# Mathematical Problem Solving (A. Schoenfeld)

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.

## 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 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.

## Principles

- Successful solution of mathematics problems depends up on a combination
of resource knowledge, 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 Science and Mathematics Education.
Hillsdale
,
NJ: Erlbaum Assoc.

## Related Websites

For more on Schoenfeld’s work, see his home page at: http://gse.berkeley.edu/faculty/AHSchoenfeld/AHSchoenfeld.html

__[ INTRO ]__ [ THEORIES ] [ CONCEPTS ] [ DOMAINS ]

#