Alan Schoenfeld presents the view that understanding and teaching mathematics should be approached as a problem-solving domain. According to Schoenfeld (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 working backwards, or drawing 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.
Schoenfeld's research and theory applies primarily to college level mathematics.
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 and has point P as its point of tangency to the lines. Examples of resource knowledge include the procedure to draw a perpendicular 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 involve 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.
Schoenfeld, A. (1985). Mathematical Problem Solving. New York: Academic Press.
Schoenfeld, A. (1987). Cognitive Science and Mathematics Education. Hillsdale , NJ: Erlbaum Assoc.
For more on Schoenfeld’s work, see his home page at: http://gse.berkeley.edu/faculty/AHSchoenfeld/AHSchoenfeld.html
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