Exploring the mathematics teacher’s specialized knowledge through the sigmoid function

Daniel Martín-Cudero 1 * , Rocío Guede-Cid 1, Ana Isabel Cid-Cid 1
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1 Consolidated Research Group in STEM Education (GIESTEM), Area of Mathematics Education, Rey Juan Carlos University, Madrid, SPAIN
* Corresponding Author
EUR J SCI MATH ED, Volume 13, Issue 4, pp. 395-416. https://doi.org/10.30935/scimath/17510
Published: 04 December 2025
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ABSTRACT

This study explores the specialized knowledge mobilized by a mathematics teacher when implementing an interdisciplinary activity based on the sigmoid function in an 11th grade class. The research aims to identify the mathematical and didactic knowledge elements present during the activity and to detect potential conceptual or pedagogical limitations in the teacher’s practice. A qualitative instrumental case study was conducted within an interpretive paradigm, using non-participant classroom observations and reflective field notes as primary data sources. The intervention was conducted in a public high school in Madrid and involved fifteen science-track students across two 50-minute sessions. It consisted of a complete mathematical study of the sigmoid function and reflective reasoning tasks designed to contextualize the concept within real-world applications. The analysis was framed using the mathematics teacher’s specialized knowledge model. The findings reveal that the teacher demonstrated strong competencies in several subdomains of this model, particularly in content knowledge, inter-conceptual connections, and pedagogical strategies. Additionally, the teacher showed strong awareness of student thinking and provided targeted instructional support. However, the study identified minor limitations in the teacher’s epistemological contextualization of the mathematical content beyond its technical aspects, pointing to the need for deeper engagement with its historical and conceptual foundations. Overall, the study confirms the usefulness of the model in characterizing the complexity and integration of knowledge required for teaching interdisciplinary mathematical content effectively, although it would be necessary to refine certain knowledge elements in future work.
Keywords: mathematics teacher’s knowledge, specialized knowledge, teaching and learning of mathematics, sigmoid function

CITATION

Martín-Cudero, D., Guede-Cid, R., & Cid-Cid, A. I. (2025). Exploring the mathematics teacher’s specialized knowledge through the sigmoid function. European Journal of Science and Mathematics Education, 13(4), 395-416. https://doi.org/10.30935/scimath/17510

REFERENCES

  • Advíncula, E., Beteta, M., León, J., Torres, I., & Montes, M. (2021). El conocimiento matemático del profesor acerca de la parábola: Diseño de un instrumento para investigación [The teacher’s mathematical knowledge about the parabola: Designing a research instrument]. Uniciencia, 35(1), 190–209. https://doi.org/10.15359/ru.35-1.12
  • Aguilar-González, Á., Rodríguez-Muñiz, L. J., & Muñiz-Rodríguez, L. (2022). Las creencias y su papel en la determinación de relaciones entre elementos del conocimiento especializado del profesor de matemáticas [Beliefs and their role in determining relationships between elements of the mathematics teacher’s specialized knowledge]. In J. Carrillo, M. A. Montes, & N. Climent (Eds.), Investigación sobre conocimiento especializado del profesor de matemáticas (MTSK): 10 años de camino (pp. 109–120). Dykinson, S. L. https://doi.org/10.14679/1458
  • Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407. https://doi.org/10.1177/0022487108324554
  • Bassey, M. (2003). Case study research in educational settings. Open University Press.
  • Baumert, J., & Kunter, M. (2013). The COACTIV model of teachers’ professional competence. In M. Kunter, J. Baumert, W. Blum, U. Klusmann, S., Krauss, & M. Neubrand (Eds.), Cognitive activation in the mathematics classroom and professional competence of teachers. Results from the COACTIV project (pp. 25–48). Springer. https://doi.org/10.1007/978-1-4614-5149-5_2
  • Carrillo, J., Climent, N., Contreras, L. C., & Muñoz-Catalán, M. C. (2013). Determining specialised knowledge for mathematics teaching. In B. Ubuz, C. Haser, & M. A. Mariotti (Eds.), Proceedings of the Congress of European Research in Mathematics Education (pp. 2985–2994). ERME.
  • Carrillo, J., Climent, N., Contreras, L. C., & Ribeiro, M. (2017). Mathematics teacher’s specialized knowledge (MTSK) in the “dissecting an equilateral triangle” problem. RIPEM, 7(2), 88–107.
  • Carrillo-Yañez, J., Climent, N., Montes, M., Contreras, L. C., Flores-Medrano, E., Escudero-Ávila, D., Vasco, D., Rojas, N., Aguilar-González, A., Ribeiro, M., & Muñoz-Catalán, M. C. (2018). The mathematics teacher’s specialised knowledge (MTSK) model. Research in Mathematics Education, 20(3), 236–253. https://doi.org/10.1080/14794802.2018.1479981
  • Caviedes Barrera, S., de Gamboa Rojas, G., & Badillo Jiménez, E. (2023). Conocimiento matemático y didáctico de futuros maestros sobre el área de figuras 2D [Mathematical and didactic knowledge of future teachers on the area of 2D figures]. Avances de Investigación en Educación Matemática, 24, 1–20. https://doi.org/10.35763/aiem24.4076
  • Climent, N., Espinoza-Vásquez, G., Carrillo, J., Henríquez-Rivas, C., & Ponce, R. (2021). A lesson on Thales’ theorem viewed from the specialized teacher’s knowledge. Educación Matemática, 33(1), 98–124. https://doi.org/10.24844/EM3301.04
  • Cohen, L., & Manion, L. (2002). Métodos de investigación educativa [Educational research methods]. La Muralla.
  • D’Amore, B. (2004). Conceptualización, registros de representaciones semióticas y noética: Interacciones constructivistas en el aprendizaje de los conceptos matemáticos e hipótesis sobre algunos factores que inhiben la devolución [Conceptualization, records of semiotic representations and noetics: Constructivist interactions in the learning of mathematical concepts and hypotheses about some factors that inhibit feedback]. UNO. Revista de Didáctica de las Matemáticas, 35, 90–106.
  • de Gamboa, G., Caviedes Barrera, S., & Badillo, E. (2022). Mathematical connections and the mathematics teacher’s specialised knowledge. Mathematics, 10, Article 4010. https://doi.org/10.3390/math10214010
  • Delgado, R., Zakaryan, D., & Alfaro, C. (2022). El conocimiento de la práctica matemática [Knowledge of mathematical practice]. In J. Carrillo, M. A. Montes, & N. Climent (Eds.), Investigación sobre conocimiento especializado del profesor de matemáticas (MTSK): 10 años de camino (pp. 58–70). Dykinson, S. L. https://doi.org/10.14679/1454
  • Delgado-Rebolledo, R., & Espinoza-Vásquez, G. (2021). ¿Cómo se relacionan los subdominios del conocimiento especializado del profesor de matemáticas [How are the subdomains of the mathematics teacher’s specialized knowledge related]? In J. G. Moriel-Junior (Ed.), Anais do V Congreso Iberoeamericano sobre Conocimiento Especializado del Profesor de Matemáticas (pp. 288–295). Congresseme.
  • Dólera-Almaida, J., & Sánchez-Jiménez, E. (2024). Pedro Puig Adam y el método heurístico en la enseñanza de las matemáticas en España [Pedro Puig Adam and the heuristic method in teaching mathematics in Spain]. El Futuro del Pasado, 15, 703–723. https://doi.org/10.14201/fdp.31159
  • Escudero-Ávila, D. (2022). Conocimiento de las características del aprendizaje matemático [Knowledge of the characteristics of mathematical learning]. In J. Carrillo, M. A. Montes, & N. Climent (Eds.), Investigación sobre conocimiento especializado del profesor de matemáticas (MTSK): 10 años de camino (pp. 83–94). Dykinson, S. L. https://doi.org/10.2307/j.ctv2zp4vp1.11
  • Escudero-Ávila, D. I., Carrillo, J., Flores-Medrano, E., Climent, N., Contreras, L. C., & Montes, M. (2015). El conocimiento especializado del profesor de matemáticas detectado en la resolución del problema de las cuerdas [The specialized knowledge of the mathematics teacher detected in the resolution of the string problem]. PNA, 10(1), 53–77. https://doi.org/10.30827/pna.v10i1.6095
  • Espinoza-Vásquez, G., Verdugo-Hernández, P., & Henríquez-Rivas, C. (2025). Relationships between domain changes and connections: An analysis from the perspective of mathematical work and teacher knowledge. European Public & Social Innovation Review, 10, 1–21. https://doi.org/10.31637/epsir-2025-1460
  • Espinoza-Vásquez, G., Verdugo-Hernández, P., Zakaryan, D., Carrillo, J., & Montoya-Delgadillo, E. (2016). Hacia una relación entre el ETM y el MTSK a través del concepto de función [Towards a relationship between ETM and MTSK through the concept of function]. In Investigación en educación matemática XIX (pp. 197–206). SEIEM.
  • Flores-Medrano, E. (2022). Conocimiento de la estructura de las matemáticas [Knowledge of the structure of mathematics]. In J. Carrillo, M. A. Montes, & N. Climent (Eds.), Investigación sobre conocimiento especializado del profesor de matemáticas (MTSK): 10 años de camino (pp. 47–56). Dykinson, S. L. https://doi.org/10.2307/j.ctv2zp4vp1.8
  • Flores-Medrano, E., Montes, M. A., Carrillo, J., Contreras, L. C., Muñoz-Catalán, M. C., & Liñán, M. M. (2016). The role of MTSK as model of teachers’ knowledge in the relationship between mathematical working spaces. Bolema, 30(54), 204–221. https://doi.org/10.1590/1980-4415v30n54a10
  • Godino, J. D. (2002). Perspectiva de la didáctica de las matemáticas como disciplina científica [Perspective of the didactics of mathematics as a scientific discipline]. Universidad de Granada.
  • Godino, J. D. (2009). Categorías de análisis de los conocimientos del profesor de matemáticas [Categories of analysis of the mathematics teacher’s knowledge]. UNIÓN, Revista Iberoamericana de Educación Matemática, 20, 13–31.
  • Godino, J. D. (2010a). Marcos teóricos sobre el conocimiento y el aprendizaje matemático [Theoretical frameworks on mathematical knowledge and learning]. Universidad de Granada.
  • Godino, J. D. (2010b). Perspectiva de la didáctica de las matemáticas como disciplina tecnocientífica [Perspective of the didactics of mathematics as a techno-scientific discipline]. Universidad de Granada.
  • González-Monteagudo, J. (2001). El paradigma interpretativo en la investigación social y educativa: Nuevas respuestas para viejos interrogantes [The interpretive paradigm in social and educational research: New answers to old questions]. Cuestiones Pedagógicas, 15, 227–246.
  • Juárez-Ruiz, E., Flores-Medrano, E., Otero-Valega, K., & Tascón-Cardona, L. (2025). Levels of complexity in mathematics teachers’ knowledge connections: An approach based on MTSK and Piaget’s schemas. Education Sciences, 15(6), Article 641. https://doi.org/10.3390/educsci15060641
  • Kagan, D. M. (1990). Ways of evaluating teacher cognition: Inferences concerning the Goldilocks principle. Review of Educational Research, 60(3), 419–469. https://doi.org/10.3102/00346543060003419
  • Kyurkchiev, N., & Markov, S. (2015). Sigmoid functions some approximation and modelling aspects: Some moduli in programming environment mathematica. LAP LAMBERT Academic Publishing. https://doi.org/10.11145/j.bmc.2015.03.081
  • Lofland, J., & Lofland, L. H. (1984). Analyzing social settings. Wadsworth Publishing Company, Inc.
  • Meléndez-Cruz, J. A., Flores-Medrano, E., & Hernández-Rebollar, L. A. (2023). Specialized knowledge of the mathematics teacher when analyzing a sequence of addition of fractions. Uniciencia, 37(1), 1–19. https://doi.org/10.15359/ru.37-1.11
  • Montes, M., & Carrillo, J. (2017). Mathematic teachers’ specialized knowledge about infinity. Bolema, 31(57), 114–134. https://doi.org/10.1590/1980-4415v31n57a06
  • Muñoz-Catalán, M. C., Contreras, L. C., Carrillo, J., Rojas, N., Montes, M. A., & Climent, N. (2015). Conocimiento especializado del profesor de matemáticas (MTSK): Un modelo analítico para el estudio del conocimiento del profesor de matemáticas [Mathematics teacher specialized knowledge (MTSK): An analytical model for studying mathematics teacher knowledge]. La Gaceta de la RSME, 18(3), 589–605.
  • OECD. (2023). PISA 2022 assessment and analytical framework. OECD Publishing. https://doi.org/10.1787/dfe0bf9c-en
  • Padilla, I. A., Acevedo-Rincón, J. P., & Montes, M. A. (2023). Specialised knowledge of the mathematics teacher to teach through modelling using ICTs. Acta Scientiae, 25(1), 160–195. https://doi.org/10.17648/acta.scientiae.7363
  • Panqueban, D., Henríquez-Rivas, C., & Kuzniak, A. (2024). Advances and trends in research on mathematical working spaces: A systematic review. Eurasia Journal of Mathematics, Science and Technology Education, 20(6), Article em2450. https://doi.org/10.29333/ejmste/14588
  • Pérez-Montilla, A., & Cardeñoso, J. M. (2023). Hacia una posible configuración del conocimiento profesional del formador de docentes de matemáticas: Un análisis comparativo [Towards a possible configuration of the professional knowledge of mathematics teacher trainers: A comparative analysis]. Bolema, 37, 148–167. https://doi.org/10.1590/1980-4415v37n75a08
  • Perignat, E., & Katz-Buonincontro, J. (2019). STEAM in practice and research: An integrative literature review. Thinking Skills and Creativity, 31, 31–43. https://doi.org/10.1016/j.tsc.2018.10.002
  • Piaget, J. (1975). L’équilibration des structures cognitives [Balancing cognitive structures]. Presses Universitaires de France.
  • Ponte, J. P. D. (1994). Mathematics teachers’ professional knowledge. In J. P. da Ponte, & J. F. Matos (Eds.), Proceedings of the 18th International Conference for the Psychology of Mathematics Education (pp. 195–210). PME.
  • Schoenfeld, A., & Kilpatrick, J. (2008). Toward a theory of proficiency in teaching mathematics. In T. Wood, & D. Tirosh (Eds.), International handbook of mathematics teacher education: Vol. 2. Tools and processes in mathematics teacher education (pp. 321–354). Sense Publishers.
  • Shulman, L. S. (1986). Those who understand. Knowledge growth in teaching. Educational Researcher, 15(2), 4–14. https://doi.org/10.3102/0013189X015002004
  • Shulman, L. S. (1987). Knowledge and teaching: Foundations of a new reform. Harvard Educational Review, 57(1), 1–22. https://doi.org/10.17763/haer.57.1.j463w79r56455411
  • Sosa, L., & Reyes, A. M. (2022). Conocimiento de la enseñanza de las matemáticas [Knowledge of the teaching of mathematics]. In J. Carrillo, M. A. Montes, & N. Climent (Eds.), Investigación sobre conocimiento especializado del profesor de matemáticas (MTSK): 10 años de camino (pp. 71–82). Dykinson, S. L. https://doi.org/10.2307/j.ctv2zp4vp1.10
  • Sosa, L., Flores-Medrano, E., & Carrillo, J. (2015). Conocimiento del profesor acerca de las características de aprendizaje del álgebra en bachillerato [Teacher’s knowledge about the learning characteristics of algebra in high school]. Enseñanza de las Ciencias, 33(2), 173–189. https://doi.org/10.5565/rev/ensciencias.1522
  • Sosa, L., Flores-Medrano, E., & Carrillo, J. (2016). Conocimiento de la enseñanza de las matemáticas del profesor cuando ejemplifica y ayuda en clase de álgebra lineal [Knowledge of the teacher’s mathematics teaching when modeling and assisting in linear algebra class]. Educación Matemática, 28(2), 151–174. https://doi.org/10.24844/EM2802.06
  • Stake, R. (2007). Investigación con estudio de casos [Case study research]. Ediciones Morata.
  • Tatto, M. T., Schwille, J., Senk, S. L., Ingvarson, L., Peck, R., & Rowley, G. (2008). Teacher education and development study in mathematics (TEDS-M): Conceptual framework. Teacher Education and Development International Study Center, College of Education, Michigan State University.
  • Vasco, D., & Moriel, J. (2022). Conocimiento de los temas [Knowledge of the topics]. In J. Carrillo, M. A. Montes, & N. Climent (Eds.), Investigación sobre conocimiento especializado del profesor de matemáticas (MTSK): 10 años de camino (pp. 35–46). Dykinson, S. L. https://doi.org/10.2307/j.ctv2zp4vp1.7
  • Verret, M. (1975). Le temps des études [Study time]. Librairie Honoré Champion.
  • Zakaryan, D., & Ribeiro, M. (2018). Mathematics teachers’ specialized knowledge: A secondary teacher’s knowledge of rational numbers. Research in Mathematics Education, 21(1), 25–42. https://doi.org/10.1080/14794802.2018.1525422
  • Zakaryan, D., & Sosa, L. (2021). Knowledge of practices in mathematics of a secondary teacher in geometry classes. Educación Matemática, 33(1), 71–97. https://doi.org/10.24844/EM3301.03
  • Zamorano, T., García, Y., & Reyes, D. (2018). Educación para el sujeto del siglo XXI: Principales características del enfoque STEAM desde la mirada educacional [Education for the 21st-century subject: Main characteristics of the STEAM approach from an educational perspective]. Contextos: Estudios de Humanidades y Ciencias Sociales, (41).