Mathematics view of "Mathematics is the refinement of common sense" and its teaching enlightenment

Qian Bin (226 15 1), Jiangsu Changbao Senior High School.

For a long time, people have formed various mathematical views, and different mathematical views have different influences on mathematical teaching views. This paper discusses the mathematical view that "mathematics is the refinement of common sense" and its enlightenment to mathematics teaching.

The first is the mathematical view that "mathematics is the refinement of common sense"

"Mathematics is the refinement of common sense" is a long-standing mathematical view. This view of mathematics reveals the essence of mathematics from a specific angle, that is, the relationship between mathematics and daily (life) common sense. Its mathematical essence mainly includes the following meanings:

Mathematics comes from common sense.

Mathematics is the Refinement of Common Sense, which first answers the basic relationship between mathematics and (life) common sense, and then answers the basic philosophical question of the origin of mathematics: abstract mathematics comes from common (life) common sense. Concepts, laws and methods in mathematics can be found in life. For example, "points, lines and surfaces" in geometry come from naturally materialized points, lines and surfaces (desktops, etc.). ) in life. The concept of function comes from the interdependent relationship between variables (such as distance and time) in life. Mathematics comes from life, which fully affirms the significance of common sense to mathematics and also criticizes mathematical mysticism.

Common sense is refined into mathematics.

Mathematics originates from life, but mathematics is not equal to life. Common sense in life is bound to go through a refining process. For example, the plane in geometry comes from the plane in life (such as desktop, blackboard, horizontal plane, etc. ), but the plane in geometry is the result of various "planes" abstractions in real life. This abstraction removes the different attributes of those concrete "planes" (such as different sizes and textures) and idealizes them, and finally obtains the abstract concept of planes (no size) in geometry. Refining is a process, which is abstract, rigorous, precise and refined. Subtlety is like a bridge (from a functional point of view). Through this bridge, the common sense of life is constantly upgraded, refined and organized, and condensed into certain concepts, rules and methods (mathematics). These concepts, rules and methods become the common sense of a higher level (indirect life), and then they are refined and organized into higher-level concepts, rules and methods ... From the above analysis, we can see that the process of refinement is actually an abstract process with the characteristics of successive abstractions. It is precisely because of this continuous abstraction that mathematics is "far away" from life and makes mathematical research possible.

Mathematics is both a process and a result.

From the above, it can be seen that common sense becomes mathematics (knowledge) through refining, and the process of refining is a mathematical process, which embodies the characteristics of mathematical activities (processes), so people often call mathematics mathematical activities, and the result of activities people pursue along the refining process is visible mathematical knowledge (results), which is relatively static and we call it formal mathematics, so mathematics is the dialectical unity of process and results.

Second, "Mathematics is the refinement of common sense" in mathematics teaching.

"Mathematics is the refinement of common sense" clearly reveals the objective relationship between mathematics and common sense, that is, mathematics comes from common sense, which provides a perceptual basis for designing advance organizers with common sense. According to Vygotsky's theory of the zone of proximal development, students' learning can only be successfully completed in the zone of proximal development. Using common sense to design the advance organizer is to use the materials of the students' recent development area and then let the students know in the recent development area. Moreover, because (life) common sense is familiar to students, has affinity for students, is most easily accepted by students, and can also arouse students' interest, the organizer who gives priority to common sense design is superior. Because of this, the new round of curriculum reform particularly emphasizes the connection between mathematics and real life: "The content of mathematics curriculum must fully consider the trajectory of human activities in the process of mathematics development, be close to the real life that students are familiar with, and constantly communicate the connection between mathematics in life and mathematics in textbooks, so that life and mathematics can be integrated" (Interpretation of Mathematics Curriculum Standards for Full-time Compulsory Education, p.1kloc-0/2). The connection between life and mathematics, or one of the life-oriented significance of mathematics education, is probably to teach mathematics with the help of common sense of life.

How to design advance organizers with common sense?

First, pursue the original growth point of mathematical knowledge. A lot of mathematical knowledge in mathematics curriculum has a life prototype, which is called the original growth point of knowledge. For example, the desktop, blackboard surface and ground in life are all prototypes of "plane" in mathematics. In teaching design, we must first find the prototype related to the mathematics content we are teaching. There are usually two ways to find it, one is logical and the other is historical. The so-called logical method is to logically infer the prototype of mathematical knowledge according to its own characteristics, such as the teaching of parallelogram judgment. Now that we have learned its definition and properties, we will naturally wonder whether there are other ways to judge parallelograms logically. The so-called historical method is to find the true prototype of mathematical knowledge by consulting the historical facts of mathematical development. For example, the symbol of addition and subtraction is proved to be obtained in practice: because the wine in the barrel is sold, the horizontal line drops, and a horizontal line "-"is drawn at the falling place. When new wine is poured into the barrel to make the wine level rise, the original horizontal line is crossed out, which means that the wine in the barrel has exceeded this line, and the current addition has been formed for a long time. This prototype may not be true in history, but it has universal representative significance.

Second, find the connection point between common sense and mathematical knowledge. Finding the original growth point of mathematical knowledge provides a material basis for us to design senior organizers. However, the key to designing excellent organizers in advance is to find the connection point between common sense and mathematical knowledge. This connection point is a bridge, a bridge from common sense to abstract mathematics. In fact, this bridge is the essential connection between common sense and mathematics.

For example, adopting the "mathematical" teaching paradigm in the teaching of plane rectangular coordinate system can not only make students understand the concept of plane coordinate system well, but also make the knowledge they have learned more convenient for students to remember. In teaching, we designed the following programs:

(1) raises the question of life. Parent Zhang came to our school to attend the parent-teacher conference. At the guard's office, they asked the security guard where Li Ladder Classroom was. Li told them: Go east from here 100 meters, and then go north for 60 meters. How to express the meaning expressed by Li with pictures?

(2) Mathematicization of life problems

A. Students discuss how to express (common sense of life) with pictures (as shown in the figure (1)).

B ask the students to mark the position of the guards 50 meters west and 30 meters south in the picture (1).

C. Discuss the significance of the map: the map can show the position of a place relative to the doorman;

D. Summary of the problem: How to use numbers to represent the relative position of a point in a graph;

2m in the north is represented by +2, 2m in the south by -2, 3m in the east by +3 and 3m in the west by -3. Therefore, an ordered pair of numbers can represent the relative position of the point in the diagram.

⑶ Abstraction of mathematical concepts: Simplify graphics and get the concept of coordinate system.

Third, the conclusion

As can be seen from the above discussion, mathematics has both results (relatively static) and processes (activities), so mathematics teaching should pay attention not only to the teaching of mathematical knowledge as a result, but also to the teaching of mathematical thinking activities as a process (including "mathematization" activities), that is, "mathematization" should become a goal of mathematics teaching.

"Mathematization" should also become a paradigm of mathematics teaching. First of all, "mathematization" is a goal of mathematics teaching. Only through the process of mathematization can teachers make students feel and understand mathematization; Secondly, the application of mathematics teaching mode is helpful for students to understand formal mathematics knowledge. "Mathematics is the refinement of common sense" actually points out the law of mathematics development, which is also the law of human (group) understanding mathematics. The law of human cognition reflects the law of individual cognition to a great extent. Applying mathematical paradigm to mathematics teaching (process teaching) conforms to the law of individual mathematics learning, and students can understand and master knowledge through process teaching.

Of course, when we adopt the "mathematization" paradigm in teaching, we should also avoid the phenomenon that mathematics teaching stops at common sense of life. The new curriculum reform emphasizes the "life" of mathematics teaching, and teachers also agree with this concept very much. Therefore, in classroom teaching, teachers use a lot of common sense resources and adopt the mode of "life → mathematics" for teaching. However, due to various reasons, improper practices actually failed to complete the transition from life mathematics to school mathematics, resulting in the "taste of life" completely replacing the "taste of mathematics" it should point out. The outlook on life advocates the teaching mode from life to mathematics (on the other hand, from mathematics to life), and its purpose lies not in life but in mathematics, but in helping students learn mathematics, so the transition from life to mathematics (imperceptibly) is particularly important. As a teacher, we should study the process of mathematization, realize the smooth transition from life to mathematics, prevent the tendency of mathematics teaching to actually stop at common sense of life because of improper practice, and truly reflect the mathematical attributes of mathematics teaching.

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