Some students think that mathematics is not like English, history and geography. Words, dates, and place names are required. Mathematics depends on wisdom, skill and reasoning. I said you were only half right. Mathematics is also inseparable from memory. Imagine, elementary school addition, subtraction, multiplication and division and Divison, can you operate smoothly without memorizing the multiplication table? Although you understand that multiplication is the operation of the sum of the same addend, when you do 9*9, you add 9 9s to get 8 1, which is too uneconomical. It is much more convenient to use "998 1". Similarly, it is also made with the rules that everyone knows by heart. At the same time, there are many laws in mathematics that need to be memorized, such as law (a≠0) and so on. So, I think mathematics is more like a game. It has many rules of the game (that is, definitions, rules, formulas, theorems, etc. Whoever remembers these rules of the game will be able to play the game smoothly. Whoever violates these rules of the game will be judged wrong and sent off. Therefore, mathematical definitions, rules, formulas and theorems must be memorized, and some of them are best memorized and catchy. For example, the familiar "Three Formulas of Algebraic Multiplication", I think some of you here can recite it, while others can't. Here, I want to remind the students who can't recite these three formulas. If they can't recite it, it will cause great trouble for future study, because these three formulas will be widely used in future study, especially the factorization of senior two, in which three very important factorization formulas are all derived from these three multiplication formulas, and they are deformations in opposite directions.

Remember the definitions, rules, formulas and theorems of mathematics, and remember those that you don't understand for the time being, and deepen your understanding on the basis of memory and application to solve problems. For example, mathematical definitions, rules, formulas and theorems are just like axes, saws, Mo Dou and planers in the hands of carpenters. Without these tools, carpenters can't make furniture. With these tools, coupled with skilled craftsmanship and wisdom, you can make all kinds of exquisite furniture. Similarly, if you can't remember the definition, rules, formulas and theorems of mathematics, it is difficult to solve mathematical problems. And remember these, plus certain methods, skills and agile thinking, you can be handy in solving mathematical problems, even solving mathematical problems.

Several important mathematical ideas

1, the idea of "equation"

Mathematics studies the spatial form and quantitative relationship of things. The most important quantitative relationship in junior high school is equality, followed by inequality. The most common equivalence relation is "equation". For example, uniform motion, distance, speed and time are equivalent, and a related equation can be established: speed * time = distance. In this equation, there are generally known quantities and unknown quantities. An equation containing unknown quantities like this is an "equation", and the process of finding the unknown quantities through the known quantities in the equation is to solve the equation. We were exposed to simple equations in primary school, but in the first year of junior high school, we systematically studied the solution of one-dimensional linear equations and summarized five steps of solving one-dimensional linear equations. If you learn and master these five steps, any one-dimensional linear equation can be solved smoothly. In the second and third day of junior high school, you will also learn to solve one-dimensional quadratic equations, binary quadratic equations and simple triangular equations. In high school, we will also learn exponential equation, logarithmic equation, linear equation, parametric equation, polar coordinate equation and so on. The solution ideas of these equations are almost the same, and they are all transformed into the form of linear equations or quadratic equations in one variable by certain methods, and then solved by the familiar five steps to solve linear equations in one variable or the root formula to solve quadratic equations in one variable. Energy conservation in physics, chemical equilibrium formula in chemistry, and a large number of practical applications in reality all need to establish equations and get results by solving them. Therefore, students must learn how to solve one-dimensional linear equations and two-dimensional linear equations, and then learn other forms of equations.

The so-called "equation" idea means that for mathematical problems, especially the complex relationship between unknown quantities and known quantities encountered in reality, we are good at constructing relevant equations from the viewpoint of "equation" and then solving them.

2. The idea of "combination of numbers and shapes"

In the world, "number" and "shape" are everywhere. Everything, except its qualitative aspect, has only two attributes: shape and size, which are left for mathematics to study. There are two branches of junior high school mathematics-algebra and geometry. Algebra studies "number" and geometry studies "shape". It is a trend to learn algebra by means of "shape" and geometry by means of "number". The more you learn, the more inseparable you are from "number" and "shape". In senior high school, a course called "Analytic Geometry" appeared, which used algebra to study geometric problems. In the third grade, after the establishment of the plane rectangular coordinate system, the learning of functions can not be separated from images. Often with the help of images, the problem can be clearly explained, and it is easier to find the key to the problem, thus solving the problem. In the future mathematics study, we should pay attention to the thinking training of "combination of numbers and shapes" Any problem, as long as it is a little close to the "shape", should draw a sketch to analyze according to the meaning of the problem. This is not only intuitive, but also comprehensive, easy to find the breakthrough point, which is of great benefit to solving problems. Those who taste the sweetness will gradually develop the good habit of "combining numbers with shapes".

3. The concept of "correspondence"

The concept of "correspondence" has a long history. For example, we correspond a pencil, a book and a house to an abstract number "1", and two eyes, a pair of earrings and a pair of twins to an abstract number "2". With the deepening of learning, we also extend "correspondence" to a form, a relationship, and so on. For example, when calculating or simplifying, we will correspond the left side of the formula, A, Y and B, and then directly get the result of the original formula with the right side of the formula. This is to use the idea and method of "correspondence" to solve problems. The second and third grades will also see the one-to-one correspondence between points on the number axis and real numbers, the one-to-one correspondence between points on the rectangular coordinate plane and a pair of ordered real numbers, and the correspondence between functions and their images. The thought of "correspondence" will play an increasingly important role in future research.

The cultivation of self-study ability is the only way to deepen learning.

When learning new concepts and operations, teachers always make a natural transition from existing knowledge to new knowledge, which is the so-called "reviewing the past and learning the new". Therefore, mathematics is a subject that can be taught by itself, and the most typical example of self-study is mathematician Hua.

We listen to the teacher's explanation in class, not only to learn new knowledge, but more importantly, to subtly influence the teacher's mathematical thinking habits and gradually cultivate our own understanding of mathematics. When I went to Foshan No.1 Middle School to hold a parent-teacher conference, the words of the headmaster of No.1 Middle School made me feel a lot. He said: I teach physics and students learn physics well. I didn't teach them, but they realized it themselves. Of course, the headmaster is modest, but he explained a truth, students can't learn passively, they should learn actively. There are dozens of students in a class, and the same teacher teaches them. The difference is so great. This is the problem of learning initiative.

The stronger the self-study ability, the higher the understanding. With the growth of age, students' dependence will be weakened, while their self-learning ability will be enhanced. So we should form the habit of previewing. Before teaching a new lesson, can a teacher preview the new lesson by using the old knowledge he has learned and analyze and understand the new learning content in combination with the new regulations in the new lesson? Because mathematics knowledge is not contradictory, what you have learned will always be useful and correct, and further study of mathematics will only be deeper and broader. Therefore, solid mathematics learning in the past laid the foundation for future progress, and it is not difficult to learn new lessons by yourself. At the same time, when preparing a new lesson, it goes without saying that it is great to listen to the teacher explain the new lesson with questions when you encounter any problems that you can't solve. Why do some students always feel that they don't understand the teacher's new lesson, or "they will make mistakes when they listen to it"? It is because they did not preview, did not study with questions, did not really turn "I want to learn" into "I want to learn", and tried to turn knowledge into their own. Learn to learn, knowledge is still someone else's. The test of whether you can learn math well is whether you can solve problems. Understanding the definitions, rules, formulas and theorems related to memory is only a necessary condition for learning mathematics well, and being able to solve problems independently and correctly is the symbol of learning mathematics well.

Confidence can make you stronger.

In the exam, I always see that some students have a lot of blanks in their papers, but they haven't done a few questions at all. Of course, as the saying goes, art is bold, art is not timid. However, it is one thing to fail, and it is another thing to fail. The solution and result of a slightly more difficult math problem are not obvious at a glance. It is necessary to analyze, explore, draw, write and calculate. After tortuous reasoning or calculation, some connection between conditions and conclusions will be revealed and the whole idea will be clear. How do you know you won't do it if you don't do it? Even a teacher can't answer you immediately when he meets a difficult problem. It is also necessary to analyze and study first and find a suitable idea before teaching you. Dare not do a little more complicated questions (not necessarily difficult questions, some questions are just a little more narrative), which is a sign of lack of confidence. Self-confidence is very important in solving mathematical problems. Believe in yourself, as long as you don't go beyond your knowledge, you can always solve any problem with what you have learned. Dare to do problems and be good at doing them. This is called "strategic contempt for the enemy, tactical attention to the enemy."

When solving a specific problem, we must carefully examine the problem, firmly grasp all the conditions of the problem, and don't ignore any one. There is a certain relationship between a problem and a class of problems. We can think about the general idea and general solution of this kind of problem, but it is more important to grasp the particularity of this problem and the difference between this problem and this kind of problem. There are almost no identical problems in mathematics, and there are always one or several different conditions, so the process of thinking and solving problems is not the same. Some students and teachers can do the questions they have talked about, while others can't. They just talk about the matter and stare at some small changes in the problem, and they can't start. Of course, where to start is a tricky thing, and you may not be sure. But it is absolutely right to grasp its particularity when doing the problem. Choose one or several conditions as the starting point to solve the problem and see what can be drawn from this condition. The more you get, the better. Then choose the topic related to other conditions, conclusions or implied conditions for reasoning or calculus. There are many solutions to general problems, and all roads lead to Beijing. I believe that using the conditions of this problem, combined with the knowledge I have learned, will definitely draw a correct conclusion.

The topics of mathematics are infinite, but the ideas and methods of mathematics are limited. As long as we learn the basic knowledge well and master the necessary mathematical ideas and methods, we can successfully deal with endless problems. The topic is not to do more, the better. The ocean of topics is endless, and you will never finish reading it. The key is whether you have cultivated good mathematical thinking habits and mastered the correct mathematical problem-solving methods. Of course, doing more questions has several advantages: first, "practice makes perfect", which is very important in the case of limited examination time; One is to consolidate and memorize the learned definitions, theorems, rules and formulas by doing problems, thus forming a virtuous circle.