Abstract

Unpacking The Relation Between Spatial Abilities and Creativity in Geometry

Panagiotis Gridos & Evgenios Avgerinos & Eleni Deliyianni & Iliada Elia & Athanasios Gagatsis & Zoi Geitona

Download: FULL TEXT PDF
Download: 39, size: 0, date: 29.Sep.2021

Abstract: This study aims to examine the relation between spatial ability and creativity in Geometry. Data was collected from 94 ninth graders. Three spatial abilities were investigated: spatial visualization, spatial relations and closure flexibility. As for students’ creativity, it was examined through a multiple solution problem in Geometry focusing on three components of creativity: fluency, flexibility, and originality. The results revealed that spatial visualization predicted flexibility and originality while closure flexibility predicted all creativity components. Additionally, it was deduced that auxiliary constructions played an essential role in the problem-solution process. Finally, further study opportunities for the teaching and learning of Geometry are discussed.

Keywords: Creativity; Geometry; Geometrical Figure Apprehension; Multiple-solution Tasks; Spatial Abilities

Please Cite: 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. DOI: https://doi.org/10.31757/euer.433

References

Bingolbali, E. (2020). An analysis of questions with multiple solution methods and multiple outcomes in mathematics textbooks. International Journal of Mathematical Education in Science and Technology, 51(5), 669-687, doi: 10.1080/0020739X.2019.1606949

Blazhenkova, O., & Kozhevnikov, M. (2009). The new object-spatialverbal cognitive style model: Theory and measurement. Applied Cognitive Psychology, 23(5), 638–663. https://psycnet.apa.org/doi/10.1002/acp.1473

Buckley, J., Seery, N., & Canty, D. (2018). A heuristic framework of spatial ability: A review and synthesis of spatial factor literature to support its translation into STEM education. Educational Psychology Review.  https://doi.org/10.1007/s10648-018-9432-z.

Carroll, J.B. (1993). Human cognitive abilities: A survey of factor-analytical studies. United Kingdom: Cambridge University Press.

Clements, D. H., Sarama, J., Spitler, M. E., & Wolfe, L. C. B. (2011). Mathematical learned by young children in an intervention based on learning trajectories: a large-scale cluster randomized trial. Journal for Research in Mathematical Education, 42(2), 127–166. https://doi.org/10.5951/jresematheduc.42.2.0127.

Dindyal, J. (2015). Geometry in the early years: a commentary. ZDM - The International Journal on Mathematics Education, 47(3), 519–529. https://doi.org/10.1007/s11858-015-0700-9

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. (2005). Les conditions cognitives de l’apprentissage de la géométrie : Développement de la visualisation, différenciation des raisonnements et coordination de leurs fonctionnements. Annales de Didactique et de Sciences Cognitives, 10, 5–53.

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.

Ekstrom, R. B., French. J. W., Harman, H. H., and Derman, D. (1976). Manual for Kit of factor-referenced cognitive tests. Princeton, NJ: Educational Testing Service.

Elia, I., Van den Heuvel-Panhuizen, M., & Gagatsis, A. (2018). Geometry learning in the early years: Developing understanding of shapes and space with a focus on visualization. In V. Kinnear, M. –Y. Lai and T. Muir (Eds.), Forging Connections in Early Mathematics Teaching and Learning, Early Mathematics Learning and Development (pp. 73-94). Singapore: Springer.

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.

European Parliament and the Council (2006). Recommendation of European Parliament and of Council of 18 December 2006 on key competences for lifelong learning. https://eur-lex.europa.eu/legalcontent/EL/TXT/PDF/?uri=CELEX:32006H0962&from=EN

Fujita, T., Kondo, Y., Kumakura, H., Kunimune, S., & Jones, K. (2020). Spatial reasoning skills about 2D representations of 3D geometrical shapes in grades 4 to 9. Mathematics Education Research Journal, 32, 285– 305. https://doi.org/10.1007/s13394-020-00335-w.

Gagatsis, A., Monoyiou, A., Deliyianni, E., Elia, I., Michael, P., Kalogirou, P., Panaoura, A., & Philippou, A. (2010).

One way of assessing the understanding of a geometrical figure. Acta Didactica Universitatis Comenianae – Mathematics, 10, 35-50.

Gagatsis, A. (2011). How can we evaluate the apprehension of a geometrical figure? In S. Sbaragli (Ed.) La mathematica e la sua didattica, quarant’ anni di impegno (Mathematics and its didactics, forty years of commitment) (pp. 97-100). Bologna: Pitagora Editrice Bologna.

Gagatsis, A. (2012). The Structure of Primary and Secondary School Students’ Geometrical Figure Apprehension. In E. Avgerinos & A. Gagatsis (Eds.), Research on Mathematical Education and Mathematics Applications (pp. 11-20).  Rhodes: University of the Aegean.  

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. 231248). Chia: Universidad de la Sabana.

 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.

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 Reseach in Mathematics Education, Vol. 18, 5-16, 2021.

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, 19 (4). https://doi.org/10.1007/s10763021-10155-4.

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.

Hegarty, M., & Waller, D. A. (2005). Individual Differences in Spatial Abilities. In P. Shah (Ed.) & A. Miyake, The Cambridge Handbook of Visuospatial Thinking (p. 121–169). Cambridge University Press. https://doi.org/10.1017/CBO9780511610448.005

Herbst, P. G. (2002). Establishing a custom of proving in American school geometry: Evolution of the two-column proof in the early twentieth century. Educational Studies in Mathematics, 49(3), 283 - 312.https://doi.org/10.1023/A:1020264906740

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.  

Kalogirou, P., Elia, I., & Gagatsis A. (2009). Spatial Ability and Geometrical Figure Understanding. In Gagatsis, A., Kuzniak, A., Deliyianni, E., & Vivier, L. (Eds). Cyprus and France Research in Mathematics Education (pp. 105-118). Lefkosia: University of Cyprus.

Kalogirou, P. & Gagatsis, A. (2011). A first insight of the relationship between students’ spatial ability and geometrical figure apprehension. Acta Didactica Universitatis Comenianae – Mathematics, 11, 25 –38.  

Kalogirou, P. & Gagatsis, A. (2012). The relationship between students’spatial ability and geometrical figure apprehension. Mediterranean Journal for Research in Mathematics Education, 11(1-2), 133-146.

Kell, H. J., Lubinski, D., Benbow, C. P., & Steiger, J. H. (2013). Creativity and technical innovation: Spatial ability’s unique role. Psychological Science, 24(9), 1831–1836. https://doi.org/10.1177/0956797613478615

Kozhevnikov, M., & Hegarty, M. (2001). A dissociation between object manipulation spatial ability and spatial orientation ability. Memory & Cognition, 29, 745–756. https://doi.org/10.3758/BF03200477

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.

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., & 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.

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.

Linn, M. C., & Petersen, A. C. (1985). Emergence and characterization of sex differences in spatial ability: A metaanalysis. Child Development, 56(6), 1479–1498. https://doi.org/10.2307/1130467

Lohman, D. F. (1988). Spatial abilities as traits, processes, and knowledge. In R. J. Stenverg (Ed.). Advances in the psychology of human intelligence (pp. 181-248). Hillside, NJ: Erlbaum.

McGrew, K. S. (2009). CHC theory and the human cognitive abilities project: standing on the shoulders of the giants of psychometric intelligence research. Intelligence, 37(1), 1–10. https://doi.org/10.1016/j.intell.2008.08.004.

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, P, Gagatsis, A., Avgerinos, E., & Kuzniak, A. (2012). Approaching the Operative Apprehension of a Geometrical Figure. In E. Avgerinos & A. Gagatsis (Eds.), Research on Mathematical Education and Mathematics Applications (pp. 69-84).  Rhodes: University of the Aegean.  

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. (2014). Ambiguity in the way of looking at a geometrical figure. Revista Latinoamericana de Investigación en Matemática Educativa – Relime, 17(4-I), 165-180.

Michael – Chrysanthou, P., Gagatsis, A. (2015). The influence of the nature of geometrical figures on geometric proofs and the role of geometrical figure apprehension. In J.C. Régnier, Y. Slimani, R. Gras, I.B. Tarbout & A. Dhioubi (Eds.), Proceedings of the 8th International conference Implicative Statistic Analysis (pp. 356-368). Radès – Tunisie : A.S.I.

Mulligan, J., Woolcott, G., Mitchelmore, M., Busatto, S., Lai, J., & Davis, B. (2020). Evaluating the impact of a Spatial Reasoning Mathematics Program (SRMP) intervention in the primary school. Mathematics Education Research Journal, 32, 285–305. https://doi.org/10.1007/s13394-020-00324-z.

National Council of Teachers of Mathematics. (2000). Principles and NCTM Standards for school mathematics. Reston, VA: NCTM.

Palatnik, A., & Dreyfus, T. (2018). Students’ reasons for introducing auxiliary lines in proving situations. The Journal of Mathematical Behavior, 55, https://doi.org/10.1016/j.jmathb.2018.10.004

Palatnik, A. & Sigler, A. (2019) Focusing attention on auxiliary lines when introduced into geometric problems.

International Journal of Mathematical Education in Science and Technology, 50 (2), 202-215, DOI:10.1080/0020739X.2018.1489076

Pitta-Pantazi, D., Sophocleous, P., & Christou, C. (2013). Spatial visualizers, object visualizers and verbalizers: Their mathematical creative abilities. ZDM - The International Journal on Mathematics Education, 45(2), 199 – 213. doi:10.1007/s11858-012-0475-1

Sanchez, C. A., & Wiley, J. (2017). Dynamic visuospatial ability and learning from dynamic visualizations. In R. Lowe & R. Ploetzner (Eds.), Learning from dynamic visualization – Innovations in research and application. Berlin: Springer.

Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM - The International Journal on Mathematics Education, 3, 75–80.

Soury-Lavergne, S., & Maschietto, M. (2015). Articulation of spatial and geometrical knowledge in problem solving with technology at primary school. ZDM - The International Journal on Mathematics Education Mathematics Education, 47(3), 435–449.

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.

Taber, K.S (2018). The use of Cronbach’s alpha when developing and reporting research instruments in science education. Research in Science Education, 48, 1273–1296. https://doi.org/10.1007/s11165-016-9602-2

Torrance, E.P. (1994). Creativity: Just wanting to know. Pretoria, South Africa: Benedic books.

Tyagi, T. K. (2016). Is there a causal relation between mathematical creativity and mathematical problem-solving performance? International Journal of Mathematical Education in Science and Technology, 47(3), 388-394, DOI: 10.1080/0020739X.2015.1075612

Ünlü,      M., & Ertekin,     E. (2017).  A structural equation model for factors affecting eighth graders’ geometry achievement.       Educational         Sciences:              Theory   &            Practice,               17,          815–1846. http://dx.doi.org/10.12738/estp.2017.5.0545  

Uttal, D. H., Meadow, N. G., Tipton, E., Hand, L. L., Alden, A. R., Warren, C., & Newcombe, N. S. (2013). The malleability of spatial skills: A meta-analysis of training studies. Psychological Bulletin, 139, 352–402.

Van den Heuvel-Panhuizen, M. & Buys, K. (Eds.). (2008). Young children learn measurement and geometry. Rotterdam, the Netherlands: Sense Publishers.

Van Harpen, X. Y., & Presmeg, N. C. (2013). An investigation of relationships between students’ mathematical problem-posing abilities and their mathematical content knowledge. Educational Studies in Mathematics, 83(1), 117–132.

Velez, M., Silver, D., & Tremaine, M. (2005). Understanding visualization through spatial ability differences. VIS 05. IEEE Visualization (pp. 511-518), DOI: 10.1109/VISUAL.2005.1532836

Wai, J., Lubinski, D., & Benbow, C. P. (2009). Spatial ability for STEM domains: Aligning over 50 years of cumulative psychological knowledge solidifies its importance. Journal of Educational Psychology, 101(4), 817– 835. https://doi.org/10.1037/a0016127

Xie, F., Zhang, L., Chen, X., & Xin, Z. (2020). Is Spatial Ability Related to Mathematical Ability: A Meta-analysis. Educational Psychology Review, 32, 113–155. https://doi.org/10.1007/s10648-019-09496-y

Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal of Research in Mathematics Education, 27(4), 458–477. doi:10.2307/749877

Yilmaz, H. B. (2009). On the development and measurement of spatial ability. International Electronic Journal of Elementary Education, 1(2), 83–96 http://www.iejee.com/1_2_2009/yilmaz.pdf.