| Mathematics teacher education implements training in a diversified and multi-level's way, which is the progress of the times and an inevitable result of social development, and also the summary of Chinese 50 years' experience of mathematics teacher education development. This paper aims to conduct a comprehensive study of mathematics' "Diversification and Multi-level" training programmer at Normal University through literature review, history study and international comparisons, particularly use the sample of 2003-2007 Department of Mathematics at East China Normal University (ECNU) and also the empirical survey of large samples from four universities, with the soul purpose of providing the theoretical basis and practical case for the future development of mathematics teachers education of 21st universities.The emergence of the two concepts "Diversification" and "Multi-level," has a profound social background. With the reform and opening up for over 30 years, the employment in market will inevitably lead to the diversification of employment. The source of middle school math teacher no longer confined to mathematics department, and graduates of mathematics can engage in other work instead of staying in the education unit. Key high schools, ordinary high school and vocational high school pose different requirements for the background of the math teachers. At the same time, math teachers need to diversify in knowledge structure and professional backgrounds because of the implementation of new curriculum standards for mathematics, promotion of school-based curriculum and establishment of the optional courses. In this case, some teachers need to be good at theory of mathematics, some modeling and applications, and still others theory of mathematics education. On the other hand, increasing enrollment of higher education and increasing number of students lead to diversification in the quality of individual student. In order to respect the differences of students, the model of teaching different levels according to different requirements becomes the inevitable choice in the basic courses."Diversification" of math teacher training mentioned in this paper refers to setting a series of courses (called "the goal of elective courses", including base series, application series and Mathematics' education series) on the premise of a good foundation for mathematics. Students can select a series of studies with accordance to his/her own characteristics so as to adapt oneself to the diversified requirements caused by social development and explosive growth of mathematical knowledge."Multi-level" refers to different requirements according to the different objects. Basic courses are divided into two levels of basic requirements and higher requirements which are implemented according to different levels (such as the Department of Mathematics at ECNU, Science Base Classes are taught according to higher requirements, regular classes of students and students of local normal university to basic requirements). Compared to the past single training programmer, "diversification" is on behalf of the width, while "multi level" is on behalf of the depth, which is the horizontal and vertical reform and development.In retrospect to the past 50 years' history of China's Teacher Education Review, in particular to interpretations of four training programmers in the past 50 years of the Department of Mathematics at ECNU, we can see the trace of formation of "diversification, multi-levels". In the second half of the 20th century into the information age, the progress of the Mathematical Sciences leads to the explosion of mathematical knowledge, which causes the diversification of mathematics courses. Development of Mathematics Education makes the study of students at normal university more flexible. Therefore, to provide a wide range of optional courses is becoming a natural developing trend.The data are collected from "the school of Mathematical Sciences" of the United States, AP courses, and the Russian "school of mathematical physics". In 2003, I also paid a visit to Sworthmore College and Arcadia University in the United States. We could see that the level and task of university diversified and different models of mathematics teacher education emerged. The United States and Russia attach great importance to the top students in mathematics education, which enlightens us that it is of great importance to train mathematics teachers of highly professional knowledge and improve the overall international competition of our innovative talents.The core of this paper is about the empirical study of training programmers "diversification and multi-levels". It provides an objective basis for the implementation of the necessity and feasibility through case studies and a large sample survey.The "diversification and multi-levels" training programmer was implemented at the 2003 (graduated in 2007) students majoring mathematics and applied mathematics at ECNU. A total number of 137 students had enrolled in 2003, and 42 of them had entered Science Base Class. 71 chose teacher education series and base series, 59 chose teacher education series and application series, 7 chose based series. Among 123 graduates who found jobs, 54 entered ordinary schools, 4 entered vocational schools, 17 were in the non-educational units, including: IT companies, banks, insurance, securities, and consulting, 38 pursued further study, 10 went abroad. The proportion for ordinary schools was 43.9%, vocational schools 3.25%, non-education unit 13.82%, further study 30.89%. Students with good math background (esp. students with master's degree) are very popular in key schools. In short, the employment situation is "diversified". The training model with three different objectives (basis, application, teacher education) meets the social requirements for the math department of normal university.As for the problem of "multi-levels", we conducted four large-scale testing and investigation from 2001 to 2003 with the purpose of answering whether the "University Expansion," can guarantee the basic quality of math teaching. The universities participated were: ECNU, Hang Zhou Teachers College, Nan tong Teachers College and Sichuan Teachers College, which represented two different levels. The numbers of samples respectively were: ECNU 517, Hang Zhou Teachers College 249, Nan Tong Teachers College 402, and Sichuan Teachers College 167. Four investigations were about: 1. Questionnaire on the curriculum and the level at ECNU; 2. Examination of "mathematical analysis" at the end of the second semester from 2 universities; 3. Test of higher levels of understanding of students enrolled in 2001 from 4 colleges; 4. Survey on relevance of the result of college entrance examination and grade of basic courses at university.The survey was about the "diversification and multi-levels" training programmer and embodied "diversification" of the "the goal of elective courses." The results showed that the basic requirements of mathematics base courses can be basically met after the large-scale expansion. However, there was great gap between the colleges at the higher level, ECNU was much better than the other three, which could serve as an evidence of the essentiality of the "different levels" of teaching.Finally, the paper discussed the long-term dispute over the "normal" issue, especially on the problem of how to transform the "academic form of mathematics" to "form of education" which can be easily accepted by students. Besides, it made analysis about the arrangement of the curriculum of math education at some famous normal university in China, including ECNU.Of course, there are lots of deficiencies in this paper. For example, it lacks a detailed analysis of the condition of teachers at vocational high school. It doesn't contain the data of Mathematics Teacher Education (Junior college level) in remote ethnic minority areas. I hope that more researches will be conducted in this area, so as to provide further practice and theory for the development of teacher education.
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