Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Lester, F.K.: 1994, ‘Musings about mathematical problem solving research: 1970–1994’, Journal for Research in Mathematics Education 25, 660–675.
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.
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.
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.
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.
Polya, G.: 1957, How To Solve It; A New Aspect of Mathematical Method, 2nd ed., Doubleday, Garden City.
Schoenfeld, A.: 1985a, Mathematical Problem Solving, Academic Press, Orlando, FL.
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.
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.
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.
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.
Schoenfeld, A.H.: 1989, ‘Explorations of students’ mathematical beliefs and behavior’, Journal for Research in Mathematics Education 20, 338–355.
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.
Silver, E.A.: 1982, ‘Thinking about problem solving: Toward an understanding of metacognitive aspects of mathematical problem solving’, in Conference on Thinking, Suva, Fiji.
Simon, M.A.: 1996, ‘Beyond inductive and deductive reasoning: The search for a sense of knowing’, Educational Studies in Mathematics 30, 197–210.
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.
Strauss, A.L. and Corbin, J.M. 1990, Basics of Qualitative Research: Grounded Theory, Procedures and Techniques, Sage Publications, Newbury Park.
Vinner, S.: 1997, ‘The pseudo-conceptual and the pseudo-analytical thought processes in mathematics learning’, Educational Studies in Mathematics 34, 97–129.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Carlson, M.P., Bloom, I. The Cyclic Nature of Problem Solving: An Emergent Multidimensional Problem-Solving Framework. Educ Stud Math 58, 45–75 (2005). https://doi.org/10.1007/s10649-005-0808-x
Issue Date:
DOI: https://doi.org/10.1007/s10649-005-0808-x