Problem solving has always been concern of Psychologists,and great achievements have been made in different fields. The history of psychology on mathematical problem solving develops relatively late.Making a comprehensive study both at home and abroad,one can find many researchs and achievements have been made on mathematical problem soiving. For example,problem representation; the strategy of problem solving;the psychological stracture of ability on mathematical problem solving;problem solving transfer;metacognition on problem solving,and so on.Howevre these researches have disadvantages.they are: (1)the visual angle of study is not comprehensive.Many psychologists only show concern for psychological phenomenon and the research for rules on mathematical problem solving,but they do not concern the combination of psychological research and mathematical teaching practice.On the other hand,mathematical educationist used to make deductive ways to infer the psychology on problem solving,which lacks experimental research.(2)the level of study is low .The targets of investigate are pupils in primary schools and junior high schools,the primary materials for study is invariable mathematics,including arithmatical problems and geometrical problems, which lacks sufficient study on problem solving of advanced mathematical thinking.(3 )the methods of study require father development.Psychologiest provide a suit of comparatively integrate methods for psychological research.However mathematical educationist lack such methods. They prefer thinking to makeing experimental research.How to apply the methods of psychological research to those of mathematical problem solving and thus combine qualitative way with quantitative way requires further study.The situation mentioned above is more conspicuous at home. This leads to many investigative blanks blanks which makes it hard to construct the system of psychology on mathematical problem solving(or psychology in mathematical learning).Primary contents and results hi this thesis:1. Mathematical description of mathematical problem space.A mathematical problem is composed of the initial state,the goal state and the rules of problem solving.Based on certain rules,mathematical problem solving is a process of problem solver from initial state,passing through a series of medial states and finally reaching the goal stste.The operate based on rules of problem solving is called operator.All medial states and all operators in the course of problm solving is called problem space.The road(S 0 ,S,,......,S ? coupling initial state and goal state is called a solution in problem space.Thelength of road is said to the length of solution,the solution that length is the shortest is calledoptimizat solution,a optimizat solution is noted L(p) or L(S 0, S ?.Two problems are said to isomorphic,if exist a 1-1 mappingf: S1-S2,such that Z , ,Z 2 € S ,, Z, R1Z2 f(Z1)R2/(Z2) ,here R , ,R 2 are relation in S , ,S 2 .If twoproblems are isomorphic,then note down P, =P2 .Two problems is said to homostasis,if exista mapping f : S1 - S 2 ,such that Z, ,Z 2 S1, Z1R1Z2 = f(Z1)R2/(Z2) or f(Z1)=/Z 2).If two problems P1, P 2 are homostasis,then noted down P1 P 2.In this paper,we obtained 6 results on mathematical problem space.2. The cognitive model of mathematical problem solving.The cognitive model on mathematical problem solving is composed of problem representation; pattern recognition;problem transfer,self-controlling,as well as the solver's mathematical knowledge and strategy of problem solving.This cognitive model is a dynamic system.3. The theory in CPFS frame.The schema that all equiralent definitions to a mathematical concept is called a concept field about concept C. If a set of concepts C1, C2,.........Cn content C1R1C2R2C3R3.......Rn-1Cn, here Ri(i=1,2,....., n-1) show some mathematical abstract relations,well then it be said to a concept chain.If every concept chain in m concept chains at least intersect with other one,then this schema of co... |