Ideological Methods of High School Mathematics, Eight Ideas and Ten Methods of High School Mathemati

The theorems in the textbook, you can try to reason by yourself. This will not only improve your proof ability, but also deepen your understanding of the formula. There are also a lot of practice questions. Basically, after each class, you have to do the questions of the after-class exercises (excluding the teacher's homework).

Listening: You should grasp the main contradictions and problems in the lecture, think synchronously with the teacher's explanation as much as possible when listening to the lecture, and take notes if necessary

Reading: When reading, you should carefully scrutinize, understand and understand every concept, theorem and law, and study together with similar reference books for example problems, learn from others' strengths, increase knowledge, and develop thinking

**: To learn to think, after the problem is solved, then explore some new methods, learn to think about the problem from different angles, and even change the conditions or conclusions to find new problems

Homework: Review first and then homework, think first and then start writing, do a class of questions to understand a large piece, homework should be serious, writing should be standardized, only in this way down-to-earth, step by step, in order to learn mathematics well

In short, in the process of learning mathematics, we should realize the importance of mathematics, give full play to our subjective initiative, pay attention to small details, develop good mathematics learning habits, and then cultivate the ability to think, analyze and solve problems, and finally learn mathematics well

So, the method of thinking is a process of accumulation, the more you know, the better you learn, so memorize more and choose your own method.

1. Functions and equations.

2. The idea of combining numbers and shapes.

3. Categorize and discuss ideas.

4. Transformation and naturalization.

The eight ideas and ten methods of high school mathematics are as follows:

The eight ideas are 1. The idea of combining numbers and shapes, the idea of combining numbers and shapes is based on the intrinsic connection between the problem and the conclusion of mathematical problems, so that the quantitative relationship and graphics are cleverly and harmoniously combined, and make full use of this combination to seek solutions to problems and solve problems. Digitizing into graphs, or being able to obtain useful numbers from graphs, is the key to the idea of combining numbers and shapes.

The key to solving problems by combining mathematics with ideas is to clarify the close connection between numbers and shapes, and the number problems can be solved by using shapes, and the problems of shapes can be solved by numbers. Attention should be paid to the combination of numbers and shapes, considering the specific situation of the problem, and transforming the problem of the graphic nature into the problem of the relationship between quantities, or the problem of the relationship between quantity into the problem of the problem of the nature of the figure, so as to simplify the complex problem.

2. Transformation and demarcation of thought, the idea of naturalization, the process of turning a problem from difficult to easy, from complex to simple, and from complex to simple is called naturalization, which is the abbreviation of transformation and reduction. Universal connection and eternal development are the philosophical foundations for the transformation of the assigned ideas. In general, complex problems are transformed into simple problems through transformation; Transform intractable problems into easy-to-solve problems through transformation; Transform an unsolved problem into a solved problem.

Naturalization is not only an important problem-solving idea, but also the most basic thinking strategy, and an effective way of mathematical thinking. The so-called naturalization method is a method that uses some means to transform the problem through transformation when studying and solving relevant mathematical problems, and then achieves a solution.

The ten methods are 1. Matching method, matching method refers to the combination of a formula (including rational formula and transcendence formula) or a certain part of a formula through identity deformation into a complete flat mode or the sum of several perfect flat methods, this method is called a matching method. This method is often used in identity deformation to explore the implicit conditions in the problem, which is one of the powerful means to solve the problem.

2. Factorization, a method used in mathematics to solve higher-order unary equations. The method of factoring the number (including unknown numbers) on one side of the equation to 0 by moving it, and turning the other side of the equation into the product of several factors, and then making each factor equal to 0 to find its solution is called factorization.

Algebraic term, refers to the process and result of expressing a polynomial as the product of several polynomials, and each polynomial with order n greater than or equal to 1 on the number field p can be uniquely decomposed into the product of the irreducible polynomial on p, and the process of expressing the polynomial on p as such a product is called the factorization of the polynomial, referred to as factorization (or factorization).

Mathematics is the study of concepts such as quantity, structure, change, space, and information. Mathematics is a general means for humans to strictly describe the abstract structure and pattern of things, and can be applied to any problem in the real world, and all mathematical objects are inherently artificially defined.

The thinking methods of high school mathematics include five methods: transformation, logic, inverse, correspondence, and analogy.

1. Transformation method: Transformation thinking is not only a method, but also a kind of thinking. Transforming thinking and shouting delay refers to changing the direction of the problem from different angles to change the direction of the problem and seek the best way to make the problem simpler and clearer by changing the direction of the problem and changing the problem from different angles.

2. Logical method: Logic is the basis of all thinking. Logical thinking is the thinking process in which people observe, compare, analyze, synthesize, abstract, generalize, judge, and reason about things with the help of concepts, Danye judgments, Zheng Hu reasoning and other forms of thinking in the process of cognition.

Logical thinking, which is widely used in solving logical reasoning problems.

3. Reverse method: Reverse thinking, also known as divergent thinking, is a way of thinking backwards about commonplace things or opinions that seem to have become a foregone conclusion. Dare to "think the opposite", let the thinking develop in the opposite direction, explore deeply from the opposite side of the problem, establish new ideas, and create a new image.

4. Correspondence method: Correspondence thinking is a thinking method that establishes a direct connection between quantitative relationships (including quantity difference, quantity multiple, and quantity rate). The more common ones are general correspondence (such as the correspondence between two or more quantities and the difference multiple) and quantity rate correspondence.

5. Analogical method: Analogical thinking refers to the thinking method of comparing unfamiliar and unfamiliar problems with familiar problems or other things according to some similar properties between things, discovering the commonality of knowledge, finding its essence, and thus solving problems.

A way to develop the logic of mathematical thinking

1. Cultivate flexibility of thinking: be good at breaking away from old patterns and general constraints and find the right direction; The ability to use knowledge freely, the use of dialectical thinking to balance the relationship between things, the specific analysis of specific problems, and the ability to adapt and adjust ideas, etc., are the direct manifestations of the development of thinking flexibility.

2. Cultivate the rigor of mathematical thinking: under the premise of clear thinking, it should be steady and steady, gradually deepened, and grasp sufficient reasons as the basis for argumentation and reasoning; When practicing the test questions, he is good at paying attention to the hidden conditions in the question stem, answering the questions in detail, and writing out the solution ideas without hesitation.

3. Cultivate the profundity of mathematical thinking: Students should see the essence of mathematics through phenomena, grasp the most basic mathematical concepts, and gain insight into the connection between mathematical objects, which is the main manifestation of profound thinking.

Functions and Equations The idea of functions refers to the analysis and study of quantitative relations in mathematics from the perspective of motion changes, the establishment of functional relations or constructors, and the use of images and properties of functions to analyze.

2 The idea of equations is to analyze the equiquantity relationship in mathematics, to construct equations or systems of equations, and to analyze and solve problems by solving or using the properties of equations.

Equation thinking is the basic idea of solving various computing problems and the basis of computing power.

Numbers and shapes combine ideas.

The object of mathematical research is the relationship between quantity and spatial form, that is, the two aspects of number and shape.

Transformation and naturalization: It is an important basic mathematical idea to reduce those problems to be solved or difficult to solve into solvable problems within the scope of existing knowledge. This kind of reduction should be an equivalent transformation.