| In terms of the research on " Problem solving", there is so much home and abroad.The research abroad focuses on the developing students' ideas in innovation spirit andpractical capability by means of " Problem solving", while the existing study in ourcountry pay more attention to the problem itself, and the process of " Problem solving"and developing capacity of " Problem solving" in order to render students to master thebasic knowledge and skills closely, which is focused on the field of "Solving of theproblem". On the basis of " Problem solving", the idea of " Problem solving" andoperation system is introduced into the mathematical class teaching by the mathematicalteaching system of " Problem solving" to carry out the curriculum organization andteaching design. The mathematical curriculum should be constructed with thecomprehensive and one-whole thinking method, aiming at putting students into thecomplicated problem scenes to inspire and moving-up inner study motivation of studentsefficiently. With the help of understanding of "Problem solving" and mastering themathematical background knowledge and the inner relationships among the relativeknowledge, it can promote the effectively transferring of knowledge, skills, and ability tomake students to obtain the flexible mathematical knowledge and innovative thinkingcapability. Basically, the new research transform the traditional teaching method onstudents' disadvantages of poorer ideas on problems, passive learning, model learning,heavy remembrance burden, lower learning efficiency, and greatly strengthen thestudents' problem consciousness and ability of innovation and practice, which showsitself the powerful life force during the period of the present new curriculum carrying-outand spreading-out.The main content and research results of this paper are:1.Analyzing the possibility and necessity of introducing the " Problem solving" intothe mathematical class teaching. Generalizing the applications of " Problem solving" inthe mathematical teaching home and abroad to find out " Problem solving" having thepower of the broad application field and application values. In china, this chapterconcerning this field is still much more narrow, therefore it is necessary to advance themuch more comprehensive study. Caparisoning the new basis teaching on mathematicalcurriculum reform with the traditional mathematical curriculum, the former powers themultifunction urgently needing the operation study supports and the want of adoptingnew curriculum reform on the basis of the mathematical " Problem solving" applicationin the mathematical teaching. Therefore it shows itself great vitality.2.Proposing the basic idea on construction mathematical " Problem solving" in theteaching system. Generalizing the study field and study level on " Problem solving"home and abroad, this chapter points out the main problems existed in teachingapplication concerning mathematical " Problem solving" of our country and proposes thebasic idea on construction mathematical " Problem solving" in the teaching system.3.Studying the comprehensive and developing on the relative theory. Conducting thesystematical abstraction and the feather generalization on the theories "Problem solving","Problem teaching", "Constructivism", "The mathematical teaching is the teaching onmathematical activity", the chapter conducting the developing research with theconstruction of the comprehensive and borrowing ideas, which lay the foundation for thetheory construction on mathematical " Problem solving" in the teaching system.4.Showing the conception definition and connotative analysis on mathematical" Problem solving" in the teaching system. On the basis of the relative conception of"Problem" and the mathematical "Problem solving", the mathematical " Problemsolving" in the teaching system is defined in the chapter, and its own basic connotation iscarefully analyzed from two latitudes.5.Explaining the basic characteristics of mathematical " Problem solving" in theteaching system. On the basis of the generalization on its basic characteristics, thischapter explains the idea from the three aspects: 1) From the aspect of the curriculumaims, the curriculum structure, the teaching and learning methods, and the mathematicalapplication and so on to explain the understanding at the different levels on mathematical" Problem solving" in the teaching system;2) Analyzing the basic characteristics on themathematical " Problem solving" teaching;3) By the comparison with the teachingmethods of the exploring teaching, the discovering learning, the researching learning andso on to explore the relationships and differences among them and further understandingthe mathematical "Problem solving" teaching system and its characteristics.6.Establishing the basic theory and teaching structure of mathematical "Problemsolving" teaching system. To the great degree the traditional mathematical teachingmodel is still stagnating on the level of technical operation, lacking of the transform oneducational thinking manners on the deep level. The main reasons lie on the lacking ofthe necessary theory support, the mechanical model rendering the model being fossilizedand inflexible. The mathematical "Problem solving" teaching system from the borrowingand comprehension of the relative theories, sets up its own theory basis. It contains thecurriculum theory, the teaching theory, the learning theory, and the evaluation theory onmathematical "Problem solving" teaching, which provides the theory support on theteaching structure of mathematical "Problem solving" teaching system.The mathematical "Problem solving" teaching structure includes the following fourstages:(1)Mathematicalizing the specific problem. It means that the teachers creativedesigning the problem circumstances on the focus of subject learning content andstudents experiences, and eliciting the students' active learning motivation and theproblem consciousness, and thinking the mathematical problems existing actively, andputting forward the mathematical problems within the teachers-to-students interaction,and making clear the mathematical elements and its relationships included in problems,and constructing the problem solving plans and strategies, and expanding these targetsand relationships, and establishing the broader relation, and further establishing theknowledge scheme structure of task learning in order to realize the mathematicalizingspecific problem. The processes usually undergo the five chains: proposing problems ---suspecting ---constructing ---defining targets ---discussing.(2)Logicalizing the mathematical materials. During the process of logicalizing themathematical materials, it is unavoidable to make use of the mathematical materials ofsome relative conceptions, methods, and conclusions, this chain uncovers the connotationand denotation on the demand of form logic, and defines the conception, and makes clearits expression manners for the conclusion, and takes the logical demonstrationfurthermore.(3)Application of the logic knowledge. It includes the direct application, theindirect application, and the comprehensive application. Taking on the specific problemsituations within production, life, and neighboring subjects, and letting students sensingand experiencing the basic laws and methods of the mathematical knowledge application,during the process of solving the specific problem, it strengthens and deepens the logicalknowledge what one has learned before, and enhances the understanding onmathematical meaning, and experiences the mathematical values. The processes usuallyundergo the four chains: problem circumstances ---establishing model ---explaining ---expanding.(4)Rethinking of the task learning. It usually includes the following steps:generalization the knowledge structure and brightening the thinking manners, andinducing the problem solving fields and strategy and methods, and summing upexperience lessons and writing out the learning knowing experiences, and thecooperation communication and teachers' inspiring evaluation.
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