Mathematics, like other sciences, has two values: material value and spiritual value. Many far-sighted mathematicians and mathematicians are "most concerned about" the spirit of mathematics. M Klein, an American applied mathematician, pointed out in his masterpiece Mathematics in Western Culture: "Mathematics is a spirit and a rational spirit. It is this spirit that inspires, promotes, inspires and drives human thinking to the most perfect degree, and it is this spirit that tries to decisively influence human material, moral and social life; Trying to answer the questions raised by human existence; Strive to understand and control nature; Try your best to explore and establish the deepest and most perfect connotation of the acquired knowledge. " Therefore, it is a brand-new subject to fully understand the spirit of mathematics and its educational value, and to establish the new educational values of the integration of science and humanities.
First, the connotation and characteristics of mathematical spirit
The so-called mathematical spirit not only refers to the concentrated expression of intentional psychology such as thinking mode, behavioral norms, value orientation and ideal pursuit in human activities, but also refers to the product of continuous generalization and internalization of mathematical experience, mathematical knowledge, mathematical methods, mathematical ideas, mathematical consciousness and mathematical concepts.
Intentionality is the essential attribute of spirit, and mathematical spirit is the embodiment of mathematical spiritual attribute. There are two remarkable features:
1 comprehensiveness
Mathematical spirit is an extremely broad comprehensive category, which not only includes the subjectivity, purpose, introspection, selectivity and value of human activities in mathematical spirit, but also can further broaden the category. Specifically, cognitive activities characterized by conscious thinking forms such as concept, judgment and reasoning; Conscious spiritual production activities such as mathematical creation, mathematical problem solving and mathematical teaching; Cognitive activities such as the development, design, adjustment and decision-making of mathematical thinking; Low-level psychological activities such as feeling, perception and representation can all be included in the category of mathematical spirit.
2 hierarchy
(1) The cognitive level is mainly manifested in the objectivity, logicality and testability of mathematical cognition, which directly reflects the essential characteristics of mathematical science and internalizes it into the scientific component of mathematical spirit.
(2) On the temperament level, Merton, an American scientific sociologist, put forward six recognized scientific temperament: universalism, public ownership, selflessness, organized skepticism, individualism and emotional neutrality. The code of conduct and moral orientation of scientific workers are clarified, which is manifested as the humanistic component of mathematical spirit.
(3) Value level: "Mathematics not only pursues truth, but also pursues beauty and goodness." Seeking truth, goodness and beauty is the fusion and sublimation of scientific and humanistic components in mathematical spirit, and it is the highest realm of mathematical spirit and even scientific spirit.
Second, the existence form of mathematical spirit
1 subjective form and objective form
Mathematical spirit can be divided into subjective spirit and objective spirit according to the form of existence. Subjective spirit refers to the feelings, perceptions, representations, thinking methods, thinking rules, logical categories, emotions, wills, interests and so on that exist in the human brain as its functions and attributes. Objective spirit is the externalization and materialization of subjective spirit, and the objective spirit of mathematics forms the culture of mathematical spirit, which mainly consists of two parts: first, the content of subjective spirit is attached to or stored in certain material materials and materialized into objective spirit, such as mathematical theory, mathematical knowledge and mathematical thinking methods in papers, works, textbooks and books; Second, the mode of thinking and psychological state, that is, subjective spirit is attached to certain substances and externalized into objective spirit, such as mathematical consciousness, mathematical concept, mathematical tradition, etc., which reflects human mathematical thinking mode and intentional psychology.
2 scientific form and humanistic form
The human spirit can usually be divided into two categories: scientific spirit and humanistic spirit. Scientific spirit refers to the concentrated expression of ideal pursuit, behavior norms and value norms in understanding nature, adapting to nature and changing natural activities; Humanistic spirit refers to the concentrated expression of ideal pursuit, behavior norms and value norms in all human activities to explore and deal with the world. As Qian Xuesen said, science and humanities are two sides of the same coin.
The scientific spirit in mathematics includes: application spirit, expansion-generalization spirit, organization-systematization spirit, unified structure spirit, rigor spirit and economic thinking spirit. There is also the spirit of freedom of thought, the spirit of mathematics and so on.
The humanistic spirit in mathematics includes: the spirit of self-improvement, the spirit of seeking truth from facts and being brave in discovery, the spirit of materialist dialectics, the spirit of innovation, selfless dedication, unity and cooperation, and so on.
There is no clear boundary between the two, and the difference lies in two different forms formed by different objects that people pay attention to: when human beings know natural objects or mathematical objects, scientific spirit occurs; When human beings have self-awareness, humanistic spirit comes into being. When people reflect on themselves, the scientific spirit and humanistic spirit are integrated into the deep background of rationality, mind and emotion, which makes it difficult to distinguish. Mathematical spirit is the unity of mutual infiltration and organic integration of scientific mathematical spirit and humanistic mathematical spirit. If the mathematical spirit of scientific form has a profound influence on the achievements of thinking activities, then the mathematical spirit of humanistic form plays an encouraging, monitoring and guiding role in thinking activities.
3 individual morphology and group morphology
Judging from the subject or carrier of mathematical spirit activities and its achievements, mathematical spirit can be divided into individual spirit and group spirit. The so-called individual (mathematician or mathematician or student) spirit refers to the spiritual phenomenon that exists as the spiritual subject of the individual. It includes individual thinking activities, cognitive activities, decision-making activities and unconscious psychological activities, and also includes objective spirits such as ideological system, knowledge system, temperament and mentality, which are internalized into individual cognitive structure, thinking mode, value orientation, behavior norms and ideal pursuit through the process of learning and education. The so-called group spirit refers to the spiritual phenomenon that exists independently of the individual spirit and is generally accepted by people consciously or unconsciously, including the conscious and rational spirit embodied in the ideological system and knowledge system of mathematics, as well as the code of conduct, value orientation and mathematical tradition generally recognized by mathematicians, mathematicians and mathematics learners.
Individual spirit and group spirit are closely linked and can be transformed into each other. Any kind of individual spirit is always subordinate to a certain group spirit, which is gradually formed under its edification and influence. Similarly, the group spirit is inseparable from the individual spirit, which always exists in the individual spirit. Once the individual spirit of a certain person or school is accepted by the group, it promotes the progress and prosperity of society and becomes the spiritual wealth of all mankind.
Third, the educational value of mathematical spirit
The so-called educational value should refer to the significance and value of education for the existence or development of society or individuals. Therefore, the educational value of mathematical spirit includes social value, personal value and educational value.
1 the social value of mathematical spirit
As an "invisible mathematics"-mathematical spirit, it not only has scientific value for the survival, evolution and development of mathematics itself, but also embodies the rational spirit of human beings, so it has very important social value for the progress of human civilization. As Max Weber asserted, under the background of every major cause of every nation, there is always a certain spiritual force of the times that determines the success or failure of this cause, and it is directly related to a specific era and a specific social and cultural background. Therefore, we should further understand the profound connotation of the strategy of "rejuvenating the country through science and education": "rejuvenating the country" includes both material and spiritual civilization construction; The reason why "rejuvenating the country" depends on science is that scientific spirit and scientific thought are the most fundamental spiritual forces. This great strategy will definitely put an end to the so-called "Needham phenomenon" and build China into an ideal society with both material modernization and spiritual modernization.
2 the personal value of mathematical spirit
(1) Mathematical spirit has the humanistic value of self-expression. Because the mathematical spirit has two components: one is the spiritual component, that is, the mathematical spirit of humanistic form; Second, the mathematical component is the mathematical spirit of scientific form. The former is characterized by intentionality, reflecting people's non-cognitive psychological factors such as emotion and will, and is a non-intellectual component of mathematical spirit; The latter is characterized by research, reflecting cognitive psychological factors such as thinking mode and thinking strategy, and is the intellectual component of mathematical spirit. From the viewpoint of system theory, the former is a power system and the latter is an operating system. Moreover, the mathematical spirit composed of these two components also has a metacognitive power, which plays a monitoring and regulating role in mathematical thinking activities, promotes the development of mathematical thinking ability and the improvement of mathematical cognitive structure, and obviously transforms non-intellectual factors into intellectual factors.
(2) Mathematical spirit has the personality value of self-improvement. West Point Military Academy, known as the cradle of western famous soldiers, has set up many advanced mathematics courses. "It is precisely because mathematics learning can strictly train students' ability to grasp and adapt to military operations, and combine students' special vitality and flexibility in military operations to pave the way for students to enter the advanced military science field." Mathematics is the unity of truth, goodness and beauty, and the spirit of mathematics plays an important role in seeking truth, worshiping goodness and beauty, forming a perfect three-dimensional personality, and promoting the all-round development and lifelong sustainable development of moral, intellectual and aesthetic education.
3 the educational value of mathematical spirit
The spirit of mathematics is not only valuable to systems other than education, but also of great significance to education itself. This is the educational value of the spirit of mathematics. It includes three aspects:
(1) Knowing and understanding the spirit of mathematics will help us to change the old and establish a new concept of mathematics education. For example, change the educational concept of focusing on intellectual development and establish the educational concept of coordinated development of intelligence and non-intelligence; Change the one-sided emphasis on "mathematical attribute" and establish the educational concept that "mathematical attribute" and "spiritual attribute" penetrate and promote each other; Change the educational concept aiming at short-term utility and establish lifelong quality education; Change the educational concept of simply improving people's mathematical knowledge and ability and seek the organic integration of mathematical quality education and humanistic quality education.
(2) Carrying forward and studying the spirit of mathematics will inject new blood and add new vitality into mathematics education. In order to make students truly understand and master mathematics, it is an inevitable trend to bring the mathematical spirit including mathematical thinking method, mathematical consciousness and mathematical concepts into the purpose of mathematics education, into the content of the development of mathematics curriculum and into mathematics classroom teaching.
(3) Mathematics education at the level of mathematics spirit focuses on the continuous development and improvement of people's quality. Perhaps it represents the new direction of the future development of mathematics education.