Abstract: As one of the basic concepts commonly used in mathematics, limit is used to describe the extreme state of a variable in a certain change process, and it is a concept that a thing approaches a certain state infinitely. Extreme thinking is an important mathematical thought, an essential reflection of mathematical knowledge, and a link between image thinking and abstract thinking. Infiltrating extreme thoughts into students in the primary stage of learning mathematics knowledge can not only improve students' abstract thinking ability, but also help them master the ideas and methods of learning mathematics and benefit them for life. This paper expounds the necessity of infiltrating extreme thoughts in primary school mathematics teaching, and combined with teaching cases such as mathematical formulas, concepts, exercises and general review, discusses the ways of infiltrating extreme thoughts in primary school mathematics teaching and the problems that should be paid attention to in the process of infiltration.

Keywords: the infiltration of extreme thoughts in primary school mathematics teaching

First, the limit thought and its historical introduction

Since the establishment of calculus in17th century, the concept of infinity has become a topic of concern. The concept of infinitesimal is the foundation of calculus. When studying the motion change of an object, it is regarded as an infinitely reducible quantity at first, then it is greater than zero, and at the same time it is regarded as zero and ignored, that is, it is considered as zero. In order to eliminate this contradiction, mathematicians have made unremitting exploration for a long time. 19th century French mathematician Cauchy fully expounded the concept and theory of limit. In Cauchy's thought, a function does not directly approach the limit, but must go through an expression containing infinitesimal. He regards infinitesimal as a variable with zero as the limit, and clarifies the fuzzy understanding that infinitesimal is "like zero and non-zero" in the process of change. Its value can be non-zero, but its changing trend is "zero" and it can be infinitely close to zero. Cauchy's limit theory is a potentially infinite process, and the completion of the limit is real infinity. It can be seen that potential infinity and real infinity in Cauchy's theory are unified to some extent, but Cauchy's definition of limit still has many loose points. After further improvement by Wilstras, it was finally refined into "ε-δ" language.

Second, the necessity of infiltrating extreme thoughts in primary school mathematics teaching

Mathematics learning in primary schools is relatively simple, and students may forget what they have learned less than two years after leaving school. However, the mathematical ideas and methods they have learned will be kept in mind, and they can play a role at any time in their future work or life. Therefore, it is the key for students to master knowledge by constantly infiltrating mathematical ideas and methods into students.

In primary school mathematics textbooks, there are many knowledge points related to extreme thinking, such as natural numbers, odd and even numbers, cyclic decimals, and geometric concepts involving infinite extensibility, such as straight lines, rays, edges of angles, lengths of parallel lines, etc. If teachers can consciously dig out the extreme thoughts and methods contained in the teaching process and properly infiltrate them into students, they will not only enable students to master knowledge points and develop their thinking, but also enable students to play a role at any time in their future life and work.

Third, an important way to infiltrate the idea of limit in primary school mathematics teaching

Primary school students are in the stage of physical and mental development, which is the stage of transforming image thinking into abstract thinking. Their understanding of extreme thoughts is limited, but it does not mean that the infiltration of extreme thoughts should be diluted in the teaching process. In the teaching process, teachers can use the process of deducing formulas, learning new concepts and practicing review to infiltrate students and improve their abstract thinking ability.

(A) In the process of deriving the formula, the concept of permeability limit

In primary school mathematics teaching, there will be many mathematical formulas, some of which are derived by extreme thinking, and teachers can use this process to infiltrate students imperceptibly. The most typical example of deducing the formula by using the limit idea is the area of the circle.

Case 1: teaching "the area of a circle"

In the teaching of "derivation of the formula of circle area", teachers often ask students to fold a circle in half continuously. In the process of continuous folding, students can find that the more times they fold, the closer the folded figure is to a triangle. After unfolding, the circle is divided into several approximately isosceles triangles along the crease. The two waists of an isosceles triangle are the radius of a circle, and the base is a part of the circumference of the circle. In this link, students can feel the process from curve to straight line and understand the mathematical thinking method from approximate segmentation to infinite subdivision.

In the process of formula derivation, the limit segmentation idea of "turning curves into straight lines" and "turning circles into squares" is adopted. On the basis of finite division, let students imagine the final state of infinite subdivision, so that students can not only remember the formula, but also penetrate the limit idea of infinite approximation into their minds.

(B) in the process of learning new concepts, penetrate the idea of limit.

The new concept is new knowledge for primary school students, and it is a process from scratch. It is also helpful for students to know and understand the technical terms in mathematics and lay a certain foundation for future study. Some new concepts contain some radical ideas, and teachers can properly infiltrate students while teaching to help them better understand the new concepts.

Case 2: Teaching "Concept of Cyclic Decimal"

In the teaching of the concept of circulating decimal, the concept is strong, and at the same time, this new lesson also contains the idea of limit. Before talking about the concept of cyclic decimal, teachers often ask students to discuss: 0.999… or 1 which is bigger? Students who have studied the equation may set 0.999… as x, then 10x=9.99…, 10x=x+9, 9x=9, then x= 1, then 0.999…= 1. Then students who have never studied the equation can find rules in some formulas: 1-0.9=0. 1, 1-0.99=0.0 1,1-0.999 = 0.001. At this time, students can find from these formulas that every time the decimal part of 9 increases by one place, its value will increase by one more 0, so if the decimal part of 0.999 has an infinite number of 9…, then the final result will be infinitely close to 0.

(C) Infiltrating extreme thoughts in the process of practice

Mathematics learning must be inseparable from practice, and practice is the consolidation and training of what you have learned. But in practice, teachers often ignore the cultivation of students' mathematical thinking methods, and the formation of mathematical thinking methods needs continuous accumulation and application. Therefore, to cultivate students' mathematical thinking methods, not only teachers need to infiltrate imperceptibly in the new curriculum teaching process, but also need to constantly consolidate and train in the practice process.

It can be seen intuitively from the figure that with the increase of fractional denominator, the space divided by the square is getting smaller and smaller, while the area of the blank part is getting larger and larger, so that the area of the square is getting closer and closer to 1, so when there are infinite items added, the result is close to 1.

(4) Infiltrating extreme thoughts in the general review process.

General review is to gather the relatively independent and scattered knowledge points learned before, sort out the knowledge points through review, induction and summary, form a knowledge network, clarify the relationship between concepts, and make mathematics knowledge more complete, organized and systematic in students' minds.

Case 4: Teaching "Arrangement and Review of Plane Graphics"

In this lesson, the teacher lists the plane figures that students have learned, including rectangle, square, parallelogram, triangle, trapezoid and circle, and analyzes their characteristics. If we comb the trapezoid area formula with the help of limit thought, how can we deduce the area formulas of other figures? The area formula of trapezoid: S= (upper bottom+lower bottom) × height ÷2. Assuming that the upper bottom of the trapezoid is infinitely close to 0, the obtained figure is similar to a triangle, and S= lower bottom × height ÷2, that is, the area formula of the triangle is S= (upper bottom+lower bottom) × height ÷2. Similarly, if the two sides of a rectangle tend to be perpendicular to the base, the four sides of a square tend to be equal, and the upper and lower base of a parallelogram tend to be equal, the area formula of each plane figure can be derived.

S=(a+b)h÷2

Through the construction of knowledge network system diagram, students can have a deeper understanding of the area formula of plane graphics they have learned, let them know that there is not only one way to solve problems, help them form a more complete cognitive structure, and let extreme thoughts be imprinted on their minds imperceptibly.

Fourth, pay attention to the infiltration of extreme thoughts in primary school mathematics teaching.

In primary school, students' logical thinking and abstract thinking ability are weak, while extreme thinking is very logical and abstract, which is difficult for primary school students to understand. First of all, in the teaching process, teachers should gradually penetrate and spiral from the shallow to the deep, from the concrete to the abstract, from the perceptual to the rational according to the students' understanding level and knowledge characteristics at various learning stages. Secondly, extreme thinking method is not like general mathematics knowledge, which can be mastered through several classes of study. Only through continuous step by step and repeated training can students really understand. Finally, teachers should try to dig out the knowledge points that can penetrate extreme thoughts in textbooks and integrate extreme thoughts into primary school mathematics teaching.

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