Exploring Students' Conceptual Images of Function and Their Properties: A Hermeneutic Phenomenological Study

Hana Auriel Sharon Siahaan; Muhammad Daut Siagian; Rikayanti

doi:10.31949/dm.v8i2.19093

Authors

Hana Auriel Sharon Siahaan Department of Mathematics Education, Universitas Singaperbangsa Karawang, Indonesia, Indonesia
Muhammad Daut Siagian Department of Mathematics Education, Universitas Singaperbangsa Karawang, Indonesia, Indonesia
Rikayanti Department of Mathematics Education, Universitas Singaperbangsa Karawang, Indonesia, Indonesia

DOI:

https://doi.org/10.31949/dm.v8i2.19093

Abstract

Understanding students' concept images of functions is fundamental to mathematics education because these images influence how learners interpret definitions, reason about function properties, and construct mathematical representations. Although previous studies have documented common misconceptions, relatively little is known about how students construct mathematical meaning through their lived learning experiences. Addressing this gap, this study explored students' concept images of functions using Ricoeur's hermeneutic phenomenology integrated with Tall and Vinner's concept image framework. A qualitative phenomenological design was employed, involving 30 Grade X secondary school students. Data were collected through written tasks focusing on function evaluation, function properties, and multiple representations, followed by semi-structured interviews with purposively selected participants. Data were analyzed through Ricoeur's hermeneutic process of naïve understanding, structural analysis, and appropriation to reveal the interpretive meanings underlying students' mathematical reasoning. The findings identified four major interpretive patterns: procedural, integrated, misconception, and representational concept images. While most students successfully performed routine procedures, many interpreted functions in terms of procedural routines, perceptual characteristics, and familiar examples rather than through formal structural definitions. Persistent misconceptions, including the conflation of codomain and range, domain-centered interpretations of surjectivity, and limited representational flexibility, reflected coherent interpretive structures rather than isolated conceptual errors. These findings extend Tall and Vinner's concept image theory by demonstrating that concept images are continuously reconstructed through the interpretation of mathematical experiences. The study suggests that function instruction should move beyond procedural fluency by engaging students with contrasting examples, multiple representations, reflective justification, and explicit connections between concept images and formal concept definitions to support meaningful conceptual development.

Keywords:

Concept image, Function concept, Hermeneutic phenomenology, Ricoeur, Mathematics education

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References

Adeoye-Olatunde, O. A., & Olenik, N. L. (2021). Research and scholarly methods: Semi-structured interviews. Journal of the American College of Clinical Pharmacy, 4(10), 1358–1367. https://doi.org/10.1002/jac5.1441

Ahmed, S. K. (2024). The pillars of trustworthiness in qualitative research. Journal of Medicine, Surgery, and Public Health, 2, Article 100051. https://doi.org/10.1016/j.glmedi.2024.100051

Ainsworth, S., Bibby, P., & Wood, D. (2002). Examining the effects of different multiple representational systems in learning primary mathematics. Journal of the Learning Sciences, 11(1), 25–61. https://doi.org/10.1207/S15327809JLS1101_2

Ayeh, I. G. (2025). Students' mathematics conceptual challenges: Exploring students' thinking, understanding, and misconceptions in functions and graphs. European Journal of Science and Mathematics Education, 13(3), 191–206. https://doi.org/10.30935/scimath/16596

Bingolbali, E., & Monaghan, J. (2008). Concept image revisited. Educational Studies in Mathematics, 68(1), 19–35. https://doi.org/10.1007/s10649-007-9112-2

Bouncken, R. B., Czakon, W., & Schmitt, F. (2026). Purposeful sampling and saturation in qualitative research methodologies: Recommendations and review. Review of Managerial Science, 20, 579–615. https://doi.org/10.1007/s11846-025-00881-2

Brod, M., Pohlman, B., & Waldman, L. T. (2023). Qualitative research and content validity. In F. Maggino (Ed.), Encyclopedia of quality of life and well-being research. Springer. https://doi.org/10.1007/978-3-031-17299-1_3848

Carpenter, T. P. (2013). Conceptual knowledge as a foundation for procedural knowledge. In J. R. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 113–132). Routledge. https://doi.org/10.4324/9780203063538-5

Daniel, K. L., Bucklin, C. J., Leone, E. A., & Idema, J. (2018). Towards a definition of representational competence. In K. L. Daniel (Ed.), Towards a framework for representational competence in science education (Models and Modeling in Science Education, Vol. 11, pp. 1–25). Springer. https://doi.org/10.1007/978-3-319-89945-9_1

Davidson, S., & Vallée, M. A. (2016). Hermeneutics and phenomenology in Paul Ricoeur. Springer. https://doi.org/10.1007/978-3-319-33426-4

Denzin, N. K. (2019). A call to a critical interpretive interactionism. In Critical and cultural interactionism (pp. 45–60). Routledge. https://doi.org/10.4324/9781315141640-4

Dodgson, J. E. (2019). Reflexivity in qualitative research. Journal of Human Lactation, 35(2), 220–222. https://doi.org/10.1177/0890334419830990

Douglas, H. (2022). Sampling techniques for qualitative research. In M. R. Islam, N. A. Khan, & R. Baikady (Eds.), Principles of social research methodology. Springer. https://doi.org/10.1007/978-981-19-5441-2_29

Elia, I., Panaoura, A., Gagatsis, A., Deliyianni, E., & Spyrou, P. (2008). Exploring different aspects of the understanding of function: Toward a four-facet model. Canadian Journal of Science, Mathematics and Technology Education, 8(1), 49–69. https://doi.org/10.1080/14926150802152277

Geniusas, S. (2015). Between phenomenology and hermeneutics: Paul Ricoeur's philosophy of imagination. Human Studies, 38(2), 223–241. https://doi.org/10.1007/s10746-014-9339-8

Hiebert, J., & Lefevre, P. (2013). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. R. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Routledge. https://doi.org/10.4324/9780203063538-1

Kleiner, I. (1989). Evolution of the function concept: A brief survey. The College Mathematics Journal, 20(4), 282–300. https://doi.org/10.1080/07468342.1989.11973245

Kublikowski, I. (2023). Reflections on analysis in qualitative research: The hermeneutic circle in Ricoeur. Paidéia (Ribeirão Preto), 33, e3319. https://doi.org/10.1590/1982-4327e3319

Küçük, B. (2011). Identifying the secondary school students' misconceptions about functions. Procedia - Social and Behavioral Sciences, 15, 3837–3842. https://doi.org/10.1016/j.sbspro.2011.04.382

Leonardo, Z. (2003). Interpretation and the problem of domination: Paul Ricoeur's hermeneutics. Studies in Philosophy and Education, 22(3–4), 329–350. https://doi.org/10.1023/A:1025129716403

Martínez-Planell, R., & Trigueros, M. (2012). Students' understanding of the general notion of a function of two variables. Educational Studies in Mathematics, 81(3), 365–384. https://doi.org/10.1007/s10649-012-9408-8

Moore-Russo, D., Conner, A., & Rugg, K. I. (2011). Can slope be negative in 3-space? Studying concept image of slope through collective definition construction. Educational Studies in Mathematics, 76(1), 3–21. https://doi.org/10.1007/s10649-010-9277-y

Olympiou, G., Zacharia, Z. C., & de Jong, T. (2013). Making the invisible visible: Enhancing students' conceptual understanding by introducing representations of abstract objects in a simulation. Instructional Science, 41(3), 575–596. https://doi.org/10.1007/s11251-012-9245-2

Pande, P., & Chandrasekharan, S. (2017). Representational competence: Towards a distributed and embodied cognition account. Studies in Science Education, 53(1), 1–43. https://doi.org/10.1080/03057267.2017.1248627

Parhizgar, Z., Dehbashi, A., Liljedahl, P., & Alamolhodaei, H. (2022). Exploring students' misconceptions of the function concept through problem-posing tasks and their views thereon. International Journal of Mathematical Education in Science and Technology, 53(12), 3261–3285. https://doi.org/10.1080/0020739X.2021.1937732

Pierce, R., Stacey, K., Wander, R., & Ball, L. (2011). The design of lessons using mathematics analysis software to support multiple representations in secondary school mathematics. Technology, Pedagogy and Education, 20(1), 95–112. https://doi.org/10.1080/1475939X.2010.534869

Pietilä, A.-M., Nurmi, S.-M., Halkoaho, A., & Kyngäs, H. (2020). Qualitative research: Ethical considerations. In H. Kyngäs, K. Mikkonen, & M. Kääriäinen (Eds.), The application of content analysis in nursing science research (pp. 49–69). Springer. https://doi.org/10.1007/978-3-030-30199-6_6

Prihandhika, A., & Perbowo, K. S. (2024). The review of concept image and concept definition: A hermeneutic phenomenological study on the derivative concepts. International Journal of Didactic Mathematics in Distance Education, 1(1), 13–23. https://doi.org/10.33830/ijdmde.v1i1.7610

Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346–362. https://doi.org/10.1037/0022-0663.93.2.346

Sajka, M. (2003). A secondary school student's understanding of the concept of function: A case study. Educational Studies in Mathematics, 53(3), 229–254. https://doi.org/10.1023/A:1026033415747

Schwarz, B. B., & Hershkowitz, R. (1999). Prototypes: Brakes or levers in learning the function concept? The role of computer tools. Journal for Research in Mathematics Education, 30(4), 362–389. https://doi.org/10.2307/749706

Skovsmose, O. (2005). Meaning in mathematics education. In J. Kilpatrick, C. Hoyles, O. Skovsmose, & P. Valero (Eds.), Meaning in mathematics education (Mathematics Education Library, Vol. 37, pp. 1–16). Springer. https://doi.org/10.1007/0-387-24040-3_6

Star, J. R., & Stylianides, G. J. (2013). Procedural and conceptual knowledge: Exploring the gap between knowledge type and knowledge quality. Canadian Journal of Science, Mathematics and Technology Education, 13(2), 169–181. https://doi.org/10.1080/14926156.2013.784828

Suddick, K. M., Cross, V., Vuoskoski, P., Galvin, K. T., & Stew, G. (2020). The work of hermeneutic phenomenology. International Journal of Qualitative Methods, 19, 1–14. https://doi.org/10.1177/1609406920947600

Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169. https://doi.org/10.1007/BF00305619

Tan, H., Wilson, A., & Olver, I. (2009). Ricoeur's theory of interpretation: An instrument for data interpretation in hermeneutic phenomenology. International Journal of Qualitative Methods, 8(4), 1–15. https://doi.org/10.1177/160940690900800401

Vinner, S. (1983). Concept definition, concept image and the notion of function. International Journal of Mathematical Education in Science and Technology, 14(3), 293–305. https://doi.org/10.1080/0020739830140305

Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356–366. https://doi.org/10.5951/jresematheduc.20.4.0356