How could the Presentation of a Geometrical Task Influence Student Creativity?
Journal of Research in Science, Mathematics and Technology Education, Volume 5, Issue 1, January 2022, pp. 93-116
OPEN ACCESS VIEWS: 439 DOWNLOADS: 319 Publication date: 15 Jan 2022
OPEN ACCESS VIEWS: 439 DOWNLOADS: 319 Publication date: 15 Jan 2022
This study aims to investigate high school students’ geometry learning by focusing on mathematical creativity and its relationship with visualisation and geometrical figure apprehension. The presentation of a geometrical task and its influence on students’ mathematical creativity is the main topic investigated. The authors combine theory and research in mathematical creativity, considering Roza Leikin’s research work on MultipleSolution Tasks with theory and research in visualisation and geometrical figure apprehension, mainly considering Raymond Duval’s work. The relations between creativity, visualization and geometrical figure apprehension are examined through four Geometry Multiple-Solution Tasks given to high school students in Greece. The geometrical tasks are divided into two categories depending on whether their wording is accompanied by the relevant figure or not. The results of the study indicate a multidimensional character of relations among creativity, visualization and geometrical figure apprehension. Didactical implications and future research opportunities are discussed.
Creativity, Geometrical figure, Geometry, Multiple-solution tasks, Visualisation.
Geitona, Z., Gagatsis, A., Elia, I., Deliyianni, E., & Gridos, P. (2022). How could the Presentation of a Geometrical Task Influence Student Creativity?. Journal of Research in Science, Mathematics and Technology Education, 5(1), 93-116. https://doi.org/10.31756/jrsmte.514
- Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52(3), 215–241.
- Bicer, A. (2021). Multiple representations and mathematical creativity. Thinking Skills and Creativity, Vol. 42, https://10.1016/j.tsc.2021.100960
- Brousseau, G. (1984). The crucial role of the didactical contract in the analysis and construction of situations in teaching and learning mathematics. In H. G. Steiner (Ed.), Theory of mathematics education (pp. 110-119). Belfield, Germany: IDMUB.
- Carroll, J.B. (1993). Human cognitive abilities: A survey of factor-analytical studies. United Kingdom: Cambridge University Press.
- Chiu, M. S. (2009). Approaches to the teaching of creative and non-creative mathematical problems. International Journal of Science and Mathematics Education, 7, 55-79.
- Cohen, J., Cohen, P., West, S., & Aiken, L. (2003). Applied multiple regression/correlation analysis for the behavioral sciences. (3rd ed.). Mahwah, NJ: Lawrence.
- Confrey, J., & Smith, E. (1991). A framework for functions: Prototypes, multiple representations and transformations.
- In R. G. Underhill (Ed.), Proceedings of the 13th annual meeting of the North American Chapter of The International Group for the Psychology of Mathematics Education (pp. 57-63). Blacksburg: Virginia Polytechnic Institute and State University.
- Dindyal, J. (2015). Geometry in the early years: a commentary. ZDM Mathematics Education, 47(3), 519–529.
- Duval, R. (1995). Geometrical Pictures: Kinds of Representation and Specific Processings. In R. Sutherland & J. Mason (Εds.), Exploiting Mental Imagery with Computers in Mathematics Education, (p.142-157). Germany: Springer.
- Duval, R. (1999). Representations, vision and visualization: Cognitive functions in mathematical thinking. Basic issues for learning. In F. Hitt & M. Santos (Eds.), Proceedings of the 21st Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education. (Vol. 1, pp. 3–26). Morelos, Mexico.
- Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1), 103–131.
- Duval, R. (2014). The first crucial point in geometry learning: Visualization. Mediterranean Journal for Research in Mathematics Education, 13(1-2), 1-28.
- Elia., I., Gagatsis, A., & Gras, R. (2005). Can we “trace” the phenomenon of compartmentalization by using the I.S.A.? An application for the concept of function. In R. Gras, F. Spagnolo & J. David (Eds.), Proceedings of the Third International Conference I.S.A. Implicative Statistic Analysis (pp. 175-185). Palermo, Italy: Universita degli Studi di Palermo.
- Elia, I., Gagatsis, A., & Demetriou, A. (2007). The effects of different modes of representation on the solution of onestep additive problems. Learning and Instruction, 17, 658-672.
- Elia, I., Panaoura, A., Eracleous, A., & Gagatsis, A. (2007). Relations between secondary pupils’ conceptions about functions and problem solving in different representations. International Journal of Science and Mathematics Education, 5, 533-556.
- Elliot, J. & Smith, I.M. (1983). An international dictionary of spatial tests. Windsor, United Kingdom: The NFERNelson Publishing Company, Ltd.
- Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.), Advanced mathematical thinking (pp. 42–53). Dordrecht, Netherlands: Kluwer.
- Gagatsis, A. (2015). Explorando el rol de las figuras geométricas en el pensamiento geométrico. In B. D’Amore & M.I. Fandiño Pinilla (Eds), Didáctica de la Matemática - Una mirada internacional, empírica y teórica (pp. 231-248). Chia: Universidad de la Sabana.
- Gagatsis, A., & Geitona Z. (2021). A multidimensional approach to students’ creativity in geometry: spatial ability, geometrical figure apprehension and multiple solutions in geometrical problems. Mediterranean Journal for Research in Mathematics Education, Vol. 18, 5-16, 2021.
- Gagatsis, Α. & Kalogirou, P. (2013). Development of spatial ability and geometric figure apprehension. Nicosia:
- University of Cyprus (in Greek).
- Gagatsis, A. & Shiakalli, M. (2004). Ability to translate from one representation of the concept of function to another and mathematical problem solving. Educational Psychology, 24(5), 645-657.
- Gagatsis, Α., Christodoulou, T. & Elia, Ι. (2013). How young students deal with recognition and conversion tasks of graphs. Acta Didactica Universitatis Comenianae – Mathematics, 13, 1-16.
- Gagatsis, A., Deliyianni, E., Elia, I., Panaoura, A., & Michael-Chrysanthou, P. (2016). Fostering Representational Flexibility in the Mathematical Working Space of Rational Numbers. Bolema: Boletim de Educação Matemática, 30(54), 287.
- 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(3-4), 89-128.
- Gras R., Suzuki E., Guillet F., & Spagnolo F. (2008). Statistical implicative analysis. Germany: Springer.
- Gridos, P., Gagatsis, A., Deliyianni, E., Elia, I., & Samartzis, P. (2018). The relation between spatial ability and ability to solve with multiple ways in geometry. Paper presented in: ISSC 2018 - International Conference on Logics of image: Visual Learning, Logic and Philosophy of Form in East and West, Crete, Greece. Research Gate.
- Gridos, P., Avgerinos, E., Mamona-Downs, J. & Vlachou, R. (2021). 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. https://doi.org/10.1007/s10763021-10155-4
- Gridos, P., Avgerinos, E., Deliyianni, E., Elia, I., Gagatsis, A., Geitona, Z. (2021). Unpacking The Relation Between Spatial Abilities and Creativity in Geometry. The European Educational Researcher, 4(3), 307-328.
- Guilford, J. P. (1967). The nature of human intelligence. New York: McGraw-Hill.
- Gutiérrez, A. (1996). Visualization in 3-dimensional geometry: In search of a framework. In L. Puig & A. Guttierez (Eds.), Proceedings of the 20th conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 3–19). Valencia: Universidad de Valencia.
- Halpern, D.F. (2000). Sex differences and cognitive abilities. Mahwah, NJ: Erlbaum.
- Haylock, D. W. (1987). A framework for assessing mathematical creativity in school children. Education Studies in Mathematics, 18(1), 59–74.
- Hiebert, J. & Carpenter, T. P. (1992). Learning and teaching with understaning. In. D. A. Ggrouws (Ed.), Handbook for research on mathematics teaching and learning (pp. 65-97). New York, NY: Macmillan.
- Hsu H. (2007). Geometric calculations are more than calculations. In J.H Woo, H.C Lew, K.S Park, D.Y Seo (Eds.) Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 57–64). Seoul, Korea.
- Kaufman, J. C., & Beghetto, R. A. (2009). Beyond big and little: The four C model of creativity. Review of General Psychology, 13, 1-12.
- Krutetskii, V. A. (1976). The Psychology of Mathematical Abilities in Schoolchildren. Chicago: University of Chicago Press.
- Leikin, R. (2007). Habits of mind associated with advanced mathematical thinking and solution spaces of mathematical tasks. In D. Pitta-Pantazi & G. Philippou (Eds.), Proceedings of the Fifth Conference of the European Society for Research in Mathematics Education (pp. 2330-2339). Larnaka, Cyprus: University of Cyprus and ERME.
- Leikin, R. (2009). Exploring mathematical creativity using multiple solution tasks. In R. Leikin, A. Berman and B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 129–145). Rotterdam, the Netherlands: Sense Publisher.
- Leikin, R. (2011). Multiple-solution tasks: From a teacher education course to teacher practice. ZDM - The International Journal on Mathematics Education. 43(6), 993-1006.
- Leikin, R. (2014). Challenging Mathematics with multiple solution tasks and mathematical investigations in geometry. In Li Y., Silver E., & Li S. (Eds.), Transforming Mathematics Instruction. Advances in Mathematics Education (pp 59-80). Cham, Heidelberg, New York, Dordrecht, London: Springer.
- Leikin, R., & Elgrabli, H. (2015). Creativity and expertise: The chicken or the egg? Discovering properties of geometry figures in DGE. In K. Krainer, & N. Vondrova (Eds.), Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education (pp. 1024–1031). Prague, Czech Republic: ERME.
- Leikin, R., Levav-Waynberg, A., Gurevich, I, & Mednikov, L. (2006). Implementation of multiple solution connecting tasks: Do students’ attitudes support teachers’ reluctance? Focus on Learning Problems in Mathematics, 28, 1-22.
- Leikin, R., & Levav-Waynberg, A. (2007). Exploring mathematics teacher knowledge to explain the gap between theory-based recommendations and school practice in the use of connecting tasks. Educational Studies in Mathematics, 66, 349-371.
- Levav-Waynberg, A., & Leikin, R. (2009). Multiple solutions to a problem: A tool for assessment of mathematical thinking in geometry. In The sixth conference of the European Society for Research in Mathematics Education- CERME-6.
- Levav-Waynberg, A., & Leikin, R. (2012a). The role of multiple solution tasks in developing knowledge and creativity in geometry. Journal of Mathematics Behavior, 31, 73-90.
- Levav-Waynberg, A., & Leikin, R. (2012b). Using multiple solutions tasks for the evaluation of students’ problemsolving performance in geometry. Canadian Journal of Science Mathematics and Technology Education, 12(4), 311-333.
- Lohman, D. F. (1988). Spatial abilities as traits, processes, and knowledge. In R. J. Sternberg (Ed.), Advances in the psychology of human intelligence (Vol. 40, pp. 181-248). Hillsdale, NJ: Erlbaum.
- Mackworth, N. H. (1965). Visual noise: causes tunnel vision. Psychonomic Science, 3, 67-68.
- Mann, E. (2006). Creativity: The essence of mathematics. Journal for the Education of the Gifted, 30(2), 236-260.
- Michael, P., Gagatsis, A, Avgerinos, E., & Kuzniak, A. (2011). Middle and High school students’ operative apprehension of geometrical figures. Acta Didactica Universitatis Comenianae – Mathematics, 11, 45 –55.
- Michael – Chrysanthou, P., & Gagatsis, A. (2013). Geometrical figures in task solving: an obstacle or a heuristic tool? Acta Didactica Universitatis Comenianae – Mathematics, 13, 17- 30.
- Michael–Chrysanthou, P., & Gagatsis, A. (2015). Ambiguity in the way of looking at a geometrical figure. Revista Latinoamericana de Investigación en Matemática Educativa – Relime, 17(4-I), 165-180.
- National Council of Teachers of Mathematics. (2000). Principles and NCTM Standards for school mathematics. Reston, VA: NCTM.
- Nicolaou, S., Gagatsis, A., Panaoura, A., Deliyianni, E., Elia, I., & Televantou, I. (2020). Economic sciences students’ understanding on representation tasks concerning the concept of function. In J.-C. Régnier, R. Gras, M. Henry, R. Couturier, & G. Brousseau (Eds.), Analyse Statistique Implicative. Cadre théorique en relation étroite et au service de multiples disciplines (pp. 281-302). Besançon: Université Bourgogne Franche-ComtéBesançon: icar.
- Palatnik, A., & Dreyfus, T. (2018). Students’ reasons for introducing auxiliary lines in proving situations. The Journal of Mathematical Behavior, https://doi.org/10.1016/j.jmathb.2018.10.004
- Panaoura, G., Gagatsis, A., Lemonides, Ch., (2007). Spatial abilities in relation to performance in geometry tasks. In Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education (CERME 5) (pp.1062-1071).
- Sawyer, K. (2015). A call to action: The challenges of creative teaching and learning. Teachers College Record, 117(10), 1-34.
- Sierpinska, A. (1992). On understanding the notion of function. In E. Dubinsky, &
- G. Harel (Ed.), The concept of function: Aspects of epistemology and pedagogy (pp. 25-58). United States: Mathematical Association of America.
- Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM, 3, 75–80.
- Singh, B. (1987). The development of tests to measure mathematical creativity. International Journal of Mathematical Education in Science and Technology, 18(2), 181-186.
- Stupel, M., & Ben-Chaim, D. (2017): Using multiple solutions to mathematical problems to develop pedagogical and mathematical thinking: A case study in a teacher education program. Investigations in Mathematics Learning, 9(2), 86-108.
- Torrance, E.P. (1994). Creativity: Just wanting to know. Pretoria, South Africa: Benedic books.
- Tzefriou, E., Santorinaiou, P., Deliyianni, E. & Elia, I. (2020). Exploring geometrical figure apprehension: the impact of didactic contract. Mediterranean Journal of Research in Mathematics Education, Vol. 18, 58-75.
- Van den Heuvel-Panhuizen, M. & Buys, K. (Eds.). (2008). Young children learn measurement and geometry. Rotterdam, the Netherlands: Sense Publishers.
- Yakimanskaya, I. S. (1991). The development of spatial thinking in school children. (Soviet Studies in Mathematics Education, Vol. 3). Reston, USA: NCTM.
This work is licensed under a Creative Commons Attribution 4.0 International License.