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Abstract
Spatial thinking plays an important role in solving conic section problems. When students encounter problems involving shapes like parabolas, ellipses, and hyperbolas, they need not only grasp the algebraic representations but also visualize how a plane intersects a cone in three-dimensional space. Unlike previous studies that focused on psychometric tasks disconnected from classroom content, this study integrates spatial thinking directly into mathematical problem-solving, specifically in conic sections. The study examines undergraduate students' spatial thinking skills on conic sections, based on spatial thinking categories and sex differences. Twenty-five undergraduate students (4 males and 21 females) from the Mathematics Education Department at Universitas Muhammadiyah Malang, enrolled in an analytical geometry course and participated in a spatial thinking test. Additionally, four students (2 males and 2 females), representing both high and low spatial thinking abilities, were interviewed to provide deeper insights into students' spatial thinking. The results show that students with high spatial thinking abilities demonstrated strong mental visualization skills but had minor difficulties and errors in representing detailed components. Moreover, students with lower spatial thinking abilities have difficulties in visualizing complex objects and often misinterpret spatial representations. Furthermore, male students provided limited written explanations of spatial object characteristics, whereas female students faced difficulties in accurately drawing 3D spatial objects but compensated with effective 2D representations and verbal explanations. In conclusion, the findings show that the difficulties of spatial thinking are influenced by spatial thinking categories and gender differences, offering rich information to design more effective mathematics learning.
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Copyright (c) 2025 Mayang Dintarini, Mega Teguh Budiarto, Yusuf Fuad
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References
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- Forndran, A., Lowrie, T., & Harris, D. (2019). Strategy Use in Mathematics Assessment: Does Spatial Reasoning Matter? In G. Hine, S. Blackley, & A. Cooke (Eds.), Mathematics Education Research: Impacting Practice: Proceedings of the 42nd annual conference of the Mathematics Education Research Group of Australasia (pp. 276–283). MERGA. https://researchprofiles.canberra.edu.au/en/publications/strategy-use-in-mathematics-assessment-does-spatial-reasoning-mat
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- Hasanah, U., Putrawangsa, S., Purwanta, E., Setiawati, F. A., & Purnomo, Y. W. (2024). Primary Student Spatial Reasoning Abilities: Progression and Challenges. Mathematics Education Journal, 18(3), 329–348. https://doi.org/10.22342/jpm.v18i3.pp329-348
- Hegarty, M. (2010). Components of Spatial Intelligence. In B. H. Ross (Ed.), Psychology of Learning and Motivation (pp. 265–297). Elsevier Inc. https://doi.org/10.1016/S0079-7421(10)52007-3
- İbili, E., Çat, M., Resnyansky, D., Şahin, S., & Billinghurst, M. (2019). An Assessment of Geometry Teaching Supported with Augmented Reality Teaching Materials to Enhance Students ’ 3D Geometry Thinking Skills. International Journal of Mathematical Education in Science and Technology, 51(2), 224–246. https://doi.org/10.1080/0020739X.2019.1583382
- Linn, M. C., & Petersen, A. C. (1985). Emergence and Characterization of Sex Differences in Spatial Ability : A Meta-Analysis. Child Development, 56(6), 1479–1498. http://www.jstor.org/stable/1130467 .
- Lowrie, T., Harris, D., Logan, T., & Hegarty, M. (2018). The Impact of a Spatial Intervention Program on Students’ Spatial Reasoning and Mathematics Performance. Journal of Experimental Education, 89(2), 259–277. https://doi.org/10.1080/00220973.2019.1684869
- Lowrie, T., Resnick, I., Harris, D., & Logan, T. (2020). In search of the mechanisms that enable transfer from spatial reasoning to mathematics understanding. Mathematics Education Research Journal, 32(2), 175–188. https://doi.org/10.1007/s13394-020-00336-9
- Miles, M. B., Huberman, A. M., & Saldaña, J. (2014). Qualitative Data Analysis: A Methods Sourcebook (3rd ed.). SAGE Publications Inc.
- Nagy-Kondor, R. (2017). Spatial Ability : Measurement and Development Definition of Spatial Ability. In M.S. Khine (Ed.), Visual-spatial Ability in STEM Education (pp. 35–58). Springer International Publishing Switzerland. https://doi.org/10.1007/978-3-319-44385-0
- Newcombe, N. S., & Shipley, T. F. (2015). Studying Visual and Spatial Reasoning for Design Creativity. Studying Visual and Spatial Reasoning for Design Creativity. https://doi.org/10.1007/978-94-017-9297-4
- Pavlovičová, G., Bočková, V., & Laššová, K. (2022). Spatial Ability and Geometric Thinking of the Students of Teacher Training for Primary Education. TEM Journal, 11(1), 388–395. https://doi.org/10.18421/TEM111-49
- Pozas, M., Letzel, V., & Schneider, C. (2020). Teachers and differentiated instruction : exploring differentiation practices to address student diversity. Journal of Research in Special Educational Needs, 20, 217–230. https://doi.org/10.1111/1471-3802.12481
- Pradana, L. N., & Sholikhah, O. H. (2023). Spatial Reasoning Construction: The Way to Use It to Solve Geometric Problems. Jurnal Pendidikan Matematika, 17(2), 209–224.
- Reilly, D., Neumann, D. L., & Andrews, G. (2017). Gender Differences in Spatial Ability: Implications for STEM Education and Approaches to Reducing the Gender Gap for Parents and Educators. In M. S. Khine (Ed.), Visual-spatial Ability in STEM Education (p. 3). Springer International Publishing Switzerland.
- 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
- Sorby, S. (2016). Developing 3D spatial skills for engineering students Developing 3D spatial skills for engineering students *. Australasian Journal of Engineering Education, 13(1).
- The National Research Council. (2006). Learning To Think Spatially. The National Academic Press.
- Unal, H., Jakubowski, E., & Corey, D. (2009). Differences in learning geometry among high and low spatial ability pre-service mathematics teachers. International Journal of Mathematical Education in Science and Technology, 40(8), 997–1012. https://doi.org/10.1080/00207390902912852
- Utomo, D. P., & Syarifah, D. L. (2021). Examining mathematical representation to solve problems in trends in mathematics and science study: Voices from Indonesian secondary school students. International Journal of Education in Mathematics, Science and Technology, 9(3), 540–556. https://doi.org/10.46328/IJEMST.1685
- Velez, M. C., Silver, D., & Tremaine, M. (2005). Understanding Visualization through Spatial Ability Differences. IEEE Visualization, 511–518.
- Wei, W., Chen, C., Zhou, X., & Bisiacchi, P. S. (2016). Spatial Ability Explains the Male Advantage in Approximate Arithmetic. Frontiers in Psychology, 7(March), 1–9. https://doi.org/10.3389/fpsyg.2016.00306
- Zhou, Y., Yi, F., Dong, B., Zhang, G., Zhang, Y., & Xu, T. (2024). Mental imagery scaffolding: The effects of detail richness and text load on geography learning. Education and Information Technologies, 16929–16956. https://doi.org/10.1007/s10639-024-12540-2
References
Albarracín, L., Hernàndez-Sabaté, A., & Gorgorió, N. (2021). Designing Levels of a Video Game to Promote Spatial Thinking. International Journal of Mathematical Education in Science and Technology, 53(11), 3138–3150. https://doi.org/10.1080/0020739X.2021.1953626
Atit, K., Power, J. R., Pigott, T., Lee, J., Geer, E. A., Uttal, D. H., Ganley, C. M., & Sorby, S. A. (2022). Examining the Relations Between Spatial Skills and Mathematical Performance : A Meta-analysis. Psychonomic Bulletin & Review, 29, 699–720.
Atit, K., Uttal, D. H., & Stieff, M. (2020). Situating Space: Using a Discipline-focused Lens to Examine Spatial Thinking Skills. In Cognitive Research: Principles and Implications (Vol. 5, Issue 19, pp. 1–16). https://doi.org/10.1186/s41235-020-00210-z
Baiduri, B., Faluthi, N. R., & Ummah, S. K. (2024). Student’S Visual Thinking Ability in Learning Linear Programs Using Geogebra Assisted Liveworksheets. Prima: Jurnal Pendidikan Matematika, 8(1), 131. https://doi.org/10.31000/prima.v8i1.10320
Bintoro, H. S. (2021). The Spatial Thinking Process of the Field-Independent Students based on Action-Process-Object-Schema Theory. European Journal of Educational Research, 10(4), 1807–1823. https://doi.org/10.12973/EU-JER.10.4.1807
Buckley, J., Seery, N., & Canty, D. (2019). Investigating the use of spatial reasoning strategies in geometric problem solving. International Journal of Technology and Design Education, 29(2), 341–362. https://doi.org/10.1007/s10798-018-9446-3
Campos, J. S. G., Sánchez-Navarro, J., & Arnedo-Moreno, J. (2019). An empirical study of the effect that a computer graphics course has on visual-spatial abilities. International Journal of Educational Technology in Higher Education, 16(41), 1–21.
Cohen, L., Manion, L., & Morrison, K. (2018). Research Methods in Education (8th ed.). Routledge.
Dintarini, M., Budiarto, M. T., & Fuad, Y. (2024). From Concept to Assessment : Developing a GeoGebra Assisted Mathematics Test to Measure Semiotic-Spatial Thinking using Abstraction in Context Framework. ICEMT ’24: Proceedings of the 2024 8th International Conference on Education and Multimedia Technology, September, 162–170. https://doi.org/10.1145/3678726.3678738
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/http://doi.org/10.22342/jme.v15i3.pp717-734
Flick, U. (2018). Triangulation. In N. K. Denzin & Y. S. Lincoln (Eds.), The SAGE Handbook of Qualitative Research (5th ed., Vol. 195, Issue 5, pp. 777–804). https://doi.org/10.1007/s11229-017-1319-x
Forndran, A., Lowrie, T., & Harris, D. (2019). Strategy Use in Mathematics Assessment: Does Spatial Reasoning Matter? In G. Hine, S. Blackley, & A. Cooke (Eds.), Mathematics Education Research: Impacting Practice: Proceedings of the 42nd annual conference of the Mathematics Education Research Group of Australasia (pp. 276–283). MERGA. https://researchprofiles.canberra.edu.au/en/publications/strategy-use-in-mathematics-assessment-does-spatial-reasoning-mat
Gagnier, K. M. (2020). Unpacking the Black Box of Translation: A framework for infusing spatial thinking into curricula. Cognitive Research: Principles and Implications, 5(1). https://doi.org/10.1186/s41235-020-00222-9
Geel, M. Van, Keuning, T., Frèrejean, J., Dolmans, D., Merriënboer, V., & Visscher, A. J. (2019). Capturing the complexity of differentiated instruction. School Effectiveness and School Improvement, 30(1), 51–67. https://doi.org/10.1080/09243453.2018.1539013
Gummer, E. S., & Mandinach, E. B. (2015). Building a conceptual framework for data literacy. Teachers College Record, 117(4), 1–22. https://www.scopus.com/inward/record.uri?eid=2-s2.0-84929413862&partnerID=40&md5=31844a49fdaf7ff4db20dc4730e160d7
Harris, D., Lowrie, T., Logan, T., & Hegarty, M. (2020). Spatial Reasoning, Mathematics, and Gender: Do Spatial Constructs Differ in their Contribution to Performance? British Journal of Educational Psychology, 9(1), 409–441. https://doi.org/10.1111/bjep.12371
Hasanah, U., Putrawangsa, S., Purwanta, E., Setiawati, F. A., & Purnomo, Y. W. (2024). Primary Student Spatial Reasoning Abilities: Progression and Challenges. Mathematics Education Journal, 18(3), 329–348. https://doi.org/10.22342/jpm.v18i3.pp329-348
Hegarty, M. (2010). Components of Spatial Intelligence. In B. H. Ross (Ed.), Psychology of Learning and Motivation (pp. 265–297). Elsevier Inc. https://doi.org/10.1016/S0079-7421(10)52007-3
İbili, E., Çat, M., Resnyansky, D., Şahin, S., & Billinghurst, M. (2019). An Assessment of Geometry Teaching Supported with Augmented Reality Teaching Materials to Enhance Students ’ 3D Geometry Thinking Skills. International Journal of Mathematical Education in Science and Technology, 51(2), 224–246. https://doi.org/10.1080/0020739X.2019.1583382
Linn, M. C., & Petersen, A. C. (1985). Emergence and Characterization of Sex Differences in Spatial Ability : A Meta-Analysis. Child Development, 56(6), 1479–1498. http://www.jstor.org/stable/1130467 .
Lowrie, T., Harris, D., Logan, T., & Hegarty, M. (2018). The Impact of a Spatial Intervention Program on Students’ Spatial Reasoning and Mathematics Performance. Journal of Experimental Education, 89(2), 259–277. https://doi.org/10.1080/00220973.2019.1684869
Lowrie, T., Resnick, I., Harris, D., & Logan, T. (2020). In search of the mechanisms that enable transfer from spatial reasoning to mathematics understanding. Mathematics Education Research Journal, 32(2), 175–188. https://doi.org/10.1007/s13394-020-00336-9
Miles, M. B., Huberman, A. M., & Saldaña, J. (2014). Qualitative Data Analysis: A Methods Sourcebook (3rd ed.). SAGE Publications Inc.
Nagy-Kondor, R. (2017). Spatial Ability : Measurement and Development Definition of Spatial Ability. In M.S. Khine (Ed.), Visual-spatial Ability in STEM Education (pp. 35–58). Springer International Publishing Switzerland. https://doi.org/10.1007/978-3-319-44385-0
Newcombe, N. S., & Shipley, T. F. (2015). Studying Visual and Spatial Reasoning for Design Creativity. Studying Visual and Spatial Reasoning for Design Creativity. https://doi.org/10.1007/978-94-017-9297-4
Pavlovičová, G., Bočková, V., & Laššová, K. (2022). Spatial Ability and Geometric Thinking of the Students of Teacher Training for Primary Education. TEM Journal, 11(1), 388–395. https://doi.org/10.18421/TEM111-49
Pozas, M., Letzel, V., & Schneider, C. (2020). Teachers and differentiated instruction : exploring differentiation practices to address student diversity. Journal of Research in Special Educational Needs, 20, 217–230. https://doi.org/10.1111/1471-3802.12481
Pradana, L. N., & Sholikhah, O. H. (2023). Spatial Reasoning Construction: The Way to Use It to Solve Geometric Problems. Jurnal Pendidikan Matematika, 17(2), 209–224.
Reilly, D., Neumann, D. L., & Andrews, G. (2017). Gender Differences in Spatial Ability: Implications for STEM Education and Approaches to Reducing the Gender Gap for Parents and Educators. In M. S. Khine (Ed.), Visual-spatial Ability in STEM Education (p. 3). Springer International Publishing Switzerland.
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
Sorby, S. (2016). Developing 3D spatial skills for engineering students Developing 3D spatial skills for engineering students *. Australasian Journal of Engineering Education, 13(1).
The National Research Council. (2006). Learning To Think Spatially. The National Academic Press.
Unal, H., Jakubowski, E., & Corey, D. (2009). Differences in learning geometry among high and low spatial ability pre-service mathematics teachers. International Journal of Mathematical Education in Science and Technology, 40(8), 997–1012. https://doi.org/10.1080/00207390902912852
Utomo, D. P., & Syarifah, D. L. (2021). Examining mathematical representation to solve problems in trends in mathematics and science study: Voices from Indonesian secondary school students. International Journal of Education in Mathematics, Science and Technology, 9(3), 540–556. https://doi.org/10.46328/IJEMST.1685
Velez, M. C., Silver, D., & Tremaine, M. (2005). Understanding Visualization through Spatial Ability Differences. IEEE Visualization, 511–518.
Wei, W., Chen, C., Zhou, X., & Bisiacchi, P. S. (2016). Spatial Ability Explains the Male Advantage in Approximate Arithmetic. Frontiers in Psychology, 7(March), 1–9. https://doi.org/10.3389/fpsyg.2016.00306
Zhou, Y., Yi, F., Dong, B., Zhang, G., Zhang, Y., & Xu, T. (2024). Mental imagery scaffolding: The effects of detail richness and text load on geography learning. Education and Information Technologies, 16929–16956. https://doi.org/10.1007/s10639-024-12540-2