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The objective of this study is to determine students’ learning barriers in understanding integral concepts through their thinking processes with the perspective of APOS theory (Action, Process, Object-Scheme). This research applied qualitative research with a case study method. The samples of this research were 19 civil engineering students who had contracted calculus courses and who had been given a written test. The results of the written test were divided into three different categories – they are comprehension ability is high (score ), medium (60  Score <  ) and low (score < 60). Deep   interviews were conducted with three representative students who took the written test and met the criteria    for each group. The results of the interviews showed that students in the high category still had conceptual ontogenic learning obstacle despite passing the APOS path. Students in the moderate category be able to reach the encapsulation stage but they had not been able to de-encapsulate it to the process. They had conceptual and epistemological ontogenical learning obstacles. Whereas the low category students had the tendency to only reach the action stage and had difficulty in doing initial de-encapsulation due to lack of learning experience in the prerequisite materials. The learning obstacles they experienced were psychological and conceptual-ontogenical learning obstacles. Aggregately, the students tended to experience conceptual ontogenic learning obstacle. The result of this study is expected to be used as a basis for designing a hypothetical learning trajectory in future research.


APOS Theory Case Study Integral Learning Obstacle

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How to Cite
Nurhayati, L., Suryadi, D., Dasari, D., & Herman, T. (2023). Integral (antiderivative) learning with APOS perspective: A case study. Journal on Mathematics Education, 14(1), 129–148.


  1. Adeniji, S. M., & Baker, P. (2022). Worked-examples instruction versus Van Hiele teaching phases: A demonstration of students’ procedural and conceptual understanding. Journal on Mathematics Education, 13(2), 337–356.
  2. Aebi, M. F., & Linde, A. (2015). The epistemological obstacles in comparative criminology: A special issue introduction. In European Journal of Criminology (Vol. 12, Issue 4, pp. 381–385). SAGE Publications Sage UK: London, England.
  3. Afgani, M. W., Suryadi, D., & Dahlan, J. A. (2019). The enhancement of pre-service mathematics teachers’ mathematical understanding ability through ACE teaching cyclic. Journal of Technology and Science Education, 9(2), 153–167.
  4. Albertini, D., Bernardini, A., & Sarti, A. (2021). Antiderivative Antialiasing Techniques in Nonlinear Wave Digital Structures. Journal of the Audio Engineering Society, 69(7/8), 448–464.
  5. Anaya, I. J. J., Leal, J. E. F., & Parada Rico, S. E. (2022). An approach to the Fundamental Theorem of Calculus through Realistic Mathematics. Educacion Matematica, 34(1), 280–305.
  6. Antonio Rivera-Figueroa, V. G.-B. (2019). On conceptual aspects of calculus: a study with engineering students from a Mexican university. International Journal of Mathematical Education in Science and Technology, 50(6), 883–894.
  7. Bajo-Benito, J. M., Sánchez-Matamoros García, G., & Gavilán-Izquierdo, J. M. (2021). The Use of Logical Implication as an Indicator of Understanding the Concept of Number Sequences. Eurasia Journal of Mathematics, Science and Technology Education, 17(12), 1–12.
  8. Baker, B., Cooley, L., & Trigueros, M. (2000). A Calculus Graphing Schema. Journal for Research in Mathematics Education, 31(5), 557–578.
  9. Brousseau, G. (1997). Theory of Didactical situations in mathematics. Dordrecht: Kluwer Academic Publishers.
  10. Brousseau, G. (2002). Epistemological obstacles, problems, and didactical engineering. Theory of Didactical Situations in Mathematics: Didactique Des Mathématiques, 1970–1990, 79–117.
  11. Brown, L. (2008). The incidence of study-related stress in international students in the initial stage of the international sojourn. Journal of Studies in International Education, 12(1), 5–28.
  12. Carter, N., Bryant-Lukosius, D., DiCenso, A., Blythe, J., & Neville, A. J. (2014). The Use of Triangulation in Qualitative Research. Oncology Nursing Forum, 41(5), 545–547.
  13. Díaz-Berrios, T., & Martínez-Planell, R. (2022). High school student understanding of exponential and logarithmic functions. The Journal of Mathematical Behavior, 66(1), 100953–100963.
  14. Dubinsky, E. A. I. O. A. F. S. R. W. K. (2014). Apos theory: A framework for research and curriculum development in mathematics education. Springer.
  15. Edyta Nowinska. (2014). A Cognitive Theory Driven New Orientation of Indonesian Lessons. Journal on Mathematics Education, 5(2), 170–190.
  16. Etikan, I. (2016). Comparison of Convenience Sampling and Purposive Sampling. American Journal of Theoretical and Applied Statistics, 5(1), 1–4.
  17. Feagin, J. R., Orum, A. M., & Sjoberg, G. (1991). A case for the case study. UNC Press Books.
  18. Ferdianto, F., & Hartinah, S. (2020). Analysis of the Difficulty of Students on Visualization Ability Mathematics Based on Learning Obstacles. International Conference on Agriculture, Social Sciences, Education, Technology and Health (ICASSETH 2019), 227–231.
  19. Flyvbjerg, B. (2011). Case study. The Sage Handbook of Qualitative Research, 4, 301–316.
  20. Greco, J., & Turri, J. (2017). Virtue epistemology. In Stanford Encyclopedia of Philosophy.
  21. Huberman, M., & Miles, M. B. (2002). The qualitative researcher’s companion. Sage.
  22. John W. Creswell, J. D. C. (2018). Research Design: Qualitative, Quantitative, and Mixed Methods Approaches (5th ed.). SAGE Publications.
  23. Johnson, P., Almuna, F., & Silva, M. (2022). The role of problem context familiarity in modelling first-order ordinary differential equations. Journal on Mathematics Education, 13(2), 323–336.
  24. Zingiswa, J. (2013). Mathematics begins with direct human experience. An APOS approach to conceptual understanding of a mathematical concept. 602–615.
  25. Kemp, A., & Vidakovic, D. (2023). Students’ understanding and development of the definition of circle in Taxicab and Euclidean geometries: an APOS perspective with schema interaction. Educational Studies in Mathematics, 112(3), 567–588.
  26. Kilpatrick, J. (2001). Understanding mathematical literacy: The contribution of research. Educational Studies in Mathematics, 47(1), 101–116.
  27. Maulida, A. S., Dasari, D., & Suryadi, D. (2020). Pupils Inter-dialogue in the Context of Problem Solving Polyhedron Geometry in Junior High School: Phenomenological Studies. Journal of Physics: Conference Series, 1521(3).
  28. Moru, E. K. (2007). Talking with the literature on epistemological obstacles. For the Learning of Mathematics, 27(3), 34–37.
  29. Moskal, B. M. (2000). Scoring Rubrics: What, When and How? Practical Assessment, Research, and Evaluation, 7, 3.
  30. Nedelcu, D., Malin, T. C., Gillich, G. R., Barbinta, C. I., & Iancu, V. (2020). Displacement and velocity estimation of the earthquake response signals measured with accelerometers. IOP Conference Series: Materials Science and Engineering, 997(1), 012051.
  31. Nguyen, D.-H., & Rebello, N. S. (2011). Students’ difficulties with integration in electricity. Physical Review Special Topics - Physics Education Research, 7(1), 1–11.
  32. Paul, Ernes O. S. J. P. van B. M. B. R. M. L. K. R. M. (2016). The philosophy of mathematics education: Vol. VII (1st ed.). Springer Cham.
  33. Prihandhika, A., Prabawanto, S., Turmudi, T., & Suryadi, D. (2020). Epistemological Obstacles: An Overview of Thinking Process on Derivative Concepts by APOS Theory and Clinical Interview. Journal of Physics: Conference Series, 1521(3).
  34. Sulistiawati, A. S. (2019). Investigating The Learning Obstacle and The Self Confidence of Students College in Material Understanding Ability of Linear Algebra Course. International Journal of Scientific and Technology Research, 8(9), 401–411.
  35. Suriyah, P., Waluya, S. T. B., & Dwijanto, I. R. (2022). Construction Of Mathematics Problem-Based on APOS Theory To Encourage Reflective Abstraction Viewed From Students’ Creative Thinking Profile. Journal of Positive School Psychology, 6(9), 1290–1309.
  36. Suryadi, D. (2013). Didactical design research (DDR) to improve the teaching of mathematics. Far East Journal of Mathematical Education, 10(1), 91–107.
  37. Thomas, G. (2021). How to do your case study. How to Do Your Case Study, 1–320.
  38. Tokgoz, E. (2022). An Analysis of Conceptual Integral Knowledge of STEM Majors. 2022 ASEE Annual Conference & Exposition.
  39. Wahba, E. (2022). Derivation of the differential continuity equation in an introductory engineering fluid mechanics course. International Journal of Mechanical Engineering Education, 50(2), 538–547.
  40. Yin, R. K. (2012). Case study methods. In H. Cooper, P. M. Camic, D. L. Long, A. T. Panter, D. Rindskopf, & K. J. Sher (Eds.), APA handbook of research methods in psychology, Vol. 2. Research designs: Quantitative, qualitative, neuropsychological, and biological (pp. 141–155). American Psychological Association.

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