Mathematical Problem Posing pp 141-174 | Cite as

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

## 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

- Amabile, T. M. (1989).
*Growing up creative: Nurturing a lifetime of creativity*. Buffalo, NY: Creative Education Foundation Press.Google Scholar - Baer, J. (1993).
*Divergent thinking and creativity: A task-specific approach*. Hillsdale, NJ: Lawrence Erlbaum.Google Scholar - Baer, J. (1996). The effects of task-specific divergent-thinking training.
*Journal of Creative Behavior, 30*, 183–187.CrossRefGoogle Scholar - Baer, J. (1998). The case for domain specificity in creativity.
*Creativity Research Journal, 11*, 173–177.CrossRefGoogle Scholar - 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 - 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 - Brown, S. I., & Walter, M. I. (1983/1990).
*The art of problem posing*. Hillsdale, NJ: Lawrence Erlbaum.Google Scholar - 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 - Csikszentmihalyi, M. (1996).
*Creativity, flow, and the psychology of discovery and invention*. New York, NY: Harper Collins.Google Scholar - 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 - Deák, G. O. (2004). The development of cognitive flexibility and language abilities.
*Advances in Child Development and Behavior, 31*, 271–327.Google Scholar - 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 - Dillon, J. T. (1982). Problem finding and solving.
*Journal of Creative Behavior, 16*, 97–111.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.),
*Advanced mathematical thinking*(pp. 42–53). Dordrecht, The Netherlands: Kluwer Academic.Google Scholar - 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 - 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 - 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 - Fasko, D. (2001). Education and creativity.
*Creativity Research Journal, 13*(3–4), 317–327.CrossRefGoogle 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*(Doctoral dissertation). Stanford University, Stanford, CA. Retrieved May 12, 2011, from http://gradworks.umi.com/3382938.pdf - 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 - Gardner, H. (2006).
*Five minds for the future*. Boston, MA: Harvard Business School Press.Google Scholar - Gardner, H., Csikszentmihalyi, M., & Damon, M. (2001).
*Good work: When excellence meets ethics*. New York, NY: Basic Books.Google Scholar - Getzels, J. W. (1975). Problem-finding and the inventiveness of solutions.
*The Journal of Creative Behavior, 9*(1), 12–18.CrossRefGoogle Scholar - Getzels, J. W. (1979). Problem finding: A theoretical note.
*Cognitive Science, 3*(2), 167–172.CrossRefGoogle Scholar - Getzels, J. W., & Csikszentmihalyi, M. (1976).
*The creative vision: A longitudinal study of problem finding in art*. New York, NY: Wiley.Google Scholar - Getzels, J. W., & Jackson, P. W. (1962).
*Creativity and intelligence: Explorations with gifted students*. New York, NY: Wiley.Google Scholar - 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 - 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 - Guilford, J. P. (1967). Creativity: Yesterday, today, and tomorrow.
*Journal of Creative Behavior, 1*, 3–14.CrossRefGoogle Scholar - Hadamard, J. (1945/1954).
*Psychology of invention in the mathematical field*. Princeton, NJ: Princeton University Press.Google Scholar - Hargreaves, A. (2003).
*Teaching in the knowledge society: Education in the age of insecurity*. New York, NY: Teachers College Press.Google Scholar - Haylock, D. W. (1987). A framework for assessing mathematical creativity in schoolchildren.
*Educational Studies in Mathematics, 18*(1), 59–74.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - NAEP Mathematics Framework for the 2011 National Assessment of Educational Progress. (2012). Retrieved October 2012, from http://www.nagb.org/publications/frameworks.html
- National Council of Teachers of Mathematics. (2000).
*Principles and standards of school mathematics*. Reston, VA: Author.Google Scholar - 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 - 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 - 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 - 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 - Poincaré, H. (1913).
*The foundations of science*. New York, NY: The Science Press.Google Scholar - Poincaré, H. (1948).
*Science and method*. New York, NY: Dover.Google Scholar - Prouse, H. L. (1967). Creativity in school mathematics.
*The Mathematics Teacher, 60*, 876–879.Google Scholar - Scott, W. A. (1962). Cognitive complexity and cognitive flexibility.
*Sociometry, 25*, 405–414.CrossRefGoogle Scholar - Sfard, A. (1998). On two metaphors for learning and the dangers of choosing just one.
*Educational Researcher, 27*(2), 4–13.CrossRefGoogle Scholar - Silver, E. A. (1994). On mathematical problem posing.
*For the Learning of Mathematics, 14*(1), 19–28.Google Scholar - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - Singh, B. (1988).
*Teaching-learning strategies and mathematical creativity*. Delhi, India: Mittal.Google Scholar - Smilansky, J. (1984). Problem solving and the quality of invention: An empirical investigation.
*Journal of Educational Psychology, 76*(3), 377–386.CrossRefGoogle Scholar - 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 - Sriraman, B. (2009). The characteristics of mathematical creativity.
*The International Journal on Mathematics Education [ZDM], 41*, 13–27.CrossRefGoogle Scholar - Sternberg, R. J., & Lubart, T. I. (1991). Creating creative minds.
*Phi Delta Kappan, 72*, 608–614.Google Scholar - 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 - 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 - Torrance, E. P. (1974).
*Torrance tests of creative thinking: Norms, technical manual*. Lexington, MA: Ginn.Google Scholar - 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 - 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 - 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