Using ideas developed recently in the theory of understanding language, an analysis is given of the process of understanding mathematical concepts. An analysis also is given of procedural knowledge in mathematics, based on recently developed theories of problem solving including production systems and automaton models. A suggestion is made as to how these theories might be synthesized to provide an analysis of performance of mathematical skills with understanding. Two examples of such performance are described, in the form of theories that are realized as computer programs, and there is discussion of ways in which these theories succeed and fail in representing procedural knowledge with understanding.

Understanding and procedural knowledge in mathematics instruction 1

Sketch-based pruning of a solution space within a formal geometric constraint solver

The naive learner's knowledge may be characterized in terms of rules and higher‐order rules that are not adequate individually for solving desired problems, but which by proper combination can be made so. The neophyte's knowledge is likened to knowing explicit rules for solving particular classes of problems. The master's, by way of contrast, corresponds to knowing rules consisting of more efficient procedures which operate on relatively complex structures. Processes of transition from naive to neophyte and from neophyte to master are shown to be explicable in terms of interactions among appropriate higher and lower‐order rules governed by a single universal control mechanism. Incidental comparisions arc made to other cognitive theories.

Problem solving in schools and beyond: Transitions from the naive to the neophyte to the master

Structural Approach to Instructional Problems.

Picture processing: 1974

https://files.eric.ed.gov/fulltext/ED081640.pdf

Wallace H. Wulfeck

Papers

Higher order rule characterization of heuristics for compass and straight edge constructions in geometry