Harmony is an important concept of China traditional culture. "Harmony" means peace, harmony, harmony and coordination.
The meaning of. "Combination" means cooperation, symmetry, combination and unity. Harmonious thought thinks that the whole thing
The world is a harmonious whole, and all elements of the universe, nature, society and spirit are harmonious.
In the optimized structure. The mathematical culture system is a perfect and harmonious optimized structure. Mathematical culture
The book is rich in the history of mathematical development, mathematical philosophy, mathematical methods, mathematical aesthetic education and other important contents.
Enrich harmonious thoughts. Its concrete manifestations are overall systematicness, balance stability and orderly symmetry.
First, the overall system
1. Compatibility of Mathematical Axiom System
The axiomatic system of mathematics is compatible, independent and complete. Of these three basic requirements,
Among them, the most important thing is compatibility. Compatibility means no contradiction or harmony, which means that axioms cannot
They contradict each other, and the true propositions they derive cannot contradict each other. The compatibility of axiomatic systems is mathematics.
The foundation of system harmony is also the basic requirement.
In addition to each branch of mathematics itself to form a compatible axiom system, mathematics also needs each branch.
Coordinate with each other and not contradict each other. Some systems also form a close isomorphic relationship, in
The compatibility between different mathematical systems is consistent. For example, Euclidean geometry and non-Euclidean geometry (Ross
Parallel axioms in geometry and Riemannian geometry are mutually negative propositions, and non-Euclidean geometry can be constructed in Euclidean geometry.
So it can be said that as long as Euclidean geometry is not contradictory, then non-Euclidean geometry is also spearless.
Shield.
2. Integrity of mathematical operation system
The arithmetic rules, formulas and conclusions of mathematics are complete and accurate. Especially numbers.
Learn operation language, which completely integrates written language, symbolic language and image language.
, mutual confirmation, mutual interpretation, mutual transformation, to achieve perfection. When expanding the digital system
When establishing new theories and operations and broadening the original operations and relationships, we should try our best to keep the original luck.
If there is any inconsistency between calculation and relationship, provisions must be made to coordinate the new system with the original system.
3. Rigidity of mathematical reasoning system
In our daily mathematical activities, the reduction to absurdity is often used. In this method, not only
We should use the axioms and theorems of the system as well as the knowledge of other branches. Throughout the reasoning process,
We want harmony. For example, one of the three famous problems in ancient Greece was to turn a circle into a square, that is, to make a circle with a given surface.
A square with equal products. It needs to be proved that squares with equal areas cannot be made with compasses and rulers.
The transcendence of using the number "=".
The appearance of "=" in mathematical equations and analytical formulas is the embodiment of harmony.
Second, equilibrium stability.
"Harmony Thought" holds that everything in the world is in a state of balance, harmony and order. all
Things and elements are interdependent, mutually inclusive and complementary, and are in the process of comprehensive and three-dimensional interaction. And number
The balance and stability of learning well embodies the harmonious thought.
1. Balance and stability of mathematical development
Compared with other disciplines, an important feature of mathematical science is the accumulation and development of history.
The equilibrium stability of. In other words, the main mathematical theories are always inheriting and developing the original theories.
Based on them, they will not overthrow the original theory, but will always tolerate the original one.
Theory. For example, the geocentric theory in astronomy was replaced by Heliocentrism, and the particle of light in physics.
Quantum theory is replaced by wave theory, phlogiston theory in chemistry is replaced by oxidation theory and so on. , and mathematics
This has never happened before. It's like a math historian H? Hankel said: "In
In most disciplines, the buildings of one generation are demolished by the next generation, and the creation of one person is destroyed by another.
People only destroy mathematics, and every generation adds a floor to the ancient buildings. "Mathematics this.
A balanced stability is the source of infinite vitality and strong vitality of mathematics.
2. Mathematics learning process is balanced and stable.
People's learning process of knowledge contains a certain cognitive structure. Students study mathematics.
The process of cognition is nothing more than a relatively stable assimilation process of "assimilation-adaptation-balance"
It is to bring new knowledge into the existing cognitive structure and enrich the original knowledge system.
Rich. Adaptation means that new knowledge cannot be integrated into the original cognitive structure, so it is necessary to improve the original cognitive structure.
Line transformation and improvement, so as to establish a new cognitive structure. Balance means that after assimilation and integration, there is one.
In the consolidation stage, the understanding and internalization of knowledge are balanced and stable. People are interested in mathematics.
Knowledge learning is carried out in the cycle of "assimilation-adaptation-balance"
Exhibition.
3. Balance and stability of mathematical methods
Mathematical method is a regular procedure and means in the process of understanding mathematical objects, which makes theory work.
In the intermediary of practice, various methods exist harmoniously in the same mathematical body. For example, commonly used
Mathematical thinking methods: observation, analysis, synthesis, abstraction, conjecture, analogy, induction and deduction; There are also commonly used
Mathematical problem solving methods: comparison method, construction method, model method, construction method, reduction method and mapping method.
Inverse method, geometric transformation method, axiomatic method, etc. These methods, whether in elementary mathematics, also
It is in advanced mathematics; It is widely used in geometry and algebra.
It is always in a state of balance and stability, and will not mutate due to changes in time, space and disciplines.
The idea and method of geometric transformation is to study geometric objects and them from the viewpoint of motion and change.
Keywords interrelation, discuss invariant relation, invariant relation, variable relation, variable,
Find the pattern from it. In the process of solving the problem, the relevant parts of the graph are transformed to change the irregularity into regularity.
Then, turn the general into the special, and turn the unfavorable conditions into favorable conditions.
Third, orderly symmetry.
"Everything must be harmonious", and "harmony" means symmetry, combination and unity. The whole world is not only harmonious and reasonable,
And the symmetry of yin and yang.
1. The beauty of order and symmetry in mathematics
The symmetry studied in elementary mathematics can be described by a graph and a formula.
Partial relation can also describe the relationship between two graphs and formulas. The transformation of graph and formula shows that
The beauty of symmetry in mathematics.
Graphic symmetry can be called narrow sense symmetry, such as central symmetry, axial symmetry and rotational symmetry.
Graphics is a kind of symmetry of graphics position. Show a kind of symmetrical beauty.
Symmetry also exists in many concepts and methods, propositions, formulas and laws, which can also be called one.
Species symmetry.
In mathematics, many concepts are positive and negative, which complement each other and appear in pairs. Like math.
Addition and subtraction, multiplication and division, multiplication and root, differentiation and integration, etc. Can be considered as one yin and one yang.
Symmetry; Subtracting a negative number can become adding a positive number, and division can become multiplication, so.
Is unified and orderly. In binary operation, it is embodied in exchange law, association law and distribution law.
Its symmetry.
2. Orderly structure of mathematical problem solving process
Examining the process of solving mathematical problems from a cultural perspective is a mathematical strategy, mathematical logic, mathematical methods,
The organic combination of mathematical knowledge, mathematical skills and stylization is the unity of orderly structure. compare
For example, the basic steps in solving equations are: denominator, brackets, shift term merging, and dividing both sides by unknowns.
Coefficient of number. This is a harmonious and orderly structure. If this orderly structure is destroyed, a solution will appear.
Problems and obstacles. From the perspective of thinking process, it is an orderly process of "observation-association-transformation"
Cheng. Observation is the basis of association, and the characteristics of a given topic can be understood through observation; Lenovo is the bridge of transformation.
Liang, looking for a solution to the problem in association; Transformation is the means to solve the problem, and the solution is determined in the transformation.
So as to finally solve the problem.
Mathematics embodies the harmony between whole and part, form and content, result and process.
The spirit and connotation of harmonious thought. We use the idea of "harmony" to re-understand mathematics and give full play to the role of mathematical culture.
The educational function in teaching can effectively cultivate students' scientific literacy and cultural literacy.
References:
Qi. Mathematical culture [M]. Changsha: Hunan Education Press, 199 1.
[2] Zhang Weizhong. Mathematics culture and mathematics curriculum [M]. Shanghai: Shanghai Education Press, 1999.
[3] Zheng Yuxin. Mathematical culture [M]. Chengdu: Sichuan Education Press, 200 1.
[4] Li Wenlin. Course of History of Mathematics [M]. Higher education press.