Problem modification as a tool for detecting cognitive flexibility in school children
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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.
KeywordsProblem posing Cognitive flexibility Creativity Change in cognitive framing
Mathematical Subject Classification97C30
The authors thank the three anonymous reviewers for their valuable comments on an early version of this paper.
- 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.Google Scholar
- 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.Google Scholar
- Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.), Advanced mathematical thinking (pp. 42–53). Dordrecht: Kluwer.Google Scholar
- 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.Google Scholar
- 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.
- 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.Google Scholar
- Johnson, P. (1998). Analytic induction. In G. Symon & C. Cassell (Eds.), Qualitative methods and analysis in organizational research. London: Sage.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- Maier, N. (1970). Problem solving and creativity in individuals and groups. Belmont, CA: Brooks/Cole.Google Scholar
- Runco, M. A. (1994). Problem finding, problem solving, and creativity. Norwood, NJ: Ablex.Google Scholar
- Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- Sriraman, B. (2004). The characteristics of mathematical creativity. The Mathematics Educator, 14(1), 19–34.Google Scholar
- 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.Google Scholar
- Torrance, E. P. (1974). Torrance tests of creative thinking. Bensenville, IL: Scholastic Testing Service.Google Scholar
- 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.Google Scholar