Mathematical paper

First, the meaning and function of mathematical skills

Skill is a way of action or psychological activity to successfully complete a task. It is an almost automatic, complex and relatively perfect action system formed through purposeful and planned exercises. Mathematical skills are actions or intellectual activities to successfully complete a certain mathematical task. It is usually manifested in the coordination of a series of actions and the automation of activities when completing a certain mathematical task. This kind of coordinated action and automatic activity is formed through repeated practice on the basis of existing mathematical knowledge and experience. For example, the multiplication skill multiplier you learn is a two-digit number, which is formed through many practical calculations on the basis of mastering its algorithm. Mathematical skills are closely related to mathematical knowledge and ability, and there are essential differences. The difference between them lies in: skill is the generalization of action and action mode, reflecting the proficiency of action itself and action mode; Knowledge is a summary of experience, which reflects people's understanding of the regularity of mutual connection between things; Ability is a summary of some stable psychological characteristics to ensure the smooth completion of activities, which embodies the individual characteristics of learners in mathematics learning activities. The relationship between the three can be clearly reflected from the role of mathematical skills.

The role of mathematics skills in mathematics learning can be summarized as the following aspects:

First, the formation of mathematical skills helps to understand and master mathematical knowledge;

Second, the formation of mathematical skills can further consolidate mathematical knowledge;

Thirdly, the formation of mathematical skills is helpful to solve mathematical problems;

Fourthly, the formation of mathematical skills can promote the development of mathematical ability;

Fifth, the formation of mathematical skills helps to stimulate students' interest in learning;

Sixth, arouse their learning enthusiasm.

Second, the classification of mathematical skills

Pupils' mathematical skills can be divided into two types according to their own nature and characteristics: operational skills (also known as motor skills) and mental skills (also known as intellectual skills).

Mathematical operation skills of length. Operation skill refers to the skill to realize the action of mathematical task activity mode mainly through the movement or operation of external body. It is an external operation activity mode composed of various local actions according to a certain program. For example, students' skills formed by measuring the degree of angle, measuring the length of objects and drawing geometric figures are such external operation skills. Operational skills have some characteristics that are obviously different from mental skills: first, they are explicit, that is, operational skills are an explicit way of activity; The second is objectivity, that is, the object of operation skill activities is physical objects or muscles; Wang is not a minimalist. As far as the structure of actions is concerned, every action of operational skills must be executed, which cannot be omitted or merged. This is an expansion project. If you draw a circle with compasses, determine the radius, determine the center of the circle, and rotate the compasses around the center with one foot, you can neither omit nor combine them, and you must perform the task in detail.

2. Mathematical psychological skills. Mathematical psychological skills refer to the way of psychological activities to successfully complete mathematical tasks. It is a cognitive activity with the help of internal speech, including psychological components such as perception, memory, thinking and imagination, and thinking is its main activity component. For example, primary school students' skills in oral calculation, pen calculation, solving equations and solving application problems are more mathematical mental arithmetic skills. Mathematical mental skills are also formed through acquired learning and training, which is different from human instinct. In addition, mathematical psychological skills are a legal way of psychological activities. "The so-called legal way of activity means that the elements of action and their order should reflect the requirements of the objective law of the activity itself, not arbitrary." These characteristics reflect the essence of mathematical psychological skills and mathematical operation skills. As a cognitive activity with thinking as the main activity component, mathematical psychological skills also have personality characteristics different from mathematical operation skills, which are mainly reflected in the following three aspects.

First, the concept of the object of litigation. The direct object of mathematical psychological skills is not the physical object itself, but the subjective image of this object in people's minds. For example, the direct object of mental activity of abdication subtraction oral calculation within 20 years is the concept of "addition and subtraction" or other calculation methods, rather than some materialized object.

Second, the implementation process of the action is hidden. The action of mathematical mental skills is completed by internal words, and its execution is carried out inside the mind. The change of the subject is very implicit, and it is difficult to observe it directly from the outside. For example, what we can directly know is the calculation result reflected by the students' external language, while the students' internal psychological activities are invisible during the calculation.

Third, the action structure is simple. The action of mathematical mental arithmetic skills does not have to be done exactly like arithmetic activities, nor does it have to be said exactly like a foreigner. Its activity process is a highly compressed and simplified automation process. Therefore, the action components in mathematical psychological skills can be merged, omitted and simplified. For example, when students master the oral arithmetic of carry addition within 20, they simply don't realize the actions of "looking at large numbers", "trying to make up numbers", "dividing decimals" and "making up ten", and the whole calculation process is compressed into a simple process that blurts out.

Third, the formation process of mathematical skills

1. The formation process of mathematical operation skills.

As an explicit operation activity, the formation of mathematical operation skills roughly goes through the following four basic stages.

(1) Action in the orientation stage. This is the initial stage of the formation of computing skills, mainly because learners establish a directional image of computing activities to complete a certain mathematical task in their minds. Including clear learning objectives, stimulate learning motivation, understand the relevant knowledge of mathematical skills, know the operating procedures and action essentials of skills, and the final result of activities. To sum up, this stage is mainly about understanding what to do and how to do it. For example, drawing an angle, this stage is mainly to understand how many degrees an angle needs to be drawn (that is, to know what to do) and the steps of drawing an angle (that is, how to do it), so as to make a specific orientation for the operation of drawing an angle. The function of action orientation is to initially establish a self-regulation mechanism operating in the mind; Through the understanding of "what to do" and "how to do it", the procedures and steps of implementing mathematical activities are clearly defined, so as to ensure that the activity mode of its actions can be better grasped in operation.

(2) the decomposition stage of the action. This is the initial stage when operational skills enter practical learning. The method is to decompose the whole set of actions of a certain mathematical skill into several individual actions, and the students imitate the exercises in turn under the teacher's demonstration, so as to master the activity mode of local actions. If the compass is used to draw a circle according to a given radius, the whole operation process can be decomposed into three local actions at this stage: ① open the two feet of the compass and set the distance between the two feet according to the given radius; (2) Fix a foot on a point with a needle tip and determine the center of the circle; ③ Rotate your feet around the center of the circle with the tip of a pencil and draw a circle. By practicing these three continuous local movements in turn, you can master the essentials of drawing a circle. At this stage, students learn mainly by imitation, on the one hand, according to the teacher's demonstration imitation; On the other hand, it can also be imitated according to the text description of the relevant operation rules, such as imitating the expression of each action mode according to the geometric drawing rules. Imitation is not necessarily passive and mechanical, "imitation can be intentional or unintentional; It can be regenerative or creative. " ② Imitation is one of the formation of mathematical operation skills.