Function thought refers to analyzing, reforming and solving problems with the concept and nature of function. The idea of equation is to start with the quantitative relationship of the problem, transform the conditions in the problem into mathematical models (equations, inequalities or mixed groups of equations and inequalities) with mathematical language, and then solve the problem by solving equations (groups) or inequalities (groups). Sometimes, functions and equations are mutually transformed and interrelated, thus solving problems.

Descartes' equation thought is: practical problem → mathematical problem → algebraic problem → equation problem. The universe is full of equality and inequality. We know that where there are equations, there are equations; Where there is a formula, there is an equation; The evaluation problem is realized by solving equations ... and so on; The inequality problem is also closely related to the fact that the equation is a close relative. Column equation, solving equation and studying the characteristics of equation are all important considerations when applying the idea of equation.

Function describes the relationship between quantities in nature, and the function idea establishes the mathematical model of function relationship by putting forward the mathematical characteristics of the problem, so as to carry out research. It embodies the dialectical materialism view of "connection and change". Generally speaking, the idea of function is to use the properties of function to construct functions to solve problems, such as monotonicity, parity, periodicity, maximum and minimum, image transformation and so on. We are required to master the specific characteristics of linear function, quadratic function, power function, exponential function, logarithmic function and trigonometric function. In solving problems, it is the key to use the function thought to be good at excavating the implicit conditions in the problem and constructing the properties of resolution function and ingenious function. Only by in-depth, full and comprehensive observation, analysis and judgment of a given problem can we have a trade-off relationship and build a functional prototype. In addition, equation problems, inequality problems and some algebraic problems can also be transformed into functional problems related to them, that is, solving non-functional problems with functional ideas.

Function knowledge involves many knowledge points and a wide range, and has certain requirements in concept, application and understanding, so it is the focus of college entrance examination. The common types of questions we use function thought are: when encountering variables, construct function relations to solve problems; Analyze inequality, equation, minimum value, maximum value and other issues from the perspective of function; In multivariable mathematical problems, select appropriate main variables and reveal their functional relationships; Practical application of problems, translation into mathematical language, establishment of mathematical models and functional relationships, and application of knowledge such as functional properties or inequalities to solve them; Arithmetic, geometric series, general term formula and sum formula of the first n terms can all be regarded as functions of n, and the problem of sequence can also be solved by function method. The basic knowledge of middle school mathematics is divided into three categories: one is the knowledge of pure numbers, such as real numbers, algebraic expressions, equations (groups), inequalities (groups), functions and so on. One kind is about pure form knowledge, such as plane geometry, solid geometry and so on. One is the knowledge about the combination of numbers and shapes, which is mainly embodied in analytic geometry.

The combination of numbers and shapes is a mathematical thinking method, which includes two aspects: "helping numbers with shapes" and "helping shapes with numbers". Its application can be roughly divided into two situations: one is to clarify the relationship between numbers with the help of the vividness and intuition of shapes, that is, to use shapes as a means and numbers as the purpose, such as using the image of functions to intuitively explain the nature of functions; Or to clarify some properties of a shape with the help of the accuracy and rigor of numbers, that is, to use numbers as a means and shape as the purpose, for example, to accurately clarify the geometric properties of curves with equations of curves.

Engels once said: "Mathematics is a science that studies the relationship between quantity and spatial form in the real world." The combination of numbers and shapes is based on the internal relationship between the conditions and conclusions of mathematical problems, which not only analyzes its algebraic significance, but also reveals its geometric intuition, so as to skillfully and harmoniously combine the accurate description of numbers with the intuitive image of spatial forms, make full use of this combination, find a solution to the problem, make the problem difficult and easy, simplify the complex, and thus solve it. "Number" and "shape" are a pair of contradictions, and everything in the universe is the unity of their contradictions. Mr. Hua said: "A few is less intuitive, and a few is difficult to be nuanced. The combination of numbers and shapes is good in all aspects and everything is harvested. "

The essence of the combination of numbers and shapes is to combine abstract mathematical language with intuitive images. The key is the mutual transformation between algebraic problems and figures, which can make algebraic problems geometric and algebraic. When analyzing and solving problems by combining numbers and shapes, we should pay attention to three points: first, we should thoroughly understand the geometric meaning of some concepts and operations and the algebraic characteristics of curves, and analyze the geometric meaning and algebraic meaning of conditions and conclusions in mathematical topics; The second is to set parameters reasonably, use parameters reasonably, establish relationships, and transform numbers from numbers to shapes. The third is to correctly determine the range of parameters.

Some knowledge of mathematics itself can be regarded as the combination of numbers and shapes. For example, the definition of acute trigonometric function is defined by right triangle; Define trigonometric function of any angle with rectangular coordinate system or unit circle.

The role of mathematical thinking in human civilization

1, Mathematics and Natural Science:

In the field of astronomy, Kepler put forward three laws of celestial motion on the basis of Tycho Bray's observation: (a) Planets orbit the sun in an elliptical orbit, and the sun is at a focus of this ellipse.

(b) The area swept radially from the sun to the planet at the same time is f (as shown in the figure).

(c) The square of the period of revolution of a planet around the sun is directly proportional to the cube of the semi-long axis of the elliptical orbit. ..

Kepler is the first person in the world to describe the motion of celestial bodies with mathematical formulas. He changed astronomy from static geometry in ancient Greece to dynamics. This law proves the mathematical principle at the core of Pythagoreanism. In fact, the mathematical structure of phenomena provides the key to understanding phenomena.

Einstein's theory of relativity is a great revolution in physics and even the whole universe. Its core content is the change of time and space view. Newtonian mechanics holds that time and space are irrelevant. Einstein's view of time and space holds that time and space are interrelated. What prompted Einstein to make this great contribution is still the way of thinking in mathematics. Einstein's concept of space was prepared by German mathematician Riemann 50 years before the birth of the theory of relativity.

In biology, mathematics makes biology rise from empirical science to theoretical science, and from qualitative science to quantitative science. Their combination and mutual promotion have produced and will continue to produce many wonderful results. The problems of biology have contributed to the birth and development of biomathematics, a major branch of mathematics. Now biomathematics has become a complete subject. Its new application in biology has the following three aspects: life science, physiology and brain science.

2. Mathematics and Social Sciences

If we say that in natural science, we use more mathematical formulas and computing power; Then in the field of social science, the role of mathematical thought can be better reflected.

With the help of mathematical thought, we should first invent some basic axioms, and then draw the theorem of human behavior from these axioms through strict mathematical deduction. How is the axiom formed? With experience and thinking. In the field of sociology, people will accept axioms only if there is enough evidence to show that they conform to human nature. Speaking of social science, let's not talk about the role of mathematics in the political field. Hume once said: "Politics can be transformed into a science". In the axioms of political science, Locke's social contract theory is of great significance. It is not only the representative of the Renaissance, but also promotes the progress of the whole society. Compared with the civilization of feudal society, western bourgeois civilization has made a lot of progress, but it will be replaced by socialist and capitalist civilization. * * * The theory of "the liberation of all mankind" put forward by producers-seeking happiness for the people, "serving the people" and "Theory of Three Represents" should and will become the basic axioms of the government.

Democracy is indispensable in politics, and the most direct manifestation of democracy is election. Mathematics plays an important role in the distribution of votes. The distribution of votes is fair first, but how can it be fair? 1952, the mathematician Arrow proved an amazing theorem-Arrow's impossibility theorem, that is, it is impossible to find a fair and reasonable election system. In other words, only relatively reasonable, not absolutely reasonable. It turns out that there is no "fairness" in the world! Arrow's impossibility theorem is a milestone in the application of mathematics to social science.

In economics, the extensive and in-depth application of mathematics is one of the most profound changes in current economics. The development of modern economics puts forward higher requirements for its own logic and rigor, which makes the combination of economics and mathematics inevitable. First of all, rigorous mathematical methods can ensure the reliability of reasoning in economics and improve the efficiency of discussing problems. Secondly, objective and rigorous mathematical methods can resist preconceived prejudice in economic research. Third, data analysis in economics needs mathematical tools, and mathematical methods can solve quantitative analysis in economic life.

In other social sciences, such as demography, ethics, philosophy, etc. The equivalent transformation of mathematical thought is also an important thinking method to transform the problem of unknown solution into the problem that can be solved within the existing knowledge. Through continuous transformation, unfamiliar, irregular and complex problems are transformed into familiar, standardized and even simple problems. Over the years, the idea of equivalent conversion has been everywhere in the college entrance examination. We should constantly cultivate and train our consciousness of transformation, which will help to strengthen our adaptability in solving mathematical problems and improve our thinking ability and skills. Transformation includes equivalent transformation and non-equivalent transformation. Equivalence transformation requires that causality in the transformation process is sufficient and necessary to ensure that the result after transformation is still the result of the original problem. The process of non-equivalent transformation is sufficient or necessary, so the conclusion needs to be revised (for example, the unreasonable equivalent rational equation needs root test), which can bring people a bright spot of thinking and find a breakthrough to solve the problem. In application, we must pay attention to the different requirements of equivalence and non-equivalence, and ensure its equivalence and logical correctness when realizing equivalence transformation.

C.A. Yatekaya, a famous mathematician and professor at Moscow University, once said in a speech entitled "What is problem solving" to the participants of the Mathematical Olympiad: "Solving a problem means turning it into a solved problem". The problem-solving process of mathematics is the transformation process from unknown to known, from complex to simple.

The equivalent transformation method is flexible and diverse. There is no unified model for solving mathematical problems by using the thinking method of equivalent transformation. Can be converted between number, shape and shape, number and shape; Equivalent conversion can be carried out at the macro level, such as the translation from ordinary language to mathematical language in the process of analyzing and solving practical problems; It can realize transformation within the symbol system, which is called identity deformation. The elimination method, method of substitution, the combination of numbers and shapes, and the problem of evaluation domain all embody the idea of equivalent transformation. We often carry out equivalent transformation among functions, equations and inequalities. It can be said that the equivalent transformation is to raise the algebraic deformation of identity deformation to keep the truth of the proposition unchanged. Because of its diversity and flexibility, we should reasonably design the ways and methods of transformation and avoid copying the questions mechanically.

When implementing equivalent transformation in mathematical operations, we should follow the principles of familiarity, simplification, intuition and standardization, that is, we should turn the encountered problems into familiar ones to deal with; Or turn more complicated and tedious problems into simpler ones, such as from transcendence to algebra, from unreasonable to rational, from fractions to algebraic expressions and so on. Or more difficult and abstract problems are transformed into more intuitive problems to accurately grasp the problem-solving process, such as the combination of numbers and shapes; Or from non-standard to standard. According to these principles, mathematical operations can save time and effort in the process of transformation, just like pushing the boat with the current, often infiltrating the idea of equivalent transformation, which can improve the level and ability of solving problems. When solving some mathematical problems, sometimes there will be many situations, which need to be classified and solved one by one, and then integrated solutions. This is the classified discussion method. Classified discussion is a logical method, an important mathematical thought and an important problem-solving strategy, which embodies the idea of breaking the whole into parts and the method of sorting out. The mathematical problems of classified discussion ideas are obviously logical, comprehensive and exploratory, and can train people's thinking order and generality, so they occupy an important position in the college entrance examination questions.

The main reasons for classified discussion are as follows:

① Classify and define the mathematical concepts involved in the problem. For example, the definition of |a| can be divided into three situations: a>0, a=0 and a<0. This kind of classified discussion questions can be called conceptual.

② Mathematical theorems, formulas, operational properties, laws, limited scope or conditions involved in the problem, or given by classification. For example, the formula of the sum of the first n terms of geometric series can be divided into two cases: q= 1 and q≠ 1. This kind of classified discussion questions can be called natural type.

③ When solving problems with parameters, we must discuss them according to the range of parameters. For example, solving the inequality ax> at 2 am>0, a=0 and a.

In addition, some uncertain quantities, the shape or position of uncertain figures, uncertain conclusions, etc. It is mainly discussed through classification to ensure their integrity and make them deterministic.

When discussing classification, we should follow the following principles: determination of classification objects, unification of standards, no omission and repetition, scientific classification, clear priority and no skipping discussion. The most important one is "no leakage and no weight".

When answering classified discussion questions, our basic methods and steps are as follows: first, we must determine the scope and the whole discussion object; Secondly, determine the classification standard, correct and reasonable classification, that is, the standard is unified, no duplication is missed, and the classification is mutually exclusive (no repetition); Then discuss it step by step, and get the stage results by classification; Finally, a summary is made and a comprehensive conclusion is drawn.