The Cognitive Analysis Of Mathematical Problem Solving And Its Use In Teaching

Mathematical problem solving(MPS)is one of the goals of mathematical education.The great mathemation G.Polya said the nature of mathematics was problem solvong (PS).He gave us a table of how to solving problem,which included the typical thinking in MPS. But it's not enough because we cannot solve mathematical problem just in Polya's way and the cognitive psychology can help us in MPS.In China some reseachers have done some useful thing in this aspect.This article have three part.Part one tells us the opinions of Dewey and Polya and gives some examples of mathematical problem and the ways in their solving processes. In this part we can find the four process of PS and some essential factors of PS.The factors of PS include the principls of learning and memory,the principls of metacognition and obtain and the cognitive tactics,etc. The process of PS includes reading and express,mathematical modeling and select operator,transfering and using operators and valuating.Part two gave the analysis of MPS in the personal constructivism and social constructivism eyes.It inscribes the obtain of intelligence ability,the mathematical cognitive construction,the improving way of conjunction and the analysis of abstract degree,mathematical intuition,creative thinking and mathematical method,etc.And we get the conclution that personal construction is the foundation of the social construction in MPS. The process of MPS and co-learning is the transforming and construction from situated objective mathematics to subjective mathematics.Part three and part four constructs a MPS teaching model and gives the aim,me process of its operation and the ralation of students,teachers and mathematical problem.In this part we also can find a positive research about MPS in a new way.Some records of a mathematical problem in the college intrence test help us in the analysis of MPS and MPS teaching process.So it's a good way in valuating the teaching model and to support the thoery which is told in this article.In the conclusion the article has some creative ideas.Firstly, give the two tables of personal and social construction in MPS.The second,construct a MPS teaching model.The third,improved the Polya's way in MPS and give a positive research.