Analysis of Concept Construction and Student Errors on the Topic of Double Integral Based on APOS Theory

Yarman Yarman; Yerizon Yerizon; Fitrani Dwina; Dewi Murni; Kelly Angelly Hevardani

doi:10.22342/jpm.v18i3.pp367-386

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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

APOS Theory Double Integral Erros Genetic Decomposition

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Yarman, Y., Yerizon, Y., Dwina, F., Murni, D., & 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

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References

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Borji, V., & Font, V. (2019). Exploring Students’ Understanding of Integration by Parts: A Combined Use of APOS and OSA. EURASIA: Journal of Mathematics, Science and Technology Education, 15(7), 1–13. https://doi.org/10.29333/ejmste/106166
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Dominguez, A., Barniol, P., & Zavala, G. (2017). Test of Understanding Graphs in Calculus: Test of Students’ Interpretation of Calculus Graphs. EURASIA Journal of Mathematics, Science and Technology Education, 13(10), 6507–6531. https://doi.org/10.12973/ejmste/78085
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
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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
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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
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References

Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Fuentes, S. R., Trigueros, M., & Weller, K. (2014). APOS Theory: A Framework for Research and Curriculum Development in Mathematics Education. New York: Springer. http://dx.doi.org/10.1007/978-1-4614-7966-6

Artigue, M. (1991). Analysis in Advanced Mathematical Thinking.

Avgerinos, E., & Remoundou, D. (2021). The Language of Rate of Change in Mathematics. European Journal of Investigation in Health, Psychology and Education, 11, 1599–1609. https://doi.org/10.3390/ejihpe11040113

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

Bansilal, S., & Mkhwanazi, T. W. (2021). Pre-service Student Teachers’ Conceptions of the Notion of Limit. International Journal of Mathematical Education in Science and Technology, 1–19. https://doi.org/10.1080/0020739X.2020.1864488

Bansilal, S., Brijlall, D., & Trigueros, M. (2017). An APOS Study on Pre-service Teachers’ Understanding of Injections and Surjections. Journal of Mathematical Behavior, 48, 22–37. https://doi.org/10.1016/j.jmathb.2017.08.002

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, 1–13. https://doi.org/10.1016/j.heliyon.2021.e07012

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

Borji, V., & Font, V. (2019). Exploring Students’ Understanding of Integration by Parts: A Combined Use of APOS and OSA. EURASIA: Journal of Mathematics, Science and Technology Education, 15(7), 1–13. https://doi.org/10.29333/ejmste/106166

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

Borji, V., Alamolhodaei, H., & Radmehr, F. (2018). Application of the APOS-ACE Theory to Improve Students’ Graphical Understanding of Derivative. EURASIA Journal of Mathematics, Science and Technology Education, 14(7), 2947–2967. https://doi.org/10.29333/ejmste/91451

Bressoud, D. M. (2021). The Strange Role of Calculus in the United States. ZDM–Mathematics Education, 53(3), 521–533. https://doi.org/10.1007/s11858-020-01188-0

Bressoud, D., Burn, H., Hsu, E., Mesa, W., Rasmussen, C., & White, N. (2014). Successful Calculus Programs: Two-year Colleges to Research Universities. USA: NCTM.

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

Chapell, K. K., & Kilpartrick, K. (2003). Effects of Concept-based Instruction on Students’ Conceptual Understanding and Procedural Knowledge of Calculus. PRIMUS, 13(1), 17–37. https://doi.org/10.1080/10511970308984043

Cline, K., Parker, M., Zullo, H., & Stewart, A. (2013). Addressing Common Student Errors with Classroom Voting in Multivariable Calculus. PRIMUS, 23(1), 60–75. https://doi.org/10.1080/10511970.2012.697098

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

Dominguez, A., Barniol, P., & Zavala, G. (2017). Test of Understanding Graphs in Calculus: Test of Students’ Interpretation of Calculus Graphs. EURASIA Journal of Mathematics, Science and Technology Education, 13(10), 6507–6531. https://doi.org/10.12973/ejmste/78085

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

Marsitin, R. (2017). Koneksi Matematis dan Berpikir Kreatif dalam Pembelajaran Matematika dengan Teori APOS [Mathematical Connection and Creative Thinking in Mathematics Learning with APOS Theory]. Al-Khwarizmi: Jurnal Pendidikan Matematika dan Ilmu Pengetahuan Alam, 5(1), 87–100. https://doi.org/10.24256/jpmipa.v5i1.268

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

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Analysis of Concept Construction and Student Errors on the Topic of Double Integral Based on APOS Theory

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