Abstract
This paper presents the results of an experiment in which fourth to sixth graders with above-average mathematical abilities modified a given problem. The experiment found evidence of links between problem posing and cognitive flexibility. Emerging from organizational theory, cognitive flexibility is conceptualized through three primary constructs: cognitive variety, cognitive novelty, and changes in cognitive framing. Among these components, changes in cognitive framing could be effectively detected in problem-posing situations, giving a relevant indication of students’ creative potential. The students’ capacity to generate coherent and consistent problems in the context of problem modification may indicate the existence of a strategy of functional type for generalizations, which seems to be specific to mathematical creativity.
Similar content being viewed by others
References
Brown, S. I., & Walter, M. I. (1993). Problem posing in mathematics education. In S. I. Brown & M. I. Walter (Eds.), Problem posing: Reflection and applications (pp. 16–27). Hillsdale, NJ: Lawrence Erlbaum Associates.
Csikszentmihalyi, M. (1994). The domain of creativity. In D. H. Feldman, M. Csikszentmihalyi, & H. Gardner (Eds.), Changing the world: A framework for the study of creativity (pp. 135–158). Westport, CT: Praeger.
Eisenhardt, K. M., Furr, N. R., & Bingham, C. B. (2010). Microfoundations of performance: Balancing efficiency and flexibility in dynamic environments. Organization Science, 21(6), 1263–1273.
Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.), Advanced mathematical thinking (pp. 42–53). Dordrecht: Kluwer.
Freiman, V., & Sriraman, B. (2007). Does mathematics gifted education need a working philosophy of creativity? Mediterranean Journal for Research in Mathematics Education, 6(1–2), 23–46.
Furr, N. R. (2009). Cognitive flexibility: The adaptive reality of concrete organization change. Ph.D. dissertation, Stanford University. http://gradworks.umi.com/33/82/3382938.html. Accessed 28 Jan 2013.
Goncalo, J. A., Vincent, L., & Audia, P. G. (2010). Early creativity as a constraint on future achievement. In D. Cropley, J. Kaufman, A. Cropley, & M. Runco (Eds.), The dark side of creativity (pp. 114–133). Cambridge: Cambridge University Press.
Haylock, D. (1997a). Recognizing mathematical creativity in schoolchildren. ZDM—The International Journal on Mathematics Education, 29(3), 68–74.
Haylock, D. (1997b). A framework for assessing mathematical creativity in school children. Educational Studies in Mathematics, 18(1), 59–74.
Jay, E. S., & Perkins, D. N. (1997). Problem finding: The search for mechanism. In M. Runco (Ed.), The creativity research handbook (pp. 257–293). New Jersey: Hampton Press.
Johnson, P. (1998). Analytic induction. In G. Symon & C. Cassell (Eds.), Qualitative methods and analysis in organizational research. London: Sage.
Kenderov, P. (2006). Competitions and mathematics education. In M. Sanz-Solé, J. Soria, J. L. Varona, & J. Verdera (Eds.), Proceedings of the international congress of mathematicians (ICM) (Vol. 3, pp. 1583–1598). Zürich: EMS.
Kontorovich, I., Koichu, B., Leikin, R., & Berman, A. (2011). Indicators of creativity in mathematical problem posing: How indicative are they? In Proceedings of the 6th international conference CMG (pp. 120–125). Latvia: Latvia University.
Krems, J. F. (1995). Cognitive flexibility and complex problem solving. In P. A. Frensch & J. Funke (Eds.), Complex problem solving: the European perspective (Chap. 8). Hillsdale, NJ: Lawrence Erlbaum Associates.
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). Rotterdam: Sense Publishers.
Maier, N. (1970). Problem solving and creativity in individuals and groups. Belmont, CA: Brooks/Cole.
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, 1097–1119.
Runco, M. A. (1994). Problem finding, problem solving, and creativity. Norwood, NJ: Ablex.
Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28.
Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM—The International Journal on Mathematics Education, 29(3), 75–80.
Singer, F. M. (2012). Exploring mathematical thinking and mathematical creativity through problem posing. In R. Leikin, B. Koichu, & A. Berman (Eds.), Exploring and advancing mathematical abilities in high achievers (pp. 119–124). Haifa: University of Haifa.
Singer, F. M., Pelczer, I., & Voica, C. (2011). Problem posing and modification as a criterion of mathematical creativity. In M. Pytlak, T. Rowland, & E. Swoboda (Eds.), Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education (pp. 1133–1142). Rzeszów, Poland: University of Rzeszów.
Singer, F. M., & Voica, C. (2011). Creative contexts as ways to strengthen mathematics learning. In M. Anitei, M. Chraif, & C. Vasile (Eds.), Proceedings PSIWORLD 2011. Procedia-Social and Behavioral Sciences (Vol. 33, pp. 538–542). Available at http://dx.doi.org/10.1016/j.sbspro.2012.01.179.
Singer, F. M., & Voica, C. (2012). A problem-solving conceptual framework and its implications in designing problem-posing tasks. Educational Studies in Mathematics, 1–18. doi:10.1007/s10649-012-9422-x.
Spiro, R. J., Feltovich, P. J., Jacobson, M. J., & Coulson, R. L. (1992). Cognitive flexibility, constructivism, and hypertext: Random access instruction for advanced knowledge acquisition in ill-structured domains. In T. M. Duffy & D. H. Jonassen (Eds.), Constructivism and the technology of instruction (pp. 57–75). Hillsdale, NJ: Lawrence Erlbaum.
Sriraman, B. (2004). The characteristics of mathematical creativity. The Mathematics Educator, 14(1), 19–34.
Stoyanova, E., & Ellerton, N. F. (1996). A framework for research into students’ problem posing. In P. Clarkson (Ed.), Technology in mathematics education (pp. 518–525). Melbourne: Mathematics Education Research Group of Australasia.
Torrance, E. P. (1974). Torrance tests of creative thinking. Bensenville, IL: Scholastic Testing Service.
Voica, C., & Singer, F.M. (2012). Problem modification as an indicator of deep understanding. Paper presented at Topic Study Group 3, Activities and Programs for Gifted Students, ICME-12, Seoul, Korea.
Yuan, X., & Sriraman, B. (2011). An exploratory study of relationships between students’ creativity and mathematical problem posing abilities. In B. Sriraman & K. Lee (Eds.), The elements of creativity and giftedness in mathematics (pp. 5–28). Rotterdam: Sense Publishers.
Acknowledgments
The authors thank the three anonymous reviewers for their valuable comments on an early version of this paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Voica, C., Singer, F.M. Problem modification as a tool for detecting cognitive flexibility in school children. ZDM Mathematics Education 45, 267–279 (2013). https://doi.org/10.1007/s11858-013-0492-8
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11858-013-0492-8