How to understand the abstractness, precision, and breadth of applications of primary school mathema

You don't have to understand what you want to do so much. It is better not to study certain subjects for why, because it is difficult to get a satisfactory answer. For example, if you learn English, ask why this word is used this way, why is it this structure? It's all conventional.

And elementary school math isn't abstract, is it? It is much easier to understand than physical chemistry and other science subjects.

Mathematics is a discipline that studies quantitative relations and spatial forms in the objective world. Compared with other disciplines, mathematics has three distinct characteristics, namely, abstraction, precision, and wide application.

The abstraction of mathematics is manifested in the following aspects: First, mathematical abstraction abandons all the concrete properties of real objects and only retains the relationship of quantity and spatial form. Second, mathematical abstraction is rich in hierarchy and is gradually improved.

Third, not only are mathematical concepts abstract and speculative, but mathematical methods are also abstract and speculative.

The precision of mathematics is manifested in the accuracy of mathematical definitions, the logical rigor of mathematical reasoning and the certainty of conclusions, which are unique characteristics of mathematics since its inception. As the Soviet Union mathematics educatorAleksandrov said:

Mathematical reasoning is carried out with such precision that it is indisputable and certain to everyone who understands it. In modern mathematics, this rigor is further strengthened.

The wide range of applications of mathematics is also one of the most striking features of mathematics. There are three main aspects: First, in production, daily life and social life, we often apply the most ordinary mathematical concepts and conclusions almost all the time.

Second, the development of all modern science and technology is inseparable from mathematics, and "almost any improvement in technology is inseparable from more or less complex calculations." Thirdly, almost all modern branches of science make substantial use of mathematics, and "both the natural and social sciences make extensive use of mathematical tools in the development of their own theories." Especially in today's era, with the rapid development of science and technology, the trend of scientific mathematics is becoming more and more obvious, and modern science is developing in the direction of extensive application of mathematics.

Therefore, it is particularly important to improve the effectiveness of the use of mathematical abstraction methods, so that students can establish correct mathematical knowledge through mathematical abstraction. 1.When mathematics is abstracted, it is necessary to give full play to the role of appearances.

Appearance is an advanced form of perceptual cognition, which is the transition and bridge from concrete perception to abstract thinking, so in the process of concept formation, calculation laws and formulas, it is very important to establish typical representations that can highlight the commonality of things, which provides a basis for further high-level abstract generalization. For example, when understanding parallelograms, in order to facilitate the abstract generalization of its essential features such as "two groups of opposite sides are equal" and "two groups of opposite sides are parallel to each other", students can be provided with the following typical figures, after fully perceiving, observing and comparing, thinking about the commonalities of these figures, and then abstractly generalizing. There is not necessarily only one type of typical figure here, but it can be varied, which helps students to establish a richer representation of the parallelogram.

However, most of the current primary school mathematics textbooks do not give rectangles and squares in the materials presented in the lesson on recognizing parallelograms (perhaps taking into account the reasons of students' cognitive laws), so it often leads students to have a one-sided understanding, that is, the four corners of the parallelogram cannot be right angles, which is caused by the incomplete representation provided. In order to avoid such problems, it is necessary to consider comprehensiveness when choosing appearances. 2. Mathematical abstraction should grasp the opportunity and abstract and summarize in time.

After fully perceiving concrete things and forming appearances, it is necessary to grasp the opportunity and abstract and summarize them in time, so that the perceptual understanding can rise to the rational understanding and improve the students' thinking ability. Think about it, if you don't abstract and summarize in time, then the students' thinking level will inevitably stay on the superficial, superficial, and fragmentary external phenomena, and the understanding of things will not be able to go deeper. For example, when learning about the thread segments, ask the students to "straighten the thread" and find that when the two ends of the yarn are tightened, the middle section is straight.

Then, students are guided to imagine the state of the straightened yarn without looking at the real thing, and draw the image formed in their minds, so as to abstract the concept of the line segment. The abstract generalization here is based on the students' full operation and imagination, and the timing is appropriate and timely. 3.

Mathematical abstraction should pay attention to the hierarchy. Primary school students' abstract ability is gradually developed with age, from extracting the external characteristics of things to extracting the essential characteristics of things, from the lower level of abstraction with the help of concrete things to the higher level of abstraction with the help of representations or mathematical concepts, and this development requires the guidance and guidance of teachers. For example, when studying axisymmetric figures, the teacher first abstracts some concrete axisymmetric objects into axisymmetric patterns, then abstracts them into concrete axisymmetric figures, and finally abstracts the concept that "shapes that completely coincide after folding in half are called axisymmetric figures".

Another example is the teaching of the commutative law of addition

Answer] :(1) Abstraction is the process of extracting the essential attributes of things from the mind and discarding their non-essential attributes. Abstraction is carried out on the basis of analyzing, synthesizing, comparing, and generalizing the properties of things, and it is a method of thinking that attacks and understands the essence of things and grasps the internal laws of things.

Abstraction is one of the basic characteristics of mathematics, and the abstract nature of mathematics is reflected in the fact that the object of its study is to completely abandon all the concrete content of concrete things and only consider the relationship between their quantities and the form of empty questions (or the structure determined by the axiomatic system).

2) The abstraction of mathematics can be summarized into the following categories: Not only are mathematical concepts abstract, but mathematical methods are also abstract, and abstract symbols are used extensively. The abstraction of mathematics is a step-by-step abstraction.

The next abstraction is the concrete background of the previous abstract material. A high degree of abstraction necessarily has a high degree of generalization.

3) First of all, we should focus on cultivating students' abstract thinking skills. The so-called abstract thinking ability refers to the ability to think without concrete images, using concepts, judgments, reasoning, etc. According to the different degrees of abstract thinking, it can be divided into empirical abstract thinking and theoretical abstract thinking.

In teaching, we should focus on the development of theoretical abstract thinking, because only those who have fully developed theoretical abstract thinking can analyze and synthesize various things well and have the ability to solve problems.

Secondly, it is necessary to cultivate students' observation ability and improve their ability to draw and correct scattered images and generalize. In teaching, you can use physical teaching aids, use the combination of numbers and shapes, and algebra with shapes. For example, when talking about the properties of logarithmic functions, you can first draw an image, and observe the image to abstract the relevant properties.

Summary. Hello 1, some proofs of abstraction such as arithmetic laws, spatial geometry. It is manifested as the pure quantity of thinking about things, the extensive use of abstract symbols, not only the concept of numbers is abstract, but also the mathematical method is abstract, giving examples of how the abstractness, logic, and practicality of mathematical qualities are reflected in junior high school functions.

Hello 1, some proofs of abstraction such as arithmetic laws, spatial geometry. It is manifested as the pure quantity of thinking about things, and the extensive use of abstract symbols, not only the concept of the number of wild fibers is abstract, but also the method of counting vertical spine is abstract and logical, such as the creation of situations when learning new content, examples, variant training, classroom burial town small ant knots, etc., can be done by students, and it will naturally become a habit. Once the habit is formed, it lays a good foundation for the student's learning and rough habits, and points out the direction.

3. The usefulness of the actual sale is reflected in all aspects of life, such as the need to use addition, subtraction, multiplication and division when going to the supermarket to buy things, drawing drawings and measuring dimensions to build a house, and the display of the driving speed of the rubber dial when driving are all used in mathematics.

Expansion, the classroom needs to be carefully preset by the teacher, and grasp the generation in the pre-Wu design, so as to better control the classroom, highlight the cautious position of the main body of the student, and improve the teaching efficiency of the classroom.

Abstraction can be summarized in the following three points:

1) Not only are mathematical concepts abstract, but mathematical methods are also abstract, and abstract symbols are used extensively.

2) The abstraction of mathematics is abstracted gradually, and the next abstraction is the concrete background of the previous abstract material.

3) A high degree of abstraction necessarily has a high degree of generalization.