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Competencies in Problem-Solving Domains

Schoenfeld's Schema

Based on the research on learning in mathematics, Alan Schoenfeld presents the following theoretical frame characterizing competencies in problem-solving domains:

I. The knowledge base

II. Problem solving strategies

For example: analogy, auxiliary elements, decomposing and recombining, induction, specialization, variation, working backwards (See Polya)

III. Monitoring and control (Metacognition)

Knowing how and when to use resources and strategies effectively and efficiently

IV. Beliefs and affects

Typical student beliefs about the nature of mathematics

V. Practices

Becoming a good mathematical problem solver – becoming a good thinker in any domain – may be as much a matter of acquiring the habits and dispositions of interpretation and sense-making as of acquiring any particular set of skills, strategies, or knowledge. If this is so, we may do well to conceive of mathematics education less as an instructional process (in the traditional sense of teaching specific, well-defined skills or items of knowledge), than as a socialization process. (Resnick, 1989, p. 58)


Schoenfeld, A. (1992). Learning to think mathematically: Problem solving, metacognition, and making sense in mathematics. In D. A. Grouws (Ed.), Handbook of research in mathematics teaching and learning (pp. 334-370). NY: Macmillan Publishing Co.

Pólya, G. (1945; 2nd edition, 1957). How to solve it. Princeton: Princeton University Press.

Pólya, G. (1954). Mathematics and plausible reasoning (Volume 1, Induction and analogy in mathematics; Volume 2, Patterns of plausible inference). Princeton: Princeton University Press.

Pólya, G. (1962,1965/1981). Mathematical Discovery (Volume 1, 1962; Volume 2, 1965). Princeton: Princeton University Press. Combined paperback edition, 1981. New York: Wiley.

Resnick, L. (1989). Treating mathematics as an ill-structured discipline. In R. Charles & E. Silver (Eds.), The teaching and assessing of mathematical problem solving , pp. 32-60. Reston, VA: National Council of teachers of Mathematics.

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