1, the abstraction of the physical layer

This level of abstraction is actually based on the existing life experience and social reality. The first step is abstraction, that is, abstraction is based on physical objects, and it ends when it has just surpassed physical objects and has not completely separated from physical objects. For example, in the section of "Power of Rational Numbers" in the first volume of the seventh grade, words and pictures are used to show the process of cell division. Every 30 minutes, the cell will divide from 1 to 2. How many cells can this cell divide from 1 after 5 hours? Abstracting mathematical problems from such an interesting process can quickly stimulate students' interest in learning. In the section of "Rich Graphic World" in the first volume of grade seven, the textbook provides several pictures to guide students to feel the rich and colorful graphic world, and through giving various physical models, students can know five kinds of geometric bodies: cylinder, cone, cube, cuboid and sphere. The second volume of the eighth grade, The Rotation of Graphics, presents a rotating Ferris wheel, which instantly brings students into the rotating situation to feel the rotation, and then thinks about what kind of graphic movement can be called the rotation of graphics. These are typical direct abstractions with the help of "objects". In these processes, through the designed situation and the teacher's intentional guidance, students carefully observe the objects in the picture, think about the power of rational numbers, the inherent essential attributes of geometry and graphics, and form their own preliminary understanding of these knowledge.

2. Semi-symbolic abstraction

This stage is actually a simple stage and a further development on the basis of physical abstraction. At this point, the related attributes have been extracted and abstracted from the physical objects, but they are not completely separated from the physical objects, or more accurately, some attributes are separated from the physical objects, and key attributes have begun to appear. For example, in the section of "Monomial Multiplication Polynomials" in the second volume of the seventh grade, the textbook requires a painting with a length of x meters and a width of mx meters, with 1/8x meters left and right. What is a drawing area? Then, two algorithms are given. Through different expressions of the same area, we can get: x (mx-1/4x) = mx2-1/4x2. At this time, the related properties of monomial multiplication polynomial have been given. In the section of "congruence of figures", after students have learned what congruence figures are, the teaching material presents a number of figures with different shapes, which requires students to find congruence figures from them, which is also the second abstraction at the physical intuitive level. In this process, the key attribute that congruent graphics can completely overlap is highlighted. What students have to do is to find out the graphs that can completely overlap according to the concept of congruent graphs.

3. Symbol abstraction

This abstract level belongs to the symbolic stage of mathematical abstraction and has typical stages and levels. Accurately speaking, the abstraction at the symbol level is to remove the specific content and express a class of simplified things with concepts, figures, symbols and relationships. For example, in the section of "Merging Similar Items" in the first volume of the seventh grade, we observe four groups of algebraic expressions, find out their * * * similarity, then summarize the concepts of similar items, and then get the law of merging similar items. In this process, students observe algebraic expressions and explore the law of merging similar items, while trying to express their findings in words, which is the abstraction at the symbolic level. In the teaching of Pythagorean Theorem, the first volume of the eighth grade, students are first given a preliminary feeling of the special relationship between the three sides of a right triangle through exploration activities, and then guided to accurately express such special relationship in language. Finally, symbols are given to the three sides of the right triangle, and Pythagorean theorem is described in symbolic language. Such a way of expressing a thing with etiquette concepts, figures and symbols is a typical symbol level abstraction. In this process, students should first form an intuitive understanding of Pythagorean theorem by observing the "seal", explore the quantitative relationship between the areas of two small right-angled triangles and the areas of large right-angled triangles through analysis, speculation and trial-and-error, and finally get the special relationship between the three sides of right-angled triangles through analysis and reasoning. This process can make students know and understand Pythagorean theorem more deeply after the exploration process. In the section of "Similar Polygons" in the first volume of the ninth grade, after students have a preliminary intuitive feeling of similar graphics, they summarize the definition of similar graphics by observing and analyzing the internal characteristics of five groups of graphics with different shapes. Students participate in the whole process from the initial understanding of similar graphics to the in-depth understanding of similar graphics, which is very beneficial for students to fully understand similar graphics.

4. Abstraction at the formal level

This level of abstraction belongs to the universal stage of mathematical abstraction, that is, rules, patterns or models are established through assumptions and reasoning, and specific things can be explained in a general sense. Abstraction at this stage often exists in primary and secondary schools. For example, the second volume of the seventh grade "Binary linear equations" established a new mathematical model from monism to dualism on the basis of the previous section "Binary linear equations", and the study in this section mainly focused on solving the problem of "chickens and rabbits in the same cage". After establishing the model, it is a typical formal abstraction to apply the model to the solution of general problems. For another example, in the demonstration of the fillet theorem in the second volume of the ninth grade, the relationship between the fillet and the central angle is obtained by guessing and reasoning, and then the students are guided to use this relationship to solve some specific problems. In this process, students should first form an understanding of the concept of circumferential angle, then boldly guess the quantitative relationship between the central angle and the circumferential angle on the basis of measuring the central angle and the degrees of the circumferential angle of the same circle, and then gradually form ideas and methods to prove this relationship under the guidance of teachers, and finally skillfully apply this theory to solving practical problems. In the section "Properties of similar triangles" in the first volume of the ninth grade, through in-depth analysis and exploration, a method to prove the perimeter ratio of similar triangles and similar polygons is obtained, and then students are guided to try to solve the area ratio and height ratio of similar triangles and similar polygons by using the obtained method. In this process, students not only learn the methods to solve problems, but also know that the learned methods can be used to solve other problems, which is in line with the goal of attaching importance to "process and method" proposed by the new curriculum standard.

Generally speaking, at present, the most commonly used abstractions in junior high school textbooks are physical abstraction, symbolic abstraction in texts, physical and semi-symbolic abstraction in exercises, and formal abstraction in mathematical activities.