A friend who drives a taxi argued with me that he hardly needs math in his life and work now, and even has a meter to keep accounts. He asked me, "What's the use of middle school mathematics?" . In my usual educational practice, students often ask, "Why do you want to study mathematics? What's the use of learning mathematics? " . I found that, as a math teacher, although I often teach students that math learning is very important and useful, I don't seem to know what math is for. Every time I speak, all I can do is "Mathematics is the basis of learning other subjects well" and "Mathematics can cultivate people's thinking". It is no wonder that some students will think that "learning mathematics has no value except coping with exams" after entering the society.

This actually involves the value of mathematics education. In fact, even among mathematicians, the understanding of mathematical value often hovers between "useful" and "useless". There is no doubt that all the development of mathematics is more or less based on reality psychologically. However, once the theory appears in the actual demand, it will inevitably gain the power of development and transcend the limitations of direct practicality. It can be seen that if we look at mathematics from a purely practical point of view, some mathematics is really useless, or at least for some people, or for a certain stage. For example, cardano (150 1 ~ 1576) was probably the first to use imaginary numbers. When he talked about the solution of cubic equation in a book 1545, he accidentally used imaginary numbers. Although others used it more and more later, they always found it incomprehensible. It was not until the19th century that people clearly understood the imaginary number-complex number with the help of Gaussian geometric explanation. Hamilton (W.R. 1805 ~ 1865), a physicist and mathematician who has made great contributions to the study of complex numbers, thinks that complex numbers have definite mechanical significance, and this mechanical explanation is of great significance to electrodynamics and relativity. ③

Of course, what I am discussing now is mainly the value of mathematics education in the basic education stage. Let me talk about the value of basic education first.

People have a variety of needs, not only material and material needs, but also spiritual and cultural needs. There are many ways to meet people's material and spiritual needs, but improving quality through education is the basic way. People receive education not only to meet their spiritual needs now, but more importantly, to lay a foundation for their ability to obtain economic benefits and meet material and spiritual needs in the future through the improvement of their own quality. Therefore, people need education, and education can meet people's needs.

Education meets people's needs, which not only promotes people's development and value, but also realizes and promotes the value of education itself. Educational value is the usefulness of educational function to meet people's educational needs, and it is the educational significance and function formed by people choosing educational function according to their own educational needs. The fundamental function of education is to meet the needs of individuals for knowledge, ability, virtue and society for qualified citizens and professionals. ⑤ It can be seen that the value of basic education lies in meeting the educational needs of people at this age, learning basic knowledge for people, improving the basic ability of asking questions, thinking and solving problems, cultivating a noble moral foundation, and laying a comprehensive quality foundation for becoming qualified talents in society. Basic education lays the foundation for human development, thus promoting human development and social development. Therefore, the value of basic education is the unity of the attribute that basic education meets people's needs and social needs. Because basic education is in a specific stage of education, its value is unique. Its uniqueness lies in that, from the concept of quality education, the value of basic education lies in improving people's quality and promoting their all-round development; From the perspective of brain science, basic education develops human potential and promotes human growth and development. ⑥ Besides, educational activities are essentially a kind of value activities. This is a basic point for us to understand educational activities and analyze the value of basic education. ⑦

As far as mathematics is concerned, it has been an extremely important part of basic education since ancient times. Socrates, an ancient Greek philosopher, attached great importance to mathematics education. In Plato's Republic, Socrates imagined that the king of philosophy ruled the society, and personal wisdom was used to control passion. The educational goal of cultivating the king of philosophy is to let students understand the ultimate truth. First of all, for practical purposes and to make students think about the abstract meaning of numbers, let them learn mathematics. The purpose is to help them discover the truth, not to teach them the truth. Socrates believes that the study of the abstract meaning of mathematics can help people shift their attention from the image of existence to the light of "goodness". Since then, the study of plane geometry and solid geometry has also played the same role. These two courses are not only practical, but also help students to look at the abstract world of "goodness". ⑧

Therefore, as an integral part of education, the value of mathematics education is self-evident. Mathematical value not only has its practical use, but also the value brought by its special nature to human spiritual needs. The vast majority of people engaged in mathematics are based on the needs of human material life, but in this process, they do have a spiritual need: the need of rational life. Pet-name ruby and this profound spirit needs to lead mankind to make unremitting efforts to find the ultimate truth of the world and self.

From this point of view, we should not discuss the value of mathematics education, but what value it has in modern education and how do we understand these values?

This question is actually difficult to answer. As I said at the beginning, it may not have a ready-made standard answer for people to use.

For example, each of us learns a language, some just learn to speak, some learn to read, some better people can write some articles, some better people can create complex and profound literary works, and some even specialize in the grammatical rules in the language. I think math education is similar. Formal mathematics, like spelling and grammar, is the correct application of accepted rules. This may be valuable only to professionals, such as mathematicians and people engaged in mathematics education, and only they can understand it. Meaningful mathematics is like newspapers and magazines used to tell interesting stories. Of course, these stories must be true. By receiving certain mathematics education, most of us can understand some mathematics, explain and solve some problems in life with mathematics, or understand some problems explained with mathematics. For example, many of us can read financial statements and understand complex data curves. The best mathematics should be like literature. The story comes from the real life in front of you, which makes you invest your energy and feelings. Taking part in this course puts forward very high requirements for mathematics education, that is, the connection between mathematicians' mathematics and ordinary people's mathematics. It requires mathematics professionals to interpret the parts related to reality in formal mathematics language in a language that the public likes and understands, so as to influence the public's understanding of reality with mathematics.

It can be seen that the value of mathematics education is not an absolute and static category. Marxist values hold that value is not a simple object attribute, nor a simple subject demand, but the degree to which the object attribute meets the subject demand. The value of mathematics education is also reflected in the needs and satisfaction of different learners for mathematics. ⑾ Therefore, the value of mathematics education is different from the object of mathematics education, and even the object of mathematics education in different periods is different.

For the taxi driver mentioned at the beginning, the value of mathematics education to him is to learn to read and to settle accounts when the machine is not working. Of course, it is also possible that he needs some math besides work, such as recording and calculating family expenses. In other words, for most people engaged in occupations that have little to do with mathematics, the value of mathematics education is to let them learn some basic computing skills.

As the basic link of national education, as the initial stage of lifelong education for the whole people, and as the foundation of the educational pyramid, basic education has the characteristics of the foundation of educational level, the comprehensiveness of educational objects and the comprehensiveness of educational contents. Therefore, for students of basic education, the value of mathematics education is different. Because they still don't know what kind of career they will be engaged in in in the future and in which field they will develop. From a temporary practical point of view, mathematics education can really help them get good grades in exams, and then gradually climb a higher social ladder. In the long run, some of them may be engaged in occupations with high correlation with mathematics in the future, such as software engineering departments, or financial personnel, or even professionals who study mathematics. Therefore, in the stage of basic education, it is necessary for them to learn more comprehensive basic knowledge and basic ability of mathematics. Their need for mathematics education, or the value of mathematics education to them, is to lay a good knowledge foundation for their future research and innovation work. Through long-term and arduous mathematics study, they can understand the process of human mathematics development and the history of mathematics knowledge generation and development, and cultivate good scientific research spirit quality.

Discussing the value of mathematics education can also start from the essence of mathematics. As an expression of human thinking, mathematics embodies people's initiative, meticulous reasoning and the will to pursue perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Although different traditions can emphasize different aspects, it is the interaction of these opposing forces and their joint efforts that constitute the life, use and lofty value of mathematical science. (13) The extension of mathematical culture as a course form should include the knowledge of mathematical history; A story that reflects mathematicians' spirit of seeking truth, goodness, beauty, wisdom, innovation and exploration; Reflect the emergence, development and essence of important mathematical concepts; Important mathematical concepts, ideas and methods that can be extended to the direction of mathematical application; Mathematical thinking and methods to deal with problems; The embodiment of the great role of mathematical science in the development of human society and economy. ⒁

Generally speaking, the content of mathematics education can be summarized into two forms, namely, the form of subject knowledge and the form of culture, so the value of mathematics education can also be analyzed from two aspects: scientific value and cultural value.

As far as the development of mathematics itself is concerned, mathematics education in the basic education stage is of course very important. The development of mathematics itself needs a large number of people who study mathematics, study mathematics and love mathematics. They learn the comprehensive knowledge of various fields of mathematics in the basic education stage, understand the latest development of various branches of mathematics, and gradually set foot on the platform of further innovation of mathematics through accumulation.

Mathematics education is also of great value to the development of other related disciplines. The development of physics is closely related to mathematics, and the idea of mathematics is the core idea of many physical theories. Since the introduction of mathematical analysis, youth economics has developed qualitatively and become a real science. Many research methods in chemistry also have a very important relationship with mathematics; Even now, many scientific researches in other disciplines rely on statistical analysis methods in mathematics. It can be said that mathematics has become the language and tool of modern science and technology. ⒂ It can be seen that mathematics education has an important influence on the development of almost all disciplines and science and technology in the world, and its value is reflected by the application of mathematics knowledge and mathematics ability by mathematics learners and users in various fields.

Of course, it should be noted that when students study mathematics, they get not only mathematical knowledge that can be directly applied in the future. Mathematics has also cultivated various mathematical thinking abilities that can be transferred to various practical fields and research fields, such as logical thinking ability, spatial imagination ability, comprehensive analysis ability and induction and deduction ability.

Besides the scientific value of mathematics education, it also has important cultural value. Mathematics Curriculum Standard for Full-time Compulsory Education points out that "Mathematics is a kind of human culture, and its content, ideas, methods and language are important components of modern civilization". According to "Mathematics Curriculum Standard (Experiment) for Ordinary Senior High School", "Generally speaking, mathematics culture is manifested in the aspects that have a great influence on human development in the process of origin, development, perfection and application of mathematics. It includes not only the subtle influence on people's ideas, thoughts and ways of thinking, but also the function of training people's thinking and developing people's creative thinking, as well as the enterprising spirit and lofty realm that human beings can achieve in the process of understanding and developing mathematics. " ⒃

As a cultural presentation, the value of mathematics education lies in establishing the firm belief that human beings pursue the ultimate truth and providing eternal spiritual motivation and cultural foundation for human development. Although all of us hope for human progress and scientific development, it is impossible for most people to take science as their lifelong career. Most people have a relationship with mathematics for a practical purpose in their lives. But this spirit of pursuing truth is of great value to all people, even to the whole mankind. This rational exploration of human beings has an eternal theme, that is, "know the universe and know human beings themselves."

As we all know, mathematics learning is very difficult, hard and particularly boring. From solving a math problem to writing a math paper, it is a process of striving for the correct method and the final answer. Take the mathematical papers created by mathematicians as an example. Most papers are flowers that will not bear fruit. The function of some papers is to hone a method and clarify some details; Some papers are firewood to carry kindling. Thousands of people have joined the team of mathematical research, most of them have not achieved remarkable results, and only a few have won. But if there are not so many people who devote their lives tirelessly regardless of gains and losses, it is impossible to imagine the progress of science and culture. ⒄ The realization of human pursuit of truth is approaching step by step in such a wave of progress. Although no one can tell us where the destination is or whether it exists, we human beings will move forward forever with the support of this belief. Mathematics, the "free creation of human cognition", bears indelible achievements in the process of educating human beings to inherit culture and stick to their beliefs.

Mathematics pursues a completely certain and reliable knowledge; Mathematics, as an integral part of human culture, constantly pursues the simplest and deepest cosmic roots beyond human senses; Mathematics studies not only the universe but also itself. ⒅

Because of these important characteristics of mathematics, the value of mathematics education lies in cultivating thinking ability, especially abstract thinking ability, and even improving human intelligence. In the utopia envisaged by Socrates mentioned above, the ultimate goal of attaching importance to mathematics education is actually to learn philosophy, which is also the highest goal of education in Socrates' view. Plato divided mathematics into two parts, one part is vulgar and vulgar, serving daily and material needs; The other part is advanced, whose purpose is to sublimate the soul, transcend all kinds of dirty utilitarian thoughts and serve the fundamental purpose of philosophy, that is, to achieve perfection. Many philosophers representing human wisdom are mathematicians themselves. Aristotle, the originator of logic, actually deduced his logic from mathematics. Aristotle's hypothesis reasoning (modus ponens) is that "if proposition A is true and A contains proposition B, then B is also true", which is exactly mathematical reasoning. Russell defined natural numbers with logical concepts, and his logical axioms are actually mathematical axioms. In addition, the exploration of the universe by people who test human wisdom is also closely related to mathematics. From the initial attempt to describe the universe with a geometric model, to Newton's description of celestial gravity with calculus, and then to Einstein's theory of relativity, every leap in understanding is the promotion of human wisdom in mathematics.

According to the characteristics of basic education, such as hierarchy, universality of objects and comprehensiveness of contents, the special purpose of basic education can be summarized as cultivating students' creative thinking ability and critical thinking ability on the basis of analyzing basic education materials at home and abroad. Judging from the development of mathematics, the value of mathematics education in this respect is irreplaceable. Euclid's Elements of Geometry not only made great contributions to the development of geometry, but also had an irreversible impact on physics and many other sciences. But the development of mathematics itself has not stopped. Mathematics, which has become the "free creation of human cognition", constantly denies and subverts itself in the process of development, and then is reborn again and again, contributing to the development of human wisdom and the ultimate goal of human understanding of the universe and self. Starting from Euclidean geometry, mathematics with critical spirit and innovative essence experienced the birth of non-Euclidean geometry, which directly changed human's view of time and space. Similar to non-Euclidean geometry breaking the parallel axiom, the birth of imaginary number broke the stereotype that multiplication must be suitable for exchange rate in dominant mathematics. In modern times, Condell's proof subverts the foundation of mathematics. Condell got two theorems in the article "On the Formal Indefinite Proposition of Mathematical Principles and Their Systems". One is that compatibility will inevitably lead to incompleteness. The second is that the compatibility of the above systems is undecidable. In this way, since there is an undecidable proposition, it can be affirmed or denied without causing contradictions. So we can take this proposition and its opposite proposition as axioms and get two different systems, just like Euclidean geometry and non-Euclidean geometry. If an axiom is added, there will be a new undecidable proposition and a new "non-Euclidean geometry". In this way, everyone used to think that they were pursuing deterministic truth. Where is the "truth" now? (2 1) This innovative and critical feature of mathematics is not only of great value to students who receive mathematics education, but also of core value to the whole human beings to reflect on themselves and critically think about their thoughts and behaviors.

In a word, the value of mathematics education can be seen from the different needs of the educated objects at different stages, which can be the daily application of mathematics, and can also be the basis for continuing to study mathematics or other disciplines. The value of mathematics education can also be seen from different periods of education. It can not only be the emphasis on basic knowledge, skills and thinking methods of mathematics in the basic education stage, but also the mathematical thinking, research methods and innovative mathematical conclusions when mathematics is studied and studied as a science to a higher degree. The value of mathematics education can also be seen from the scientific and cultural attributes of mathematics. As a part of science, mathematics needs valuable mathematics education for its own development and contribution to other disciplines. Mathematics, as a culture, its educational value is reflected in its contribution to human progress, leading the development of human wisdom and enterprising spirit. Especially the critical and creative features of mathematics make it contribute to every stage of human progress. The cultural value of mathematics education is deposited in the hearts of every generation through various forms of educational content at all stages of mathematics education. Mankind has created mathematics, and mathematics has also shaped future people and human beings through education.