First, the emergence of mathematical education theory

As a social phenomenon, mathematics education has gone through thousands of years. In this historical process, great development and changes have taken place in the content, organizational form and scale of mathematics education, from which a theoretical discipline with mathematics education as the research object was born. Swiss educator J.H.Pestalozzi first proposed to separate the mathematics education process from the education process and study it as an independent science. He first put forward the term "mathematics teaching method" in the book "Intuition of Numbers" published by 1803. Therefore, it is generally believed that the theoretical system of mathematics education was founded at the beginning of19th century.

Hu Ruiji, a scholar who specializes in mathematics teaching methods, 19 17 lives in Peking University. In the 1940s, the Commercial Press also published books on mathematics teaching methods written by China people themselves. After the founding of the People's Republic of China, the mathematics teaching method has been systematically developed through the input of Soviet educational literature. The research of mathematics education theory in China has gone through three stages: from mathematics teaching method to mathematics textbook teaching method, and then to the establishment of mathematics pedagogy. Every stage has undergone revolutionary changes in the scope, purpose, characteristics and means of research. Mathematics pedagogy is a new interdisciplinary subject involving mathematics, pedagogy and thinking science. Although there were many works on mathematics pedagogy in China in 1980s, the theoretical research level of mathematics education was improved day by day, and a theoretical system was gradually formed, but mathematics pedagogy is still in the stage of theoretical construction and teaching experiment, which needs to be developed and improved. Now, first of all, discuss the research object, characteristics, structure and research methods of mathematical pedagogy respectively.

Second, the research object of mathematical pedagogy

In a broad sense, mathematics pedagogy should study all issues related to mathematics education, such as the interaction between society and mathematics education, the accomplishment and cultivation of mathematics teachers, the compilation and evaluation of mathematics textbooks, the study of students' learning rules, the selection and application of mathematics teaching methods, the discussion of the organizational form of mathematics teaching, the application of modern technical means, the role and cultivation of mathematics language, the structure and cultivation of mathematical thinking, the significance and cultivation of mathematical ability, the essence and law of mathematics teaching process, mathematics education and training, etc.

Here, the teaching process should be the core of many problems, and mathematics pedagogy should first pay attention to the problems related to the teaching process.

Teaching process, especially mathematics teaching process, is a control process that teachers use a series of means (teaching materials, teaching AIDS and technical means), a process of information exchange and transmission between teachers and students, and is completed by cooperative activities between teachers and students. As shown in figure 0- 1- 1:

Teachers, students and courses are the three basic elements of communication system. Teachers and students are the subjects of communication and acceptance, and knowledge is the object of this communication system. In the process of teaching, teachers are the organizers and leaders of teaching, and their understanding, mastery and application of teaching rules determine the quality of teaching. Therefore, what is the law of mathematics teaching should be regarded as an important content. In this way, mathematics teaching theory should be regarded as one of the research objects of mathematics pedagogy. Textbooks and courses that reflect the teaching content and requirements are the norms of knowledge and skill structure and the main basis for implementing teaching. What principles and laws should be followed in curriculum setting and textbook compilation to meet the requirements of cultivating people? Therefore, mathematics curriculum theory should also be one of the research objects of mathematics pedagogy. The teaching process needs students' active participation, and students have to go through a complicated psychological process to learn mathematics, which has its own laws. What are these laws? We should study them. Therefore, mathematics learning theory should also be one of the research objects of mathematics pedagogy.

To sum up, the main research objects of mathematics pedagogy should be mathematics teaching theory, mathematics curriculum theory and mathematics learning theory, the so-called "three theories"

At the Third International Conference on Mathematics Education (ICME3- 1976), H.Bauersfeld described three research objects of mathematics education: curriculum, teaching and learning. Later, in an editorial entitled "Research on Mathematics Education-Triangle", Tom Keelen of the United States vividly compared them to the three vertices of the triangle, corresponding to three kinds of people: curriculum designers, teachers and students. There are three research directions in mathematics education, namely curriculum theory, teaching theory and learning theory.

These three aspects are closely related, intertwined and interrelated, so it is difficult to study them separately. Their relationship is equivalent to the sides of a triangle, and the study of one vertex will also play a role in the study of the other two vertices.

This triangle has an "interest center", which is the practical experience of children and adults in learning mathematics. Researchers should make effective use of these experiences and make their own research improve them directly or indirectly.

The triangle should be internal and external, and the research of teaching design, teaching analysis classroom activities and teaching experience all belong to the "internal" of the triangle of mathematics education research. Mathematics, psychology, pedagogy, philosophy, thinking science, technical means, symbols and language all belong to the "outside" of the triangle of mathematical education research.

From what has been discussed above, we can draw the following conclusions:

(1) The research object of mathematics pedagogy is closely related to mathematics curriculum theory, mathematics teaching theory and mathematics learning theory.

(2) The three theories are based on practical experience, and the research results will directly or indirectly enrich and improve these experiences. This shows that mathematics pedagogy is a theoretical discipline with strong practicality, and the purpose of learning mathematics pedagogy is to improve the quality of learning mathematics.

(3) Mathematics pedagogy involves mathematics, philosophy, pedagogy, psychology, thinking science and other comprehensive disciplines.

(4) The research methods of mathematics pedagogy can be teaching design, teaching, classroom activity analysis, experiment and directional observation.

Third, the characteristics of mathematics pedagogy

Mathematics pedagogy mainly has the characteristics of comprehensiveness, practicality, scientificity and education.

1. comprehensive

Mathematics pedagogy is a comprehensive subject involving mathematics, pedagogy, psychology and thinking science. The so-called comprehensiveness is not the random patchwork and combination of these disciplines, but the use of the principles, conclusions, ideas, viewpoints and methods of these disciplines to solve the problems of mathematics education itself from the characteristics of mathematics and mathematics teaching.

The study of mathematics education must have certain mathematical literacy. The higher the attainments of mathematics, the more you can grasp the essence of mathematics. It is in this sense that the research of mathematics education can not be separated from mathematics for a moment. But it is worth pointing out that mathematics education is not the natural result of mathematics, because mathematics education has its own regularity.

Mathematics learning is a special cognitive process, of course, it must obey the general cognitive laws. However, the object of mathematics learning has its own characteristics (such as abstraction, high generality, close causal connection of knowledge, etc.) In this way, mathematics learning has its particularity. The comprehensiveness of mathematics education is the high unity of generality and particularity. This unity does not simply take particularity as a general positive example, but derives some inevitable conclusions suitable for mathematics education from the particularity of mathematics education under the guidance of general theory, thus fully enriching the general conclusions.

The comprehensive characteristics of mathematical pedagogy require us to pay attention to the development of disciplines closely related to mathematical pedagogy. For example, in psychology, the cognitive psychology school put forward the view that the structure of mathematical thinking is similar to that of mathematical science, and the teaching theory absorbed many viewpoints such as system theory, information theory and cybernetics. This should arouse our attention and research. With the development of mathematics education, the ideas and viewpoints of some new disciplines will also be introduced into the research field of mathematics education.

Step 2 be practical

The practicality of mathematical pedagogy is manifested in the following three aspects:

First of all, mathematics pedagogy should be based on extensive practical experience. Mathematics education practice has always been the source of mathematics education research. Without practice, mathematics education will become passive water and a tree without roots. Just discussing from theory to theory can't solve practical teaching problems.

Second, the problem of mathematical pedagogy research comes from practice. Take curriculum theory as an example. There are many unsolved problems that need to be studied by mathematics pedagogy, such as how to evaluate the traditional mathematics content in primary and secondary schools. How to understand the modernization of mathematics textbooks? How to embody the characteristics of quality education in mathematics textbooks is an urgent problem to be solved at present, and it is also a problem that mathematics education should study.

Thirdly, mathematics pedagogy should be able to guide practice and test theory through practice. The understanding of practicality should not be too narrow. Because of the different levels of theories, their directness to practical guidance will be different.

science

The scientific nature of mathematics pedagogy is generally reflected in the fact that mathematics education should conform to the general law of the development of mathematics education, the development trend of things and the reality.

The general law of mathematics education exists objectively. The question is whether it has been recognized by people and how deep it is. Because of the different understanding depth and angle, people may have different views on the same issue, which is very natural. Mathematics education is not like mathematics. Although there are different methods for the same problem, the correct conclusion is unique. However, mathematics education is different. There may be many ways to deal with the same problem, and all of them may get different and ideal results. This is a characteristic of the scientific nature of mathematics education.

The objective laws are endless, and people's understanding is endless. People's understanding is always limited by the development of science and technology, cultural background and personal conditions at that time, so there are always some limitations. With the development of the times, the understanding of a certain problem will also develop, and some have the necessity of re-understanding. For example, after the emergence and introduction of computers into teaching, it is necessary to re-understand the choice of teaching content, the application of teaching methods and the form of teaching organization.

Any formalism or absolutism is unscientific. Some people regard a certain teaching method as the best, or a certain theory and practice as the best, ignoring time, place, conditions and objects, isolating the problem or isolating it from the outside world, thus making it absolute, which does not meet the scientific requirements.

The scientific nature of mathematics pedagogy is also reflected in conforming to the development trend of things and advancing with the times.

4. Educational

Mathematics pedagogy, as an educational discipline, should give full play to its training function for all kinds of mathematics educators at all levels and serve basic education. Mathematics education shoulders the heavy responsibility of cultivating talents for the four modernizations, and should play its role in cultivating normal college students with profound educational theory foundation, strong educational and teaching ability and strong innovation ability.

Fourthly, the structure of mathematical pedagogy and its related disciplines

The research objects of mathematics pedagogy are mainly mathematics learning theory, mathematics curriculum theory and mathematics teaching theory. The relationship between these three theories is shown in figure 0- 1-2:

Although the three theories are interrelated, studying one of them will inevitably affect the other two. But among these three theories, learning theory is the foundation, which provides the necessary psychological basis for curriculum theory and teaching theory, and teaching theory is the direct embodiment of learning theory and curriculum theory.

The structure of mathematical pedagogy and its related disciplines is shown in figure 0- 1-3.

Mathematical pedagogy and its related disciplines are roughly divided into three parts:

1. Basic part

These include philosophy, mathematics, the history of mathematical thought, the modern foundation of middle school mathematics, mathematical methodology, pedagogy, psychology, logic, thinking science, computer science, computer-aided teaching and so on.

Mathematics includes not only the old bases of analytic geometry, advanced algebra and mathematical analysis, but also the new three bases of topology, abstract algebra and functional analysis. In addition, there should be probability statistics, discrete mathematics, fuzzy mathematics, geometric basis, set theory and some traditional elementary mathematics. In a word, mathematics educators need extensive mathematics knowledge and a deeper understanding of a branch.

The history of mathematical thought focuses on how a mathematical concept or branch of mathematics evolved from gestation, maturity to development, from roughness to precision, and how the thought developed in this period, so as to obtain the necessary enlightenment of mathematical education research.

Modern Basis of Middle School Mathematics is a course to learn elementary mathematics from a high angle. In other words, it is to learn elementary mathematics from a modern and unified point of view, so as to have a deeper understanding of elementary mathematics.

Mathematical methodology studies and discusses the laws of mathematical development, mathematical thinking methods and discoveries, inventions and creations in mathematics from the perspective of methodology.

Pedagogy, including educational theory and teaching theory, belongs to the general law of education and teaching.

Psychology, which refers to general psychology here, mainly studies the laws of psychological activities in the process of cognition, emotion and will.

Logic includes mathematical logic and formal logic, and formal logic is its focus.

Computer science, including computer principles, several commonly used programming languages and programming methods and skills.

Computer-aided instruction includes the function of computer-aided instruction, teaching principles and the compilation of courseware.

The above is the necessary basis for the research of mathematical pedagogy, and it is mainly the core part of the following research.

2. Core part

Including mathematics curriculum theory, mathematics learning theory and mathematics teaching theory.

Enlarged part

It includes mathematics education evaluation, mathematics education history, mathematics education psychology and comparative mathematics pedagogy.

Mathematics education evaluation includes general evaluation view, mathematics curriculum evaluation, mathematics teaching evaluation and mathematics learning evaluation. Evaluation is not an end, but a means. Through evaluation, affirm achievements, find problems and put forward suggestions for further improvement. Select teaching methods and learning methods suitable for learning through evaluation.

The history of mathematics education, including the development history of mathematics education at home and abroad, especially the study of some representative figures' mathematics education thoughts, can inspire today's mathematics education, so as to make foreign things serve China and make the past serve the present.

Mathematics educational psychology takes the interaction between teachers and students in the process of mathematics education as the object, studies various psychological phenomena and their changes in the educational situation, analyzes the dependence of the physical and mental development of the educated on the educational conditions, and probes into the laws and characteristics of the formation and development of students' knowledge, skills, abilities, attitudes and personality qualities under the educational conditions.

Comparative mathematics pedagogy, which studies mathematics education in different countries, nationalities and regions in the world today; On the basis of studying their respective economic, political, philosophical and national traditions, this paper studies some similarities, development laws and general trends of education and makes scientific predictions. Its purpose is to learn from the useful experience of developing Chinese mathematics education abroad.

It can be seen that mathematics education is a subject involving a wide range of fields, so mathematics pedagogy can also be regarded as a scientific system, just as there are many branches under mathematics. This course briefly introduces the core part of the above content. For other contents, please refer to related works.

Fifth, the research methods of mathematical pedagogy.

The research method of mathematics pedagogy refers to the method used to study the phenomena and laws of mathematics education, specifically to explore the relationship between the internal elements of mathematics education and other things, as well as the changes and laws between the quality and quantity of mathematics education.

General educational research methods, such as observation, literature, investigation, statistics, behavior research, comparison, analysis, experiment and experience summary, are all applicable to the research of mathematics education.

However, as far as the current situation is concerned, the research methods of mathematics education should pay attention to the following points:

1. the unity of theory and practice

Mathematics pedagogy is a theoretical science with strong practicality. From the perspective of development, theoretical research and experimental research should be further combined, complement each other and use each other to promote the in-depth development of mathematics education research.

The disconnection between theoretical research and experimental research in mathematics education is manifested in two aspects: on the one hand, in the past, the research methods of mathematics education mostly used speculative methods, that is, on the basis of one's own experience, or related literature, or seeing related phenomena in mathematics education, thinking independently, or demonstrating a certain topic, or putting forward one's own views or judgments, which were basically limited to theoretical exposition and still had a certain distance from actual mathematics teaching. On the other hand, the mathematics education carried out by actual teaching workers lacks further theoretical research.

In the research of mathematics education, we advocate seeking truth from facts, integrating theory with practice and proceeding from reality. Any separation between theory and practice is not conducive to the research of mathematics education.

2. Unity of part and whole

All the parts and problems involved in mathematical pedagogy are interdependent and interrelated. We can only solve the problems one by one, but the problems to be solved are under the whole and related to other problems under the whole. So we must consider its relationship with the whole and other parts.

The unity of the part and the whole is actually the application of systematic methods. The so-called systematic method is to treat the object as a systematic method. By studying the interrelation and interaction between the whole and parts in the system, it dialectically combines analysis and synthesis, so as to correctly understand the problem as a whole or solve it reasonably.

The system method has the following two main characteristics:

First of all, the systematic approach emphasizes the study of the integrity of things.

All kinds of objects, events and processes in the world are not randomly piled up, but an organic whole composed of various components according to laws. The essence of the whole thing only exists in the interrelation of its constituent elements, and the sum of isolated parts cannot reflect the essence and motion law of the whole.

Second, the systematic method emphasizes the dialectical combination of analysis and synthesis.

Analysis is a method to understand the whole by breaking it down into parts, aspects and elements, and synthesis is a method to connect all parts, aspects and elements into a whole. In the system method, analysis and synthesis are organically combined, and analysis should be oriented by synthesis, and synthesis should be based on analysis, and the bridge between analysis and synthesis is the internal relationship among the components of the system.

Attach importance to the application of systematic method in mathematics education research

3. Unity of qualitative and quantitative.

Everything is the unity of quality and quantity, and the aspects of material and quantity are interrelated and restricted. When we know something, we must first know its nature, that is, make a so-called qualitative analysis. Things have not only qualitative aspects, but also quantitative aspects. On the basis of understanding its essence, we should also grasp its quantitative aspect, that is, make a quantitative analysis of its attributes, that is, make a so-called quantitative analysis, so as to accurately judge the changes of things. If we only do qualitative analysis and not quantitative analysis, then our understanding of things may not be comprehensive.

In the past, the research on mathematics education was mostly qualitative analysis, from theory to theory, lacking further quantitative description. It is not easy to grasp teaching, and the application of teaching theory is not convincing. We believe that qualitative analysis is the beginning of revealing the law of mathematics education and the basis of quantitative analysis; Quantitative analysis is the continuation and deepening of revealing mathematical laws, and it is the further precision of qualitative analysis. If both qualitative analysis and quantitative analysis are carried out, we can not only grasp the law of mathematics education qualitatively, but also describe the law of mathematics teaching quantitatively. In the research of mathematics education, the unity of qualitative analysis and quantitative analysis is the direction of our efforts.

Dialectical materialism is the philosophical basis of mathematics education. Specifically, materiality and dialectics are the philosophical basis of mathematics education.

Substantialness is generally manifested in two aspects: first, the practicality of mathematics education, and the unity of theory and practice of mathematics education research. Mathematics education is based on extensive practical experience, educational theory should be endowed with vitality by educational practice, and educational thought should follow the footsteps of educational practice; Secondly, considering that mathematics education must be based on China's national conditions, all thoughts, theories and methods that are not in line with China's national conditions are lifeless.

Dialectically speaking, it is manifested in three aspects: first, all thoughts, theories and methods are conditional and interrelated; Secondly, theory and practice, part and whole, qualitative analysis and quantitative analysis are dialectical. Moreover, teaching and learning, teachers and students, heredity, education, environment, collectivization education and personalized education are all dialectical unity, and only by dialectical treatment can we achieve the expected results; Thirdly, mathematics education is dynamic, so are the ideas, theories and methods of mathematics education, which develop with the development of the times.

Defining materiality and dialectics and developing mathematics pedagogy on this basis will make mathematics education move forward in the right direction and road.