Can Heuristics Be Taught?
The Elements of a Theory and a Report on the Teaching
of General Mathematical Problem-Solving Skills
Alan H. Schoenfeld
Can students be taught general strategies that truly enhance their abilities to
solve mathematical problems? Or are the heuristics described by Polya and
others merely a description of the actions of accomplished problem solvers? Are
they essentially valueless as prescriptions for problem solving? While many
mathematicians are convinced that they employ heuristics, there is little evidence that general problem-solving skills can be taught.I offered a course based on the applications of heuristics to mathematics majors at the University of California, Berkeley. This article presents the rationale
for heuristics and notes some questions about their effectiveness in the teaching
of problem solving. I offer some suggestions regarding these questions, and
describe the course I used to implement these suggestions. I discuss what we can
and cannot expect students to assimilate--heuristics they can learn to use and
obstacles that prevent them from employing others effectively.
SECTION 1. Problem Solving in Perspective: Theory and Practice
George Polya How to Solve It
was published in 1945. That and his subsequent work laid the foundations for the study of general strategies for problem
solving in mathematics, focusing on the broad strategies he called "heuristics."
Definitions vary, but the following is compatible with Polya's usage:
A heuristic is a general suggestion or strategy, independent of subject matter,
that helps problem solvers approach, understand, and/or efficiently marshal
their resources in solving problems.
Examples of heuristics are: "draw a diagram if possible," "try to establish
subgoals," and "exploit analogous problems"; a more complete list is given in Section 3. A rationale for the study and teaching of heuristics is the following:
|1. ||Through the course of his career, a problem solver develops an idiosyncratic
style and method of problem solving. A systematic use of these strategies
may take years to develop fully.|
|2. ||In spite of these idiosyncracies, there is a surprising degree of homogeneity in
the approaches of expert problem solvers.|