"Mathematics is the mother of all sciences" and "Mathematics is the gymnastics of thinking". It is a science that studies numbers and shapes, and it is everywhere. To master technology, we must first learn mathematics well, and to climb the peak of science, we must learn mathematics well. What are the characteristics of mathematics compared with other disciplines? What is its corresponding way of thinking? What kind of subjective conditions and learning methods does it require us to have? This lecture will briefly explain the characteristics, ideas and learning methods of mathematics. I. Characteristics of Mathematics (I) The rigidity, abstraction and extensive application of mathematics The so-called rigor of mathematics means that mathematics has strong logic and high proficiency, which is generally embodied in an axiomatic system. What is the axiomatic system? It refers to selecting a few undefined concepts and propositions without logical proof, and deducing some theorems to make them a mathematical system. In this respect, the ancient Greek mathematician Euclid is a model, and his Elements of Geometry studies most problems in plane geometry on the basis of several axioms. Here, even the most basic and commonly used original concepts cannot be described intuitively, but must be confirmed or proved by axioms. There are still some differences in rigor between middle school mathematics and mathematics science. For example, the continuous expansion of several sets in middle school mathematics, the expansion operation law of several sets is not strictly deduced, but obtained by default. From this point of view, middle school mathematics is still far from rigorous, but to learn mathematics well, we must not relax the requirements for rigor and ensure the scientific content. For example, arithmetic progression's general term is summed up through the recursion of the previous items, but it needs to be strictly proved by mathematical induction to be confirmed. The abstraction of mathematics is manifested in the abstraction of spatial form and quantitative relationship. In the process of abstraction, it abandons the specific characteristics of more things, so it has a very abstract form. It shows a high degree of generality and symbolizes the concrete process. Of course, abstraction must be based on concreteness. As for the wide application of mathematics, it is well known. Only in the past teaching and learning, we often paid too much attention to the abstract meaning of theorems and concepts, but sometimes gave up their wide application. If abstract concepts and theorems are compared to bones, then the extensive application of mathematics is like flesh and blood, and the lack of any one will affect the integrity of mathematics. The purpose of increasing the application space of mathematics knowledge and research-based learning in the new high school mathematics textbook is to cultivate students' ability to solve practical problems by applying mathematics. Second, the characteristics of high school mathematics often lead to students' inability to adapt to mathematics learning after entering high school, which in turn affects their enthusiasm for learning and even their grades plummet. Why is this happening? Let's take a look at the changes in high school mathematics and junior high school mathematics. 1, theoretical strengthening 2, curriculum increase 3, difficulty increase 4, requirement improvement 3, mastery of mathematical thinking High school mathematics is closer to advanced mathematics in learning methods and thinking methods. Learning it well requires us to master it from the height of methodology. When we study mathematical problems, we should always use materialist dialectical thinking to solve mathematical problems. Mathematical thought is essentially a reflection of the application of materialist dialectics in mathematics. The mathematical thoughts that should be mastered in middle school mathematics learning are: set and correspondence, initial axiom, combination of numbers and shapes, movement, transformation and transformation. For example, the concepts of sequence, linear function and straight line in analytic geometry can be unified with the concept of function (special correspondence). For another example, the concepts of number, equation, inequality and sequence can also be unified into the concept of function. Let's look at the following example of solving problems with a "contradictory" point of view. Given that the moving point Q moves on the circle x2+y2= 1, fix the point P (2 2,0) and find the locus of the midpoint of the straight line PQ. Analyzing this problem, P, Q and M are mutually restricted, and the movement of Q drives the movement of M; The main contradiction is the movement of point Q, whose trajectory follows the equation x02+y02 =1①; Secondary contradiction: m is the midpoint of the straight line PQ, and the coordinates (x, y) of m can be expressed by the midpoint formula with the coordinates of point Q, and X = (x0+2)/22y = y0/2③ Obviously, the trajectory can be obtained by substituting x0 and y0 in the elimination problem. Mathematical thinking method is different from problem-solving skills. In proving or solving, it can be said that solving problems by induction, deduction and method of substitution is a technical problem, and mathematical thinking is a guiding general thinking method. When solving a problem, from the overall consideration, how to start, what are the methods? It is a common problem under the guidance of mathematical thinking methods. With mathematical ideas, we should master specific methods, such as method of substitution, undetermined coefficient method, mathematical induction, analysis, synthesis and induction. Only under the guidance of problem-solving thought and flexible use of specific problem-solving methods can we really learn mathematics well. It is often difficult to make mathematics learning enter a higher level by mastering specific operation methods without considering problems from the perspective of problem-solving thinking, which will bring great trouble for further study in universities in the future. In terms of specific methods, commonly used are: observation and experiment, association and analogy, comparison and classification, analysis and synthesis, induction and deduction, general and special, finite and infinite, abstraction and generalization. If you want to win the battle, you can't just fight bravely, not afraid of death or suffering. You must formulate tactics and strategies that have a bearing on the overall situation. When solving mathematical problems, we should also pay attention to solving the problem of thinking strategy, and often think about what angle to choose and what principles to follow. Generally speaking, the general idea adopted to solve problems is a principled thinking method, a macro guidance and a general solution. Mathematical thinking strategies often used in middle school mathematics include: controlling complexity with simplicity, combining numbers with shapes, combining advance and retreat, turning life into practice, turning difficulties into difficulties, turning difficulties into advances, turning static into dynamic, combining and bringing out the best in each other. If you have correct mathematical thinking methods, appropriate mathematical thinking strategies, rich experience and solid basic skills, you will certainly learn high school mathematics well. Fourth, the improvement of learning methods is in the strange circle of exam-oriented education. Every teacher and student can't help falling into the ocean of problems. Teachers pay attention to a certain kind of questions, and the college entrance examination can't do it. Students are afraid to do one less question. In case the loss is too heavy, in such an atmosphere, the cultivation of learning methods is often ignored. Every student has his own method, but what kind of learning method is correct? Is it necessary to "read a lot of questions" to improve your level? Reality tells us that it is a very important issue to boldly improve learning methods. (1) Learn to listen and read. We listen to teachers and read textbooks or materials every day at school, but do we listen and read correctly? Let's talk about listening (listening, classroom learning) and reading (reading textbooks and related materials). What students learn is often indirect knowledge, abstract and formal knowledge, which is refined on the basis of previous exploration and practice, and generally does not include the process of exploration and thinking. So be sure to listen to the teacher, concentrate and think positively. Find out what the content is. How to analyze it? What is the reason? In what way? Is there a problem? Only in this way can we understand the teaching content. The process of attending classes is not a process of passive participation. On the premise of attending class, we need to analyze: what thinking method is used here, and what is the purpose of doing so? Why can the teacher think of the shortest way? Is there a more direct way to solve this problem? "Learning without thinking is useless, thinking without learning is dangerous", so we must have positive thinking and participation in the process of listening to classes, so as to achieve the highest learning efficiency. Reading mathematics textbooks is also a very important way to master mathematics knowledge. Only by reading and reading mathematics textbooks can we master mathematics language well and improve our self-study ability. We must change the bad tendency of using textbooks as dictionaries to look up formulas without reading books. When reading textbooks, we should also strive for the guidance of teachers. When reading the content of the day or the content of a unit or a chapter, we should consider comprehensively and have a goal. For example, to learn the arcsine function, we should ask the following questions through reading: (1) Does every function have an inverse function? If not, when does a function have an inverse function? (2) Under what circumstances does a sine function have an inverse function? If so, how to express its inverse function? (3) What is the relationship between the image of sine function and the image of arcsine function? (4) What are the properties of the arcsine function? (5) How to find the value of the arcsine function? (2) Learn to think Einstein once said, "Developing the general ability of independent thinking and independent judgment should always be put in the first place." Being diligent and good at thinking is the most basic requirement for us to learn mathematics. Generally speaking, we should try our best to do the following two things. 1, good at finding and asking questions 2, good at reflection and reverse seeking.