Shen, Huairou District, Beijing

Mathematics education in primary and secondary schools in China has been undergoing reform, and the method of reform is the six-word policy of "infiltration, addition and deletion". Realizing the modernization of mathematics education has always been the goal of mathematics teaching reform. It should be an important task for our primary school mathematics teachers to infiltrate some modern mathematics ideas into their minds as soon as possible on the basis of conforming to the reality and acceptance of modern children. We have entered the era of knowledge informationization. Children should learn digital computer knowledge in advance if they want to adapt to the future development of high technology. In my opinion, mathematical ideas such as set are the most important foundation for the development of mathematics. It is necessary to properly infiltrate the study in primary school, form perceptual knowledge from childhood, and let students get in touch with modern mathematics as soon as possible.

Let's talk about how to infiltrate modern mathematical ideas into primary school mathematics from the perspective of set thought. I. Concept definition (1) Academic definition 1. the collective thinking

Collection refers to the gathering of scattered people or things. In mathematics, set refers to a set of mathematical elements with certain * * * properties. Things that are certain and distinguishable within a certain range, as a whole, are called sets, or elements for short. The so-called set thought refers to the idea of set knowledge contained in applied mathematics thought, which is called set thought.

2. Modern mathematical thought

The so-called mathematical thought refers to people's essential understanding of mathematical theory and content, which directly dominates the practical activities of mathematics. The so-called modern mathematical thought refers to the spatial form and quantitative relationship of the real world reflected in human consciousness, as well as the result of thinking activities. Mathematical thought is the essential understanding after summarizing mathematical facts and theories; The thought of basic mathematics is the basic, summative and most extensive mathematical thought embodied or should be embodied in basic mathematics. They contain the essence of traditional mathematical thought and the basic characteristics of modern mathematics, and are historically developed. Through the cultivation of mathematical thinking, the ability and talent of mathematics will be greatly improved. Mastering mathematical thought means mastering the essence of mathematics.

3. Primary school mathematics

Primary school mathematics is the basis of learning advanced mathematics and the necessary knowledge of learning discrete mathematics and fuzzy mathematics, so primary school mathematics is similar to elementary mathematics but not completely elementary mathematics. The explanation of elementary mathematics in the tool book is a general term for the branches of mathematics that do not involve variables. The algebraic operation between constants and the relationship between different objects are studied to form elementary algebra and elementary geometry respectively, which are collectively called elementary mathematics. 1

The main contents of primary school curriculum, such as arithmetic, algebra, geometry, trigonometry, etc., are collectively called elementary algebra. 2

Therefore, primary school mathematics mainly takes elementary mathematics as the main learning content, and at the same time, it is interspersed with advanced mathematics thoughts.

(B) the significance of appropriate infiltration of modern mathematical ideas

Appropriate infiltration of some contents and methods, in primary school mathematics teaching, the main modern mathematics ideas are: set, correspondence, statistics and so on. Infiltrating these ideas is based on the cognitive rules of primary school students, taking the form of visual images and infiltrating them with intuitive graphics, without these mathematical terms and symbols of modern mathematics, so that students are influenced intuitively and subtly.

Second, the necessity of infiltrating modern mathematical ideas into primary school mathematics teaching

Infiltrating modern mathematics thoughts into primary school mathematics teaching plays an inestimable role. It is not only the foundation of students' thinking development, but also a good assistant for teachers' teaching.

(A) from a psychological point of view

In cognitive psychology, mathematical thinking belongs to the category of metacognition, which plays a monitoring and regulating role in cognitive activities and plays a decisive role in the cultivation of ability. The purpose of learning mathematics is "to solve problems" (in Polish), and the key to solving problems lies in finding suitable solutions. Modern mathematical ideas and methods are the guiding ideology to help build solutions. Therefore, it is an important way to cultivate students' ability to analyze and solve problems by infiltrating some basic mathematical thinking methods into students and improving their metacognition level.

(B) from the perspective of social needs

Mathematical knowledge itself is very important, but it is not the only determinant. It is the mathematical thinking method that really plays a long-term role in students' future study, life and work and benefits them for life. The future society will need a large number of talents with strong mathematical consciousness and quality. 2/KLOC-0 The fundamental goal of international mathematics education in the century is "problem solving". Therefore, infiltrate some basic mathematical thinking methods into students,

This is the requirement of the future society and the inevitable result of the development of international mathematics education.

(C) From the perspective of quality education

The fundamental task of primary school mathematics teaching is to improve students' quality in an all-round way, among which the most important factor is the quality of thinking, and the mathematical thinking method is the key to enhance students' mathematical concepts and form good thinking quality. If students' mathematical quality is regarded as a coordinate system, then mathematical knowledge and skills are like factors on the horizontal axis, and mathematical thinking method is the content on the vertical axis. Weakening or neglecting the teaching of mathematical thinking methods will not only hinder students from grasping the basic structure of mathematics from both vertical and horizontal dimensions, but also affect the development of students' ability and the improvement of mathematics quality. Therefore, infiltrating some basic mathematical thinking methods into students is a new perspective of mathematics teaching reform and a breakthrough of mathematics quality education.

3. Examples of modern mathematics thoughts contained in primary school mathematics textbooks (1) The significance of infiltrating set theory thoughts into primary school mathematics textbooks.

It plays an inestimable role in infiltrating modern mathematical ideas into primary school mathematics teaching. By observing and comparing the figures, students can gain some perceptual knowledge and deepen their understanding of basic knowledge, which is conducive to cultivating their thinking ability, developing their intelligence and further studying mathematics and modern science and technology. Set thought occupies a very important position in modern mathematics thought, and the knowledge in primary school textbooks always embodies set thought. Therefore, the infiltration of modern mathematical ideas into primary school mathematics textbooks also suggests that teachers should pay attention to the infiltration of modern mathematical ideas.

(2) Examples of modern mathematics ideas in textbooks As far as lower-grade mathematics is concerned, there are two forms: 1 and a collection of things expressed in various forms.

Example: When five apples are put on a plate, it means that five apples have been collected; A table with eight books is a collection of eight books.

2. Infiltrate the set idea in the teaching of digital recognition and operation.

Example: 10 is used to guide students to observe a series of graphs of equivalence sets. They are abstracted and summarized into the concept of natural numbers. For example, when forming the concept of radix "5", students must use discrete quantities such as physical objects, cards, counters, and set diagrams to do their own calculations. For example, count five matchsticks, name five people, and point out five trees and five chairs in the collection; In addition, we use a continuous quantity, such as measuring 5 cups of water and 5 feet of cloth, and then gradually abandon the essential things and extract the cardinal concept of "5". With the growth of knowledge, there are five points, five-year plans and one-fifth meanings to enrich its content.

Infiltrate the idea of union in addition operation and the idea of difference set in subtraction operation. 3. Let students practice several representations of sets.

For example, the idea of intersection permeates the divisibility of numbers. And begin to use braces {} 12, the factors are {1, 2, 3, 4, 6, 12} 18, and the factors are {1, 2, 3, 6, 9,/kl.

Fourth, how to infiltrate the set idea of modern mathematical thoughts (1) into the primary school mathematics classroom and the purpose of infiltrating modern mathematical methods into the primary school mathematics classroom.

Primary school students have just learned mathematics, and their knowledge and thinking methods are at the primary level. Therefore, modern mathematical ideas should be infiltrated in primary school, and students should not be exposed to academic terms and mathematical symbols directly. Students can get some perceptual knowledge through graphics, which can deepen their understanding of basic knowledge without increasing their burden. In short, it is intuitive teaching, which is a subtle form.

(B) Examples of ways to infiltrate the idea of set in teaching.

1. There is obvious infiltration of mathematical ideas in the textbook, so the method of teaching by the book is adopted.

In the first volume of primary school mathematics textbook, when learning the understanding and addition and subtraction of numbers, it is easy to penetrate modern mathematical ideas because of its small number and strong intuition. When counting, the essence is to classify the objects first, and treat each category as a set. Then only each element in the set corresponds to 1, 2, 3 in the natural number in turn, pointing to the last element, and the corresponding natural number is the number of elements in this set, that is, the total number of objects. The introduction of addition into textbooks has infiltrated the idea of union; The idea of infiltrating subsets when subtraction is introduced. The textbook draws the same kind in a circle, so that students can initially and intuitively realize that the things in the circle are a whole; When we know "0", we can deepen our understanding of "0" through the idea that there is nothing in the circle that makes students initially contact with empty sets. These are the products of students' intuition.

Tired of some emotional material. This is a mathematical idea that the textbook intentionally penetrates into the collection, and it does not need to be deliberately explained, so that students can have a perceptual knowledge. When teaching, you only need to observe and compare under the guidance of the teacher, and you have some perceptual knowledge of these contents.

2, combined with teaching materials, intuitive analysis

An exercise that often appears in textbooks: fill in the numbers in the corresponding brackets.

This kind of practice is intuitive and not difficult for students, but it is a good opportunity to infiltrate collective thinking. After completing the questions, guide the students to observe the characteristics of each box. Can you design a special box by yourself? Without telling students that this is a set, let students be influenced by the idea of set in their observation, and understand the meaning of set while designing their own boxes. At this time, it is through analysis that the set ideas are infiltrated.

3, beyond the textbook, appropriate penetration.

The relationship between square, rectangle and parallelogram and the greatest common divisor are represented by schematic diagram:

This treatment is more intuitive and easier to understand than language narration. On the premise of not changing the teaching material content, system structure and related terms and symbols, some schematic diagrams appear in an intuitive form in combination with the teaching content, so as to guide students to be unconsciously influenced through observation or practice.

Verb (abbreviation of verb) suggestion

(A) to raise awareness of infiltration

six

Mathematical concepts, laws, formulas, properties and other knowledge are clearly written in the textbook, which has a "shape", while modern mathematical ideas are implicit in the mathematical knowledge system, without a "shape", and scattered in various chapters of the textbook in an unsystematic way. Teachers don't talk, talk more and talk less, which is arbitrary. They often squeeze it out as a "soft task" because of the tight teaching time. The requirement for students is to calculate as much as they can. Therefore, as a teacher, we should first renew our ideas, constantly improve our understanding of the importance of infiltrating modern mathematical thoughts, integrate both mastering mathematical knowledge and infiltrating mathematical thoughts into teaching purposes, and integrate the requirements of modern mathematical thoughts teaching into lesson preparation. Secondly, we should study textbooks deeply and try to find out all kinds of factors that can penetrate modern mathematical thought. For each chapter and section, we should consider how to infiltrate the mathematical ideas of specific content, which mathematical ideas to infiltrate, how to infiltrate, and to what extent. It is necessary to have an overall design and put forward specific teaching requirements at different stages.

(B) Grasp the feasibility of infiltration

The teaching of modern mathematics thought must be realized through concrete teaching process. Therefore, we must grasp the opportunity of mathematical thought teaching in the teaching process-the process of concept formation, conclusion derivation, method thinking, thinking exploration and law revelation. At the same time, we should pay attention to the organic combination and natural infiltration in the teaching of mathematical ideas, consciously and subtly inspire students to understand all kinds of mathematical ideas contained in mathematical knowledge, and avoid the counterproductive practices such as mechanically copying, generalizing and being divorced from reality.

(C) pay attention to the repeatability of infiltration

Modern mathematics thought is gradually accumulated and formed in the process of enlightening students' thinking. Therefore, in teaching, we should first emphasize "reflection" after solving problems, because the mathematical ideas refined in this process are easy for students to understand and accept. Secondly, we should pay attention to the long-term nature of infiltration. It should be noted that the infiltration of students' modern mathematical thoughts can not improve students' mathematical ability overnight, but a process. Mathematical thinking must be gradually and repeatedly infiltrated, so that students can really understand it.

refer to

1, Ma, Journal of Yunnan University of Finance and Trade, s 1, 2000 2, Journal of Chuzhou Vocational and Technical College, 0 1, 2003.