Abstract
The existence and identification of functions of mathematical concepts reasoning schemata in problem solving processes are proposed in this paper. Firstly, the existence of reasoning schemata that are based on characteristics of mathematical concepts in simple reasoning problems is pointed out. Problem space of mathematical problem solving processes is not only defined by simple operators like chess, but also include rich mathematical concepts. The existence and the role of such mathematical concepts reasoning schemata have been discussed in three research papers understanding of generalized arithmetic. These are (1) student-professor problem (Clement 1982), (2) CSMS algebra test problems (Kuchemann 1981), and (3) "conceptual entity" in problem solving processes (Greeno 1982). Secondly, functions of mathematical concepts reasoning schemata in problem solving processes are identified through several concrete problems. The results show that, mathematical concepts reasoning schemata (1) help using mathematical structures in problem solving processes, (2) connect different knowledge in problem solving processes as cognitive glue (Lesgold 1988), and (3) produce elegant methods in problem solving. These results are applied to research into task analysis of nonroutine problems and task sequencing.
