Comprehend the mathematical thinking method, let the classroom bloom charm, let the students show elegant demeanour —— Thinking and practice report of infiltrating the mathematical thinking method in primary school mathematics teaching: The topics of our three reports today are: Comprehend the mathematical thinking method, let the classroom bloom charm, let the students show elegant demeanour. Zhang Jingzhong, an academician of China Academy of Sciences and a famous mathematician, once pointed out: "The mathematics learned by primary school students is very elementary and simple. But although it is simple, it contains some profound mathematical ideas. " As two clues of mathematics learning in primary schools, mathematical knowledge and mathematical thinking methods complement each other, among which mathematical thinking methods reveal the essence and development law of mathematics, which can be said to be the essence of mathematics. Let's talk about mathematical thinking methods first. 1. Why should the mathematical thinking method 1 and the basic mathematical thinking method be infiltrated into teaching? It is of great significance to the development of students. An educator once pointed out: "Mathematics, as knowledge, may be forgotten in less than two years after leaving school, but it is the brilliant spirit of mathematics, its ideas, research methods and key points that are deeply remembered in the mind, and it will benefit students for life whenever and wherever it happens." Mathematical thinking method is the soul and essence of mathematics. Mastering the scientific mathematical thinking method is of great significance for improving students' thinking quality, for subsequent mathematics learning, for other studies and even for students' lifelong development. It is the key to strengthen students' mathematical concepts and form good thinking quality to consciously infiltrate some basic mathematical thinking methods in primary school mathematics teaching. Not only can students understand the true meaning and value of mathematics, learn to think and solve problems with mathematics, but also can organically unify the learning of knowledge with the cultivation of ability and the development of intelligence. 2. Infiltrating basic mathematical thinking methods is the need to implement the spirit of new curriculum standards. Mathematics curriculum standards take "four basics": basic knowledge, basic skills, basic ideas and basic activity experience as the target system. The basic idea is one of the goals of mathematics learning, and its importance is self-evident. The new textbook presents some important mathematical thinking methods through the simplest examples in students' daily life, and uses intuitive means such as operation and experiment to solve these problems. So as to deepen students' understanding of mathematical concepts, formulas, theorems and laws and improve students' mathematical ability and thinking quality. This is an important way to realize mathematics education from imparting knowledge to cultivating students' ability to analyze and solve problems, and it is also the real connotation of the new curriculum reform of primary mathematics. Second, what mathematical ideas are infiltrated into the textbook? The most important idea in primary school mathematics is deduction and induction, which is the main line of mathematics teaching. There are also some commonly used mathematical thinking methods: corresponding thinking, which refers to grasping the relationship between two set elements. Many mathematical methods are derived from corresponding ideas. For example, students often use 10 in calculation exercises. 20×2? 30? 40? 50? The appearance of form actually reflects the corresponding thought. For example, a point on the number axis corresponds to a number, and any number can find the corresponding point on the number axis, which is perfect. Symbolic thought and mathematics have developed into a symbolic world today. Sue, a famous British mathematician, once said, "What is mathematics? Mathematics is symbols plus logic. " Symbolization means that people consciously and universally use symbolic language to express their research objects. Symbolization has permeated the whole primary school, such as Arabic numerals: 1, 2, 3, 5, 6, …+,–,,and other operational symbols; & gt,<, =, and other symbols indicating relationships; (), [] and other brackets; Letters representing numbers: x, y, z, etc. Letter expression formula: area of rectangle and square S=abS=a? Letters represent symbols of units of measurement: m\cm\dm\mm\g\km, etc. Set thought-putting a group of objects together as the scope of discussion, this is the idea of set. For example, when teaching children to recognize numbers, the first-grade textbooks enclose some pictures in a circle, which is the embryonic form of the collection that children first came into contact with, and it is also the first time to infiltrate the idea of this collection into primary school students. In the later teaching, the ideas of union set, difference set and empty set gradually emerged. Extreme Thought-China's ancient extreme thought, and Liu Hui's "circumcision", an outstanding mathematician in ancient times, are typical examples of using extreme breast thought. The idea of limit is to study the changing trend of variables in infinite change. With this idea, people's thinking can be sublimated from limited space to infinite space, from static to dynamic, from concrete to abstract. Statistical thought-the statistical thought in primary school mathematics is mainly embodied in: simple data sorting and averaging, simple statistical tables and charts. Students should be able to find mathematical problems and information from data and charts while sorting out, tabulating and drawing, and draw relevant conclusions. Hypothetical thinking-it is a thinking method that first makes some assumptions about the known conditions or problems in the topic, then calculates according to the known conditions in the topic, makes appropriate adjustments according to the contradiction in quantity, and finally finds the correct answer. Comparative thinking is one of the common thinking methods in mathematics teaching, and it is also a means to promote the development of students' thinking. Teachers are good at guiding students to compare the situation before and after the change of known quantity and unknown quantity in mathematics score application problems, which can help students find solutions quickly. Analogy-refers to the fact that based on the similarity between two types of mathematical objects, the known attributes of one type of mathematical object can be transferred to another type of mathematical object. For example, additive commutative law and multiplicative commutative law, rectangular area formula, parallelogram area formula and triangle area formula. This idea not only makes mathematical knowledge easy to understand, but also makes the memory of formulas natural and concise. Changing thinking-it is a way of thinking, one form is changed into another, and its own size remains unchanged. Such as geometric equal product transformation, homotopy transformation for solving equations, formula deformation, etc. , also commonly used in calculation. Classification idea-embodies the classification of mathematical objects and its classification standards, such as the classification of natural numbers, and triangles are divided by edges and angles. Different classification standards will have different classification results and produce new concepts. The idea of combining numbers and shapes-numbers and shapes are two main objects of mathematical research. Numbers are inseparable from shapes, and shapes are inseparable from numbers. On the one hand, abstract mathematical concepts and complex quantitative relations are visualized, visualized and simplified through graphics. On the other hand, complex shapes can be expressed by simple quantitative relations. When solving application problems, the quantitative relationship is often analyzed with the help of line segment diagram. Substitution idea is an important principle in solving equations, and one condition can be replaced by other conditions when solving problems. If the school bought four tables and nine chairs, * * * spent 504 yuan, and the price of a table and three chairs was exactly the same. What is the unit price of each table and chair? Reversible Acacia is the basic idea in logical thinking. When positive thinking is difficult to solve, we can seek a solution from conditional or problem thinking, and sometimes we can represent the line chart in reverse. For example, when a car goes from A to B, it travels 1/7 in the first hour, and the second hour is more than the first hour 16 kilometers, leaving 94 kilometers to find the distance between A and B, and changing the way of thinking-reducing the problems that are likely to be solved or indicated to be solved to one class through the conversion process, so as to solve the problems that are easy to solve, thus obtaining one. This is the so-called "transformation". But mathematical knowledge is closely related, and new knowledge is often the extension and expansion of old knowledge. There is no doubt that it is of great help for students to think through reduction in the face of new knowledge and improve their ability to acquire new knowledge independently-how to grasp the quantitative relationship in complex changes and grasp the constant quantity as the starting point. For example, there are 630 books on science and literature, 20% of which are science and technology books, and then buy some science and technology books. At this time, science and technology books account for 30%, how many books to buy? The thinking method of mathematical model is based on a concrete life prototype in the real world, and makes full use of the processes of observation, experiment, operation, comparison and analysis to simplify and assume it. It is a way of thinking that transforms practical problems in life into mathematical problem models. Cultivating students to understand and deal with the surrounding or mathematical problems from a mathematical perspective is the highest realm of mathematics and the goal pursued by students with high mathematical literacy. These mathematical thinking methods are the essence and essence of mathematics. Only by mastering methods and forming ideas can students benefit for life. Let me share with you my study and practice of mathematical thinking methods. Third, let the classroom show the charm of ideas. First of all, talk about preparing lessons: when preparing lessons, we should study teaching materials, define goals, design plans, and fully explore mathematical thinking methods. If teachers don't know what kind of thinking method is suitable for teaching materials before class, then classroom teaching can't be targeted. Therefore, when preparing lessons, we should not only look at the basic knowledge and skills of mathematics written directly in the textbook, but also study the textbook in depth, creatively use the textbook, explore the mathematical thinking methods hidden in the textbook, clearly write out which mathematical thinking methods are infiltrated into the teaching objectives, design mathematical activities to implement all aspects of teaching presupposition, and realize the organic integration of mathematical thinking methods in the formation of mathematical knowledge. In fact, every textbook has the infiltration of mathematical thinking methods, and we select representative units in each textbook. This is just the tip of the iceberg compared with all the teaching contents. To this end, when I study textbooks, I often ask myself a few more reasons to internalize the arrangement of textbooks into my own teaching ideas, such as: how to let students experience the process of knowledge generation and development? How can we arouse students' deep thinking about mathematics? How to stimulate students' initiative to explore new knowledge? How to infiltrate mathematical thinking methods in time according to teaching materials, and so on. Only when I have a well-thought-out plan can I infiltrate the corresponding mathematical ideas into students. Type 2: creating situations, establishing models, explaining applications and infiltrating mathematical thinking methods. Mathematics is an organic combination of knowledge and thinking methods. There is no mathematical knowledge that does not contain mathematical thinking methods, and there is no mathematical thinking method that is divorced from mathematical knowledge. This requires teachers to infiltrate mathematical thinking methods in the process of revealing the formation of mathematical knowledge in classroom teaching, and at the same time, they can teach students mathematical knowledge and get the enlightenment of mathematical thinking methods. Teachers actively infiltrate mathematical thinking methods in the classroom, which embodies teachers' great wisdom in teaching and opens up a broad new world for students' learning. Different teaching contents and different classes can appropriately infiltrate mathematical thinking methods according to their different characteristics. Take the following three classes as an example. (1) new teaching: exploring the occurrence and formation of knowledge and infiltrating mathematical thinking methods. For example, in the class of triangle classification, the teacher provides students with triangle learning tools, so that students can try triangle classification in group cooperation first. Students will start with the characteristics of the angles and sides of a triangle, and look, compare, measure, divide, think, find features and abstract, which will be * * * in comparison. In this way, students have experienced the process of triangle classification, infiltrated the idea of classification and set, enriched the experience of classification activities, formed the basic strategy of classification and developed the ability of induction. In mathematics teaching, solving problems is the most basic form of activity. Any problem, from putting forward to solving, needs specific mathematical knowledge, but it really depends on mathematical thinking methods. Therefore, in the process of exploring and discovering mathematical problems, we should seriously explore mathematical thinking methods. For example, when I was teaching "Tree Planting" in the third grade, I first asked: On one side of a road with a length of 100 meters, if you plant a tree every 2 meters, how many trees can you plant? Faced with this challenging problem, students speculated one after another, some said to plant 50 trees, and some said to plant 5 1 tree. How many? Can we start with "planting two or three trees ..." and find out the rules first? With the questions raised, the students were lost in thought. If you look at five trees with your fingers open, there is a "gap" (blackboard writing) between every two trees. A * * *, how many gaps are there? The students answered thoughtfully that there were four. If you plant 6 trees and 7 trees, what is the relationship between the number of trees and the number of intervals? So I inspired students to put a pendulum, draw a picture and discuss it, and found the quantitative relationship between the number of trees and the number of intervals when planting at both ends (number of trees = number of intervals+1), which successfully solved the above problems. Then change the question to "plant only one head, plant several trees without planting both heads", and the students look for the answer with interest in the same way. The above-mentioned problem-solving process conveys a strategy to students: when encountering complex problems, they might as well retreat to simplicity, then find the law from the study of simple problems and finally solve complex problems. Through such problem-solving activities, students can feel the important role of thinking methods in solving problems by infiltrating the thinking methods of exploration, induction and mathematical modeling. Therefore, teachers should consider the design of mathematical problems from the perspective of mathematical thinking methods, try to arrange some problems that will help students deepen the experience of mathematical thinking methods, and pay attention to guiding students to communicate after solving problems in order to deepen their understanding of problem-solving methods. ② Practice course: the consolidation and application of knowledge, the infiltration of mathematical thinking methods, the consolidation of mathematical knowledge, the formation of skills, the development of intelligence and the cultivation of ability. These experiences can only be obtained through appropriate practice. The practice of practical class is different from the practice of new teaching. The exercises in the new teaching are mainly to consolidate the new knowledge just learned, and the exercises are mainly knowledge. The practice of practice class is to transform skills into abilities, improve students' ability to solve practical problems with knowledge and develop students' thinking ability. Therefore, teachers should have the teaching consciousness of mathematical thinking method, not only have the requirements of specific knowledge and skill training, but also have clear teaching requirements of mathematical thinking method in practice class. For example, in the practice class of "Multiplication Formula of 6", after the students finish the practice of thinking and calculation, let the students calculate first, and then through the exchange of their own algorithms, take "7×6+6" as an example, use pictures to demonstrate and understand the meaning of the formula, and use the idea of combining numbers and shapes to convert the formula into 8×6 for calculation and infiltration transformation, so as to understand the different forms of the two formulas, and then let the students calculate how many squares there are in each picture. Through practical operation, students began to cut and paste and spell, and then calculated rectangles with 6×3 and 4×3 to feel the charm of changing ideas. "What are we going to teach our children?" "Mathematics learning is mainly about learning thinking methods and problem-solving strategies", so we should constantly sum up and explore in the process of practice to find out the most valuable and essential thing for children-mathematical thinking methods. For example, when I was teaching "See Who's Smart" in grade four, students mainly used the following methods to calculate "1 100÷25": ① vertical calculation 21100 ÷ 25 = (1/. The results show that method ① is a general method and methods ②-⑥ are ingenious methods. Although methods ②-⑥ have their own advantages, methods ③, ④ and ⑥ use the division of numbers, method ② belongs to the equivalent transformation, and method ⑥ is similar to the "compensation" strategy in estimation, but all methods transform data into easy-to-calculate problems by grasping the characteristics of data and using the learned operation rules and properties, thus achieving the same goal. Students' evaluation and reflection on various methods is to deepen the mathematical thought behind the methods, so as to have an essential grasp of mathematical knowledge and methods. The teaching idea of "algorithm diversification" advocated by the new curriculum is to let students solve problems flexibly by summarizing and optimizing algorithms in the process of learning algorithm diversification, and finally internalize mathematical thinking methods into students' mathematical literacy. Review class: learn to organize and review knowledge, and strengthen the review of mathematical thinking methods, which is different from the teaching of new knowledge. It is to review mathematics on the basis that students have basically mastered a certain mathematical knowledge system, have some experience in solving problems and basically know some mathematical thinking methods. Mathematical thinking and methods are always hidden in mathematical knowledge, and combined with specific mathematical knowledge into an organic whole, but they can't be taught as chapters like mathematical knowledge, but permeate all primary school mathematical knowledge. Different chapters of mathematical knowledge often contain different mathematical thinking methods, and sometimes many mathematical thinking methods will be involved in the teaching of a chapter or a unit. Therefore, before the review class, teachers should be able to grasp the thinking method implied in the textbook as a whole, clarify the relationship between the knowledge before and after, achieve "looking before and looking after", and implement the infiltration of mathematical thinking method into the teaching plan. When reviewing, we should not only help students master knowledge and skills and form a good cognitive structure, but also strengthen the infiltration of mathematical thinking methods, reveal, summarize and strengthen a certain mathematical thinking method in time, and inspire its name, content and application, so that students can grasp the essence and inherent law of knowledge from the height of mathematical thinking methods and gradually realize the value of mathematical thinking methods. With the deepening of students' understanding of mathematical knowledge, mathematical thinking methods show some progress. In class summary, unit review and knowledge application, teachers should guide students to consciously check their thinking activities, reflect on how they found and solved problems, and what basic thinking methods they used, and summarize and refine some mathematical thinking methods in time, so that students can grasp the essence of knowledge from the height of mathematical thinking methods and enhance the value of classroom teaching. For example, when I was teaching "Review of the Area of Plane Graphics" in Grade Five, I asked students to write out the formulas for calculating the area of various plane graphics (rectangle, square, parallelogram, triangle, trapezoid and diamond), and then asked: How are these formulas derived? Each student chooses 1 ~ 2 kinds of graphs, demonstrates the derivation process with learning tools, and then communicates in groups. After the exchange, I pointed out: Can you organize this knowledge into a knowledge network? When students form a knowledge network (as shown below), guide students to calculate the area of these plane figures again. For example, when reviewing the area derivation of polygons, teachers can guide students to think: How are the area calculation formulas of parallelogram, triangle and trapezoid derived? What are the similarities? Let the students refine and summarize: when learning the calculation of parallelogram area, we use the cut-and-paste method to convert it into the learned rectangle for deduction; When learning to calculate the area of triangle and trapezoid, we use two identical figures to form a figure or cut a figure into a learned figure to deduce ... After a series of generalization and refinement, students come to an important way of thinking-changing ideas. Once students master mathematical thinking methods, they can not only improve their knowledge structure, but also be particularly helpful for their future study and application. Because of mastering the thinking method of mathematics, students will know how to think when facing new problems, and truly achieve a qualitative leap. (3) Homework: Mastering knowledge, forming skills, developing intelligence, and applying mathematical thinking methods to carefully design homework are also a way to infiltrate mathematical thinking methods. Designing homework, designing some topics containing mathematical thinking methods and adopting effective practice methods not only consolidate knowledge and skills, but also organically infiltrate mathematical thinking methods, killing two birds with one stone. Therefore, teachers should pay attention to the assignment of homework, and after students finish their homework, they should seize the opportunity to comment appropriately, so that students can not only consolidate what they have learned and acquire problem-solving skills, but more importantly, understand mathematical laws and mathematical thinking methods. Another example is a sixth-grade teacher, who arranged the following after-school thinking questions. In homework evaluation, teachers should not only give answers, but more importantly, inspire students to think: how do you calculate? what do you think? What thinking method is used in it? Guide students to sum up ideas and methods by combining the above picture: analogy, mathematical modeling, limit and combination of numbers and shapes. (4) Extracurricular: cultivate interest, increase knowledge, cultivate ability and improve mathematical thinking methods. Mathematics extracurricular activities are an important supplement to in-class teaching. According to the students' learning level, lectures on the contents of mathematical thinking methods are held every year. If the infiltration of mathematical thinking methods in normal teaching is a "delicious snack", then the special lecture is a "rich meal" for students, who systematically understand the common mathematical thinking methods and applications and broaden their horizons; The infiltration of mathematical thinking methods and the combination of extracurricular practice activities can complement each other. Regular mathematical practice activities can cultivate students' practical ability and innovative consciousness, and cultivate their ability to solve problems by using mathematical thinking methods. Regular mathematical intelligence competitions not only stimulate the enthusiasm of gifted students to learn mathematics, but also examine students' mastery of mathematical thinking methods; Students' activities, such as compiling mathematical tabloids and publishing blackboard newspapers, can increase their knowledge and learn more. Various forms of extracurricular activities in mathematics make mathematical thinking methods imperceptible and guide students to enhance their understanding of mathematical thinking methods in their study and application.