Abstract
The links between the mathematical and cognitive models that interact during problem solving are explored with the purpose of developing a reference framework for designing problem-posing tasks. When the process of solving is a successful one, a solver successively changes his/her cognitive stances related to the problem via transformations that allow different levels of description of the initial wording. Within these transformations, the passage between successive phases of the problem-solving process determines four operational categories: decoding (transposing the text into more explicit relations among the data and the operating schemes, induced by the constraints of the problem), representing (transposing the problem via a generated mental model), processing (identifying an associated mathematical model based on the mental configurations suggested by the problem and own mathematical competence), and implementing (applying identified mathematical techniques to the particular situation of the problem, with the purpose of drafting a conventional solution). The study of this framework in action offers insights for more effective teaching and can be used in problem posing and problem analysis in order to devise questions more relevant for deep learning.
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References
Ashlock, R. B. (2002). Error patterns in computation: Using error patterns to improve instruction (8th ed.). Upper Saddle River, NJ: Merrill Prentice Hall.
Bloom, B. S., & Broder, L. J. (1950). Problem-solving processes of college students. Chicago: University of Chicago Press.
Brown, S., & Walter, M. (2005). The art of problem posing (3rd ed.). Mahwah, NJ: Lawrence Erlbaum Associates.
Carey, S., & Spelke, E. (1994). Domain-specific knowledge and conceptual change. In L. Hirschfeld & S. Gelman (Eds.), Mapping the mind: Domain specificity in cognition and culture (pp. 169–200). Cambridge, UK: Cambridge University Press.
Christou, C., Mousoulides, N., Pittalis, M., Pitta-Pantazi, D., & Sriraman, B. (2005). An empirical taxonomy of problem posing processes. ZDM, 37(3), 149–158.
Coben, D. (2003). Adult Numeracy: Review of research and related literature. Research review. National Research and Development Centre for Adult Literacy and Numeracy. http://www.nrdc.org.uk/publications_details.asp?ID=35#. Accessed 9 January 2011.
Dillon, J. T. (1982). Problem finding and solving. The Journal of Creative Behavior, 16(2), 97–111.
Duffin, J., & Simpson, A. (2000). When does a way of working become a methodology? The Journal of Mathematical Behavior, 19, 175–188.
Duncker, K. (1945). On problem solving. Psychological Monographs, 58(5). (Whole # 270.) Washington, DC: American Psychological Association.
Ellerton, N. F., & Clements, M. A. (1992). Implications of Newman research for the issue of “What is basic in school mathematics?”. In B. Southwell, B. Perry, & K. Owens (Eds.), Space: The first and final frontier (pp. 276–284). Sydney: MERGA. Proceedings of the 20th annual conference of the Mathematics Education Research Group of Australasia.
Jay, E. S., & Perkins, D. N. (1997). Problem finding: The search for mechanism. In M. A. Runco (Ed.), The creativity research handbook (pp. 257–293). Cresskill, New Jersey: Hampton Press.
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.
Mason, J. (1998). Researching from the Inside in mathematics education. In A. Sierpinska & J. Kilpatric (Eds.), Mathematics education as a research domain: A search for identity (pp. 357–377). Norwell, MA: Kluwer.
Newman, M. A. (1977). An analysis of sixth-grade pupils' errors on written mathematical tasks. Victorian Institute for Educational Research Bulletin, 39, 31–43.
Newman, M. A. (1983). Strategies for diagnosis and remediation. Sydney: Harcourt, Brace Jovanovich.
Noveanu, G. N., & Singer, F.M. (2008). Romania. In I.V.S. Mullis & M. O. Martin (Eds.) TIMSS 2007 Encyclopedia a guide to mathematics and scienece education around the world, vol 2. (481–490). IEA, TIMSS & PIRLS International Study Center, Lynch School of Education, Boston College.
Pólya, G. (1945). How to solve it. Princeton University Press.
Pelczer, I., Singer, F. M., & Voica, C. (2011). An analysis of relevant hints in problem solving. In B. Ubuz (Ed.), Developing mathematical thinking, vol 1 (p. 370). Ankara, Turkey: PME. Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education.
Rowland, T., Heal, C., Barber, P., & Martyn, S. (1998). Mind the “gaps”: Primary teacher trainees' mathematics subject knowledge. In E. Bills (Ed.), Proceedings of the British Society for Research in Learning Mathematics day conference at Birmingham (pp. 91–96). Coventry, UK: University of Warwick.
Rowland, T., Huckstep, P., & Thwaites, A. (2003). The knowledge quartet. In J. Williams (Ed.), Proceedings of the British Society for Research into Learning Mathematics 23(3), 97–103.
Ryan, J., & Williams, J. (2000). Mathematical discussions with children: Exploring methods and misconceptions as a teaching strategy. Manchester, UK: Centre for Mathematics Education, University of Manchester.
Schoenfeld, A. H. (1980). Teaching problem solving skills. The American Mathematical Monthly, 87(10), 794–805.
Silver, E. A. (Ed.). (1985). Teaching and learning mathematical problem solving: Multiple research perspectives. Hillsdale, NJ: Erlbaum.
Silver, E., 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.
Singer, F. M. (2007). Beyond conceptual change: Using representations to integrate domain-specific structural models in learning mathematics. Mind, Brain, and Education, 1(2), 84–97.
Singer, F. M. (2009). The dynamic infrastructure of mind—A hypothesis and some of its applications. New Ideas in Psychology, 27, 48–74.
Singer, F. M., & Voica, C. (2008). Operational categories for assessing problem solving (in Romanian). ROMAI Educational Journal, 3, 36–41.
Singer, F. M., Ellerton, N., Cai, J., & Leung, E. (2011). Problem posing in mathematics learning and teaching: a research agenda. In B. Ubuz (Ed.), Developing mathematical thinking, vol 1 (pp. 137–166). Ankara, Turkey: PME. Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education.
Singer, F. M., Pelczer, I., & Voica, C. (2011). Problem posing and modification as a criterion of mathematical creativity, In In T. Rowland & E. Swoboda (Eds.) Proceedings of the 7th Conference of the European Society for Research in Mathematics Education (CERME 7) University of Rzeszów, Poland, 9–13 February, 2011.
Spelke, E. (2003). What makes us smart? Core knowledge and natural language. In D. Gentner & S. Goldin-Meadow (Eds.), Language in mind (pp. 277–311). Cambridge, MA: MIT Press.
Stehlikova, N. (2003). Emergence of mathematical knowledge structures. Introspection. In N. Pateman, B. Dougherty & T. Zilliox (Eds.), Proceedings of the 27th International Group for the Psychology of Mathematics Education, Honolulu, Hawaii, Volume 4, 251–258.
Voica, C., & Singer, M. (2012). Creative contexts as ways to strengthen mathematics learning. In M. Aniţei, M. Chraif & C.Vasile (Guest Eds.), Procedia—Social and Behavioral Sciences, PSIWORLD 2011, Volume 33, 538–542.
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The paper was partially supported by the project POSDRU/17/1.1/G/37412.
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Singer, F.M., Voica, C. A problem-solving conceptual framework and its implications in designing problem-posing tasks. Educ Stud Math 83, 9–26 (2013). https://doi.org/10.1007/s10649-012-9422-x
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DOI: https://doi.org/10.1007/s10649-012-9422-x