Mathematics Teaching Efficiency

Teaching should serve for the study. Teaching efficiency not only embodies short-term study result, but also embodies long-term study result. The short-term study result is acquiring and grasping the knowledge. The long-term study result is what student acquires. Mathematics teaching efficiency is not decided by the teacher who wants to teach student what, but decided by the student who acquires something finally. We can understand the concept of teaching efficiency from two dimensions. The first dimension is about the time that student used. This dimension means that the student can make full use of the time and take an active part in mathematics learning. The second dimension means the result ofmathematics teaching, which embodies short-term study result-cognitive score and long-termresult-the spirit of rationality, the consciousness of efficiency, reasonable cognitive structure and theability of mathematics learning. Teaching efficiency is a relative concept. The student who uses less time and has the same result can be called high efficiency. The student who uses the same time and has better and various results can also be called high efficiency.The contribution that mathematics teaching efficiency study to educate for mathematics includes: the need of resolving realistic problem, the need of comparative study and the urgent need of the development of mathematics education. The local scholars try to define the concept of teaching efficiency from surveying and evaluation. This method more or less tries to imitate the view of natural science which about efficiency, and deduce the problem of the cognition concerning teaching efficiency. Really, the concept of teaching efficiency exists objectively, but the standard of evaluation has the subjectivity because of the view of teaching. Moreover, the factors which affect teaching efficiency are not only multiple but also complicated. So, the surveying and evaluation of teaching efficiency is impossible to attain the natural science meaning that is completely objective and pure. We can only make our efforts to make it more scientific. But the view that only quantitative analysis is scientific is one-sided and over-simplified. So we had better use "advanced, high, intermediate, low" to evaluate teaching efficiency.The main conclusion includes: (1) Mathematics teaching efficiency is not decided by the teacher who wants to teach student what, but decided by the student who acquires something finally. (2) Teaching efficiency thought is developed form the concerning about teachers' efficiency in the process of teaching, then students' efficiency in the study, and now we pay more attention to students' development efficiency. (3) Students' mathematics "double bases" and mathematics cognition base are not solid. (4) Mathematics teaching efficiency should be highly improved in China. (5) The spirit of rationality is the spirit when people can exceed outer-want in the pursuit of logic, freedom and universal principle. (6) That mathematics education makes students develop rationality spirit means that students can feel the inner-rationality of mathematics and rationality spirit of mathematicians and they can acquire the culture on the level of spirit and experience in the mathematics study. (7) The effect of mathematics teaching should include: develop students' rationality spirit, develop students' efficiency consciousness, help students construct good cognitive structure, help students have a fundamental understanding toward mathematics and help them learn how to study. (8) We should pay more attention to mathematics inner-value, which means that we should train students' thought in mathematics study. (9) Teaching of mathematics should develop students' internal construction, external construction ability and the ability to construct thinking dialectically. (10) Cognitive understanding in mathematics has three kinds: operational understanding, relative understanding and transferred understanding. (11) Although both students and teachers make efforts and use much time, students' understanding toward knowledge can not attain the fundamental level. (12) The mathematics teachers' understanding on the mathematics and education has three levels: operational, relative, creative understanding. (13) The mathematics teachers' understanding level on the mathematics and education is the main factor which affects teachers' teaching efficiency. (14) Only the students who think their mathematics learning efficiency is high have high appraisal of mathematics and their teachers. (15) With high standard of evaluation about teaching efficiency, mathematics teaching efficiency can also be improved. (16) There is no universal method that can improve the mathematics teaching efficiency. (17) The thought that confine to technology is a big problem in the study of teaching efficiency. (18) In cognitive teaching, high-efficiency teaching attaches importance to thought teaching, understanding and constructing better cognitive structure. (19) From the teaching efficiency point of view, we should analyze the advantages and defects of our country's mathematics education dialectically. (20) What ensures high-efficiency in mathematics teaching is students emotional and active thinking involvement. (21) The theoretical base of mathematics teaching efficiency is not some doctrine of mathematics education from the west, but some home and abroad related theories and some related subjects knowledge that can help us study the mathematics teaching efficiency. (22) We should use various "doctrines" to help us think deeply, not to rigidify our thought. (23) Students need advanced notion and good habits to improve mathematics study efficiency. (24) Mathematics teaching would rather inspire the want to grasp knowledge than inspire the want to recognize how to do it.