Main Article Content
Abstract
Integral learning is particularly challenging for students, primarily due to misconceptions predominantly caused by students' lack of understanding about functions, limits, and derivatives. Therefore, this research aims to investigate students’ thinking processes when solving double integral using Action, Process, Object, and Schema (APOS) theory, with a focus on past errors. In order to achieve the objective, a descriptive qualitative method was adopted. Data was collected from tests, interviews, and relevant documentation, and tested for validity using triangulation methods. The obtained results showed that high-ability students understood APOS stages in solving double integral. However, at the object stage, a lack of thoroughness in simplifying algebra led to misunderstandings. Medium-Ability Student (MS) was observed to successfully reach APOS stages when solving double integral using polar coordinates. Low-Ability Student (LS), on the other hand, showed inadequate understanding at the process stage, as evidenced by the failure to correctly draw the area and set integral boundaries. During the course of this investigation, process errors were found to be commonly associated with the calculations of double integral. In order to address these issues, Genetic Decomposition (GD) should be designed for other calculus topics, and error classification expanded to enhance the effectiveness of lectures.
Keywords
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Artigue, M. (1991). Analysis in Advanced Mathematical Thinking.
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Bangaru, S. P., Michel, J., Mu, K., Bernstein, G., Li, T. M., & Ragan-Kelley, J. (2021). Systematically Differentiating Parametric Discontinuities. ACM Transactions on Graphics, 40(4), 1–18. https://doi.org/10.1145/3450626.3459775
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Bikner-Ahsbahs, A., & Prediger, S. (2014). Networking of Theories as a Research Practice in Mathematics Education. Switzerland: Springer. https://doi.org/10.1007/978-3-319-05389-9
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Borji, V., & Martínez-Planell, R. (2023). On Students’ Understanding of Volumes of Solids of Revolution: An APOS Analysis. Journal of Mathematical Behavior, 70, 1–20. https://doi.org/10.1016/j.jmathb.2022.101027
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Brijlall, D., & Ndlazi, N. J. (2019). Analyzing Engineering Students’ Understanding of Integration to Propose a Genetic Decomposition. Journal of Mathematical Behavior, 55, 1–12. https://doi.org/10.1016/j.jmathb.2019.01.006
Burgos, M., Bueno, S., Godino, J. D., & Pérez, O. (2021). Onto-semiotic Complexity of the Definite Integral: Implications for Teaching and Learning Calculus. REDIMAT – Journal of Research in Mathematics Education, 10(1), 4–40. https://doi.org/10.17583/redimat.2021.6778
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Díaz-Berrios, T., & Martínez-Planell, R. (2022). High School Student Understanding of Exponential and Logarithmic Functions. Journal of Mathematical Behavior, 66, 1–20. https://doi.org/10.1016/j.jmathb.2022.100953
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Dubinsky, E. (2013). Using a Theory of Learning in College Mathematics Courses. MSOR Connections, 1(2), 10–15.
Ergene, Ö. & Özdemir, A. Ş. (2020). Investigating Pre-service Elementary Mathematics Teachers’ Perception of Integral. Journal of Educational Sciences, 51(51), 155–176. https://doi.org/10.15285/maruaebd.622149
Ergene, Ö. (2014). Investigation of Personal Relationship in Integral Volume Problems Solving Process within Communities of Practices. Dissertation. Marmara University.
Ergene, Ö., & Özdemir, A. Ş. (2022). Understanding the Definite Integral with the Help of Riemann Sums. Participatory Educational Research, 9(3), 445–465. https://doi.org/10.17275/per.22.75.9.3
Fernandez, A., & Mohammed, P. (2021). Hermite‐Hadamard Inequalities in Fractional Calculus Defined Using Mittag‐Leffler Kernels. Mathematical Methods in the Applied Sciences, 44(10), 8414–8431. https://doi.org/10.1002/mma.6188
Ferrini-Mundi, J., & Graham, K. (1994). Research in Calculus Learning: Understanding of Limits, Derivatives and Integrals. In J. J. Kaput & E. Dubinsky (eds.), Research Issues in Undergraduate Mathematics Learning, (pp.31–45). Washington DC: MAA.
Figueroa, A. P., Possani, E., & Trigueros, M. (2017). Matrix Multiplication and Transformations: An APOS Approach. Journal of Mathematical Behavior, 1–15. https://doi.org/10.1016/j.jmathb.2017.11.002
Font, V., Trigueros, M., Badillo, E., & Rubio, N. (2016). Mathematical Objects through the Lens of Two Different Theoretical Perspectives: APOS and OSA. Educational Studies in Mathematics, 91(1), 107–122. https://doi.org/10.1007/s10649-015-9639-6
Frank, K., & Thompson, P. W. (2021). School Students’ Preparation for Calculus in the United States. ZDM–Mathematics Education, 53(3), 549–562. https://doi.org/10.1007/s11858-021-01231-8
García-Martínez, I., & Parraguez, M. (2017). The Basis Step in the Construction of the Principle of Mathematical Induction Based on APOS Theory. Journal of Mathematical Behavior, 46, 128–143. https://doi.org/10.1016/j.jmathb.2017.04.001
Harel, G. (2017). The Learning and Teaching of Linear Algebra: Observations and Generalizations. The Journal of Mathematical Behavior, 46, 69–95. https://doi.org/10.1016/j.jmathb.2017.02.007
Hong, D. S., Choi, K. M., Hwang, J., & Runnalls, C. (2017). Integral Students’ Experiences: Measuring Instructional Quality and Instructors’ Challenges in Calculus 1 Lessons. International Journal of Research in Education and Science, 3(2), 424–437. https://doi.org/10.21890/ijres.327901
Kiat, S. E. (2005). Analysis of Students’ Difficulties in Solving Integration Problems. The Mathematics Educator, 9(1), 39–59.
Lam, T. T., Guan, T. E., & Luen, T. C. (2021). Fallacies About the Derivative of the Trigonometric Sine Function. The Mathematician Educator, 2(1), 1–10.
Li, V. L., Julaihi, N. H., & Eng, T. H. (2017). Misconceptions and Errors in Learning Integral Calculus. Asian Journal of University Education, 13(1), 17–39. https://ir.uitm.edu.my/id/eprint/21914
Maharaj, A. (2014). An APOS Analysis of Natural Science Students’ Understanding of Integration. REDIMAT – Journal of Research in Mathematics Education, 3(1), 54–73. https://doi.org/10.4471/redimat.2014.40
Mahir N. (2009). Conceptual and Procedural Performance of Undergraduate Students in Integration. International Journal of Mathematical Education in Science and Technology, 40(2), 201–211. https://doi.org/10.1080/00207390802213591
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Martínez-Planell, R., & Trigueros, M. (2019). Using Cycles of Research in APOS: The Case of Functions of Two Variables. Journal of Mathematical Behavior, 55, 1–22. https://doi.org/10.1016/j.jmathb.2019.01.003
Martínez-Planell, R., & Trigueros, M. (2020). Students’ Understanding of Riemann Sums for Integrals of Functions of Two Variables. Journal of Mathematical Behavior, 59, 1–26. https://doi.org/10.1016/j.jmathb.2020.100791
Martínez-Planell, R., & Trigueros, M. (2021). Multivariable Calculus Results in Different Countries. ZDM – Mathematics Education, 53, 695–707. https://doi.org/10.1007/s11858-021-01233-6
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