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1、数学 - 重要数学思想与方法（Mathematics - important mathematical ideas and methods）First, the combination of figures and shapesThe combination of thinking is one of the four important methods of thinking in middle school mathematics, the algebraic problem of nature, sometimes we study the geometry of the problem
2、 could be solved (to help shape); or for geometric problems based on the number of relations with the corresponding graphics to solve the problems (in number this method helps to shape), to solve the problem that combination of number and shape.The purpose of the combination of 1. figures and severa
3、l figures is to make use of the vividness and intuition of form, to give play to the standardization and strictness of the thinking, and to complement each other, and to strengthen the strengths and circumvent weaknesses.2. Engels defines mathematics in this way: mathematics is the science of quanti
4、ty relations and spatial forms of the real world. That is to say, the combination of number and shape is the essential feature of mathematics. Everything in the universe is not a harmonious unity of numbers and shapes. Therefore, the thought of combining numbers and shapes in mathematics learning fu
5、lly grasps the essence and soul of mathematics.The essence of the combination of 3. numbers is that the nature of geometric figure reflects the quantitative relation, and the quantity relation decides the nature of geometric figure.4. Mr. Hua Luogeng once pointed out: the number of missing less intu
6、itive, a shape to the details; the combination of a variety of good separation, separation of all non. The combination of application as a mathematical way of thinking can be divided into two kinds: some properties of shape to clarify or by the number of accurate and intuitive geometric shape or wit
7、h the help of some to clarify the relationship between the number of.5., the number as a means of combination of figures, mainly reflected in the analytic geometry, the college entrance examination questions over the years on this aspect of the examination (that is, the use of algebraic methods to s
8、tudy geometric issues). And figure as a means of combination of numbers in the college entrance examination objective reflected in the problem.We should grasp the following 6. points: the problem solving method combined with shape (1) research on the distance, angle or area, can start directly from
9、the geometry of the model can be solved; (2) for the study of function, equation or inequality (maximum) problem, the function of the image (as a function of zero solution that is the key point for the vertex), knowledge transfer and comprehensive application.Two. Mathematical ideas of classificatio
10、n discussionThe idea of classified discussion is a kind of important mathematics thought, when the object cannot be studied when the needs of the research object classification, and then for each kind of study is given for each class, the final results to get the whole integrated all kinds of questi
11、ons.The 1. mathematical problems of classification discussion need to use the idea of classified discussion to solve, the cause of classification discussion can be summed up as following: (1) the mathematical concepts involved are classified discussion; (2) the use of mathematical theorems and formu
12、las, or operational properties, classification rule is given; (3) the mathematical problem for the conclusion of a variety of circumstances or a variety of possibilities; (4) with parametric mathematical problems, different values of these parameters lead to different results; (5) complex or non gen
13、eral math problems, need to adopt problem-solving strategies to solve the classification discussion.The 2. category discussion is a logical method and has a very wide range of applications in middle school mathematics. According to different standards can have different classification methods, but t
14、he classification must proceed from the same standard, do not repeat, do not miss, contain all kinds of circumstances, at the same time, to facilitate the study of the problem.Three, function and equation thoughtThe idea of function and equation is a way of thinking to solve the problem of the relat
15、ion between variables or unknowns by the view and method of function and equation. It is an important mathematical idea.1. function thought: some of the variables in the process of change are expressed by functional relation, and the relationship between these variables is studied, and finally the p
16、roblem is solved. This is the idea of function;2. application function, problem solving, establish the relationship between variables is a critical step, can be roughly divided into the following two steps: (1) according to the established function relationship between variables, the problem is tran
17、sformed into a function corresponding to the problem; (2) according to the needs of the constructor, use the knowledge function to solve the problem.3. equation thought: in some process, often need according to some requirements, determine the values of certain variables, then often lists these vari
18、ables or equations (equations), by equation (or equations) are obtained, which is the equation thinking.4. function and equation is two with mathematical concepts closely linked, mutual penetration between them, a lot of problems to solve equations with knowledge and method of the function, a lot of
19、 problems but also need to use the function equation method of support, the dialectical relationship between function and equation, the formation of the function equation of thought.Four, the thought of transformation and transformationThe idea of transformation and transformation means that when we
20、 study and solve some mathematical problems, we should adopt some means to transform the problem through transformation so as to achieve the goal of solution.In general, complex problems are often transformed into simple ones by means of change, which transforms difficult questions into easily solve
21、d problems, and transforms the unsolved ones into the ones that have been solved.1. general principles of normalization and transformation are as follows: to target simple principle; the principle of harmony and unity (return towards the problem to be solved in the form of performance tends to be ha
22、rmonious, the unity in quantity and form, the relationship between the direction of the problem, the condition and conclusion more uniform and right.) Thirdly, the principle of formalization; the formal principle of standardization; formalization of the problem to be formalized in the form of the st
23、andard of the problem. A standard form is a mathematical model that has been established. As a quadratic function of y=ax2+bx+c (a = 0); the low level (the principle of solving math problems, should be in high dimensional space to be solved problem into low dimensional space, high frequency problem
24、into low frequency problems, to solve the problem of multiple small problem. This is because low level issues are more intuitive, specific, and simple than high-level ones.2. reduction and transformation strategies are: transforming the known and unknown (known conditions often contain rich content,
25、 to explore the hidden conditions, the known conditions toward the clear direction of the transformation, such as the comprehensive method; for the new problem, an unknown by Lenovo, looking into the way, or known transformation from the conclusion such as manpower analysis). The conversion of posit
26、ive and negative (in dealing with a problem, in accordance with the customary way of thinking from the positive thinking and difficult, or even impossible, with the method of reverse thinking to solve, can often achieve breakthrough results). Thirdly, the combination of number and form (the combinat
27、ion of figures and shapes) is a combination of abstract mathematical language and intuitive graphics, which can make many concepts and relations intuitive and visualized, and help to explore the ways of solving problems. General and special transformation. Transforming complex and unfamiliar problem
28、s into simple and familiar problems is one of the most important principles in solving mathematical problems. High school mathematics involving most of the thought of transformation, such as algebraic and transcendental equation in three-dimensional space plane, complex real problems, in order to re
29、alize the transformation, and correspondingly produce many mathematical methods, such as elimination method, substitution method and image method, method of undetermined coefficient, distribution method. Through the use of these mathematical methods, students can fully appreciate the position and fu
30、nction of mathematical thinking in the field of mathematics.II mathematical method1, matching methodRefers to a method with algebraic form into an algebraic or several square form, distribution method is mainly applicable to the two items related to the function, equation, equality and inequality ar
31、e discussed, and discussed for proof and two times curve.2 、 undetermined coefficient methodThe method of undetermined coefficients is to solve some mathematical problems with certain determinacy by introducing some undetermined coefficients into equations. The main theoretical basis for the method
32、of undetermined coefficients is that:(1) the necessary and sufficient condition of the polynomial f (x) =g (x) is that for any value a, there is f (a) = g (a);(2) polynomial f (x) = g (x) is the necessary and sufficient conditions: two the polynomial coefficients corresponding to equal much;Two, the
33、 procedure of applying the undetermined coefficient method is:(1) determine the analytic formula (or equation of the equation) for the given problem with undetermined coefficients;(2) according to the same condition, a set of equations with undetermined coefficients is listed;(3) solving the equatio
34、n or eliminating the undetermined coefficients, thus solving the problem;Three, the method of undetermined coefficients is mainly applied to: the analytic formula of the function, the equation of the curve, the factorization, etc.3, change element methodThe change element method refers to the introd
35、uction of one or more new variables to replace some of the original variables (or algebraic expressions), to new variables to find the results, and then return to the original variables. By introducing new elements, the element method links the dispersed conditions, or shows implicit conditions, or
36、links the conditions with the conclusions, or becomes familiar ones. The theoretical basis is equivalent substitution. In senior high school mathematics, the change element method mainly has the following two kinds:(1) the whole change the Yuan: Yuan for type; (2) triangular variables, to style for
37、yuan.;In addition, there are symmetric substitution, mean substitution, universal substitution, substitution method is widely used, such as the solution of the equation, inequality, inequality proof, and the range of the function for the series, the general and etc., it is also widely used in analyt
38、ic geometry. When using the meta change method, we should pay attention to the constraints of the new capital and the strategy of overall replacement.4, vector methodVector method is a method of using vector knowledge to solve problems. The following common knowledge is used in solving problems:(1)
39、the necessary and sufficient conditions for the geometric representation of vectors and the collinearity of two vectors; (2) the fundamental theorem and theory of plane vectors; (3) dealing with the problems of length, angle and vertical by means of the scalar product of vectors;(4) the distance for
40、mula between two points, the definite score point formula and the translation formula of the line segment.5. Analysis and synthesis(1) analysis is starting from the verification results, the gradual introduction of it can make the conditions for the establishment of a known fact, until now; analysis
41、 method is a kind of direct result reason method.(2) the synthesis method is the conclusion that the required certificate is gradually introduced from the proved conclusion and formula. The synthesis method is a kind of direct proof method of narrative fluency.(3) analysis method and synthesis metho
42、d are the two most basic methods to prove mathematical problems. The analysis result reason analysis method, clear, easy to find ways of problem solving, but the written form requirement is high, not easy to clear narrative, so the analysis method and comprehensive method are often used interchangea
43、bly. The analysis method and the synthesis method are widely used. Almost all questions can be solved by these two methods.6, the reduction to absurdityThe reduction to absurdity is an important method of mathematical proof, because proposition p is contrary to its negation, non Ps truth and false,
44、so it is necessary to prove a proposition as true, so long as its negation is false. This proof from the proof of contradictory propositions (i.e., the negation of propositions) is false, and then proves that the proposition is true.The general procedure of proof by rebuttal is.:(1) set the hypothes
45、is that the conclusion of the proposition does not hold water, that is, the opposite of the hypothesis;(2) fallacy: starting from the conditions of the proposition and the conclusions drawn, after the correct reasoning and reasoning, we can get the contradictory results; (3) the conclusion is that t
46、here is a contradiction to judge the hypothesis is not correct, so that the conclusion is correct.Two, the scope of the application of rebuttal:(1) a proposition with little known conditions or a few conclusions that can be derived from known conditions;(2) the opposite of the conclusion is a propos
47、ition that is more specific and simpler than the original conclusion, especially the proposition that the conclusion is definite (no, impossible, impossible), and so on;(3) a proposition involving various kinds of infinite conclusions;(4) the proposition that at the most (less) and several is the conclusion;(5) existence proposition; (6) unique proposition; (7) the converse theorem of some theorems;(8) inequalities, such as indefinite or difficult to direct proof of general relations.On the basis of the three logical apagoge is the law of contradiction and law of excluded middle.