I have only one purpose in introducing these situations, that is, to persuade those students who are afraid of mathematics not to give up mathematics. The basic knowledge of mathematics is not difficult to learn, and I believe everyone can learn it well. We should establish self-confidence, believe in ourselves and believe that we can narrow the gap with other students through our efforts!
Some students may want to ask, so how to work hard? Can you recommend some good methods that are effective and not difficult to learn? Of course, let me talk about how to operate to really learn math well.
First, don't think you understand what you should remember and recite.
Some students think that mathematics is not like English and social politics, but depends on words, dates, names and places. 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, it is not cost-effective to add 9 9s to get 8 1. 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 still many laws to remember in mathematics. For example, when simplifying the secondary root, it is stipulated: "Unless otherwise specified, the letters in the root of this chapter are all positive numbers." Wait a minute. 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, we are familiar with "multiplication formula, root formula" and "trigonometric function value of special angle". I think some of our classmates can recite, while others can't. Here, I want to remind those students who can't recite these formulas. If they can't recite these formulas, it will cause great trouble for future study, because these formulas and data will be used in a large number in future study.
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.
Second, understand several important mathematical ideas: 1 and 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. Learn to understand one-dimensional quadratic equation, two-dimensional quadratic equation and simple triangular equation in the second and third day of junior high school. 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 one-dimensional quadratic equations to lay a good foundation for learning 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. Anything, peel off its two eyes, a pair of earrings and twins correspond to an abstract number "2"; With the deepening of learning, we also extend "correspondence" to a form, a relationship, and so on. For example, in the simplified evaluation calculation, we substitute letters or a whole value into the formula and directly calculate the result of the original formula. For example, in the third grade, we comprehensively studied the angles related to the circle, and the quantitative relationship among the central angle, the circumferential angle and the tangential angle must "correspond" to the same arc to be established. This is to use the idea and method of "correspondence" to solve problems. We also saw 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. In short, the idea of "correspondence" will play an increasingly important role in future research.
4. The concept of "transformation"
The most fundamental way to solve mathematical problems is to "turn the difficult into the easy, simplify the complex, and turn the unknown into the known", that is, through certain mathematical thinking, methods and means, a complex mathematical problem is gradually transformed into a well-known simple mathematical form, and then it is solved through familiar mathematical operations.
For example, if our school wants to expand its campus, it needs to requisition land from a village. And a village gave an irregular piece of land, how to measure its area? Firstly, the actual terrain is drawn into paper graphics according to a certain proportion by using suitable measuring tools, and then the paper graphics are divided into several trapezoid, rectangle and triangle. Using the learned area calculation method, the sum of the areas of these figures is calculated, and the total area of this irregular terrain is obtained. Here, we transform the irregular graphics that can't be calculated into regular graphics that can be calculated, thus solving the problem of land survey. In addition, all kinds of multivariate equations and higher-order equations mentioned above can be finally transformed into linear equations or quadratic equations with one variable by means of elimination and simplification, and then solved by known steps or formulas.
The thought of "transformation and substitution" is the most important thinking habit in solving problems. In the face of difficult problems and unfamiliar problems, we must first think of "transformation" and be able to "transform" at all times. Usually, teachers should pay more attention to how to solve problems, and how to "make the difficult easy, simplify the complicated, and turn the unknown into the known". Students should also exchange more experiences of "successful transformation", deeply understand the true meaning of "transformation" and earnestly master the thinking and skills of "transformation".
Thirdly, 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 Zhejiang Institute of Education for a meeting at the end of last year, I was deeply moved by Vice President Wu's words. He said: I teach physics, but I often go out and my classmates learn physics well. I didn't teach them, but they realized it themselves. Of course, Vice President Wu is modest, but he explained a truth: students should not study passively, but should study 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, teachers should be able to use the old knowledge they have learned to preview the new lesson 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 incomprehensible when listening to the teacher's new class?
The feeling of "understanding as soon as you listen, making mistakes as soon as you do it" is because you didn't preview, didn't study with questions, didn't really turn "I want to learn" into "I want to learn", and tried to turn knowledge into your 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.
Fourth, self-confidence can make you stronger.
In previous exams, I always saw that some students had many blanks in their papers, but they didn't do 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 conditions, including implied conditions. Then, from what you want to know, from what you know to what you know, we can build a bridge between what you know and what you need to know, forming a channel from what you know to what you want, so that the problem can be solved smoothly. In fact, there is a certain * * * between a problem and a kind of problem. 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 follow the trend, and if the problem changes slightly, they will be in a daze and have no way to 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, the so-called "all roads lead to Rome". 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 with constant changes. The topic is not to do more, the better. The ocean of topics is endless, you can never finish reading them, but you can't help doing them. The key is a "degree". To a certain extent, I still encourage students to "do more and practice more, because practice makes perfect;" See more and think more to be well informed. "In this way, through intensive training, we can cultivate our good mathematical thinking habits and master the correct mathematical problem-solving methods. Then I went to the senior high school entrance examination, because there are many kinds of questions, I can "extrapolate, practice makes perfect", which speeds up the speed and saves time, which is especially important in the senior high school entrance examination with limited examination time.
Solving math problems requires rich knowledge and self-confidence. Without self-confidence, you will be afraid of difficulties and give up; With self-confidence, we can forge ahead, not give up easily, study harder, and hope to overcome difficulties, reach the other side of success and create our own brilliant tomorrow.