teachers and the concept of function.

This systematic literature review reveals key theoretical and methodological directions for research on teaching the concept of function in continuing teacher education. The study highlights the importance of using tasks, managing the transition between arithmetic and algebraic knowledge, and effectively using didactic-pedagogical resources. Future research could explore the development of more comprehensive assessment tools, investigate the long-term impact of continuing education programs, and examine the role of collaborative learning in promoting deeper understanding of the function concept.

This review is limited by its focus on articles from four specific databases, potentially excluding relevant information from other sources such as books and conference proceedings. The sample size, while substantial, might not fully represent the diversity of approaches to teaching the function concept in all educational contexts. Further research could expand the scope to include a broader range of publications and geographical areas.