The purpose of this study was the construction of a theoretical model for analyzing mathematical problem solving. A list of general problem-solving abilities was generated through a literature search. This list was narrowed to eight basic abilities pertinent to mathematics problem solving. Each of these was operationally defined and exemplified in a document sent to experts in mathematics education and research mathematicians for their comments and validity assessments. No basic changes were suggested. A collection of problems which are not amenable to solution by a single use of an algorithm was then presented to mathematics students from grade seven through graduate school. These students were asked to "think aloud" during the taped problem-solving sessions. Analysis of these protocols showed that the eight abilities comprising the model were necessary and sufficient. Applications of the model to analysis of problem solving difficulty and errors, and to the teaching of problem solving, are suggested. (SD)