Representational Difficulty Across Cognitive Stages in Quadratic Functions: An APOS–Newman Analysis

Hesty Marwani Siregar; Turmudi; Nurjanah; Darhim

doi:10.23947/2334-8496-2026-14-2-317-329

Authors

Hesty Marwani Siregar Universitas Pendidikan Indonesia – Faculty of Mathematics and Science Education, Bandung, Indonesia https://orcid.org/0000-0003-3405-6144
Turmudi Universitas Pendidikan Indonesia – Faculty of Mathematics and Science Education, Bandung, Indonesia https://orcid.org/0000-0001-7976-211X
Nurjanah Universitas Pendidikan Indonesia – Faculty of Mathematics and Science Education, Bandung, Indonesia https://orcid.org/0000-0001-9274-0804
Darhim Universitas Pendidikan Indonesia – Faculty of Mathematics and Science Education, Bandung, Indonesia https://orcid.org/0000-0003-4796-5697

DOI:

https://doi.org/10.23947/2334-8496-2026-14-2-317-329

Keywords:

mathematical representation, quadratic function, APOS, Newman error analysis, cognitive stages, asymmetrical representational development

Abstract

This study aims to identify the types of students’ mathematical representation errors using Newman’s error classification and to analyze their relationships with the stages of cognitive development in the Action, Process, Object, and Schema theory. The study employed a qualitative descriptive approach involving 42 tenth-grade students from a senior high school. Data were collected through a mathematical representation ability test and in-depth interviews, and were analyzed qualitatively by mapping the types of errors and students’ cognitive development stages. The results indicate that students’ mathematical representation ability remains low, with symbolic representation the most challenging aspect across all ability levels. Comprehension and transformation errors were dominant among low-ability students, most of whom were at the Action stage. Moderate-ability students showed an emerging uneven pattern of cognitive attainment, with Schema-level attainment in the visual representation and, to a limited extent, in the verbal representation, whereas no Schema attainment was observed in the symbolic representation. Meanwhile, high-ability students demonstrated Schema-level attainment in the visual and verbal representations, whereas their symbolic performance remained at the Object stage because of persistent process-skill difficulties. These findings indicate that cognitive development is specific to the type of representation and does not occur uniformly across representations. This pattern of uneven cognitive attainment across representational forms is conceptualized in this study as asymmetrical representational development. The implications of this study suggest that the teaching of quadratic functions should emphasize coordination among visual, symbolic, and contextual representations through multi-representation approaches and gradual scaffolding to consistently support students’ conceptual understanding.

Downloads

Download data is not yet available.

👁 Abstract Views: 0📥 PDF Downloads: 0

References

Altindis, N., & Fonger, N. L. (2025). A Culturally Inclusive Mathematics Learning Environment Framework: Supporting Students’ Representational Fluency and Covariational Reasoning. Education Sciences, 15(8), 980–1002. https://doi.org/10.3390/educsci15080980

Ayala-Altamirano, C., Pinto, E., Molina, M., & Cañadas, M. C. (2022). Interacting with Indeterminate Quantities through Arithmetic Word Problems: Tasks to Promote Algebraic Thinking at Elementary School. Mathematics, 10(13), 2229–2246. https://doi.org/10.3390/math10132229

Baye, M. G., Ayele, M. A., & Wondimuneh, T. E. (2021). Implementing GeoGebra integrated with multi-teaching approaches guided by the APOS theory to enhance students’ conceptual understanding of limit in Ethiopian Universities. Heliyon, 7(5), 1–13. https://doi.org/10.1016/j.heliyon.2021.e07012

Bintoro, H. S., Sukestiyarno, Y. L., Mulyono, & Walid. (2021). The Spatial Thinking Process of the Field-Independent Students Based on Action-Process-Object-Schema Theory. European Journal of Educational Research, 10(4), 1807–1823. https://doi.org/10.12973/EU-JER.10.4.1807

Borji, V., Martínez-Planell, R., & Trigueros, M. (2024). Students’ Understanding of Riemann Sums and Double Integrals: The Case of Task Design in APOS Theory. International Journal of Research in Undergraduate Mathematics Education, 1–29. https://doi.org/10.1007/s40753-024-00250-6

Castro, E., Cañadas, M. C., Molina, M., & Rodríguez‑Domingo, S. (2022). Difficulties in semantically congruent translation of verbally and symbolically represented algebraic statements. Educational Studies in Mathematics, 109, 593–609. https://doi.org/10.1007/s10649-021-10088-3

Erath, K., Ingram, J., Moschkovich, J., & Prediger, S. (2021). Designing and enacting instruction that enhances language for mathematics learning: a review of the state of development and research. ZDM – Mathematics Education, 53, 245–262. https://doi.org/10.1007/s11858-020-01213-2

Faradiba, S. S., Sa’dijah, C., Parta, I. N., & Rahardjo, S. (2019). Looking Without Seeing: The Role of Metacognitive Blindness of Student with High Math Anxiety. International Journal of Cognitive Research in Science, Engineering and Education (IJCRSEE), 7(2), 53–65. https://doi.org/10.5937/IJCRSEE1902053F

Fathonah, S., Cahyono, E., Haryani, S., Sarwi, S., & Lestari, N. H. (2024). Application of Multirepresentation-Based Creative Problem-Solving Learning Models to Improve Critical and Creative Thinking Skills for Students. International Journal of Cognitive Research in Science, Engineering and Education (IJCRSEE), 12(1), 185–200. https://doi.org/10.23947/2334-8496-2024-12-1-185-200

Hariyani, M., Suherman, Andriani, M., & Herawati. (2023). The Importance of Mathematical Representation Ability for Elementary School Students: A Literature Review and Its Implications. Syekh Nurjati International Conference on Elementary Education (SICEE), 1, 38–46. https://dx.doi.org/10.24235/sicee.v1i0.14579

Ke, L., Sadler, T. D., Zangori, L., & Friedrichsen, P. J. (2021). Developing and Using Multiple Models to Promote Scientific Literacy in the Context of Socio‑Scientific Issues. Science & Education, 30, 589–607. https://doi.org/10.1007/s11191-021-00206-1

Kolar, V. M., & Hodnik, T. (2021). Mathematical Literacy from the Perspective of Solving Contextual Problems. European Journal of Educational Research, 10(1), 467–483. https://doi.org/10.12973/eu-jer.10.1.467

Korstjens, I., & Moser, A. (2018). Series: Practical guidance to qualitative research. Part 4: Trustworthiness and publishing. European Journal of General Practice, 24(1), 120–124. https://doi.org/10.1080/13814788.2017.1375092

Li, S., Shen, Y., Jiao, X., & Cai, S. (2022). Using Augmented Reality to Enhance Students’ Representational Fluency: The Case of Linear Functions. Mathematics, 10(10), 1718–1738. https://doi.org/10.3390/math10101718

Luthfiani, U., Saragih, S., & Suanto, E. (2025). Development of learning based teaching materials on quadratic function materials to improve students’ mathematical representation skills. JTMT: Journal Tadris Matematika, 6(2), 29–38. https://doi.org/10.47435/jtmt.v6i2.4191

Mainali, B. (2021). Representation in Teaching and Learning Mathematics. International Journal of Education in Mathematics, Science, and Technology (IJEMST), 9(1), 1–21. https://doi.org/10.46328/ijemst.1111

Nurrahmawati, Sa’dijah, C., Sudirman, & Muksar, M. (2021). Assessing Students’ Errors in Mathematical Translation: From Symbolic to Verbal and Graphic Representations. International Journal of Evaluation and Research in Education (IJERE), 10(1), 115–125. https://doi.org/10.11591/ijere.v10i1.20819

Pramesti, C. A., Shoffa, S., & Soemantri, S. (2024). Analysis of Class X MAN Surabaya Students’ Mistakes in Solving Function Questions Quadratics Based on Learning Styles and Newman Procedures. JOELI: Journal of Educational and Learning Innovation, 1(2), 147–160. https://doi.org/10.72204/3e8ha171

Rahman, T. F. binti A., & Foad, M. S. bin M. (2021). Quadratic Functions in Additional Mathematics and Mathematics: An Analysis on Students’ Errors. Academic Journal of Business and Social Sciences, 5(1), 1–16.

Rahmawati, A. N. L., Sa’dijah, C., & Sudirman, S. (2025). Exploring high school students’ mathematical representation abilities: Translation from verbal to symbolic using higher-order thinking skills. TEM Journal, 14(3), 2442–2453. https://doi.org/10.18421/TEM143-48

Santos-Trigo, M. (2024). Problem solving in mathematics education: tracing its foundations and current research-practice trends. ZDM – Mathematics Education, 56, 211–222. https://doi.org/10.1007/s11858-024-01578-8

Şefik, Ö., Uzun, Ö. E., & Dost, Ş. (2021). Content Analysis of the APOS Theory Studies on Mathematics Education Conducted in Turkey and Internationally: A Meta-Synthesis Study. Necatibey Faculty of Education Electronic Journal of Science and Mathematics Education, 15(2), 404–428. https://doi.org/10.17522/balikesirnef.1020526

Selden, A., & Vidakovic, D. (2022). Remembering Ed Dubinsky and his Visionary Work. International Journal of Research in Undergraduate Mathematics Education, 8, 421–436. https://doi.org/10.1007/s40753-022-00201-z

Siller, H., Nitzan‑Tamar, O., & Kohen, Z. (2023). Scaffolding practices for modelling instruction in STEM‑related contexts: insights from expert and novice teachers. ZDM – Mathematics Education, 55, 1351–1364. https://doi.org/10.1007/s11858-023-01529-9

Siregar, H. M., & Solfitri, T. (2019). An Analysis of Students’ Errors in Solving Indefinite Integral Problems Viewed From Gender Differences. Journal of Research on Mathematics Instruction (JRMI), 1(1), 17–24. https://doi.org/10.33578/jrmi.v1i1.12

Sopamena, P., Kaliky, S., Sehuwaky, N., Kasliyanto, & Juhaevah, F. (2021). Student Thinking Process in Solving Mathematical Problems Based on APOS. Matematika Dan Pembelajaran, 9(2), 31–46. http://dx.doi.org/10.33477/mp.v9i2.2384

Stahl, N. A., & King, J. R. (2020). Expanding Approaches for Research: Understanding and Using Trustworthiness in Qualitative Research. Journal of Developmental Education, 44(1), 26–28.

Tiew, N. N. A. M. F., Hoon, T. S., Singh, P. S. A., Walida, S. El, & Faradiba, S. S. (2023). An Analysis of Students’ Understanding of Algebraic Concepts. Jurnal Pendidikan Sains Dan Matematik Malaysia, 13(2), 86–95. https://doi.org/10.37134/jpsmm.vol13.2.8.2023

Umam, K., & Susandi, A. D. (2022). Critical Thinking Skills: Error Identifications on Students’ with APOS Theory. International Journal of Evaluation and Research in Education, 11(1), 182–192. https://doi.org/10.11591/ijere.v11i1.21171

Wilkie, K. J. (2024). Coordinating Visual and Algebraic Reasoning with Quadratic Functions. Mathematics Education Research Journal, 36, 27–63. https://doi.org/10.1007/s13394-022-00426-w

Yarman, Dwina, F., Murni, D., Yerizon, & Hevardani, K. A. (2024). Analysis of Concept Construction and Student Errors on the Topic of Double Integral Based on APOS Theory. Jurnal Pendidikan Matematika, 18(3), 367–386. https://doi.org/10.22342/jpm.v18i3.pp367-386