Mathematical Creativity: Student Geometrical Figure Apprehension in Geometry Problem-Solving Using New Auxiliary Elements
Muhammad Muzaini
,
Sri Rahayuningsih
,
Muhammad Ikram
,
Fathimah Az-Zahrah Nasiruddin
The definition of creativity among professional mathematicians and the definition of mathematical creativity in the classroom context are significantl.
- Pub. date: February 15, 2023
- Pages: 139-150
- 515 Downloads
- 815 Views
- 0 Citations
The definition of creativity among professional mathematicians and the definition of mathematical creativity in the classroom context are significantly different. The purpose of this study was to investigate the relationship between students’ mathematical creativity (i.e., cognitive flexibility) and figure apprehension when solving geometric problems with novel auxiliary features such as straight lines and curved lines. In other words, this study determined if geometry knowledge influenced mathematical creativity (cognitive flexibility) in problem-solving. Grade-12 students participated in the intervention. The high school that is the research topic attempts to equip students with academic abilities and is, except for vocational schools, the most popular form of high school among all other types. Such a school was chosen for the study so that a significant proportion of students in Makassar could be represented. In this study, we discovered a relationship between cognitive flexibility and the geometric ability of pupils while solving problems involving auxiliary lines. This indicates that the usage of auxiliary lines as a reference for developing pupils’ creative thinking skills must be advocated. In addition, good geometric abilities (e.g., visual thinking, geometrical reasoning) will encourage pupils to generate various problem-solving concepts. This finding contributes significantly to future research by focusing on auxiliary lines.
cognitive flexibility geometrical figure apprehension mathematical creativity new auxiliary elements
Keywords: Cognitive flexibility, geometrical figure apprehension, mathematical creativity, new auxiliary elements.
References
Brunkalla, K. (2009). How to increase mathematical creativity- an experiment let us know how access to this document benefits you. The Mathematics Enthusiast, 6(1-2), 257–266. https://doi.org/10.54870/1551-3440.1148
Chen, L., Dooren, W. V., & Verschaffel, L. (2015). Enhancing the development of Chinese fifth- graders’ problem-posing and problem- solving abilities, beliefs, and attitudes: A design experiment. In F. M. Singer. N. F. Ellerton & J. Cai (Eds.), Mathematical problem posing: From research to effective practice (pp. 309–329). Springer. https://doi.org/10.1007/978-1-4614-6258-3_15
Clements, D. H., Sarama, J., Spitler, M. E., Lange, A. A., & Wolfe, C. B. (2011). Mathematics learned by young children in an intervention based on learning trajectories: A large-scale cluster randomized trial. Journal for Research in Mathematics Education, 42(2), 127–166. https://doi.org/10.5951/jresematheduc.42.2.0127
Duval, R. (1995). Geometrical pictures: Kinds of representation and specific processings. In R. Sutherland & J. Mason (Eds.), Exploiting mental imagery with computers in mathematics education (pp. 142–157). Springer. https://doi.org/10.1007/978-3-642-57771-0_10
Duval, R. (1999). Representation, vision and visualization: Cognitive functions in mathematical thinking. Basic issues for learning. In F. Hitt & M Santos (Eds.), Proceedings of the Twenty First Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 3-26). ERIC. https://files.eric.ed.gov/fulltext/ED466379.pdf
Duval, R. (2017). Understanding the mathematical way of thinking-The registers of semiotic representations. Springer. https://doi.org/10.1007/978-3-319-56910-9
Eisenhardt, K. M., Furr, N. R., & Bingham, C. B. (2010). Crossroads microfoundations of performance: Balancing efficiency and flexibility in dynamic environments. Organization Science, 21(6), 1263–1273. https://doi.org/10.1287/orsc.1100.0564
Elia, I., van den Heuvel-Panhuizen, M., & Kolovou, A. (2009). Exploring strategy use and strategy flexibility in non-routine problem solving by primary school high achievers in mathematics. ZDM – Mathematics Education, 41, 605–618. https://doi.org/10.1007/s11858-009-0184-6
Furr, N. R. (2009). Cognitive flexibility: The adaptive reality of concrete organization change (Publication No. 3382938) [Doctoral dissertation, Stanford University]. ProQuest Dissertations and Theses Global. https://bit.ly/PROQUEST-3382938
Gagatsis, A., Michael-Chrysanthou, P., Deliyianni, E., Panaoura, A., & Papagiannis, C. (2015). An insight to students’ geometrical figure apprehension through the context of the fundamental educational thought. Communication & Cognition, 48, 89-128. https://l24.im/Etxj
Goncalo, J. A., Vincent, L. C., & Audia, P. G. (2010). Early creativity as a constraint on future achievement. In D. H. Cropley, A. J. Cropley, J. C. Kaufman & M. A. Runco (Eds.), The dark side of creativity (pp. 114–133). Cambridge University Press. https://doi.org/10.1017/CBO9780511761225.007
Gridos, P., Avgerinos, E., Mamona-Downs, J., & Vlachou, R. (2022). Geometrical figure apprehension, construction of auxiliary lines, and multiple solutions in problem solving: Aspects of mathematical creativity in school geometry. International Journal of Science and Mathematics Education, 20, 619–636. https://doi.org/10.1007/s10763-021-10155-4
Gridos, P., Gagatsis, A., Elia, I., & Deliyianni, E. (2019). Mathematical creativity and geometry: The influence of geometrical figure apprehension on the production of multiple solutions. In U. T. Jankvist, M. van den Heuvel-Panhuizen, & M. Veldhuis (Eds.), Eleventh Congress of the European Society for Research in Mathematics Education (pp.156–176). Springer. https://hal.science/hal-02402180/document
Haylock, D. (1997). Recognising mathematical creativity in schoolchildren. ZDM – Mathematics Education, 29, 68–74. https://doi.org/10.1007/s11858-997-0002-y
Hsu, H.-Y., & Silver, E. A. (2014). Cognitive complexity of mathematics instructional tasks in a Taiwanese classroom: An examination of task sources. Journal for Research in Mathematics Education, 45(4), 460–496. https://doi.org/10.5951/jresematheduc.45.4.0460
Hulsizer, H. (2016). Student-produced videos for exam review in mathematics courses. International Journal of Research in Education and Science, 2(2), 271–278. https://bit.ly/3HGvgQJ
Kim, N. J., Vicentini, C. R., & Belland, B. R. (2022). Influence of scaffolding on information literacy and argumentation skills in virtual field trips and problem-based learning for scientific problem solving. International Journal of Science and Mathematics Education, 20, 215–236. https://doi.org/10.1007/s10763-020-10145-y
Leikin, R. (2009). Exploring mathematical creativity using multiple solution tasks. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 129–145). Brill. https://doi.org/10.1163/9789087909352_010
Leikin, R., & Lev, M. (2013). Mathematical creativity in generally gifted and mathematically excelling adolescents: What makes the difference? ZDM - Mathematics Education, 45, 183–197. https://doi.org/10.1007/s11858-012-0460-8
Levav-Waynberg, A., & Leikin, R. (2012). The role of multiple solution tasks in developing knowledge and creativity in geometry. The Journal of Mathematical Behavior, 31(1), 73–90. https://doi.org/10.1016/j.jmathb.2011.11.001
Mamona-Downs, J. (2008). On development of critical thinking and multiple solution tasks. In R. Leikin, A. Levav-Waynberg, & M. Applebaum (Eds.), Proceedings of the International Research Workshop of the Israel Science Foundation (pp. 77–79). Tel Aviv University.
Michael-Chrysanthou, P., & Gagatsis, A. (2014). Ambiguity in the way of looking at geometrical figures. Revista Latinoamericana de Investigacion en Matematica Educativa, 17(4-1), 165-179. https://doi.org/10.12802/relime.13.1748
Michael, S., Gagatsis, A., Elia, I., Deliyianni, E., & Monoyiou, A. (2009). Operative apprehension of geometrical figures of primary and secondary school students. In Gagatsis & S. Grozdev (Eds.), Proceedings of the 6th Mediterranean Conference on Mathematics Education (pp. 67–78).University of Cyprus.
Orion, N., & Hofstein, A. (1994). Factors that influence learning during a scientific field trip in a natural environment. Journal of Research in Science Teaching, 31(10), 1097–1119. https://doi.org/10.1002/tea.3660311005
Polya, G. (2004). How to solve it: A new aspect of mathematical method (Vol. 85). Princeton University Press.
Presmeg, N. (2020). Visualization and learning in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 900-904). Springer. https://doi.org/10.1007/978-3-030-15789-0_161
Rahayuningsih, S., Nurhusain, M., & Indrawati, N. (2022). Mathematical creative thinking ability and self-efficacy: A mixed-methods study involving indonesian students. Uniciencia, 36(1), 318–331. http://bit.ly/3DphXSg
Rahayuningsih, S., Sirajuddin, S., & Ikram, M. (2021). Using open-ended problem-solving tests to identify students’ mathematical creative thinking ability. Participatory Educational Research, 8(3), 285–299. https://doi.org/10.17275/per.21.66.8.3
Senk, S. L. (1985). How well do students write geometry proofs? The Mathematics Teacher, 78(6), 448–456. https://doi.org/10.5951/MT.78.6.0448
Sheffield, L. J. (2009). Developing mathematical creativity - Questions may be the answer. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 87–100). Brill. https://doi.org/10.1163/9789087909352_007
Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM – Mathematics Education, 29, 75–80. https://doi.org/10.1007/s11858-997-0003-x
Singer, F. M., Ellerton, N., & Cai, J. (2013). Problem-posing research in mathematics education: New questions and directions. Educational Studies in Mathematics, 83, 1–7. https://doi.org/10.1007/s10649-013-9478-2
Singer, F. M., Voica, C., & Pelczer, I. (2017). Cognitive styles in posing geometry problems: Implications for assessment of mathematical creativity. ZDM – Mathematics Education, 49(1), 37–52. https://doi.org/10.1007/s11858-016-0820-x
Spiro, R. J., Feltovich, P. J., Coulson, R. L., Jacobson, M., Durgunoglu, A., Ravlin, S., & Jehng, J.-C. (1992). Knowledge acquisition for application: Cognitive flexibility and transfer of training in III-structured domains. Illinois University at Urbana Center for the Study of Reading.
Sriraman, B. (2019). Uncertainty as a catalyst for mathematical creativity. In Proceedings of the 11th International Conference on Mathematical Creativity and Giftedness (pp. 32–51). WTM-Verlag für wissenschaftliche Texte und Medien.
Sriraman, B., & Lee, K.-H. (Eds.). (2011). The elements of creativity and giftedness in mathematics. Sense Publishers. https://doi.org/10.1007/978-94-6091-439-3
Tahmir, S., Faruddin, J., Abbas, N. F., & Matematika, J. (2018). Deskripsi pemahaman geometri siswa smp pada materi segiempat berdasarkan teori van hiele ditinjau dari gaya kognitif siswa [Description of junior high school students' understanding of geometry on quadrilateral material based on van hiele theory in terms of students' cognitive style]. IMED: Issues in Mathematics Education, 2(1), 23–34. https://l24.im/2W6R
Torrance, E. P. (1994). Creativity: Just wanting to know. Benedic books.