Abstract
This study analyzed the intra-mathematical connections activated by pre-service mathematics teachers when explaining exponential and logarithmic functions in a Colombian university context. The research was theoretically grounded in the Extended Theory of Connections and the Onto-semiotic perspective, which allow the analysis of relationships established between mathematical concepts, procedures, meanings, and representations during instructional processes. Using an exploratory qualitative comparative case-study design involving two purposively selected pre-service mathematics teachers, the study was conducted in four stages: lesson design, case selection, participant observation in a Didactics of Calculus course, and thematic and onto-semiotic analysis through the networking between ETC and OSA. The findings revealed an interconnected network of mathematical connections, including instruction-oriented, meaning, feature, part–whole, procedural, implication, reversibility, and representational connections. These connections showed how the PMTs coordinated mathematical objects, processes, and representations when explaining exponential and logarithmic functions. Particularly, algebraic, graphical, tabular, and verbal representations supported the construction of inverse relationships. Reversibility emerged as the most structurally complex connection and was manifested through recognition, procedural, and structural forms. Rather than confirming a meta-connection, the findings suggest that reversibility may function as an integrative connection, coordinating meanings, procedures, properties, representations, and logical reasoning. The study concludes that the analysis of mathematical connections provides a valuable framework for strengthening didactic-mathematical knowledge in initial teacher education and understanding how pre-service teachers communicate mathematical meanings during teaching processes.
- Abdurahmonov, H. (2025). Logarithmic functions. Academic Journal of Science, Technology and Education, 1(5), 50–51. https://integrumpublication.org/index.php/ajste/article/view/59
- Anugraheni, I., Gufron, A., & Purnomo, Y. W. (2025). The impact of realistic problem-based learning on mathematical connection abilities: Evidence from elementary schools in Indonesia. Cogent Education, 12(1), Article 2523078. https://doi.org/10.1080/2331186X.2025.2523078
- Araújo, S., Viseu, F., Soares, A. J., & Leite, I. (2022). El aprendizaje de las funciones logarítmicas por parte de estudiantes de 12.º grado basado en tareas de modelización [The learning of logarithmic functions by 12th-grade students based on modelling tasks]. Alteridad. Revista de Educación, 17(2), 224–243. https://doi.org/10.17163/alt.v17n2.2022.05
- Bingölbali, E., & Coşkun, M. (2016). A proposed conceptual framework for enhancing the use of making connections skill in mathematics teaching. Education and Science, 41(183), 233–249. https://doi.org/10.15390/EB.2016.4764
- Borji, V., Surynková, P., Kuper, E., & Robová, J. (2025). Using contextual problems to develop preservice mathematics teachers' understanding of exponential and logarithmic concepts. International Journal of Mathematical Education in Science and Technology, 56(6), 1022–1052. https://doi.org/10.1080/0020739X.2024.2309284
- Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77–101. https://doi.org/10.1191/1478088706qp063oa
- Businskas, A. M. (2008). Conversations about connections: How secondary mathematics teachers conceptualize and contend with mathematical connections [Doctoral dissertation, Simon Fraser University]. Summit Research Repository. http://summit.sfu.ca/item/9245
- Campo-Meneses, K. G., Font, V., & García-García, J. (2024). Criterios que orientan la práctica de un profesor al enseñar las funciones exponencial y logarítmica [Criteria that guide a teacher's practice when teaching the exponential and logarithmic functions]. Educação e Pesquisa, 50, Article e267455. https://doi.org/10.1590/S1678-4634202450267455es
- Campo-Meneses, K. G., Font, V., García-García, J., & Sánchez, A. (2021). Mathematical connections activated in high school students' practice solving tasks on the exponential and logarithmic functions. EURASIA Journal of Mathematics, Science and Technology Education, 17(9), Article em1998. https://doi.org/10.29333/ejmste/11126
- Campo-Meneses, K. G., & García-García, J. (2020). Explorando las conexiones matemáticas asociadas a la función exponencial y logarítmica en estudiantes universitarios colombianos [Exploring the mathematical connections associated with the exponential and logarithmic functions in Colombian university students]. Educación Matemática, 32(3), 209–240. https://doi.org/10.24844/em3203.08
- Campo-Meneses, K. G., & García-García, J. (2021). La comprensión de las funciones exponencial y logarítmica: Una mirada desde las conexiones matemáticas y el Enfoque Ontosemiótico [Understanding the exponential and logarithmic functions: A view from mathematical connections and the Onto-Semiotic Approach]. PNA, 16(1), 25–56. https://doi.org/10.30827/pna.v16i1.15817
- Campo-Meneses, K. G., & García-García, J. (2023). Conexiones matemáticas identificadas en la clase sobre funciones exponencial y logarítmica [Mathematical connections identified in the class on exponential and logarithmic functions]. Bolema: Boletim de Educação Matemática, 37(76), 849–871. https://doi.org/10.1590/1980-4415v37n76a22
- Campo-Meneses, K. G., & García-García, J. (2025). Comprensión matemática evidenciada por estudiantes de secundaria sobre las funciones exponencial y logarítmica [Mathematical understanding evidenced by secondary school students about the exponential and logarithmic functions]. AIEM – Avances de Investigación en Educación Matemática, 27, 179–201. https://doi.org/10.35763/aiem27.6430
- Campo-Meneses, K. G., García-García, J., & Font, V. (2023). Mathematical connections associated with the exponential and logarithmic functions promoted in the mathematics curriculum. International Journal of Instruction, 16(4), 17–36. https://doi.org/10.29333/iji.2023.1642a
- Codogni, E. (2025). Aproximación a las funciones exponencial y logarítmica a través del análisis didáctico [An approach to the exponential and logarithmic functions through didactic analysis]. MLS Pedagogy, Culture and Innovation, 2(2). https://www.mlsjournals.com/Pedagogy-Culture-Innovation/article/view/4160
- Díaz-Berríos, T., & Martínez-Planell, R. (2022). High school student understanding of exponential and logarithmic functions. The Journal of Mathematical Behavior, 66, Article 100953. https://doi.org/10.1016/j.jmathb.2022.100953
- Díaz-Berríos, T., & Martínez-Planell, R. (2025). La construcción mental de funciones exponenciales usando multiplicación repetida [The mental construction of exponential functions using repeated multiplication]. Educación Matemática, 37(2), 138–170. https://doi.org/10.24844/EM3702.05
- Dolores-Flores, C., & García-García, J. (2017). Conexiones intramatemáticas y extramatemáticas que se producen al resolver problemas de cálculo en contexto: Un estudio de casos en el nivel superior [Intra-mathematical and extra-mathematical connections that occur when solving calculus problems in context: A case study at a higher level]. Bolema: Boletim de Educação Matemática, 31(57), 158–180. https://doi.org/10.1590/1980-4415v31n57a08
- Drijvers, P., Godino, J. D., Font, V., & Trouche, L. (2013). One episode, two lenses: A reflective analysis of student learning with computer algebra from instrumental and onto-semiotic perspectives. Educational Studies in Mathematics, 82(1), 23–49. https://doi.org/10.1007/s10649-012-9416-8
- Eli, J. A., Mohr-Schroeder, M. J., & Lee, C. W. (2011). Exploring mathematical connections of prospective middle-grades teachers through card-sorting tasks. Mathematics Education Research Journal, 23(3), 297–319. https://doi.org/10.1007/s13394-011-0017-0
- Eli, J. A., Mohr-Schroeder, M. J., & Lee, C. W. (2013). Mathematical connections and their relationship to mathematics knowledge for teaching geometry. School Science and Mathematics, 113(3), 120–134. https://doi.org/10.1111/ssm.12009
- Escobar, N. V. (2014). Elementos históricos para la enseñanza de la función logarítmica en la educación básica [Historical elements for the teaching of the logarithmic function in basic education]. Revista Brasileira de História da Matemática, 14(29), 83–115.
- Faz, V. V. L., & Navas, R. A. C. (2026). Impacto de un laboratorio matemático en la comprensión de funciones exponenciales y logarítmicas en estudiantes de bachillerato [Impact of a mathematics laboratory on high school students' understanding of exponential and logarithmic functions]. Ciencia Latina Revista Científica Multidisciplinar, 10(3), 4309–4322. https://doi.org/10.37811/cl_rcm.v10i3.24473
- Font, V. (2007). Una perspectiva ontosemiótica sobre cuatro instrumentos de conocimiento que comparten un aire de familia: Particular/general, representación, metáfora y contexto [An onto-semiotic perspective on four instruments of knowledge that share a family resemblance: Particular/general, representation, metaphor and context]. Educación Matemática, 19(2), 95–128.
- Font, V., Godino, J. D., & Gallardo, J. (2013). The emergence of objects from mathematical practices. Educational Studies in Mathematics, 82(1), 97–124. https://doi.org/10.1007/s10649-012-9411-0
- García-García, J., & Dolores-Flores, C. (2018). Intra-mathematical connections made by high school students in performing calculus tasks. International Journal of Mathematical Education in Science and Technology, 49(2), 227–252.https://doi.org/10.1080/0020739X.2017.1355994
- García-García, J., & Dolores-Flores, C. (2021). Pre-university student s' mathematical connections when sketching the graph of derivative and antiderivative functions. Mathematics Education Research Journal, 33(1), 1–22. https://doi.org/10.1007/s13394-019-00286-x
- Godino, J. D., & Batanero, C. (1994). Significado institucional y personal de los objetos matemáticos [Institutional and personal meaning of mathematical objects]. Recherches en Didactique des Mathématiques, 14(3), 325–355.
- Godino, J. D., Batanero, C., & Font, V. (2007). The onto-semiotic approach to research in mathematics education. ZDM—The International Journal on Mathematics Education, 39(1–2), 127–135. https://doi.org/10.1007/s11858-006-0004-1
- Kidron, I., & Bikner-Ahsbahs, A. (2015). Advancing research by means of the networking of theories. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Approaches to qualitative research in mathematics education: Examples of methodology and methods (pp. 221–232). Springer. https://doi.org/10.1007/978-94-017-9181-6_9
- Koştur, M., & Yılmaz, A. (2017). Technology support for learning exponential and logarithmic functions. Ihlara Eğitim Araştırmaları Dergisi, 2(2), 50–68.
- Ledezma, C., Rodríguez-Nieto, C. A., & Font, V. (2024). The role played by extra-mathematical connections in the modelling process. AIEM – Avances de Investigación en Educación Matemática, 25, 81–103. https://doi.org/10.35763/aiem25.6363
- Loi, N. H., Khanh, T. L. C., & Tien, L. V. (2020). Connecting mathematics and practice: A case study of teaching exponential functions. European Journal of Education Studies, 7(12), 612–624. https://doi.org/10.46827/ejes.v7i12.3473
- Mhlolo, M. K. (2012). Mathematical connections of a higher cognitive level: A tool we may use to identify these in practice. African Journal of Research in Mathematics, Science and Technology Education, 16(2), 176–191. https://doi.org/10.1080/10288457.2012.10740738
- Mulqueeny, E. (2012). How do students acquire an understanding of logarithmic concepts? (Publication No. 3529853) [Doctoral dissertation, Kent State University]. ProQuest Dissertations & Theses Global.
- National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics.
- Okoye-Ogbalu, I., & Nnadozie, V. (2024). Subject competency and teacher knowledge: An exploration of second-year pre-service mathematics teachers' difficulties in solving logarithmic problems using basic rules for logarithm. Journal of Mathematics and Science Teacher, 4(1), Article em054. https://doi.org/10.29333/mathsciteacher/13887
- Pino-Fan, L., Godino, J. D., & Font, V. (2015). Una propuesta para el análisis de las prácticas matemáticas de futuros profesores sobre derivadas [A proposal for the analysis of prospective teachers' mathematical practices on derivatives]. Bolema: Boletim de Educação Matemática, 29(51), 60–89. https://doi.org/10.1590/1980-4415v29n51a04
- Pino-Fan, L., Godino, J. D., & Font, V. (2018). Assessing key epistemic features of didactic-mathematical knowledge of prospective teachers: The case of the derivative. Journal of Mathematics Teacher Education, 21(1), 63–94. https://doi.org/10.1007/s10857-016-9349-8
- Porras, J. A. D. (2025). TikTok: Tecnología educativa para el aprendizaje significativo de ecuaciones exponenciales y logarítmicas en la educación secundaria [TikTok: Educational technology for the meaningful learning of exponential and logarithmic equations in secondary education]. Revista Ensayos Pedagógicos, 20(1), 1–44. https://doi.org/10.15359/rep.20-1.12
- Prediger, S., Bikner-Ahsbahs, A., & Arzarello, F. (2008). Networking strategies and methods for connecting theoretical approaches: First steps towards a conceptual framework. ZDM—The International Journal on Mathematics Education, 40(2), 165–178. https://doi.org/10.1007/s11858-008-0086-z
- Radford, L. (2008). Connecting theories in mathematics education: Challenges and possibilities. ZDM—The International Journal on Mathematics Education, 40(2), 317–327. https://doi.org/10.1007/s11858-008-0090-3
- Realpe Vidal, C. L. (2022). La enseñanza de la función exponencial, apoyada en la teoría del aprendizaje significativo crítico, en los estudiantes del grado 9° de la IER Botero, sede Santa Gertrudis [The teaching of the exponential function, supported by the theory of critical meaningful learning, among 9th-grade students at IER Botero, Santa Gertrudis campus] [Master's thesis, Universidad Nacional de Colombia]. Repositorio Institucional UNAL. https://repositorio.unal.edu.co/handle/unal/81697
- Rodríguez, M. A., Ledezma, C. A., Vergara, A. S., & Gregori, P. (2021). Reconstrucción cognitiva de los conceptos centrales de la función exponencial: Un estudio de enfoque mixto [Cognitive reconstruction of the central concepts of the exponential function: A mixed-methods study]. Formación Universitaria, 14(6), 149–164. https://doi.org/10.4067/S0718-50062021000600149
- Rodríguez-Nieto, C. A., & Font, V. (2025). Mathematical connections promoted in multivariable calculus' classes and in problems-solving about vectors, partial and directional derivatives, and applications. Eurasia Journal of Mathematics, Science and Technology Education, 21(4), Article em2619. https://doi.org/10.29333/ejmste/16187
- Rodríguez-Nieto, C. A., Font, V., Borji, V., & Rodríguez-Vásquez, F. M. (2022). Mathematical connections from a networking of theories between extended theory of mathematical connections and onto-semiotic approach. International Journal of Mathematical Education in Science and Technology, 53(9), 2364–2390. https://doi.org/10.1080/0020739X.2021.1875071
- Rodríguez-Nieto, C. A., González, H. A. C., Arenas-Peñaloza, J., Schnorr, C. E., & Font, V. (2024). Onto-semiotic analysis of Colombian engineering students' mathematical connections to problems-solving on vectors: A contribution to the natural and exact sciences. Eurasia Journal of Mathematics, Science and Technology Education, 20(5), Article em2438. https://doi.org/10.29333/ejmste/14450
- Rodríguez-Nieto, C. A., Pabón-Navarro, M. L., Cantillo-Rudas, B. M., Sudirman, & Font, V. (2025). The potential of ethnomathematical and mathematical connections in the pre-service mathematics teachers' meaningful learning when problems-solving about brick-making. Infinity Journal, 14(2), 419–444. https://doi.org/10.22460/infinity.v14i2.p419-444
- Rodríguez-Nieto, C. A., Rodríguez-Vásquez, F. M., & Font, V. (2022) . A new view about connections: The mathematical connections established by a teacher when teaching the derivative. International Journal of Mathematical Education in Science and Technology, 53(6), 1231–1256. https://doi.org/10.1080/0020739X.2020.1799254
- Rodríguez-Nieto, C. A., Rodríguez-Vásquez, F. M., & García-García, J. (2021a). Exploring university Mexican students' quality of intra-mathematical connections when solving tasks about derivative concept. Eurasia Journal of Mathematics, Science and Technology Education, 17(9), Article em2006. https://doi.org/10.29333/ejmste/11160
- Rodríguez-Nieto, C. A., Rodríguez-Vásquez, F. M., & García-García, J. (2021b). Pre-service mathematics teachers' mathematical connections in the context of problem-solving about the derivative. Turkish Journal of Computer and Mathematics Education, 12(1), 202–220. https://doi.org/10.16949/turkbilmat.797182
- Silva, N. G. (2026). La pelota, ¿dejará de rebotar? Modelación de la función exponencial [Will the ball stop bouncing? Modelling the exponential function]. Revista Chilena de Educación Matemática, 18(1), 38–54. https://doi.org/10.46219/rechiem.v18i1.225
- Sudirman, Rodríguez-Nieto, C. A., Hidayat, R., Isnawan, M. G., Pauzan, M., Yumiati, Martadiputra, B. A. P., & Faizah, S. (2026). Operationalizing didactical situation-based online learning to support eighth-grade students' mathematical reasoning and understanding in geometry: Participatory design research. Journal on Mathematics Education, 17(1), 43–68. https://doi.org/10.22342/jme.v17i1.pp43-68
- Sureda, P., & Otero, M. (2013). Estudio sobre el proceso de conceptualización de la función exponencial [Study on the process of conceptualization of the exponential function]. Educación Matemática, 25(2), 89–118.
- Vicuña Espinoza, J. M. (2026). Impacto del material didáctico interactivo en el aprendizaje de las funciones exponenciales: Evidencia desde el bachillerato ecuatoriano [Impact of interactive teaching material on learning exponential functions: Evidence from Ecuadorian high school]. Revista Social Fronteriza, 6(4), Article e1187. https://doi.org/10.59814/resofro.2026.6(4)1187
APA 7th edition
In-text citation: (Rodríguez-Nieto et al., 2026)
Reference: Rodríguez-Nieto, C. A., Cortina-Gutiérrez, J. A., Galindo Illanes, M. K., Sudirman, S., Chamorro Manríquez, D. D., Breda, A., & Morales-Beltrán, I. R. (2026). Pre-service mathematics teachers’ intra-mathematical connections: An onto-semiotic analysis of explanations of exponential and logarithmic functions.
European Journal of STEM Education.
https://doi.org/10.20897/ejsteme/19497
AMA 10th edition
In-text citation: (1), (2), (3), etc.
Reference: Rodríguez-Nieto CA, Cortina-Gutiérrez JA, Galindo Illanes MK, et al. Pre-service mathematics teachers’ intra-mathematical connections: An onto-semiotic analysis of explanations of exponential and logarithmic functions.
European Journal of STEM Education. 2026.
https://doi.org/10.20897/ejsteme/19497
Chicago
In-text citation: (Rodríguez-Nieto et al., 2026)
Reference: Rodríguez-Nieto, Camilo Andrés, Jeiner Antonio Cortina-Gutiérrez, Maritza Katherine Galindo Illanes, Sudirman Sudirman, Denise Deyanira Chamorro Manríquez, Adriana Breda, and Iván Ramiro Morales-Beltrán. "Pre-service mathematics teachers’ intra-mathematical connections: An onto-semiotic analysis of explanations of exponential and logarithmic functions".
European Journal of STEM Education (2026).
https://doi.org/10.20897/ejsteme/19497
Harvard
In-text citation: (Rodríguez-Nieto et al., 2026)
Reference: Rodríguez-Nieto, C. A., Cortina-Gutiérrez, J. A., Galindo Illanes, M. K., Sudirman, S., Chamorro Manríquez, D. D., Breda, A., and Morales-Beltrán, I. R. (2026). Pre-service mathematics teachers’ intra-mathematical connections: An onto-semiotic analysis of explanations of exponential and logarithmic functions.
European Journal of STEM Education.
https://doi.org/10.20897/ejsteme/19497
MLA
In-text citation: (Rodríguez-Nieto et al., 2026)
Reference: Rodríguez-Nieto, Camilo Andrés et al. "Pre-service mathematics teachers’ intra-mathematical connections: An onto-semiotic analysis of explanations of exponential and logarithmic functions".
European Journal of STEM Education, 2026.
https://doi.org/10.20897/ejsteme/19497
Vancouver
In-text citation: (1), (2), (3), etc.
Reference: Rodríguez-Nieto CA, Cortina-Gutiérrez JA, Galindo Illanes MK, Sudirman S, Chamorro Manríquez DD, Breda A, et al. Pre-service mathematics teachers’ intra-mathematical connections: An onto-semiotic analysis of explanations of exponential and logarithmic functions. European Journal of STEM Education. 2026.
https://doi.org/10.20897/ejsteme/19497