Self-confidence is very important in solving mathematical problems. Believe in yourself, as long as you don't go beyond your knowledge, no matter which problem you have studied, you can always solve it 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 of them. 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, and more importantly, we should grasp the particularity of this kind of problem and the difference between this kind of problem and this kind of problem. There are almost no identical mathematical problems, and there are always one or several different conditions, so the thinking and problem-solving processes are not the same.

Second, cultivate the thinking ability of "equation"

Mathematics studies the spatial form and quantitative relationship of things. The most important quantitative relationship is equality, followed by inequality. The most common equivalence relation is "equation". For example, in constant-speed motion, there is an equal relationship among distance, speed and time, and a related equation can be established: speed × time = distance. In such an 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 seventh grade, we systematically studied the solution of the linear equation with one yuan, and summarized the five steps of solving the linear equation with one yuan. If you learn and master these five steps, any one-dimensional linear equation can be solved smoothly. In the eighth and ninth grades, you will learn to solve quadratic equations in one variable, quadratic equations in two variables and fractional equations. In high school, you will also learn exponential equation, logarithmic equation, linear equation, parametric equation and polar coordinate equation. The ideas and methods of solving these equations are almost the same, and they are all transformed into linear equations or quadratic equations of one variable by certain methods, and then they are solved by the familiar five steps of solving linear equations of one variable or the root formula of solving quadratic equations of one variable. Energy conservation in physics, chemical equilibrium in chemistry and a large number of practical applications in reality all need to establish equations and get results by solving them. Therefore, it is necessary to teach the solutions of linear equation and quadratic equation well, so that students can learn this part well and then learn other forms of equations well. The so-called "equation" thinking 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 perspective of "equation" and then solving them.

Third, cultivate the thinking ability 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 extend correspondence to a relationship, a form and so on. For example, when calculating or simplifying the factorization factor, we need to use the square difference formula, where A on the left side of the formula corresponds to X+2 and B corresponds to Y, and then we can directly get the factorization result (x+2+y)(x+2-y) by using the one on the right side of the formula. This is to use the idea and method of "correspondence" to solve problems. In middle school mathematics, we will see one-to-one correspondence between points on the number axis and real numbers, 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.

Fourthly, cultivate the ability of "transforming" thinking in mathematics.

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 the campus area, it needs to requisition land from the town. The town gave an irregular piece of land. How to measure its area? Firstly, the actual terrain is drawn into paper graphics with a small flat instrument (level or theodolite if possible), then the paper graphics are divided into several trapezoid, rectangle and triangle, and the sum of the areas of these graphics is calculated by the learned area calculation method, and then the total area of this irregular terrain is obtained. Here we convert uncountable irregular figures into the sum or difference of the areas of countable regular figures, 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 "reduction", and then solved by known steps or formulas.

Fifth, cultivate the ability of "combination of numbers and shapes"

"Number" and "shape" are everywhere. Everything, stripped of qualitative aspects, has only two attributes: shape and size, which can be left to mathematics to study. There are two branches of junior high school mathematics-algebra and geometry. Algebra studies "number" and geometry studies "shape". But algebra should be learned 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, there is a special course to study geometric problems by algebraic method, which is called Analytic Geometry. After the establishment of plane rectangular coordinate system, the study of function 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 mathematics learning, we should attach importance to the thinking training of "combination of numbers and shapes" As long as any problem is a little marginal to "shape", it is necessary to draw a sketch and analyze it 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".