A, construction skills:

Its basic form is to construct a new mathematical form with known conditions as raw materials and conclusions as the direction, so that the problem can be solved simply in this form. Common ones are structure diagrams, equations, identities, functions, counterexamples, drawers and algorithms.

Second, the drawing skills:

Its basic form is RMI principle. Let R represent a set of relational structures (or systems) of the original image, which contains the original image to be determined. Let R represent a mapping, through which the original image structure R is mapped into an image relational structure R*, which naturally contains images of unknown original images.

If there is a definite method, it is determined by inversion, that is, inverse mapping. This principle is embodied in logarithmic calculation, substitution, introduction of coordinate system, design of mathematical model, construction of generating function and so on. Establishing corresponding relationships to solve problems also belongs to this skill.

Third, recursive skills:

If there is a definite relationship between the former and the latter, then we can recursively get the results at any time from some (several) initial conditions and solve the problem by recursive method, which is related to mathematical induction (but not predicting the conclusion) and infinite descent method. The key is to find out the recursive relationship between the former proposition and the latter proposition.

Fourth, identification skills:

When the "black box of mathematics" is too complicated, it can be divided into several small black boxes to decipher one by one, that is, the parts with the same nature are classified into one category, forming a very distinctive method in mathematics-distinguishing situations or classifying. Without correct classification, it is impossible to master mathematics.

Sometimes, a problem can be arranged into a series of small goals in stages, so that once the previous situation is proved, it can be used to prove the later situation. This is the so-called climbing procedure. For example, solving Cauchy function equation is to turn integers into natural numbers, then rational numbers into integers, and finally real numbers into rational numbers.

Distinguishing the situation not only distinguishes the difficulty of the problem, but also attaches a known condition to the classification standard itself, so the difficulty of solving each sub-problem is greatly reduced.