**Mathematical Problem Solving (A. Schoenfeld) **

*Overview:*

Alan Schoenfeld presents the view that understanding and teaching mathematics should be approached as a problem-solving domain. According to Schoenfel d (1985), four categories of knowledge/skills are needed to be successful in mathematics: (1) Resources - proposition and procedural knowledge of mathematics, (2) heuristics - strategies and techniques for problem solving such as wor king backwards, or dr awing figures, (3) control - decisions about when and what resources and strategies to use, and (4) beliefs - a mathematical "world view" that determines how someone approaches a problem.

Schoenfeld's theory is supported by extensive protocol analysis of students solving problems. The theoretical framework is based upon much other work in cognitive psychology, particularly the work of Newell & Simon. Schoenfeld (1987) places more emphasis on the importance of metacognition and the cultural components of learning mathematics (i.e., belief systems) than in his original formulation.

*Scope/Application:*

Schoenfeld's research and theory applies primarily to college level mathematics.

*Example:*

Schoenfeld (1985, Chapter 1) uses the following problem to illustrate his theory: Given two intersecting straight lines and a point P marked on one of them, show how to construct a circle that is tangent to both lines including point P. Examples of resour ce knowledge include the procedure to draw a pe rpendicular line from P to the center of the circle and the significance of this action. An important heuristic for solving this problem is to construct a diagram of the problem. A control strategy might invo lve the decision to construct an actual circle and line segments using a compass and protractor. A belief that might be relevant to this problem is that solutions should be empirical (i.e., constructed) rather than derived.

*Principles:*

1. Successful solution of mathematics problems depends up on a combination of resources, heuristics, control processes and belief, all of which must be learned and taught.

*References:*

Schoenfeld, A. (1985). Mathematical Problem Solving. New York: Academic Press.

Schoenfeld, A. (1987). Cognitive Scien ce and Mathematics Education. Hillsdale, NJ: Erlbaum Assoc.