Main Article Content

Abstract

In solid geometry, the concept of conic sections plays an important role in teaching graphs such as parabolas, ellipses, and hyperbolas to undergraduate students in Mathematics Education. It is understood that the abstraction process in mastering conic sections is strongly needed. This study examines the abstraction process of conic sections among third-year undergraduate Mathematics Education students (4 males and 21 females) at Universitas Muhammadiyah Malang (UMM), Indonesia. The data was collected by analyzing students' responses in a 60-minute diagnostic test using the Abstraction in Context (AiC) framework. The test consists of 3 questions, validated by 2 Professors of UMM (average score = 4.14) and 2 lecturers (average score = 4.04). The results showed that 1 male and 11 female students did not reach the construction stage of AiC. Subsequently, a student with a low diagnostic test score and the least completion of AiC stages was observed further through an interview. This student passed through all stages of abstraction with the help of DGE. We also underscored undergraduates' challenges in this process, particularly in visualizing conic section objects, spatial thinking, and employing appropriate mathematical signs. Based on these findings, further research with a broader sample is needed to explore diverse abstraction processes.

Keywords

Abstraction Conic Section Difficulties Dynamic Geometry Environment Undergraduate Students

Article Details

How to Cite
Dintarini, M., Fuad, Y., & Budiarto, M. T. (2024). Examining undergraduate students’ abstraction of conic sections in a dynamic geometry environment. Journal on Mathematics Education, 15(3), 717–734. https://doi.org/10.22342/jme.v15i3.pp717-734

References

  1. Breive, S. (2022). Abstraction and Embodiment: Exploring the Process of Grasping a General. Educational Studies in Mathematics, 110(2), 313–329. https://doi.org/10.1007/s10649-021-10137-x
  2. Budiarto, M. T., Khabibah, S., & Setianingsih, R. (2017). Construction of High School Students’ Abstraction Levels in Understanding the Concept of Quadrilaterals. International Education Studies, 10(2), 148. https://doi.org/10.5539/ies.v10n2p148
  3. Bueno, R. W. da S., Henriques, A., & Galle, L. A. V. (2023). Mathematics Preservice Teachers ’ Perceptions Regarding ICT Use in Teaching and Learning Practices. Acta Scientiae, 25(5), 227–249. https://doi.org/10.17648/acta.scientiae.7633
  4. Centina, A. Del. (2016). On Kepler ’ s System of Conics in Astronomiae Pars Optica. Archive for History of Exact Sciences, 70, 567–589. https://doi.org/10.1007/s00407-016-0175-2
  5. Dewi, I., Siregar, N., & Andriani, A. (2018). The analysis of junior high school students’ mathematical abstraction ability based on local cultural wisdom. Journal of Physics: Conference Series, 1088(1), 012076. https://doi.org/10.1088/1742-6596/1088/1/012076
  6. Dintarini, M., & Zukhrufurrohmah. (2021). Analisis Pemberian Scaffolding Melalui Online Form untuk Mengatasi Kesulitan Berpikir Spasial. ẟELT∆ Jurnal Ilmiah Pendidikan Matematika, 9(2), 221–232. http://dx.doi.org/10.31941/delta.v9i2.1414
  7. Dreyfus, T., Hershkowitz, R., & Schwarz, B. (2015). The Nested Epistemic Actions Model for Abstraction in Context: Theory as Methodological Tool and Methodological Tool as Theory. In A. B.-A. et Al. (Ed.), Approaches to Qualitative Research in Mathematics Education, Advances in Mathematics Education (pp. 185–2017). Springer Science+Business Media Dordrecht. https://doi.org/10.1007/978-94-017-9181-6_8
  8. Dubinsky, E., & Tall, D. (1991). Advanced mathematical thinking and the computer. Advanced Mathematical Thinking, 8, 231–248. https://doi.org/10.1007/0-306-47203-1_14
  9. Elgrably, H., & Leikin, R. (2021). Creativity as a function of problem-solving expertise: posing new problems through investigations. ZDM - Mathematics Education, 53(4), 891–904. https://doi.org/10.1007/s11858-021-01228-3
  10. Florio, E. (2021). Claude Mydorge Reader and Interpreter of Apollonius ’ Conics. Mathematics, 9, 261. https://doi.org/10.3390/math9030261
  11. Florio, E. (2022). The Parabola: Section of a Cone or Locus of Points of a Plane? Tips for Teaching of Geometry from Some Writings by Mydorge and Wallis. Mathematics, 10(6), 974–992. https://doi.org/10.3390/math10060974
  12. Fried, M. N. (2003). The Use of Analogy in Book VII of Apollonius ’ Conica The Use of Analogy in Book VII of Apollonius ’ Conica. Science in Context, 16(3), 349–365. https://doi.org/10.1017/S026988970300084X
  13. Fried, M. N. (2022). Edmond Halley and Apollonius : second ‑ order historical knowledge in mathematics education. ZDM – Mathematics Education, 0123456789. https://doi.org/10.1007/s11858-022-01391-1
  14. Glaister, E. M., & Glaister, P. (2006). Introducing conics without eccentricity. International Journal of Mathematical Education in Science and Technology, 37(2), 235–245. https://doi.org/10.1080/00207390500285800
  15. Gray, E. M., & Tall, D. O. (1992). Success and Failure in Mathematics: Procept and Procedure 1. A Primary Perspective. Workshop on Mathematics Education and Computers, April, 216–221.
  16. Gray, E. M., & Tall, D. O. (2007). Abstraction as a natural process of mental compression. Mathematics Education Research Journal, 19(2), 23–40. https://doi.org/10.1007/BF03217454
  17. Hartinah, S., Suherman, S., Syazali, M., Efendi, H., Junaidi, R., Jermsittiparsert, K., & Umam, R. (2019). Probing-prompting based on ethnomathematics learning model: The effect on mathematical communication skills. Journal for the Education of Gifted Young Scientists, 7(4), 799–814. https://doi.org/10.17478/jegys.574275
  18. Heath, T. (1896). Apollonius of Perga: Treatise on Conic Sections. C. J. CLAY AND SONS.
  19. Hershkowitz, R., Hadas, N., Dreyfus, T., & Schwarz, B. B. (2007). Abstracting Processes, from Individuals’ Constructing of Knowledge to a Group’s “Shared Knowledge.” Mathematics Education Research Journal, 19(2), 41–68. https://doi.org/10.1007/BF03217455
  20. Hershkowitz, R., Schwarz, B. B., & Dreyfus, T. (2001). Abstraction in Context: Epistemic Actions. Journal for Research in Mathematics Education, 32(2), 195–222. https://doi.org/10.1093/ojls/14.2.255
  21. Hodiyanto, Budiarto, M. T., Ekawati, R., Susanti, G., Kim, J., & Bonyah, E. (2024). How abstraction of a pre-service teacher in constructing relationships among quadrilaterals. Journal on Mathematics Education, 15(2), 339–362. https://doi.org/10.22342/jme.v15i2.pp339-362
  22. Hollebrands, K., & Stohl Lee, H. (2011). Introduction to dynamic geometry environments. Kendall Hunt Online, 1–22.
  23. Jupri, A. (2017). From geometry to algebra and vice versa: Realistic mathematics education principles for analyzing geometry tasks. AIP Conference Proceedings, 1830. https://doi.org/10.1063/1.4980938
  24. Jupri, Al, & Drijvers, P. (2016). Student Difficulties in Mathematizing Word Problems in Algebra. Eurasia Journal of Mathematics, Science & Technology Education, 12(9), 2481–2502. https://doi.org/10.12973/eurasia.2016.1299a
  25. Komala, E. (2018). Analysis of Students’ Mathematical Abstraction Ability By Using Discursive Approach Integrated Peer Instruction of Structure Algebra Ii. Infinity Journal, 7(1), 25-34. https://doi.org/10.22460/infinity.v7i1.p25-34
  26. Kouropatov, A., & Dreyfus, T. (2014). Learning the integral concept by constructing knowledge about accumulation. ZDM - Mathematics Education, 46(4), 533–548. https://doi.org/10.1007/s11858-014-0571-5
  27. Memnun, D. S., Aydın, B., Özbilen, Ö., & Erdoğan, G. (2017). The abstraction process of limit knowledge. Kuram ve Uygulamada Egitim Bilimleri, 17(2), 345–371. https://doi.org/10.12738/estp.2017.2.0404
  28. Memnun, D. S., Sevindik, F., Beklen, C., & Dinç, E. (2019). Analysis of the Abstraction Process of Continuity Knowledge. World Journal of Education, 9(2), 141. https://doi.org/10.5430/wje.v9n2p141
  29. Mitchelmore, M. C., & White, P. (2004). Abstraction in mathematics learning. Mathematics Education Research Journal, 19(2), 1–9. https://doi.org/10.1007/BF03217452
  30. Munson, J. (2019). After eliciting : Variation in elementary mathematics teachers ’ discursive pathways during collaborative problem solving. Journal of Mathematical Behavior, August, 100736. https://doi.org/10.1016/j.jmathb.2019.100736
  31. Ng, O. L., Shi, L., & Ting, F. (2020). Exploring differences in primary students’ geometry learning outcomes in two technology-enhanced environments: dynamic geometry and 3D printing. International Journal of STEM Education, 7(1), 1–13. https://doi.org/10.1186/s40594-020-00244-1
  32. Nurhasanah, F., Kusumah, Y. S., Sabandar, J., & Suryadi, D. (2017). Mathematical Abstraction: Constructing Concept of Parallel Coordinates. Journal of Physics: Conference Series, 895(1), 012076. https://doi.org/10.1088/1742-6596/895/1/012076
  33. Ozmantar, M. F. (2005). An investigation of the formation of mathematical abstractions through scaffolding. Univeristy of Leeds, July.
  34. Ozmantar, M. F., & Monaghan, J. (2007). A Dialectical Approach to the Formation of Mathematical Abstractions. Mathematics Education Research Journal, 19(2), 89–112. https://doi.org/10.1007/BF03217457
  35. Pegg, J., & Tall, D. (2005). The fundamental cycle of concept construction underlying various theoretical frameworks. ZDM - International Journal on Mathematics Education, 37(6), 468–475. https://doi.org/10.1007/BF02655855
  36. Priatna, N., Martadiputra, B. A. P., & Wibisono, Y. (2018). Developing geogebra-assisted reciprocal teaching strategy to improve junior high school students’ abstraction ability, lateral thinking and mathematical persistence. Journal of Physics: Conference Series, 1013(1), 012142. https://doi.org/10.1088/1742-6596/1013/1/012142
  37. Putra, Z. H., Afrillia, Y. M., Dahnilsyah, & Tjoe, H. (2023). Prospective elementary teachers ’ informal mathematical proof using GeoGebra : The case of 3D shapes. Journal on Mathematics Education, 14(3), 449–468. https://doi.org/10.22342/jme.v14i3.pp449-468
  38. Rich, K. M., & Yadav, A. (2020). Applying Levels of Abstraction to Mathematics Word Problems. TechTrends, 64(3), 395–403. https://doi.org/10.1007/s11528-020-00479-3
  39. Rich, K. M., Yadav, A., & Zhu, M. (2019). Levels of abstraction in students’ mathematics strategies: What can applying computer science ideas about abstraction bring to elementary mathematics ? Journal of Computers in Mathematics and Science Teaching, 38(3), 267–298. https://www.learntechlib.org/p/210229/
  40. Ron, G., Dreyfus, T., & Hershkowitz, R. (2010). Partially correct constructs illuminate students ’ inconsistent answers. Educational Studies in Mathematics, 75, 65–87. https://doi.org/10.1007/s10649-010-9241-x
  41. Ron, G., Dreyfus, T., & Hershkowitz, R. (2017). Looking back to the roots of partially correct constructs: The case of the area model in probability. Journal of Mathematical Behavior, 45, 15–34. https://doi.org/10.1016/j.jmathb.2016.10.004
  42. Salinas, P., & Pulido, R. (2017). Understanding the Conics through Augmented Reality. Eurasia Journal of Mathematics, Science and Technology Education, 13(2), 341–354. https://doi.org/10.12973/eurasia.2017.00620a
  43. Sfard, A. (1991). On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin. Educational Studies in Mathematics, 22(1), 1–36. https://doi.org/10.1007/BF00302715
  44. Skemp, R. R. (1987). The Psychology of learning mathematics. In The Psychology of Learning Mathematics: Expanded American Edition (pp. 1–218). Lawrence Erlbaum Associates. https://doi.org/10.4324/9780203396391
  45. Turgut, M. (2019). Sense-Making Regarding Matrix Representation of Geometric Transformations in R2 : A semiotic mediation perspective in a dynamic geometry environment. ZDM - Mathematics Education, 51(7), 1199–1214. https://doi.org/10.1007/s11858-019-01032-0
  46. Yao, X., & Manouchehri, A. (2019). Middle school students’ generalizations about properties of geometric transformations in a dynamic geometry environment. Journal of Mathematical Behavior, 55(September 2018), 100703. https://doi.org/10.1016/j.jmathb.2019.04.002

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