Exploring the role of cognitive style and APOS-based scaffolding in developing students’ advanced mathematical thinking on function limits
##plugins.themes.bootstrap3.article.sidebar##
##plugins.themes.bootstrap3.article.main##
Abstract
This qualitative case study explored how field dependence-independence shaped students’ access to Action-Process-Object-Schema (APOS)-based scaffolding and how stage-specific support contributed to the qualitative development of advanced mathematical thinking (AMT) on function limits. In this study, developing denotes the observable emergence or reorganization of mathematical constructions during scaffolded activity, not a statistically measured learning gain. The bounded classroom case involved 24 mathematics education students at an Islamic university in Sumatra, Indonesia. The Group Embedded Figures Test classified 6 students as field-independent (FI) and 18 as field-dependent (FD); four information-rich cases, comprising 2 FI and 2 FD students, were selected for cross-case analysis. APOS-based scaffolding was implemented in four 135-minute Calculus I sessions and examined further through stimulated-recall interviews. Initial written solutions provided the primary evidence, post-prompt explanations and revisions documented students’ responses to Action-, Process-, Object-, and Schema-oriented support, and classroom observation records verified the enactment and gradual fading of the planned scaffolds. FI-1 constructed a complete Schema and required only fading prompts. FI-2 moved from correct but disconnected Objects toward an integrated Schema after a Schema-oriented prompt. FD-1 corrected an invalid limit value after Process-oriented prompts stabilized the transformation sequence. FD-2 moved from direct substitution and 0/0 toward guided algebraic Actions and an emerging Process after explicit Action-oriented cues. The findings indicate that cognitive style influenced task entry and the amount of external structure used, whereas APOS identified the construction that support needed to target. The observed shifts support a diagnostic account of AMT development while avoiding causal or long-term claims.
##plugins.themes.bootstrap3.article.details##
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
The author is responsible for acquiring the permission(s) to reproduce any copyrighted figures, tables, data, or text that are being used in the submitted paper. Authors should note that text quotations of more than 250 words from a published or copyrighted work will require grant of permission from the original publisher to reprint. The written permission letter(s) must be submitted together with the manuscript.References
Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Roa Fuentes, S., Trigueros, M., & Weller, K. (2014). APOS theory: A framework for research and curriculum development in mathematics education. Springer. https://doi.org/10.1007/978-1-4614-7966-6
Bansilal, S., & Mkhwanazi, T. W. (2022). Pre-service student teachers’ conceptions of the notion of limit. International Journal of Mathematical Education in Science and Technology, 53(8), 2083–2101. https://doi.org/10.1080/0020739x.2020.1864488
Baxter, P., & Jack, S. (2008). Qualitative case study methodology: Study design and implementation for novice researchers. The Qualitative Report, 13(4), 544–559. https://doi.org/10.46743/2160-3715/2008.1573
Borg, A., & Fahlgren, M. (2026). Students’ development of instrumented action schemes for numerically determining limits using programming. Digital Experiences in Mathematics Education, 12(1), 95–127. https://doi.org/10.1007/s40751-025-00177-w
Borji, V., Martínez-Planell, R., & Trigueros, M. (2022). Student understanding of functions of two variables: A reproducibility study. The Journal of Mathematical Behavior, 66, 100950. https://doi.org/10.1016/j.jmathb.2022.100950
Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77–101. https://doi.org/10.1191/1478088706qp063oa
Chien, T. M., Leong, K. E., & Chua, K. H. (2026). Design-based research on instructional lessons for differential calculus incorporating Desmos. Infinity Journal, 15(2), 345–370. https://doi.org/10.22460/infinity.v15i2.p345-370
Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K., & Vidakovic, D. (1996). Understanding the limit concept: Beginning with a coordinated process scheme. The Journal of Mathematical Behavior, 15(2), 167–192. https://doi.org/10.1016/s0732-3123(96)90015-2
DiNapoli, J., & Miller, E. K. (2022). Recognizing, supporting, and improving student perseverance in mathematical problem-solving: The role of conceptual thinking scaffolds. The Journal of Mathematical Behavior, 66, 100965. https://doi.org/10.1016/j.jmathb.2022.100965
Dubinsky, E. (2014). Actions, Processes, Objects, Schemas (APOS) in Mathematics Education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 8–11). Springer. https://doi.org/10.1007/978-94-007-4978-8_3
Hattie, J., & O’Leary, T. (2025). Learning styles, preferences, or strategies? An explanation for the resurgence of styles across many meta-analyses. Educational Psychology Review, 37(2), 31. https://doi.org/10.1007/s10648-025-10002-w
Hofer, S. I., & Reinhold, F. (2025). Scaffolding of learning activities: Aptitude-treatment-interaction effects in math? Learning and Instruction, 99, 102177. https://doi.org/10.1016/j.learninstruc.2025.102177
Kusmaryono, I., Ubaidah, N., & Basir, M. A. (2020). The role of scaffolding in the deconstructing of thinking structure: A case study of pseudo-thinking process. Infinity Journal, 9(2), 247–262. https://doi.org/10.22460/infinity.v9i2.p247-262
Listiawati, E., Juniati, D., & Ekawati, R. (2025). Understanding mathematics prospective teachers' comprehension of function derivatives based on APOS theory: Insights from low mathematics anxiety levels. Infinity Journal, 14(2), 483–512. https://doi.org/10.22460/infinity.v14i2.p483-512
Lithner, J. (2008). A research framework for creative and imitative reasoning. Educational Studies in Mathematics, 67(3), 255–276. https://doi.org/10.1007/s10649-007-9104-2
Lithner, J. (2017). Principles for designing mathematical tasks that enhance imitative and creative reasoning. Zdm, 49(6), 937–949. https://doi.org/10.1007/s11858-017-0867-3
Martínez-Planell, R., Trigueros, M., & Borji, V. (2024). The role of topology in the construction of students’ optimization schema for two-variable functions. The Journal of Mathematical Behavior, 73, 101106. https://doi.org/10.1016/j.jmathb.2023.101106
Mirin, A. (2024). Where we see one function, they see two. International Journal of Mathematical Education in Science and Technology, 55(9), 2107–2135. https://doi.org/10.1080/0020739x.2022.2123285
Mirlanda, E. P., & Pujiastuti, H. (2018). Kemampuan penalaran matematis: Analisis berdasarkan gaya kognitif siswa [Mathematical reasoning ability: An analysis based on students' cognitive styles]. Symmetry: Pasundan Journal of Research in Mathematics Learning and Education, 3(2), 56–67. https://doi.org/10.23969/symmetry.v3i2.1252
Nowell, L. S., Norris, J. M., White, D. E., & Moules, N. J. (2017). Thematic analysis: Striving to meet the trustworthiness criteria. International journal of qualitative methods, 16(1), 1–13. https://doi.org/10.1177/1609406917733847
Santosa, C. A. H. F., & Filiz, M. (2025). Investigating the limit of peer collaboration: Insight from worked-example in multivariable calculus. Infinity Journal, 14(2), 461–482. https://doi.org/10.22460/infinity.v14i2.p461-482
Setiyani, S., Gunawan, G., Waluya, S. B., Sukestiyarno, Y. L., Cahyono, A. N., Mutia, M., & Yunianto, W. (2025). The role of scaffolding in shaping reflective mathematical thinking of dependent field students in numeracy problems. Infinity Journal, 14(2), 531–550. https://doi.org/10.22460/infinity.v14i2.p531-550
Tall, D. (1991). Advanced mathematical thinking. Kluwer Academic Publishers.
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
Thompson, P. W., & Harel, G. (2021). Ideas foundational to calculus learning and their links to students’ difficulties. ZDM – Mathematics Education, 53(3), 507–519. https://doi.org/10.1007/s11858-021-01270-1
Witkin, H. A., Moore, C. A., Goodenough, D. R., & Cox, P. W. (1977). Field-dependent and field-independent cognitive styles and their educational implications. Review of Educational Research, 47(1), 1–64. https://doi.org/10.3102/00346543047001001
Witkin, H. A., Oltman, P. K., Raskin, E., & Karp, S. A. (1971). A manual for the embedded figures tests. Consulting Psychologists Press.
Yin, R. K. (2018). Case study research and applications: Design and methods (6th ed.). Sage Publication.