High-level Mathematical Concepts Of Cognitive Development

The cognition of mathematical concept has always been concern of mathematics education psychologists, and great achievements have been made in different fields. The history of cognition on advanced mathematical concept develops relatively late. Making a comprehensive study both at home and abroad, one can find many research and achievements have been made on advanced mathematical concept. For example, there are many special concepts, such as 'concept definition', 'concept domain', 'concept image' and so on. However these researches have defects. They are: (1) the visual angle of study is not comprehensive. Many mathematicians only show concern for concept structure and logical foundation, 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 cognition of concept, which lacks deep research. (2)the methods of study require father development. Psychologists provide a suit of comparatively integrate methods for psychological research. However mathematical educationist lack such methods. They prefer thinking to making experimental research. How to apply the methods of psychological research to cognition of mathematical advanced concept and combine qualitative way and quantitative way is a key study. If the situation mentioned above is not solved, it is hard to construct the system of cognition on mathematical concept (or psychology in mathematical education). Primary contents and results in this thesis: 1. Topology psychology structure of advanced mathematical concept(TPS). 2. The experimental investigation on problem representation. (l)There exist differences on the cognition of advanced mathematical concept among students of different ages. The individual' s knowledge of mathematics and the level of thinking would influence problem mental representation. (2)There exist obvious differences between the man and the woman. The cause is the difference of cognitive way. (3)The mental representation has influence on advanced mathematical concept.(4) Problem representation help to establish advanced mathematical concept. 3.Through the experimental investigation, we study the relevant problems, for example, reflective abstraction & encapsulation. In a word, the cognitive process of advanced mathematical concept is a senior thinking. The process is not only an individual' s constructing knowledge, but also is a social constructing knowledge.