| Getting the goals of mathematics education requirements of the role of mathematics teaching to the depths of the human mind:the development of mathematical ability, improve the sense of function and enhance the spiritual character. This requires in mathematics teaching, along with the formation and development of mathematical knowledge, some of the excellent growth and honed psychological characteristics, organizational characteristics of the dynamic nature of these experiences of subjects can be accumulated and integrated through a link with the combination of these characteristics are generated to movement and the concept of restructuring and reorganization in motion the process of the formation of new materials to absorb the concept of regeneration. Resides in the spirit of awareness of the continued existence of the whole or renewable content of the concept of being; driven by the concept of exercise and mental vitality of the spirit embodied in these factors in determining when the process of integration and separation. It is this concept of regeneration, productivity of thinking occurred in the subject, the formation of the creative function of consciousness.Chapter I, the goal of modern math education at several levels should have the following:(1) access to mathematical knowledge; (2) the development of mathematical ability; (3) penetration of mathematical concepts; (4) enhance the spiritual character. Penetration of the teaching of mathematical concepts, that is, the level of teaching mathematical concepts. The formation of mathematical knowledge in mathematicians thinking mathematical problems contain rich ideas, the ideas in these, the mathematical concept is one of the most important objectives; improve educated spiritual character is the ultimate goal of mathematics education, it is through subject to master mathematical knowledge, mathematical concepts through the penetration of such means to achieve their goals. Contributed to the development of the main spiritual character, from the highest level of mathematics teaching reflects the tremendous improvement in the quality of their function. We would study three questions:(1) mathematical concepts and new several core objectives of mathematics curriculum set by the relationship;(2) preliminary research on outside the process of mathematical concepts from minds;(3) penetration of the concept by teaching design method of linear equation with one unknown for classroom.Chapterâ…¡, the literature on mathematical concepts mainly base on Mr. Zhang Naida' monograph and paper, in the mid 90s of last century, Guo Boxiang had been reviewed math concepts, made a key reference here. Nearly 15 years of mathematical concepts research weakened, either from the practice made a number of simple questions, or the help of Mr. Zhang Naida monograph with the teaching practice in their own specific problems examples of mathematical concepts, no one theory raised new problems. Few foreign literature, only to find one or two articles. These are many literature on the design of mathematics teaching, so we have chosen some influential researchers in the article, based on the structure of materials were reviewed and commentary.Chapterâ…¢, About research methods. The methods we used are mainly speculative,so it is the highlight of the literature method. From the philosophical to the mathematical concept of the concept of evolution, then we define mathematical concepts, are obtained by the speculation. Because the concept of the invisible but omnipresent approximate, mathematical concepts are mathematical thinking activity in the brain of intent to start dynamic mechanism. We can work analysis, sound thinking and other means to investigate and interview. Finally, penetration in the test design method of teaching mathematical concepts, we mainly on the knowledge of linear equation with one unknown teaching classes using empirical comparison of two teaching class experiments.Chapterâ…£, we discussed the debate of concepts in the history of philosophy, and to discuss the core concept of "mathematical concept" is defined:people's basic views of mathematics and general knowledge. Mathematical concepts in a systematic way of acting on the problem, mathematical concept of the system can be regarded as the spirit of the mathematical (rational spirit of exploration), the traditional mathematics (mathematics culture on the individual's "Immersion" and the mathematical community to set standards of behavior for the individual constraints) and the basic idea of mathematics (including basic math resulting mathematical methods and the basic attitude of the main form of fixed potential) posed by the knowledge system. This recognition system, culminating in the spirit of the body structure of the initiative and the logic of peacekeeping, means that the main mathematical way of thinking to consider the issue, deal with the problem of self-consciousness and thinking habits. On this basis, analysis of the characteristics of mathematical concepts:(1) subjectivity and objectivity; (2) intellectual and cognitive; (3) static and dynamic; (4) hierarchical and systematic. These four characteristics of mathematical concepts, mathematical ability of people to form a subjective basis, it is configured with the internal thinking materials (expressed as a static mathematical knowledge) and external thinking materials (expressed as the main system of mathematical problems faced by the unresolved mathematical problems), and to combine the two to form a problem space, the main problem-solving process, the obtained mathematical knowledge, the development of thinking skills, mathematical concepts formed to optimize the psychological structure, to enhance the spiritual character. Further study the concept of the structure of mathematical knowledge and mathematical knowledge to the mathematical structure of the system:"one body and the two side" structure. Mathematical concepts into mathematical cognitive structure for the power system.Of mathematical concepts and the proposed new course out of the relationship between a number of educational objectives, curriculum objectives of the new division to human communication and integration. Reveals the mathematical concepts and mathematical skills, mathematical thinking, mathematics, life value judgments, mathematical understanding, mathematical problem solving of a series of mathematical education in the project objectives to inter-relate, and access to a range of results.Chapterâ…¤. About the externalization of mathematics concept. The concept of spiritual intelligence is reflected in the structure of a sense of practical problems with the intention, it is clear from the vague to the pity, if not externalization, that it is only out of a personality, and this is characterized by individuality; empirical and discontinuity, which is not conducive to persistence of the people long-term impact. Only externalization of mathematics concept is to a community of all mankind, to contribute to mankind. Outside the mathematical concept of great significance, it also involves a complex process, we do a preliminary study. The inquiry process is mainly how to use language to express vague concepts, this is a difficult place, because we ordinary discourse domain, which involves emotional and intuitive specified foreign objects, is the "meaning" and "can refers to "a combination, and externalization of mathematical concepts are highly abstract, not in the direct experience of people involved in composition, therefore, must choose refined mathematical symbols, the process needs to be used outside of mathematical concepts within the process of symbolic expression. Limited to the level of, not as in-depth study. Conclusion of this chapter, we also answered why the mathematical concept becoming the target system into mathematics teaching.Chapter VI. About the design of mathematics teaching method. Mathematics teaching design based on two points, first, the characteristics of mathematical knowledge, a high degree of abstraction, rigor and wide application; the second is the psychological development of students in the age characteristics process, the characteristics of mathematical knowledge and student characteristics of psychological development of the unity of the two constitutes the basis for the design of mathematics teaching. Design methods of teaching mathematics, mathematical knowledge is to open, then find the means to meet the psychological characteristics of students, the role of mathematics in children's psychological knowledge to ensure the effectiveness of the teaching of mathematical knowledge. Place to ensure the effectiveness of psychological knowledge, design methods of teaching mathematics there are many limitations discussed here, which focus on three main limitations:balance of the macro process and the micro process; balance of the logical process and the mental processes; balance of the giving of teachers and the creation of students.Chapter VII. The main access to the front of the results of a series of theoretical mathematics teaching in the practical application. We chose the knowledge of linear equation with one unknown as a starting point, through the ages in this specific equation of knowledge for students to learn specific difficult analyzed to determine the design of macro and micro teaching instructional design and its reasonable to integrate the two way process. Practical emphasis to the specific micro teaching method designed to achieve penetration through the design of micro teaching mathematical concepts: That is the essence of the purpose of this article.Chapter VIII. This chapter is to test the design methods of teaching effectiveness about penetration of mathematical concepts related to mathematical theory of education. The program has three points:First, the two classes were compared using tests started this equation the idea of knowledge through teaching a class using conventional means, and the other classes, a penetration theory to guide the design of mathematical concepts of teaching. Test method is tested through the papers,21 qustions in a title, inserted into the five concepts must have some math to solve the problem, as tested by the penetration of the teaching of mathematical concepts is more effective. Second, work analysis, the use of students to analyze text responses, collect data, to explore the concept of the role of penetration. Third, in the above two interviews on the basis of students, guide students to explore the concept of the psychological process of mathematical knowledge occurs. We conclude that the study of the characteristics of students in mathematical cognition, whereby characteristics of mathematical knowledge, mathematical knowledge of the properties as possible from among those simulated by the reduction of knowledge of how the original concept inside out into the mathematical knowledge, teaching knowledge at the same time, penetration of these ideas, students form a mathematical way of thinking. This series of mathematics education goal can be achieved. Chapter IX. The study is to more theoretical about the characteristics of the mathematics knowledge and in teaching mathematics at the same time, and we research on the meaning of penetration of mathematical concepts and specific instructional design methods. On the mathematical knowledge brought into the concept of how the creators of mathematical knowledge in the development of specific out not to make a more depth research, this study is one of the more regrettable, for the externalization of the process of mathematical concepts is just a tiny exploration, etc., which constitute the subject of further study. This study reveals the development of mathematical concepts to solve mathematical problems for students to enhance the quality plays a very important role, but also to improve the efficiency of effective mathematics teaching methods, with our depth understanding of this topic, will re-understanding of the purpose of mathematics education, mathematics education, as rich domain, which really makes the new generation of mathematical knowledge for quality improvement from a possibility into reality.
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