| The research of Larkin (1977, 1979), De Groot (1965), Chase and Simon (1973), Dee Lucas and Larkin (1986), Lucas (1972) and Kalman (1980) provided the framework for (a) motivational questions to programs investigated and (b) theorizing about a distribution that models novices' generation of mathematics concepts when solving problems. The problem solving processes of six novices (i.e., undergraduate seniors), six intermediates (i.e., beginning graduate students) and six specialists (i.e., advanced graduate students) in mathematical analysis were examined for patterns in their organization of mathematics knowledge.;One purpose of this research was to investigate differences among novices, intermediates and specialists in mathematics on the prime information features of mathematics problems that they frequently use to decide on what is essential for obtaining solutions. The other purpose was to study the organization and recollection of advanced mathematics problem information that related to performance on problem solving. Specific questions were: (1) Do these groups of subjects differ when compared on how they generate problem information? (2) Are students' problem-solving performances at problem solving related to (a) the generation and use of mathematics theorems and (b) the generation and use of mathematics concepts? (3) Does any meaningful pattern fit students' generation of mathematics concepts during problem solving? (4) Can a coding system be developed that reliably transforms the problem solving activities?;All 18 subjects were taped recorded as they independently solved three undergraduate level mathematical analysis problems. The intermediates and specialists also solved three graduate level mathematical analysis problems. A relatively reliable coding scheme was developed.;Analysis of variance (ANOVA) of the coded data indicated that the novices were more likely than the intermediates and the specialists to generate mathematics concepts when solving problems that require the use and development of theorems and that the specialists were more likely than the novices or the intermediates to generate relevant theorems. Students' generation of mathematics theorems was found to be strongly related to problem-solving performance while their generation of mathematics concepts was not. The novice-generation of mathematics concepts did not reject the null hypothesis for Kolmogorov-Smirnov goodness-of-fit test for Poisson distribution. It was concluded that when novices solve problems, they generate mathematics concepts that are not meaningfully connected with others in problem contexts. The conclusions are tentative as a result of limitations of the research. The study provided clues that could form a base for other future investigations suggested. |
