Schoenfeld defines a hierarchy of four elements of “knowledge and behavior necessary for an adequate characterization of mathematical problem-solving performance”:
Resources
Heuristics
Control
Belief systems
Schoenfeld defines resources as follows:
Mathematical knowledge possessed by the individual that can be brought to bear on the problem at hand:
Intuitions and informal knowledge regarding the domain
Facts
Algorithmic procedures
Routine nonalgorithmic procedures
Propositional knowledge about the agreed-upon rules for working in the domain.
Schoenfeld draws an analogy between resources in other fields of intellectual endeavor and those of mathematical problem solving. He notes that chess masters develop a “vocabulary” of tens of thousands of complete game configurations and that they also have learned a stereotypical response to this configuration.
In artificial intelligence (AI), this type of knowledge is stored as condition-action pairs, also called “productions.” Expert systems are often built using knowledge structured in this way.
His research showed that in mathematics, too, expert problem solvers are able to match stereotypical problem situations to stereotypical responses quickly. He also argues that resources of this type can be quite complex, even though they invoke automatic responses.
Schoenfeld defines heuristics as:
Strategies and techniques for making progress on unfamiliar problems; rules of thumb for effective problem solving:
Drawing figures
Introducing notation
Exploiting related problems
Reformulating problems
Working backwards
Testing and verification procedures
Schoenfeld believes that the heuristics used in mathematics education are actually names for families of closely related heuristics. These heuristics have to be developed in detail before they can be used by students in problem solving. He also argues that while heuristics are useful, they’re no substitute for subject-matter knowledge.
Schoenfeld defines control as:
Global decisions regarding the selection and implementation of resources and strategies:
Planning
Monitoring and assessment
Decision making
Conscious metacognitive acts
Control decisions determine which way to go and which way to abandon. They determine the use of resources, particularly time. Schoenfeld asserts that an effective control mechanism must have “periodic monitoring and assessment of solutions.”
He describes a control strategy that he used in teaching integration in a calculus. The three steps of this strategy successively apply more general, difficult, and time-consuming procedures to the problem. At each step, the form of the integrand was matched against forms to which a set of procedures was applicable.
Schoenfeld also describes a general strategy that he taught in a class on mathematical problem solving. It prescribes five phases: analysis, design, exploration, implementation, and verification. The strategy integrated many common heuristics, and the students were trained in monitoring their own progress and assessing their solutions. The evidence from his studies of these classes indicates that the students became much more adept at dealing with problems unlike those that were presented in class. He summarizes their development thus: “The most impressive result is that the students did quite well on a set of problems that had been placed on their exam precisely because I did not know how to solve them!”
Schoenfeld defines belief systems as:
One’s “mathematical world view,” the set of (not necessarily conscious) determinants of an individual’s behavior:
Self
The environment
The topic, and mathematics itself