Is Problem Posing a Tool for Identifying and Developing Mathematical Creativity?

Part of the Research in Mathematics Education book series (RME)

Abstract

The mathematical creativity of fourth to sixth graders, high achievers in mathematics, is studied in relation to their problem-posing abilities. The study reveals that in problem-posing situations, mathematically high achievers develop cognitive frames that make them cautious in changing the parameters of their posed problems, even when they make interesting generalizations. These students display a kind of cognitive flexibility that seems mathematically specialized, which emerges from gradual and controlled changes in cognitive framing. More precisely, in a problem-posing context, students’ mathematical creativity manifests itself through a process of abstraction-generalization based on small, incremental changes of parameters, in order to achieve synthesis and simplification. This approach results from a tension between the students’ tendency to maintain a built-in cognitive frame, and the possibility to overcome it, which is constrained by their need to devise mathematical problems that are coherent and consistent.

Keywords

Creativity Mathematical creativity Problem posing Cognitive flexibility Cognitive variety Cognitive novelty Cognitive framing Reframing Problem solving Coherence Mathematical consistency 

References

  1. Amabile, T. M. (1989). Growing up creative: Nurturing a lifetime of creativity. Buffalo, NY: Creative Education Foundation Press.Google Scholar
  2. Baer, J. (1993). Divergent thinking and creativity: A task-specific approach. Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
  3. Baer, J. (1996). The effects of task-specific divergent-thinking training. Journal of Creative Behavior, 30, 183–187.CrossRefGoogle Scholar
  4. Baer, J. (1998). The case for domain specificity in creativity. Creativity Research Journal, 11, 173–177.CrossRefGoogle Scholar
  5. Baer, J., & Kaufman, J. C. (2005). Bridging generality and specificity: The amusement park theoretical (APT) model of creativity. Roeper Review, 27, 158–163.CrossRefGoogle Scholar
  6. Balka, D. S. (1974). The development of an instrument to measure creative ability in mathematics. Dissertation Abstracts International, 36(01), 98. (UMI No. AAT 7515965).Google Scholar
  7. Brown, S. I., & Walter, M. I. (1983/1990). The art of problem posing. Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
  8. Cai, J., & Cifarelli, V. (2005). Exploring mathematical exploration: How two college students formulated and solved their own mathematical problems. Focus on Learning Problems in Mathematics, 27(3), 43–72.Google Scholar
  9. Csikszentmihalyi, M. (1996). Creativity, flow, and the psychology of discovery and invention. New York, NY: Harper Collins.Google Scholar
  10. Csikszentmihalyi, M., & Getzels, J. W. (1971). Discovery-oriented behavior and the originality of creative products: A study with artists. Journal of Personality and Social Psychology, 19(1), 47.CrossRefGoogle Scholar
  11. Deák, G. O. (2004). The development of cognitive flexibility and language abilities. Advances in Child Development and Behavior, 31, 271–327.Google Scholar
  12. Demetriou, A., Kui, Z. X., Spanoudis, G., Christou, C., Kyriakides, L., & Platsidou, M. (2005). The architecture, dynamics, and development of mental processing: Greek, Chinese, or Universal? Intelligence, 33(2), 109–141.CrossRefGoogle Scholar
  13. Dillon, J. T. (1982). Problem finding and solving. Journal of Creative Behavior, 16, 97–111.CrossRefGoogle Scholar
  14. Dow, G. T., & Mayer, R. E. (2004). Teaching students to solve insight problems: Evidence for domain specificity in creativity training. Creativity Research Journal, 16(4), 389–398.CrossRefGoogle Scholar
  15. Dreyfus, T., & Eisenberg, T. (1996). On different facets of mathematical thinking. In R. J. Sternberg & T. Ben-Zeev (Eds.), The nature of mathematical thinking (pp. 253–284). Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  16. 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.CrossRefGoogle Scholar
  17. English, D. L. (1998). Children’s problem posing within formal and informal contexts. Journal for Research in Mathematics Education, 29(1), 83–106.CrossRefGoogle Scholar
  18. Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.), Advanced mathematical thinking (pp. 42–53). Dordrecht, The Netherlands: Kluwer Academic.Google Scholar
  19. European Commission. (2003/2004). Working group on basic skills, progress reports 2003, 2004, European Commission interim working document for the objectives 1.2 (developing the skills for the knowledge society), 3.2 (developing the spirit of enterprise), and 3.3 (Improving foreign language learning) Annex 2 Domains of Key Competences—Definitions, Knowledge, Skills and Attitudes. Retrieved from http://europa.eu.int/comm/education/policies/2010/objectives_en.html#basic
  20. European Commission. (2005). Proposal for a recommendation of the European Parliament and of the Council on Key Competences for Lifelong Learning, Council of The European Union, 11 November 2005.Google Scholar
  21. Evans, E. W. (1964). Measuring the ability of students to respond in creative mathematical situations at the late elementary and early junior high school level. Dissertation Abstracts, 25(12), 7107. (UMI No. AAT 6505302).Google Scholar
  22. Fasko, D. (2001). Education and creativity. Creativity Research Journal, 13(3–4), 317–327.CrossRefGoogle Scholar
  23. 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
  24. Furr, N. R. (2009). Cognitive flexibility: The adaptive reality of concrete organization change (Doctoral dissertation). Stanford University, Stanford, CA. Retrieved May 12, 2011, from http://gradworks.umi.com/3382938.pdf
  25. Gardner, H. (1993). Creating minds: An anatomy of creativity as seen through the lives of Freud, Einstein, Picasso, Stravinsky, Eliot, Graham, and Ghandi. New York, NY: Basic Books.Google Scholar
  26. Gardner, H. (2006). Five minds for the future. Boston, MA: Harvard Business School Press.Google Scholar
  27. Gardner, H., Csikszentmihalyi, M., & Damon, M. (2001). Good work: When excellence meets ethics. New York, NY: Basic Books.Google Scholar
  28. Getzels, J. W. (1975). Problem-finding and the inventiveness of solutions. The Journal of Creative Behavior, 9(1), 12–18.CrossRefGoogle Scholar
  29. Getzels, J. W. (1979). Problem finding: A theoretical note. Cognitive Science, 3(2), 167–172.CrossRefGoogle Scholar
  30. Getzels, J. W., & Csikszentmihalyi, M. (1976). The creative vision: A longitudinal study of problem finding in art. New York, NY: Wiley.Google Scholar
  31. Getzels, J. W., & Jackson, P. W. (1962). Creativity and intelligence: Explorations with gifted students. New York, NY: Wiley.Google Scholar
  32. Ginsburg, H. P. (1996). Toby’s math. In R. J. Sternberg & T. Ben-Zeev (Eds.), The nature of mathematical thinking (pp. 175–282). Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  33. 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, United Kingdom: Cambridge University Press.CrossRefGoogle Scholar
  34. Guilford, J. P. (1967). Creativity: Yesterday, today, and tomorrow. Journal of Creative Behavior, 1, 3–14.CrossRefGoogle Scholar
  35. Hadamard, J. (1945/1954). Psychology of invention in the mathematical field. Princeton, NJ: Princeton University Press.Google Scholar
  36. Hargreaves, A. (2003). Teaching in the knowledge society: Education in the age of insecurity. New York, NY: Teachers College Press.Google Scholar
  37. Haylock, D. W. (1987). A framework for assessing mathematical creativity in schoolchildren. Educational Studies in Mathematics, 18(1), 59–74.CrossRefGoogle Scholar
  38. Haylock, D. (1997). Recognizing mathematical creativity in school children. International Reviews on Mathematical Education, 29(3), 68–74. Retrieved March 10, 2003, from http://www.fiz-karlsruhe.de/fix/publications/zdm/adm97
  39. Huang, R., & Leung, K. S. F. (2004). Cracking the paradox of Chinese learners: Looking into the mathematics classrooms in Hong Kong and Shanghai. In L. Fan, N. Y. Wong, J. Cai, & S. Li (Eds.), How Chinese learn mathematics: Perspectives from insiders (pp. 348–381). River Edge, NJ: World Scientific.CrossRefGoogle Scholar
  40. Jay, E. S., & Perkins, D. N. (1997). Problem finding: The search for mechanism. In M. Runco (Ed.), The creativity research handbook (pp. 257–293). Cresskill, NJ: Hampton Press.Google Scholar
  41. Jensen, L. R. (1973). The relationships among mathematical creativity, numerical aptitude and mathematical achievement. Dissertation Abstracts International, 34(05), 2168. (UMI No. AAT 7326021).Google Scholar
  42. Krems, J. F. (1995). Cognitive flexibility and complex problem solving. In P. A. Frensch & J. Funke (Eds.), Complex problem solving: The European perspective (pp. 201–218). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
  43. Leikin, R., & Lev, M. (2007). Multiple solution tasks as a magnifying glass for observation of mathematical creativity. In J.-H. Woo, H.-C. Lew, K.-S. Park, & D.-Y. Seo (Eds.), Proceedings of the 31st International Conference for the Psychology of Mathematics Education (Vol. 3, pp. 161–168). Seoul, South Korea: PME.Google Scholar
  44. Leung, S. S. (1997). On the role of creative thinking in problem posing. Zentralbatt fur Didaktik der Mathematik (ZDM), 97(2), 48–52.Google Scholar
  45. Leung, S. S., & Silver, E. A. (1997). The role of task format, mathematics knowledge, and creative thinking on the arithmetic problem posing of prospective elementary school teachers. Mathematics Education Research Journal, 9(1), 5–24.CrossRefGoogle Scholar
  46. Lim, C. S. (2007). Characteristics of mathematics teaching in Shanghai, China: Through the lens of a Malaysian. Mathematics Education Research Journal, 19(1), 77–89.CrossRefGoogle Scholar
  47. Lowrie, T. (2002). Young children posing problems: The influence of teacher intervention on the type of problems children pose. Mathematics Education Research Journal, 14(2), 87–98.CrossRefGoogle Scholar
  48. Lubart, T., & Guignard, J. (2004). The generality-specificity of creativity: A multivariate approach. In R. J. Sternberg, E. L. Grigorenko, & J. L. Singer (Eds.), Creativity: From potential to realization (pp. 43–56). Washington, DC: American Psychological Association.CrossRefGoogle Scholar
  49. NAEP Mathematics Framework for the 2011 National Assessment of Educational Progress. (2012). Retrieved October 2012, from http://www.nagb.org/publications/frameworks.html
  50. National Council of Teachers of Mathematics. (2000). Principles and standards of school mathematics. Reston, VA: Author.Google Scholar
  51. Nickerson, R. S. (1999). Enhancing creativity. In R. J. Sternberg (Ed.), Handbook of creativity (pp. 392–430). Cambridge, United Kingdom: Cambridge University Press.Google Scholar
  52. 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.CrossRefGoogle Scholar
  53. Pelczer, I., Singer, F. M., & Voica, C. (2013a). Cognitive framing: A case in problem posing. Procedia—Social and Behavioral Sciences, 78, 195–199.CrossRefGoogle Scholar
  54. Pelczer, I., Singer, F. M., & Voica, C. (2013b). Teaching highly able students in a common class: Challenges and limits. In B. Ubuz, C. Haser, & M. A. Mariotti (Eds.), Proceedings of CERME 8 (pp. 1235–1244). Antalya, Turkey: ERME.Google Scholar
  55. Poincaré, H. (1913). The foundations of science. New York, NY: The Science Press.Google Scholar
  56. Poincaré, H. (1948). Science and method. New York, NY: Dover.Google Scholar
  57. Prouse, H. L. (1967). Creativity in school mathematics. The Mathematics Teacher, 60, 876–879.Google Scholar
  58. Scott, W. A. (1962). Cognitive complexity and cognitive flexibility. Sociometry, 25, 405–414.CrossRefGoogle Scholar
  59. Sfard, A. (1998). On two metaphors for learning and the dangers of choosing just one. Educational Researcher, 27(2), 4–13.CrossRefGoogle Scholar
  60. Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28.Google Scholar
  61. Silver, E. A. (1997). Fostering creativity though instruction rich in mathematical problem solving and problem posing. International Reviews on Mathematical Education, 29, 75–80.Google Scholar
  62. Silver, E. A., & Marshall, S. P. (1989). Mathematical and scientific problem solving: Findings, issues and instructional implications. In B. F. Jones & L. Idol (Eds.), Dimensions of thinking and cognitive instruction (pp. 265–290). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
  63. Singer, F. M. (2006). A cognitive model for developing a competence-based curriculum in secondary education. In A. Crisan (Ed.), Current and future challenges in curriculum development: Policies, practices and networking for change (pp. 121–141). Bucharest, Romania: Humanitas Educational.Google Scholar
  64. Singer, F. M. (2007). Balancing globalisation and local identity in the reform of education in Romania. In B. Atweh, M. Borba, A. Barton, D. Clark, N. Gough, C. Keitel, C. Vistro-Yu, & R. Vithal (Eds.), Internalisation and globalisation in mathematics and science education (pp. 365–382). Dordrecht, The Netherlands: Springer.CrossRefGoogle Scholar
  65. 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, Israel: University of Haifa.Google Scholar
  66. Singer, F. M., Ellerton, N. F., Cai, J., & Leung, E. (2011). Problem posing in mathematics learning and teaching: A research agenda. In B. Ubuz (Ed.), Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 137–166). Ankara, Turkey: PME.Google Scholar
  67. 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 CERME 7 (pp. 1133–1142). Rzeszów, Poland: University of Rzeszów.Google Scholar
  68. Singer, F. M., & Sarivan, L. (Eds.). (2006). QUO VADIS ACADEMIA? Reference points for a comprehensive reform of higher education (In Romanian, with a synthesis in English). Bucharest, Romania: Sigma.Google Scholar
  69. 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). Retrieved from http://dx.doi.org/10.1016/j.sbspro.2012.01.179
  70. Singer, F. M., & Voica, C. (2013). A problem-solving conceptual framework and its implications in designing problem-posing tasks. ESM, 83(1), 9–26. doi: 10.1007/s10649-012-9422-x.Google Scholar
  71. Singh, B. (1988). Teaching-learning strategies and mathematical creativity. Delhi, India: Mittal.Google Scholar
  72. Smilansky, J. (1984). Problem solving and the quality of invention: An empirical investigation. Journal of Educational Psychology, 76(3), 377–386.CrossRefGoogle Scholar
  73. 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
  74. Sriraman, B. (2004). The characteristics of mathematical creativity. The Mathematics Educator, 14(1), 19–34.Google Scholar
  75. Sriraman, B. (2009). The characteristics of mathematical creativity. The International Journal on Mathematics Education [ZDM], 41, 13–27.CrossRefGoogle Scholar
  76. Sternberg, R. J., & Lubart, T. I. (1991). Creating creative minds. Phi Delta Kappan, 72, 608–614.Google Scholar
  77. Sternberg, R. J., & Lubart, T. I. (1999). The concept of creativity: Prospects and paradigms. In R. J. Steinberg (Ed.), Handbook of creativity (pp. 3–15). New York, NY: Cambridge University Press.Google Scholar
  78. 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, Australia: Mathematics Education Research Group of Australasia.Google Scholar
  79. Torrance, E. P. (1974). Torrance tests of creative thinking: Norms, technical manual. Lexington, MA: Ginn.Google Scholar
  80. Voica, C., & Singer, F. M. (2012). Problem modification as an indicator of deep understanding. Paper presented to Topic Study Group 3 “Activities and Programs for Gifted Students,” at the 12th International Congress for Mathematics Education (ICME-12). ICME-12 Pre-proceedings (pp. 1533–1542). Retrieved from http://icme12.org/upload/UpFile2/TSG/1259.pdf
  81. Voica, C., & Singer, F. M. (2013). Problem modification as a tool for detecting cognitive flexibility in school children. ZDM Mathematics Education, 45(2), 267–279. doi: 10.1007/s11858-013-0492-8.CrossRefGoogle Scholar
  82. Yuan, X., & Sriraman, B. (2011). An exploratory study of relationships between students’ creativity and mathematical problem posing abilities: Comparing Chinese and U.S. students. In B. Sriraman & K. Lee (Eds.), The elements of creativity and giftedness in mathematics (pp. 5–28). Rotterdam, The Netherlands: Sense.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Educational SciencesUniversity of PloiestiBucharestRomania
  2. 2.Department of MathematicsUniversity of BucharestBucharestRomania

Personalised recommendations