Theoretical and methodological directions on the concept of function in teaching education: a systematic literature review

The paper addresses how the concept of function is treated in the continuing education of mathematics teachers and seeks to systematize theoretical and methodological trends in this domain. Functions are central in school algebra and in representing natural phenomena, yet students and teachers face persistent difficulties, including bridging arithmetic and algebra and using definitions productively. The study’s purpose is to describe theoretical approaches and methodological strategies used in research on continuing teacher education focused on functions, in order to inform future studies and practices. Motivated by limitations of unsystematic reviews in education, the authors adopt PRISMA to provide transparent identification, selection, and analysis procedures and to synthesize implications for professional learning opportunities and classroom practice.

The paper situates the concept of function within algebra’s dimensions (equations, structural dimension, generalized arithmetic) and emphasizes its roles in modeling and connecting mathematical ideas. Prior research highlights didactical challenges: correct use of the term does not ensure broader conceptual command; classroom practice influences both initial and continuing teacher education; and ongoing formation can lead to re-signification of teachers’ knowledge of function. The review also notes barriers to the inclusion of research from developing countries in major international databases, justifying the use of Scielo in addition to Scopus, WoS, and MathEduc. Frameworks referenced include Mathematical Knowledge for Teaching (MKT), instrumental orchestration for technology integration, functional thinking categories, and perspectives on process-object and covariation/correspondence in understanding functions.

Design: Systematic literature review following PRISMA. Steps: (i) identification, (ii) screening, (iii) selection, (iv) inclusion. Databases: Scopus, Web of Science (WoS), Scielo, MathEduc. Search date: July 31, 2019. Time window: January 2009 – July 2019. Search terms (in body/full text): - English: "Teacher education"; "concept of function" OR "function concept" - Portuguese: "Formação de professores"; "conceito de função" - Spanish: "Educación del profesorado" OR "Formación docente"; "concepto de función" Query outcomes by database: Scopus=232; WoS=1; Scielo=0; MathEduc=2; total Np=235. Screening exclusions: non-articles (Ne=28), duplicates (Nr=2), unavailable files (Ni=12) → Na=193. Selection filters (by abstract/objective/method): Ns=177 excluded: initial teacher education (N1=14), basic education students’ learning (N2=17), theoretical studies/reviews (N3=6), other meanings of "function" not mathematical (N4=140). Final inclusion Ncorpus=16. Analytic categorization: Articles grouped by predominant methodological approach in continuing education contexts: (1) programs/courses using only teacher-produced data; (2) programs/courses using teacher and student data; (3) classroom practice analyses with teacher and student data (protocols/tests/interviews); (4) classroom practice analyses focusing on teachers only; (5) studies using questionnaires/assessments/interviews to profile teachers’ knowledge and strategies. Meta-synthesis produced integrative summaries and a table of theoretical–methodological directions.

- Corpus: 16 studies (from initial 235 records; after screening Na=193; excluded Ns=177). - Theoretical–methodological directions identified for continuing education on functions: 1) Mastery of didactic-pedagogical resources: selection and use of concrete materials and technologies; design choices guided by didactical variables; instrumental orchestrations when using technology. 2) Coordination of the transition from arithmetic to algebra: work with invariances and patterns, functional thinking categories, and multiple representations (tables, graphs, algebraic rules); support covariation and correspondence perspectives; adapt and align curricula across grade levels; promote modeling contexts (e.g., thermodynamics) to conceptualize functions as relationships between changing quantities. 3) Management of formative and assessment processes: extensive use of tasks (including practice-based tasks) and collaborative discussions to evaluate professional learning; diagnostic use of mathematical tasks (pre/post tests, student work) to inform instruction; instrument development to assess algebra/function knowledge for teaching. - Impact on teacher and student outcomes: - Participation in practice-based PD is associated with richer use of representations, clearer connections among them, and better anticipation of student thinking (MKT shifts noted from common to specialized content knowledge in some cases). - Programs linking teacher PD with classroom implementation show greater student gains in algebra/functions compared to non-participating teachers’ students. - Teacher approaches (conceptual vs. procedural emphasis) differentially shape students’ understanding of functions (e.g., piecewise functions), including their ability to connect representations and use definitions. - Advanced teacher knowledge of functions supports building students’ conceptual understanding "in flow" across topics and fosters intellectual need for the concept. - Technology use: Effective integration requires planned instrumental orchestrations; technology can scaffold transitions among representations and covariation reasoning but adds classroom management complexity. - Analytical frameworks and tools: Functional thinking categories, didactical variables for task design, recognition vs. heuristic strategies for graphing formulas, and multidimensional assessments of algebra knowledge for teaching provide actionable structures for PD and research.

The review’s synthesis addresses the central question—how research conceptualizes and operationalizes continuing education on functions—by converging on three actionable domains: resource mastery, arithmetic-to-algebra transition, and formative/assessment management. Collectively, the studies show that PD anchored in tasks, representations, and student thinking (MKT-informed) enhances teachers’ capacity to design, implement, and diagnose learning around functions. Inclusion of student data in PD studies evidences transfer from teacher learning to student achievement, especially in representational fluency and definition use. Technology-centered approaches are effective when guided by instrumental orchestration and coherent didactical variables. The findings are relevant for structuring PD that explicitly targets function as a hub concept connecting multiple mathematical ideas and real-world phenomena, reinforcing the need for coherent curricular trajectories and assessment tools aligned with functional thinking.

This review compiles theoretical and methodological directions for PD on the function concept, offering a consolidated agenda: (i) design PD around tasks and didactical variables that foreground multiple representations, covariation, and modeling; (ii) intentionally scaffold the arithmetic-to-algebra transition via functional thinking and patterns; (iii) embed diagnostic assessment and collaborative reflection to align instruction with student understanding. The meta-synthesis provides a reference for future empirical studies to test and refine PD designs, including interdisciplinary projects and partnerships between university-based educators and school practitioners. Future research should examine scalability of PD units, long-term impacts on classroom practices and student outcomes, and context-sensitive instrumental orchestrations for technology-rich environments.

- Only journal articles were included; books and conference proceedings were excluded. - Not all possible databases were covered; reliance on Scopus, WoS, Scielo, and MathEduc may omit relevant studies elsewhere. - Time window restricted to 2009–2019 searches (conducted July 31, 2019). - Language coverage focused on Portuguese, English, and Spanish. - Some files were unavailable (Ni=12), which may bias the corpus.