• Igor’ KontorovichEmail author
  • Boris Koichu


This paper is concerned with organizational principles of a pool of familiar problems of expert problem posers and the ways by which they are utilized for creating new problems. The presented case of Leo is part of a multiple-case study with expert problem posers for mathematics competitions. We present and inductively analyze the data collected in a reflective interview and in a clinical task-based interview with Leo. In the first interview, Leo was asked to share with us the stories behind some problems posed by him in the past. In the second interview, he was asked to pose a new competition problem in a thinking-aloud mode. We found that Leo’s pool of familiar problems is organized in classes according to certain nesting ideas. Furthermore, these nesting ideas serve him in posing problems that, ideally, are perceived by Leo as novel and surprising not only to potential solvers, but also to himself. Because of the lack of empirical research on experts in mathematical problem posing, the findings are discussed in light of research on experts in problem solving and on novices in mathematical problem posing.

Key words

experts mathematical competitions mathematical problem nesting ideas problem posing 


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  1. Carlson, M. & Bloom, I. (2005). The cyclic nature of problem solving: An emergent multidimensional problem-solving framework. Educational Studies in Mathematics, 58, 45–75.CrossRefGoogle Scholar
  2. Chi, M. T. H., Feltovich, P. J. & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices. Cognitive Science, 5, 121–152.CrossRefGoogle Scholar
  3. Crespo, S. & Sinclair, N. (2008). What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems. Journal of Mathematics Teacher Education, 11(5), 395–415.CrossRefGoogle Scholar
  4. Elder, S. (2006). Geometry: Basics and classics. Retrieved February 16, 2011, from
  5. Ericsson, K. A. (2006). The influence of experience and deliberate practice on the development of superior expert performance. In K. A. Ericsson, N. Charness, P. Feltovich & R. R. Hoffman (Eds.), Cambridge handbook of expertise and expert performance (pp. 685–706). Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
  6. Furinghetti, F. & Morselli, F. (2009). Every unsuccessful problem solver is unsuccessful in his or her own way: Affective and cognitive factors in proving. Educational Studies in Mathematics, 70, 71–90.CrossRefGoogle Scholar
  7. Galperin, G.A. & Zemliakov, A.N. (1990). Matematicheskie Biliardi [Mathematical Billiards]. Kvant.Google Scholar
  8. Koichu, B. & Andžaāns, A. (2009). Mathematical creativity and giftedness in out-of-school activities. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in Mathematics and Education of Gifted Students (pp. 285–308). Rotterdam, The Netherlands: Sense Publishers.Google Scholar
  9. Koichu, B. & Kontorovich, I. (2013). Dissecting success stories on mathematical problem posing: a case of the Billiard Task. Educational Studies in Mathematics, 83(1), 71–86.Google Scholar
  10. Konstantinov, N. N. (1997). Tournir gorodov i matematicheskaya olympiada [Tournament of the Towns and mathematical Olympiad]. Matematicheskoe Prosveschenie, 3(1), 164–174 (in Russian).Google Scholar
  11. Kontorovich, I. & Koichu, B. (2012). Feeling of innovation in expert problem posing. Nordic Studies in Mathematics Education, 17(3–4), 199–212.Google Scholar
  12. Kontorovich, I., Koichu, B., Leikin, R. & Berman, A. (2012). An exploratory framework for handling the complexity of mathematical problem posing in small groups. Journal of Mathematical Behavior, 31(1), 149–161.Google Scholar
  13. Liljedahl, P. (2009). In the words of the creators. In R. Leikin, A. Berman & B. Koichu (Eds.), Creativity in mathematics and education of gifted students (pp. 51–70). Rotterdam, The Netherlands: Sense Publishers.Google Scholar
  14. Miller, G. A. (1956). The magical number seven, plus or minus two: Some limits on our capacity for processing information. The Psychological Review, 63, 81–97.CrossRefGoogle Scholar
  15. Norman, I. & Bakar, M. N. (2011). Secondary school students’ problem posing strategies: Implications to secondary school students’ problem posing performances. Journal of Edupres, 1, 1–8.Google Scholar
  16. Pelczer, I. & Gamboa, F. (2009). Problem posing: Comparison between experts and novices. In M. Tzekaki, M. Kaldrimidou & C. Sakonidis (Eds.), Proceedings of the 33th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 353–360). Thessaloniki, Greece: PME.Google Scholar
  17. Schoenfeld, A. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grows (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). New York: Macmillan.Google Scholar
  18. Schoenfeld, A. H. & Herrmann, D. J. (1982). Problem perception and knowledge structure in expert and novice mathematical problem solvers. Journal of Experimental Psychology: Learning, Memory, and Cognition, 8(5), 484–494.Google Scholar
  19. Sharigin, I.F. (1991). Otkuda berutsia zadachi? [Where do problems come from?] Part I, Kvant, 8, 42–48; part II, Kvant, 9, 42–49 (in Russian).Google Scholar
  20. Silver, E. & Metzger, W. (1989). Aesthetic influence on expert problem solving. In D. B. McLeod & V. M. Adams (Eds.), Affect and mathematical problem solving (pp. 59–74). New York: Springer.CrossRefGoogle Scholar
  21. Silver, E. A., Mamona-Downs, J., Leung, S. & Kenney, P. A. (1996). Posing mathematical problems: An exploratory study. Journal for Research in Mathematics Education, 27(3), 293–309.CrossRefGoogle Scholar
  22. Stoyanova, E. (2005). Problem posing strategies used by years 8 and 9 students. Australian Mathematics Teacher, 61(3), 6–11.Google Scholar
  23. Thomas, D. T. (2006). A general inductive approach for analyzing qualitative evaluation data. American Journal of Evaluation, 27(2), 237–246.CrossRefGoogle Scholar
  24. Thrasher, T.N. (2008). The benefits of mathematics competitions. Alabama Journal of Mathematics, Spring–Fall, 32:59–63.Google Scholar
  25. Van Someren, M. Y., Barnard, Y. F. & Sandberg, J. A. C. (1994). The think aloud method: A practical guide to modeling cognitive processes. London: Academic.Google Scholar
  26. Wilkerson-Jerde, M. H. & Wilensky, U. J. (2011). How do mathematicians learn math?: Resources and acts for constructing and understanding mathematics. Educational Studies in Mathematics, 78, 21–43.CrossRefGoogle Scholar

Copyright information

© National Science Council, Taiwan 2013

Authors and Affiliations

  1. 1.Department of Education in Technology and ScienceTechnion—Israel Institute of TechnologyHaifaIsrael

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