Educational Studies in Mathematics

, Volume 58, Issue 1, pp 45–75 | Cite as

The Cyclic Nature of Problem Solving: An Emergent Multidimensional Problem-Solving Framework



This paper describes the problem-solving behaviors of 12 mathematicians as they completed four mathematical tasks. The emergent problem-solving framework draws on the large body of research, as grounded by and modified in response to our close observations of these mathematicians. The resulting Multidimensional Problem-Solving Framework has four phases: orientation, planning, executing, and checking. Embedded in the framework are two cycles, each of which includes at least three of the four phases. The framework also characterizes various problem-solving attributes (resources, affect, heuristics, and monitoring) and describes their roles and significance during each of the problem-solving phases. The framework’s sub-cycle of conjecture, test, and evaluate (accept/reject) became evident to us as we observed the mathematicians and listened to their running verbal descriptions of how they were imagining a solution, playing out that solution in their minds, and evaluating the validity of the imagined approach. The effectiveness of the mathematicians in making intelligent decisions that led down productive paths appeared to stem from their ability to draw on a large reservoir of well-connected knowledge, heuristics, and facts, as well as their ability to manage their emotional responses. The mathematicians’ well-connected conceptual knowledge, in particular, appeared to be an essential attribute for effective decision making and execution throughout the problem-solving process.


affect conceptual knowledge control mathematical behavior mathematics practices metacognition problem solving 


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  1. Carlson, M.P.: 1998, ‘A cross-sectional investigation of the development of the function concept’, in A.H. Schoenfeld, J. Kaput, and E. Dubinsky (eds.), Research in Collegiate Mathematics Education III, Vol. 7, American Mathematical Association, Providence, pp. 114–162.Google Scholar
  2. Carlson, M.P.: 1999a, ‘The mathematical behavior of six successful mathematics graduate students: Influences leading to mathematical success’, Educational Studies in Mathematics 40, 237–258.CrossRefGoogle Scholar
  3. Carlson, M.P.: 1999b, ‘A study of the problem solving behaviors of mathematicians: Metacognition and mathematical intimacy in expert problem solvers’, in 24th Meeting of the International Group for the Psychology of Mathematics Education, Haifa, Israel, pp. 137–144.Google Scholar
  4. DeBellis, V.A.: 1998, ‘Mathematical intimacy: Local affect in powerful problem solvers’, in Twentieth Annual Meeting of the North American Group for the Psychology of Mathematics Education, Eric Clearinghouse for Science, Mathematics and Environmental Education, pp. 435–440.Google Scholar
  5. DeBellis, V.A. and Goldin, G.A.: 1997, ‘The affective domain in mathematical problem solving’, in Twenty-first Annual Meeting of the International Group for the Psychology of Mathematics Education, Lahti, Finland, University of Helsinki, pp. 209–216.Google Scholar
  6. DeBellis, V.A. and Goldin, G.A.: 1999, ‘Aspects of affect: Mathematical intimacy, mathematical integrity’, in Twenty-third Annual Meeting of the International Group of for the Psychology of Mathematics Education, Haifa, Israel, Technion – Israel Institute of Technology, pp. 249–256.Google Scholar
  7. DeFranco, T.C.: 1996, ‘A perspective on mathematical problem-solving expertise based on the performances of male Ph.D. mathematicians’, Research in Collegiate Mathematics, II Vol. 6, American Mathematical Association, Providence, RI, pp. 195–213.Google Scholar
  8. Geiger, V. and Galbraith, P.: 1998, ‘Developing a diagnostic framework for evaluating student approaches to applied mathematics problems’, International Journal of Mathematics, Education, Science and Technology 29, 533–559.Google Scholar
  9. Hannula, M.: 1999, ‘Cognitive emotions in learning and doing mathematics’, in Eighth European Workshop on Research on Mathematical Beliefs, Nicosia, Cyprus, University of Cyprus, pp. 57–66.Google Scholar
  10. Krutetskii, V.A.: 1969, ‘An analysis of the individual structure of mathematical abilities in schoolchildren’, in J. Kilpatrick and I. Wirzup (Eds.), Soviet Studies in the Psychology of Learning and Teaching Mathematics, Vol. 2, Chicago, University of Chicago.Google Scholar
  11. Lesh, R.: 1985, ‘Conceptual analysis of problem-solving performance’, in E.A. Silver (Ed.), Teaching and Learning Mathematical Problem Solving: Multiple Research Perspectives, Lawrence Erlbaum Association, Hillsdale, pp. 309–329.Google Scholar
  12. Lesh, R. and Akerstrom, M.: 1982, ‘Applied problem solving: Priorities for mathematics education research’, in F.K. Lester and J. Garofalo (eds.), Mathematical Problem Solving: Issues in Research, Franklin Institute Press, Philadelphia, pp. 117–129.Google Scholar
  13. Lester, F.K.: 1994, ‘Musings about mathematical problem solving research: 1970–1994’, Journal for Research in Mathematics Education 25, 660–675.Google Scholar
  14. Lester, F.K., Garofalo, J. and Kroll, D.: 1989a, The Role of Metacognition in Mathematical Problem Solving, Final Report, Technical Report, Indiana University Mathematics Education Development Center, Bloomington.Google Scholar
  15. Lester, F.K., Garofalo, J. and Kroll, D.L.: 1989b, ‘Self-confidence, interest, beliefs, and metacognition: Key influences on problem-solving behavior’, in D.B. McLeod and V.M. Adams (eds.), Affect and Mathematical Problem Solving: A New Perspective, Springer-Verlag, New York, pp. 75–88.Google Scholar
  16. Mason, J. and Spence, M.: 1999, ‘Beyond mere knowledge of mathematics: The importance of knowing-to-act in the moment’, Educational Studies in Mathematics 38, 135–161.CrossRefGoogle Scholar
  17. McLeod, D.B.: 1992, ‘Research on affect in mathematics education: A reconceptualization’, in D.A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and learning, Macmillan Publishing Company, New York, NY, pp. 575–595.Google Scholar
  18. Polya, G.: 1957, How To Solve It; A New Aspect of Mathematical Method, 2nd ed., Doubleday, Garden City.Google Scholar
  19. Schoenfeld, A.: 1985a, Mathematical Problem Solving, Academic Press, Orlando, FL.Google Scholar
  20. Schoenfeld, A.H.: 1982, ‘Some thoughts on problem solving research and mathematics education’, in F.K. Lester and J. Garofalo (eds.), Mathematical Problem Solving: Issues in Research, Franklin Institute PressPhiladelphia, pp. 27–37.Google Scholar
  21. Schoenfeld, A.H.: 1983, ‘The wild, wild, wild, wild, wild world of problem solving: A review of sorts’, For the Learning of Mathematics 3, 40–47.Google Scholar
  22. Schoenfeld, A.H.: 1985, ‘Problem solving in context(s)’, in E.A. Silver (Ed.), The Teaching and Assessing of Mathematical Problem-Solving, Vol. 3, Reston, VA.Google Scholar
  23. Schoenfeld, A.H.: 1987, ‘Cognitive science and mathematics education: An overview’, in A.H. Schoenfeld (Ed.), Cognitive Science and Mathematics Education, Lawrence Erlbaum Associates, London, pp. 1–31.Google Scholar
  24. Schoenfeld, A.H.: 1989, ‘Explorations of students’ mathematical beliefs and behavior’, Journal for Research in Mathematics Education 20, 338–355.Google Scholar
  25. Schoenfeld, A.H.: 1992, ‘Learning to think mathematically: Problem solving, metacognition and sense-making in mathematics’, in D.A. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning, Macmillan Publishing Company, New York, pp. 334–370.Google Scholar
  26. Silver, E.A.: 1982, ‘Thinking about problem solving: Toward an understanding of metacognitive aspects of mathematical problem solving’, in Conference on Thinking, Suva, Fiji.Google Scholar
  27. Simon, M.A.: 1996, ‘Beyond inductive and deductive reasoning: The search for a sense of knowing’, Educational Studies in Mathematics 30, 197–210.CrossRefGoogle Scholar
  28. Stillman, G.A. and Galbraith, P.L.: 1998, ‘Applying mathematics with real world connections: Metacognitive characteristics of secondary students’, Educational Studies in Mathematics 36, 157–195.CrossRefGoogle Scholar
  29. Strauss, A.L. and Corbin, J.M. 1990, Basics of Qualitative Research: Grounded Theory, Procedures and Techniques, Sage Publications, Newbury Park.Google Scholar
  30. Vinner, S.: 1997, ‘The pseudo-conceptual and the pseudo-analytical thought processes in mathematics learning’, Educational Studies in Mathematics 34, 97–129.CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsArizona State UniversityTempeArizona

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