Analysis of students' errors in drawing function graphs in three-dimensional space based on APOS theory
DOI:
https://doi.org/10.56294/saludcyt20251833Keywords:
Errors, Genetic Decomposition, APOS Theory, Graphics, Three-dimensional SpaceAbstract
This research is aimed to investigate students' conceptual understanding of drawing graphs in three-dimensional space through the application of APOS theory. Additionally, genetic decomposition is designed for graphs to facilitate conceptual understanding and assist lecturers in planning learning activities. A qualitative descriptive method is also adopted and data collection is performed through tests, interviews, and documentation, while the validity techniques use triangulation. The results show that students with high, moderate, and low abilities have an understanding of the action, process, and object stages in drawing graphs in three-dimensional space. However, those with moderate and low abilities lack understanding at the object stage. Some reported errors are challenges in creating graphs, with instances of being erroneously constructed in two-dimensional space. There are also errors in determining the intersection point of lines with Cartesian coordinate axis, as well as in drawing and connecting traces.
References
1. Naidoo J, Paideya V. Exploring the Possibility of Introducing Supplemental Instruction at Secondary School Level. South African J Educ. 2015;35(2):1–10. DOI: https://doi.org/10.15700/saje.v35n2a1022
2. Aziz S Al, Winanda RS. Differential Calculus Mind Map as Output Concept Analysis in Preliminary Research Phase. AIP Conference Proceedings. 2023. p. 1–5. DOI: https://doi.org/10.1063/5.0122341
3. Yerizon, Arnawa IM, Fitriani N, Tajudin NM. Constructing Calculus Concepts through Worksheet Based Problem-based Learning Assisted by GeoGebra Software. HighTech Innov J. 2022;3(3):282–96. DOI: https://doi.org/10.28991/HIJ-2022-03-03-04
4. Hurrell D. Conceptual Knowledge OR Procedural Knowledge or Conceptual Knowledge AND Procedural Knowledge: Why the Conjunction is Important to Teachers. Aust J Teach Educ. 2021;46(2):57–71. DOI: https://doi.org/10.14221/ajte.2021v46n2.4
5. Aniswita, Fauzan A, Armiati. Students’ Conceptual and Procedural Knowledge on Integration: Reflections on Calculus Learning. AIP Conference Proceedings. 2023. p. 1–8. DOI: https://doi.org/10.1063/5.0122392
6. Arnon I, Cottrill J, Dubinsky E, Oktac A, Fuentes SR, Trigueros M, et al. APOS Theory: A Framework for Research and Curriculum Development in Mathematics Education. New York: Springer. 2014. DOI: https://doi.org/10.1007/978-1-4614-7966-6
7. Afgani MW, Suryadi D, Dahlan JA. Analysis of Undergraduate Students’ Mathematical Understanding Ability of the Limit of Function Based on APOS Theory Perspective. Journal of Physics: Conference Series. 2017. p. 1–7. DOI: https://doi.org/10.1088/1742-6596/895/1/012056
8. Mahir N. Conceptual and Procedural Performance of Undergraduate Students in Integration. Int J Math Educ Sci Technol. 2009;40(2):201–11. DOI: https://doi.org/10.1080/00207390802213591
9. Gaisman MT, Martínez-Planell R. Geometrical Representations in the Learning of Two-variable Functions. Educ Stud Math. 2010;73(1):3–19. DOI: https://doi.org/10.1007/s10649-009-9201-5
10. Martínez-Planell R, Gaisman MT. Graphs of Functions of Two Variables: Results from the Design of Instruction. Int J Math Educ Sci Technol. 2013;44(5):663–72. DOI: https://doi.org/10.1080/0020739X.2013.780214
11. Dubinsky E. Reflective Abstraction in Advanced Mathematical Thinking. Netherlands: Kluwer Academic Publishers. 1991. DOI: https://doi.org/10.1007/978-1-4612-3178-3_9
12. Maharaj A. An APOS Analysis of Students’ Understanding of the Concept of a Limit of a Function. Pythagoras. 2010;71:41–52. DOI: https://doi.org/10.4102/pythagoras.v0i71.6
13. Maharaj A. An APOS Analysis of Natural Science Students’ Understanding of Derivatives. South African J Educ. 2013;33(1):1–19. DOI: https://doi.org/10.15700/saje.v33n1a458
14. Ndlovu Z, Brijlall D. Pre-service Mathematics Teachers’ Mental Constructions when Using Cramer’s Rule. South African J Educ. 2019;39(1):1–13. DOI: https://doi.org/10.15700/saje.v39n1a1550
15. Oktaç A, Trigueros M, Romo A. APOS Theory: Connecting Research and Teaching. Learn Math. 2019;39(1):33–7.
16. Siyepu SW. Analysis of Errors in Derivatives of Trigonometric Functions. Int J STEM Educ [Internet]. 2015;2:1–16. DOI: https://doi.org/10.1186/s40594-015-0029-5
17. Orton A. Students’ Understanding of Integration. Educ Stud Math. 1983;14:1–18. DOI: https://doi.org/10.1007/BF00704699
18. Yerizon, Sukestiyarno, Arnellis, Suherman, Hevardani KA. Analysis of Students’ Mental Construction in Understanding the Concept of Partial Derivatives Based on Action-Process-Object-Schema Theory. Nurture. 2024;18(4):735–49. DOI: https://doi.org/10.55951/nurture.v18i4.798
19. Yarman, Dwina F, Murni D, Yerizon, Hevardani KA. Analysis of Concept Construction and Student Errors on the Topic of Double Integral Based on APOS Theory. J Pendidik Mat. 2024;18(3):367–86. DOI: https://doi.org/10.22342/jpm.v18i3.pp367-386
20. Tasman F, Fauzan A. Teacher Pre-knowledge About Realistic Mathematics Education: A Survey on Teachers in KKG Gugus 5 Bayang, Pesisir Selatan. AIP Conference Proceedings. 2023. p. 1–6. DOI: https://doi.org/10.1063/5.0122328
21. Fauzan A, Yanti WT. The Role of Mathematical Cognition-based Learning Design in Improving a Slow Learner Understanding About Numbers. AIP Conference Proceedings. 2023. p. 1–11. DOI: https://doi.org/10.1063/5.0122400
22. Susanti M, Suyanto, Jailani, Retnawati H. Problem-based Learning for Improving Problem-solving and Critical Thinking Skills: A Case on Probability Theory Course. J Educ Learn. 2023;17(4):507–25. DOI: https://doi.org/10.11591/edulearn.v17i4.20866
23. Awoniyi FC, Amponsah KD. Positive Attitudes Toward Mathematics among Senior High School Students in Cape Coast Metropolis. J Educ Learn. 2023;17(2):183–94. DOI: https://doi.org/10.11591/edulearn.v17i2.20531
24. Kariman D, Harisman Y, Sovia A, Prahmana RCI. Effectiveness of Guided Discovery-based Module: A Case Study in Padang City, Indonesia. J Math Educ. 2019;10(2):239–50. DOI: https://doi.org/10.22342/jme.10.2.6610.239-250
25. Suliani M, Juniati D, Lukito A. Mathematics Belief Impact on Metacognition in Solving Geometry: Middle School Students. J Educ Learn. 2024;18(2):286–95. DOI: https://doi.org/10.11591/edulearn.v18i2.21110
26. Rifandi R, Tasman F, Rahmi YL, Gusteti MU. A Need Analysis of Developing STEM Integrated Student Worksheet in Multivariate Calculus Course. AIP Conference Proceedings. 2023. p. 1–6. DOI: https://doi.org/10.1063/5.0122297
27. Harisman Y, Ashari J, Amam A, Harun L. The Analysis Application of the Working Backwards Strategy in Mathematic Problem Solving for Senior High School Students Between Gender. AIP Conference Proceedings. 2023. p. 1–7. DOI: https://doi.org/10.1063/5.0122627
28. Listiawati E, Juniati D. An APOS Analysis of Student’s Understanding of Quadratic Function Graph. Journal of Physics: Conference Series. 2021. p. 1–9. DOI: https://doi.org/10.1088/1742-6596/1747/1/012028
29. Martínez-Planell R, Trigueros M. Using Cycles of Research in APOS: The Case of Functions of Two Variables. J Math Behav. 2019;55:1–22. DOI: https://doi.org/10.1016/j.jmathb.2019.01.003
30. Bansilal S, Brijlall D, Trigueros M. An APOS Study on Pre-service Teachers’ Understanding of Injections and Surjections. J Math Behav. 2017;48:22–37. DOI: https://doi.org/10.1016/j.jmathb.2017.08.002
31. Şefik Ö, Dost Ş. The Analysis of the Understanding of the Three-dimensional (Euclidian) Space and the Two-variable Function Concept by University Students. J Math Behav. 2020;57:1–19. DOI: https://doi.org/10.1016/j.jmathb.2019.03.004
32. Díaz-Berrios T, Martínez-Planell R. High School Student Understanding of Exponential and Logarithmic Functions. J Math Behav. 2022;66:1–20. DOI: https://doi.org/10.1016/j.jmathb.2022.100953
33. Harel G. The Learning and Teaching of Linear Algebra: Observations and Generalizations. J Math Behav. 2017;46:69–95. DOI: https://doi.org/10.1016/j.jmathb.2017.02.007
34. Nagle C, Martínez-Planell R, Moore-Russo D. Using APOS Theory as a Framework for Considering Slope Understanding. J Math Behav. 2019;54:1–14.
35. Baye MG, Ayele MA, Wondimuneh TE. Implementing GeoGebra Integrated with Multi-teaching Approaches Guided by the APOS Theory to Enhance Students’ Conceptual Understanding of Limit in Ethiopian Universities. Heliyon. 2021;7:1–13. DOI: https://doi.org/10.1016/j.heliyon.2021.e07012
36. Martínez-Planell R, Trigueros M. Students’ Understanding of Riemann Sums for Integrals of Functions of Two Variables. J Math Behav. 2020;59:1–26. DOI: https://doi.org/10.1016/j.jmathb.2020.100791
37. Borji V, Martínez-Planell R. On Students’ Understanding of Volumes of Solids of Revolution: An APOS Analysis. J Math Behav. 2023;70:1–20. DOI: https://doi.org/10.1016/j.jmathb.2022.101027
38. Figueroa AP, Possani E, Trigueros M. Matrix Multiplication and Transformations: An APOS Approach. J Math Behav. 2017;52:77–91. DOI: https://doi.org/10.1016/j.jmathb.2017.11.002
39. García-Martínez I, Parraguez M. The Basis Step in the Construction of the Principle of Mathematical Induction Based on APOS Theory. J Math Behav. 2017;46:128–43. DOI: https://doi.org/10.1016/j.jmathb.2017.04.001
40. Umam K, Susandi D. Critical Thinking Skills: Error Identifications on Students’ with APOS Theory. Int J Eval Res Educ. 2022;11(1):182–92. DOI: https://doi.org/10.11591/ijere.v11i1.21171
41. Tatira B. Mathematics Education Students’ Understanding of Binomial Series Expansion Based on the APOS Theory. Eurasia J Math Sci Technol Educ. 2021;17(12):1–13. DOI: https://doi.org/10.29333/ejmste/11287
42. Arnawa IM, Yerizon, Nita S, Putra RT. Development of Students’ Worksheet Based on APOS Theory Approach to Improve Student Achievment in Learning System of Linear Equations. Int J Sci Technol Res. 2019;8(4):287–92.
43. Nagle C, Martínez-Planell R, Moore-Russo D. Using APOS theory as a framework for considering slope understanding. J Math Behav. 2019;54:1-14. DOI: https://doi.org/10.1016/j.jmathb.2018.12.003
44. Arnawa IM, Yanita, Yerizon, Ginting B, Nita S. Does the use of APOS theory promote students’ achievement in elementary linear algebra? Int J Instr. 2021;14(3):175–86. DOI: https://doi.org/10.29333/iji.2021.14310a
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Copyright (c) 2025 Yerizon, I Made Arnawa, Arnellis, Fridgo Tasman, Yarman, Kelly Angelly Hevardani (Author)
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