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
- What am I doing?
- Why am I doing it?
- How does it help me?
IV. Beliefs and affects
Typical student beliefs about the nature of mathematics
- Mathematics problems have one and only one right answer.
- There is only one correct way to solve any mathematics problem -- usually the rule the teacher has most recently demonstrated to the class.
- Ordinary students cannot expect to understand mathematics; they expect simply to memorize it, and apply what they have learned mechanically and without understanding.
- Mathematics is a solitary activity, done by individuals in isolation.
- Students who have understood the mathematics they have studied will be able to solve any assigned problem in five minutes or less.
- The mathematics learned in school has little or nothing to do with the real world.
- Formal proof is irrelevant to processes of discovery or invention.
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.