Keywords: infiltration of mathematical thinking methods
Mathematics teaching in primary schools should not only impart knowledge to students, but also infiltrate mathematical thinking methods in teaching. Mathematical thoughts and methods are an inseparable part of mathematical knowledge. There are many mathematical ideas and methods in primary school mathematics textbooks, such as symbolic thinking, mathematical model thinking, statistical thinking, reduction thinking, combination thinking, transformation thinking, corresponding thinking, extreme thinking, set thinking, transformation modeling thinking, guessing, verification methods, reduction to absurdity and so on. Students' learning of mathematics is not only the acquisition and repeated practice of knowledge, but also the mathematical thinking method throughout. If the basic knowledge and skills in mathematics textbooks are a bright line, then the mathematical thinking method contained in the textbooks is a dark line. Teachers should pay attention to the infiltration of mathematical thinking methods, grasp the favorable factors in the teaching content, consciously guide them, infiltrate them purposefully, selectively and timely, so that students can master mathematical thinking methods imperceptibly.
First, it is an inevitable trend to infiltrate mathematical thinking methods into teaching.
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 mathematical method refers to the way, procedure and means of a certain mathematical activity process, which has the characteristics of process, hierarchy and operability. Mathematical thought is the soul of mathematical method, and mathematical method is the manifestation and means of realization of mathematical thought. Therefore, people call it mathematical thinking method. The necessity of infiltrating mathematical thinking methods in primary school mathematics teaching mainly has the following four points:
1, the need to cultivate innovative talents. In today's world, science and technology are advancing by leaps and bounds, knowledge economy is beginning to take shape, and international competition is becoming increasingly fierce. The improvement of national quality and the construction of "talent highland" are increasingly becoming decisive factors for economic growth and social development. The importance of quality education is highlighted. Mathematics teaching should also implement quality education. China's "Mathematics Curriculum Standard for Full-time Compulsory Education" clearly points out that the mathematics curriculum in the compulsory education stage is devoted to making students understand the close relationship between mathematics and nature, human society, know the value of mathematics, and enhance their confidence in understanding and applying mathematics; Learn to use mathematical thinking to observe and analyze the real society and solve problems in daily life and other disciplines; Form a scientific spirit of being brave in exploration and innovation; Obtain important mathematical knowledge (including mathematical knowledge and experience in mathematical activities), basic thinking methods and necessary application skills necessary for future social life and further development. Innovative talents need high-quality people, and high-quality people must have excellent thinking quality, while mathematics is the science of thinking, and thinking ability is the core of mathematical ability. Infiltrating mathematical thinking methods in mathematics teaching is the most fundamental way to cultivate students' innovative consciousness.
2. The need of mathematics teaching reform. According to the investigation, it is found that the teaching of mathematical thinking methods is not paid attention to in mathematics teaching. A considerable number of teachers have not included mathematical thinking methods in their teaching objectives at all. Strengthening the teaching of mathematical thinking method is the need to further improve the quality of mathematics teaching. From the perspective of mathematics textbook system, there are two main lines running through the whole primary school mathematics textbook. One is the most basic mathematical knowledge written into the textbook, which is a bright line and has always been valued. Make sure that students learn well. The other is the cultivation of mathematical ability and the infiltration of mathematical thinking methods, which is a hidden line, rarely or not directly written into textbooks, but it is very important for the growth of primary school students and is paid more and more attention. In teaching, we should not only attach importance to the teaching of mathematical knowledge, but also ignore the teaching of mathematical thinking methods. These two lines should go hand in hand in classroom teaching, and intangible mathematical thoughts run through tangible mathematical knowledge. Attaching importance to the teaching of mathematical thinking methods will help teachers grasp the purpose of mathematics teaching as a whole, show students the essence of mathematics, the process of knowledge formation and the process of problem solving, and make teaching achieve twice the result with half the effort. At present, there is a conclusion that teaching emphasizes knowledge and ignores the process of knowledge occurrence; Pay attention to the evaluation of knowledge reaching the standard, but ignore the evaluation of the formation of mathematical thought; Pay attention to students' immediate score interests and ignore the present situation of students' long-term quality development. Some teachers don't have a deep understanding of mathematical thinking methods, and the infiltration teaching of mathematical thinking methods is difficult to achieve results in a short time in classroom teaching. Therefore, in primary school mathematics teaching, it is difficult to implement the teaching of mathematical thinking methods in a standardized and orderly manner, and it has become a forgotten and neglected "corner". If we insist on "teaching of mathematical knowledge" in mathematics teaching, it is far from cultivating mathematical thinking ability, which requires the teaching and infiltration of mathematical thinking methods. Based on the above situation, it is necessary to practice and explore the teaching of mathematical thinking methods in primary school mathematics teaching methods.
3. In cognitive psychology, thinking method 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). The key to solving problems lies in finding suitable problem-solving ideas, and mathematical thinking methods are the guiding ideology to help build problem-solving ideas. 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.
4. 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 on the horizontal axis, and mathematical thinking methods are 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.
Second, the knowledge distribution and teaching strategies of the main mathematical thinking methods in the current primary school mathematics textbooks.
No matter whether the current primary school mathematics is a new textbook or an old textbook, from the content of the textbook, mathematical models, symbolic ideas, statistical ideas, combinatorial ideas and so on are often used to solve primary school mathematics problems. These mathematical thinking methods play an important role in helping students solve practical problems.
1, a symbol of thought.
Russell, a famous British philosopher and mathematician, said, "What is mathematics? Mathematics is symbol plus logic. " There are the following kinds of symbols in primary school textbooks: (1) Individual symbols: symbols representing numbers, such as 1, 2, 3, 4…, 0; A, b, c, …, π, χ and symbols of decimals, fractions and percentages. (2) The operation symbols of numbers:+,-,× (), ⊙ (/,:). (3) Relation symbols: =, >,<, etc. (4) Combination symbols: (), [], etc. , as well as unit of measurement symbols for angles and separator symbols for vertical operations.
Because of the contradiction between the abstraction of mathematical symbols and the concreteness of primary school students' thinking habits, and because symbols are often the representatives of concepts. Therefore, teachers should pay attention to: ① Let students understand and use mathematical symbols correctly. In practical teaching, students often make the following mistakes when using these mathematical symbols. For example, teaching the lower grade text question "How much is 90 more than 60?" Pupils often use "+"when watching "more" and "-"when watching "less" because they don't understand the meaning of addition. The formula of staggered columns is "90+60". For example, the first grade text topic "Six times less than 24 is a number 180". What number is this? " Students often use x to mean "times",-to mean "less", and (180-24)×6 "is mistaken for" times ". In such an example, teachers pay attention to let students understand the connotation of symbols and correctly understand the concepts represented by symbols in teaching. If we only correct it from the solution and not from the symbolic thought, we will get twice the result with half the effort, and students will make similar mistakes in the future. ② Master the conversion between daily language and symbolic language. Mathematics teaching is actually the teaching of mathematics language. In teaching activities, students should be helped to learn the transformation between simple mathematical symbolic language and daily language, that is, the quantitative relationship or spatial form of daily language narrative should be transformed into mathematical symbolic language. On the contrary, we can also turn symbolic language into a problem and understand the quantitative relationship or spatial form reflected by abstract symbols. For example:
Xiaoying village has 75 hectares of cotton fields, 60% of which is the solution: suppose the cultivated land area of the village is
It is 60% of the cultivated land area in the village, and the total analysis is 75. What is the number? χ hectares.
What is the area of cultivated land in this village? X 60%=75
Daily language, mathematical language, symbolic language
Therefore, teachers should guide students to describe life language with mathematical language in teaching, instead of mechanically instilling mathematical symbols into students, so as to cultivate students' abstract thinking ability. (3) The idea of infiltrating variables in digital filling. At different stages, primary school mathematics textbooks have infiltrated the idea of variables at different levels and in different forms, so that students can gradually understand the idea of variables. For example: 3. □7 & gt； 3.27，45. 16 & lt； 45. 1□, it is easy for students to fill in a number in the box, but the teacher should understand that if you fill in the box with χ, it becomes a one-dimensional linear inequality. Therefore, the teacher should guide the students to continue thinking: How many numbers can fit in the box at most? This kind of thinking can make students understand the thought of variables preliminarily. (4) Infiltrating symbolic thoughts into letters and numbers. In primary school textbooks, letters are used to represent numbers, and there are algorithms, the relationship between numbers, the formulas of area and volume, and so on. For example: additive commutative law: a+b=b+a, distance = speed × time, s=vt, etc. Teachers should use letters to represent numbers step by step in teaching, and combine students' life and the original cognitive structure to construct naturally.
2. Mathematical model method.
Mr. Hua, a famous mathematician, said: "If you can't see the number, it's not intuitive. If you count too much, it's difficult to be subtle." This sentence vividly and concisely points out the dialectical relationship of interdependence and mutual restriction between form and number. Mathematical model is an approximate reflection of the spatial form and quantitative relationship of objective things. Mathematical models can be divided into broad sense and narrow sense. Broadly speaking, all mathematical concepts, formulas, theoretical systems, equations and algorithm systems can be called mathematical models. Mathematical models can be divided into three categories: ① Conceptual mathematical models, such as real numbers, functions, sets and vectors. ② Methodological models, such as various equations and formulas. ③ Structural models, such as groups, rings, domains, vector spaces, etc. The basic structure of the mathematical model to solve the problem is as follows:
practical problem
Mathematical abstraction
Simplified description of mathematical model
Calculus reasoning
Solution of mathematical model
Because of the intuitive expression of mathematical model, the essential attribute of the concept is obvious, which makes it easier for students to master. Therefore, it is helpful for primary school students to master mathematical knowledge, enhance their ability to solve problems and improve the effect of mathematics teaching by properly infiltrating mathematical model methods into primary school mathematics teaching. Mathematics teaching in primary schools generally adopts conceptual mathematics model and methodological mathematics model.
① The infiltration of set model in teaching. Triangles classified by angle can be represented by the following figure:
triangle
right triangle
Acute triangle obtuse triangle
After students understand the meaning of set diagram, they will try to express the relationship between concepts with set diagram in their later study. For example:
parallelogram
rectangle
square
When solving practical problems, teachers can also inspire students to use set diagrams to help analyze the meaning of problems and explore solutions. For example, the engineering team plans to build a 250-kilometer-long expressway. Complete 20% of the total length on the first day, 40% on the second day, and the rest on the third day. How many kilometers were built on the third day?
250km ("1")
The first day, the second day and the third day.
20% 40% ?
As can be seen from the figure, the length of the road repaired on the third day is 250km (1-20%-40%), which is easy to solve: 250x (1-20%-40%) =100 (km).
② Infiltration of equation model in teaching. The key to solving application problems with column equation is to simulate the quantitative relationship with mathematical model, that is, to represent the same quantity in two different ways according to conditions and list the relationship between known quantity and unknown quantity. In the middle and senior grades of primary schools, equations have been gradually used to solve application problems and application problems. For example, a factory used to produce 1800 machine parts every day, which is less than 10% now. How many machine parts are produced every day now?
Solution: Suppose that there are χ machine parts manufactured every day.
Now every day, the original daily manufacturing machine is made every day.
Machine parts-less than now 10%, = 1800 machine parts.
χ 10%χ 1800
So the equation is listed: χ- 10%χ= 1800. That is to say, 1800 machine parts are manufactured every day, which is equivalent to the present (1- 10%). The equation χ (1-10%) =1800 can also be listed.
③ Infiltration of geometric model in teaching. When solving application problems, if the mathematical problems of difficult problems can be transformed into related figures, the geometric model can be constructed by drawing, and then the answers will be simple and intuitive according to the nature and characteristics of the figures. For example, the wheel width of the roller is 6 meters. If it walks 200 meters a minute, how many square meters will it run in an hour?
200 meters
The wheel width is 6 meters.
As can be seen from the figure, this problem is actually to find the area of 60 rectangles with a length of 200 meters and a width of 6 meters. 6× 200× 60 = 32,000 (square meters).
④ Infiltration of formula model in teaching. Mathematical formula is not only a symbol reflecting the mathematical relationship in the objective world, but also a mathematical model abstracted from the real world. Because it abandons the individual attributes of everything, it is more typical. For example, total workload = working efficiency × working time, distance = speed × time, total output = single output × hectare, etc. Many related problems can be solved by using these abstract mathematical models. Example: "It takes 6 hours for Party A to do a job alone, and 4 hours for Party B to do a job alone. After Party A finishes 1/3, how many hours will it take for two people to cooperate? " To solve this problem, the total workload is regarded as the unit "1", the working efficiency of A is regarded as 1/6, and the efficiency of B is regarded as 1/4. According to the formula model of total workload = working efficiency × working time, the following formula is obtained: (1- 1/3) ÷.
3. Statistical thought
The basic idea of statistics is to infer the state of the whole system from the statistical characteristics of local observation data, or to judge the probability of an assertion to ensure its correctness, or to calculate the probability of wrong judgment. Statistical methods are scientific methods from "part to whole" and "special to general". The statistical thought in primary school mathematics is embodied in simple data sorting and averaging, simple statistical tables and charts. Students should be able to find some related problems and draw some conclusions from data and charts while sorting out, tabulating and drawing. In the arrangement of teaching materials, students learn the data sorting method in the middle and lower grades after understanding the slightly simple statistical idea, and further group statistical sorting method, composite bar chart and line chart according to the size of data in the senior grades. In addition to teaching according to the arrangement of textbooks, teachers can also organically infiltrate the idea of statistics in their usual teaching. For example, arrange students to collect relevant information before class. For example, the course "Reading and Writing Numbers within 100 million" allows students to collect data about numbers within 100 million in their lives. Through the collection before class and the communication in class, students not only learned to read and write these numbers, but also experienced the idea of statistics in the national education. In some classes, you can also collect data and statistics in class to serve the teaching content. For example, in the "three-step application problem" class, investigate students' subscriptions, including the number of people, unit price, quantity, newspaper types and so on. Ask questions through charts and other forms, and teach around the idea of solving three-step application problems. Through this kind of teaching, teachers consciously infiltrate statistical thoughts, and students learn mathematics in their lives, which greatly improves the effectiveness of learning. Of course, the infiltration of statistical thought in primary school mathematics can only be preliminary, involving only some of the simplest methods of sorting out sample data. As for general speculation, it only guides students to make some preliminary imagination and estimation, so as to gradually accept the influence of statistical thought and lay the foundation for further study in the future.
Step 4 turn to thinking
The idea of transformation is to transform a practical problem into a mathematical problem and a more complicated problem into a simpler one. It should be pointed out that this transformation idea is different from the general "transformation" and "transformation". It is irreversible and unidirectional.
Example 1. The fox and the weasel have a jumping competition. The fox can jump 412 meters at a time, and the weasel can jump 2/3/4 meters at a time. They only jump once a second. In the competition, there is a trap every 12 3/8 meters from the starting point. When one of them fell into the trap, how many meters did the other jump?
This is a practical problem, but through analysis, it is known that when a fox (or weasel) falls into a trap for the first time, its jumping distance is an integer multiple of its jumping distance of 4654,38+0/2 (or 2 3/4) meters, and it is also an integer multiple of the trap interval of 654,38+02 3/8 meters, that is, 4654,38+0. In view of the two situations, the problem is basically solved by calculating the number of jumps to determine who falls into the trap first. The above-mentioned thinking process is essentially to transform a practical problem into a "least common multiple" problem through analysis, that is, to transform a practical problem into a mathematical problem, which is one of the manifestations of mathematical ability.
5. Combinatorial thinking
The idea of combination is to group the studied objects reasonably and solve all possible situations without repetition or omission.
In the following multiplication formula, the same Chinese characters represent the same number, and different Chinese characters represent different numbers. Find this formula.
Xiao Ai Mathematics
× 4
──────
Learn to count and love to learn from others.
Analysis: Because the product of five digits multiplied by four is still five digits, the first digit of the multiplicand can only be 1 or 2, but if "from" = 1, the unit of the product of "Xue" ×4 should be 1, and "Xue" has no solution. So "from" = 2.
In the unit, the unit of the product of Xue× 4 is 2, and Xue = 3 or 8. But because Xue is the first digit of the product, it must be greater than or equal to 8, so Xue = 8.
In thousands, because "Xiao" ×4 can't carry to thousands, "Xiao" = 1 or 0. If "small" = 0, then the number of digits of "number" × 4+3 (carry) on the tenth bit is 0, which is impossible, so "small" = 1.
In the tenth place, the unit of "number" × 4+3 (carry) is 1, and it is deduced that "number" = 7.
The unit of "love" × 4+3 (carry) in hundreds is still "love", and hundreds must be rounded to 3, so "love" = 9.
Therefore, the multiplication formula is
2 1 9 7 8
× 4
──────
8 7 9 1 2
The above classification methods are not repeated or omitted, which embodies the combination idea.
6. In actual teaching, teachers will use different ways of thinking to teach the same teaching content because of their different understanding of teaching materials. Some teaching contents are often analyzed and answered by several mathematical thinking methods. Therefore, in teaching, teachers should fully understand the educational function of textbooks and explore their hidden mathematical thinking methods, so that students can master their context and cultivate their consciousness of consciously using mathematical thinking methods in the process of drawing conclusions, finding methods and revealing laws. In addition to the five ways of thinking listed above, transformational thinking, corresponding thinking, extreme thinking, collective thinking, associative thinking, inductive guessing, deductive transformational modeling thinking, guessing, verification and reduction to absurdity are also often used in primary school mathematics teaching, and teachers should also pay attention to conscious infiltration in teaching.
3. Infiltrate mathematical thinking methods into daily teaching.
The new curriculum standards formulated by the new round of basic education curriculum reform pay special attention to students' knowledge and skills, processes and methods, emotional attitudes and values. It is mentioned in the curriculum standard that the mathematics curriculum in the compulsory education stage should be basic, universal and developmental, so that mathematics education can be oriented to all students and everyone can learn valuable mathematics; Everyone gets the necessary math; Different people get different development in mathematics. This requires our teachers to pay attention not only to knowledge and skills, but also to skills and methods in teaching.
1, the principle of infiltrating mathematics thinking method teaching
(1) process principle.
When infiltrating mathematical thinking methods into teaching, students are not directly pointed out the applied mathematical thinking methods, but are consciously guided to understand the mathematical thinking and methods through carefully designed teaching processes. For example, when teaching additive commutative law, let students describe the "changing and unchanging truth" in the game with the language of daily life through a guessing game. Then, let the students use figures or mathematical symbols to express, and then abstract the mathematical model A+B = B+A. ..
(2) the principle of repetition.
Mathematical method belongs to the category of logical thinking, and students' understanding and mastery of it has a cognitive process of "from individual to general, from concrete to abstract, from perceptual to rational, from low to high". Then, teachers should combine infiltration and repetition in teaching. For example, in the application of teaching arithmetic, typical application problems and solving some practical problems, various mathematical models and methods such as set model, equation model, set model and formula model are repeatedly infiltrated.
(3) Systematic principle.
The infiltration of mathematical thinking methods should be from shallow to deep, not too casual. Teachers should know exactly how far a mathematical thinking method has been excavated and how far students can understand it. Therefore, when making teaching plans, teachers should fully understand what can be infiltrated into mathematical thinking methods in combination with this textbook, and then combine the follow-up teaching to sort out the teaching system of mathematical thinking methods.
(3) the principle of clarity.
Long-term repeated and vague infiltration, students will not consciously understand and use mathematical thinking methods. Therefore, in a teaching stage, teachers should consciously sum up our thinking methods when solving problems, so that students can moderately clarify the laws and application methods of mathematical thinking methods, which is conducive to future study.
2. An effective way to infiltrate mathematical thinking methods.
(1) In the process of knowledge generation, mathematical thinking methods should be infiltrated in time.
In teaching, teachers should not simply give definitions, jump to conclusions prematurely, and look for connections rigidly, which is conducive to cultivating students' ability to analyze, observe, compare, abstract and summarize logical thinking. For example, in the process of teaching "the nature of decimals", the teacher does not simply tell the students what the nature of decimals is, but reveals the nature of decimals by comparing the sizes of 0. 10 and 0. 100. There are five or six reasons why students discuss in groups that 0. 10 equals 0. 100. Some use the combination of numbers and shapes to verify; Some are verified by actual measurement; Some users' quotients are verified by analogy; Some of them are verified by reduction to absurdity.
(2) Summing up mathematical thinking methods through summary and review.
When sorting out and reviewing each unit, students should not only sort out the knowledge points of mathematics, but also recall some typical thinking methods applied to solving problems. Let students use these methods to solve practical problems.
(3) Pay attention to the comprehensive application of various mathematical thinking methods in teaching.
In the process of solving practical problems, it is often necessary to use multiple methods at the same time to be effective. Then, pay attention to guiding students' comprehensive application ability in teaching.
(4) Pay attention to summary and evaluation.
After a period of training, combined with students' homework and exams, teachers should summarize and evaluate students in time. When evaluating, we should not simply evaluate the results, but affirm them by analyzing students' problem-solving ideas and some mathematical thinking methods used. Only in this way can students' innovative ability and learning motivation be stimulated.
Some people have studied the teaching for one semester through experiments, constantly improved and summarized in the research process, and initially saw some results. From the students' achievements, we can see that students can benefit from the infiltration of mathematical thinking methods purposefully, planned and orderly in teaching, exercise their thinking ability and enhance their ability to solve problems, thus improving the teaching quality.
Four. conclusion
With the new round of curriculum reform, the infiltration of mathematical thinking methods in primary school mathematics has been placed in an important leading position. Every teacher should actively reform and try in practice. Through effective practice and research, it is feasible to infiltrate mathematical thinking methods into primary school mathematics, which is completely acceptable to students. Through purposeful, planned and orderly infiltration, students' thinking ability is enhanced, and different students get different gains. What they get is not only "fish", but also "fishing", which has positive significance and far-reaching influence on the long-term development of students. In this study, teachers have improved their mathematics literacy, improved their teaching concepts, and really improved the "people-oriented" classroom benefit and teaching quality.