A combination of numbers and shapes

Change idea

Combinatorial thinking

Equation ideas, etc.

How to infiltrate mathematical thinking methods mainly includes: first, classroom introduction, induction and infiltration.

Teacher: Students, now let's observe a set of pictures. Students should explain the characteristics of these figures in the process of observation. Give pictures of both sides of the mirror on the screen, including five-pointed stars, flowers and snowflakes. )

Health 1: These numbers on both sides of the mirror are the same, just like reflections.

Health 2: These images can overlap.

Teacher: The students all speak very well. Are these figures symmetrical with a line like a mirror?

Health: Yes.

Teacher: We call this overlapping figure folded in half along a straight line in a plane an axisymmetric figure. Then the students began to look at these pictures presented by the teacher on the blackboard to see which are axisymmetric figures.

Then, the teacher showed the students some pictures of axisymmetric graphics, and taught them to use induction and deduction's mathematical thinking method to make the process of mathematics learning easier.

Second, content expansion, Lenovo analysis

Teacher: We have just learned the basic knowledge of axial symmetry. Now, let's think about which graphs we have learned and which of these graphs are axisymmetric.

Health 1: I have studied rectangles and squares before. These two figures are both axisymmetric figures. A rectangle has two axes of symmetry and a square has four axes of symmetry.

Teacher: That's right. Students, let's think about whether the figure "circle" is axisymmetric. How many axes of symmetry does a circle have?

Life 1: The circle is an axisymmetric figure, but there seems to be countless symmetry axes of the circle.

What important mathematical thinking methods should be infiltrated into the calculation teaching in primary schools? The so-called mathematical thinking 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. Primary school mathematics textbooks are the explicit knowledge system of mathematics teaching. Among many important laws and formulas, textbooks can only see beautiful conclusions and ingenious solutions to many examples, but they can't see the psychological activity process of observing, experimenting, analyzing, inducing, abstracting or exploring reasoning through special examples. Therefore, mathematical thinking method is a tacit knowledge system in mathematics teaching, and primary school mathematics teaching should include explicit knowledge teaching and tacit knowledge teaching. If teachers only follow the arrangement of textbooks and the traditional teaching process from concepts and formulas to examples and exercises, even if teachers speak deeply and ask students to remember the conclusions and master the types and methods of solving problems, the students trained in this way can only be "knowledge-based" and "memory-based", which will completely deviate from the goal of mathematics education. 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. Mathematical knowledge itself is very important, but it is not the only determinant. What really plays a long-term role in students' future study, life and work, and benefits them for life is the mathematical thinking method. 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, it is the requirement of the future society and the inevitable result of the development of international mathematics education to infiltrate some basic mathematical thinking methods into students. 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.

On how to infiltrate mathematical thinking methods Abstract: The so-called mathematical thinking is the essential understanding of mathematical knowledge and methods, and it is the crystallization and generalization of mathematical thinking. It directly dominates the practical activities of mathematics and is the soul of solving mathematical problems. The so-called mathematical method is the expression of mathematical thinking, the means and tools to realize mathematical thinking, and the fundamental strategy and procedure to solve mathematical problems. There is no strict boundary between method and thought. However, the solution of any mathematical problem is guided by a certain mathematical thought. Therefore, mathematical thought has theoretical characteristics, while mathematical methods have practical tendency. Therefore, people are used to calling concrete and operable methods methods methods, while those abstract, extensive or framed methods are called ideas. Figuratively speaking, the method is like a key. A key can only open one lock. Mathematical thought is equivalent to the principle of making keys, and the solution of any problem is always carried out under the guidance of some thought. The process of solving problems by mathematical methods is the process of accumulating perceptual knowledge. When this accumulation reaches a certain level, there will be a leap to mathematical thinking. Once mathematical thought is formed, it plays a guiding role in mathematical methods. Therefore, people usually regard mathematical thought and method as a whole concept-mathematical thought and method.

What mathematical ideas and methods are infiltrated into the understanding of graphics? The first volume of mathematics "Understanding of 6 and 7" (1) Teaching content: "Understanding of 6 and 7" Teaching purpose: 1, knowing 6 and 7, being able to write 6 and 72 correctly, and representing various objects in life with 6 and 7. 3. Cultivate students' sense of numbers and the ability of careful observation. Teaching emphasis: 1, distinguish the cardinal meaning and ordinal meaning of 6 and 7. 2. Write numbers to cultivate students' sense of numbers. Teaching process: 1. Create a situation and display 42 pages of 6 and 7 thematic maps on the computer for students to observe carefully. 1. What's on the map? Please count the number of people, tables and chairs on the picture. Number of students enrolled. How do you count the number of people in the classroom? How else can I count? 4. How do you count the number of chairs? (Count six chairs that have been put away first, and then count 1 those that have been removed. We just counted them in the order of 1, 2, 3, 4, 5, 6 and 7. In counting, we find that after counting five, 1 is six, and after counting six, 1 is seven, seven is more than six, 1, six is more than five, 1. Second, you have observed the 1 of the new prize and know 6 and 7 very carefully. Today, we will meet our new friends 6 and 7. Title on the blackboard: Understanding 6 and 7. 2. Can you think of a learning tool that represents 6? Can you put them in your favorite graphics? Do you know how 6 came from? Students take learning tools, teachers show train of thought or other magnetic teaching AIDS, students raise their hands, and teachers choose creativity to praise and show? What number should be after 5? Show the counter demonstration, 5 dial 1 is 6. Add 1 after 6. What is it? Anti-demonstration Can you come up with 4 learning tools? And show your favorite pictures. Compared with the size, we met 5 earlier and 6 and 7 today. Do you know who is older and younger? Who has more than 5 and 6? What about 6 and 7? Can you still see who is less than who? 6 is less than 7, and vice versa. 4. Can the meaning of cardinality (1) be from small to large to 7? Starting from 7, from large to small 1? (2) Look at the goldfish picture on page 43, find the starting point and count how many bottles of goldfish there are. (Group work) (3) First find out which bottle contains six goldfish? How many bottles are there from the left? (4) Find the seventh bottle from the left, and then count how many goldfish are there in the bottle? 5. Teach the writing methods of 6 and 7 to observe the font characteristics. What does 6 look like? 6 is written in one stroke, starting from the top half of Tian Zige's pen, all the way to the bottom, and then draw a circle. What is 7 like?

How to Effectively Infiltrate Mathematical Thinking Methods Ma Ming, a famous mathematical educator in China, said: "The essence of mathematics teaching is the thinking process." Cultivating students' thinking ability is one of the purposes of mathematics teaching. In mathematics teaching, the cultivation of thinking ability depends on the solution of mathematical problems. Therefore, teachers can cultivate students' thinking quality in mathematics problem-solving teaching. The solution of mathematical problems is guided by mathematical ideas and means of mathematical methods. And mathematical method breeds mathematical thought, and mathematical thought contains mathematical thinking. Mathematical thinking method is the essence of mathematical knowledge, the soul of mathematical content, the guiding ideology and universally applicable method of mathematical activities. It enables students to understand the true meaning of mathematics, learn to think and deal with problems in mathematics, and is a magic weapon to combine learning knowledge, developing intelligence and cultivating ability. Teachers should make mathematical thinking method a link from knowledge to ability, and promote the formation and development of students' good thinking quality.

What mathematical thinking methods should be infiltrated in primary school mathematics teaching? The following mathematical thinking methods are not only easy for students to accept, but also have a good role in promoting the improvement of students' mathematical ability.

1. Change your mind

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 20 meters forward at a time, and the weasel can jump 6 meters forward at a time. They only jump once a second. During the competition, set a trap every15m 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, we know 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 20 (or 6) meters, and it is also an integer multiple of the trap interval 15 meters, that is, the "least common multiple" of 20 and 15. 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.

2. Combination of numbers and shapes

The idea of the combination of number and shape is to make full use of "shape" to express a certain quantitative relationship vividly. In other words, by making some graphs such as line segments, tree diagrams, rectangular area diagrams or set diagrams, students can correctly understand the quantitative relationship and make the questions concise and intuitive. Example 2 A glass of milk, A drank half a cup for the first time and the remaining half a cup for the second time, so he drank the remaining half a cup every time. How much is a person who drinks milk five times? If you add up the milk you drank five times, it is1/2+1/4+1/8+1/06+1/32, which is the optimal solution strategy. Let's draw a square first, assuming its area is "1". As can be seen from the figure, 1- 1/32 is what we want. Here, not only the idea of combining numbers with shapes is infiltrated into students, but also the idea of analogy is infiltrated into students.

3. Combinatorial thinking

The idea of combination is to group the studied objects reasonably and solve all possible situations one by one without repetition or omission.

4. "Function" thought

Function is one of the important concepts in modern mathematics and is widely used in modern science and technology. In primary school mathematics textbooks, the idea of function is widely infiltrated. In the first learning period, the function idea is infiltrated into it through mapping and other forms; The students in the second phase have mastered many calculation formulas, such as s=vt, which are actually some simple functional relationships. In the sixth grade, the significance of positive and negative proportions is an important content of the thought of osmotic function, because the number of positive and negative proportions reflects the dependence between two variables.

In addition, there are symbolic thinking, corresponding thinking, extreme thinking and thinking set, which should be infiltrated purposefully, selectively and timely in primary school mathematics teaching.

In addition, there are mathematical thinking methods such as set thinking, symbolic thinking and corresponding thinking.

How to Infiltrate Mathematical Thinking Methods There are two lines in ppt courseware mathematics teaching, one is the teaching of open lines, that is, mathematical knowledge, and the other is the teaching of dark lines, that is, mathematical thinking methods. Mathematical thinking method is the essence of mathematics, the link for students to form a good cognitive structure, the bridge for transforming knowledge into ability, and the carrier for cultivating students' good mathematical concepts and innovative thinking. We must attach importance to the infiltration teaching of mathematical thinking methods in teaching.

First, the definition of mathematical thinking method

Mathematical thought is the essential understanding of mathematical knowledge, methods and laws; Mathematical method is the strategy and procedure to solve mathematical problems, and it is the concrete embodiment of mathematical thought; Mathematical knowledge is the carrier of mathematical thinking method, and mathematical thinking is at a higher level than basic mathematical knowledge and common mathematical methods. It comes from basic mathematical knowledge and common mathematical methods, and has a guiding position in using basic mathematical knowledge and methods to deal with mathematical problems. For learners, the process of using mathematical methods to solve problems is the process of accumulating perceptual knowledge. When this accumulation reaches a certain level, it will make a leap and rise to mathematical thinking. Once mathematical thinking is formed, it will play a guiding role in mathematical methods. Therefore, people usually regard mathematical thoughts and methods as a whole concept-mathematical thoughts and methods.

Second, the main mathematical thinking methods that should be infiltrated in junior high school.

In junior high school mathematics teaching, at least the following main mathematical thinking methods should be infiltrated into students:

1. Thinking method of classified discussion

Classification is a way of thinking by comparing the similarities and differences of the essential attributes of mathematical objects, and then dividing mathematical objects into different categories according to certain attributes. Classified discussion is not only an important mathematical thought, but also an important mathematical method, which can overcome the one-sidedness of thinking and prevent missing solutions.

2. Analogical thinking method

Analogy is a form of reasoning based on two or two objects having some common properties, which is called the most creative way of thinking.

3. The thinking method of combining numbers and shapes

The thinking method of combining number and shape refers to a thinking strategy that combines number (quantity) and shape to analyze, study and solve problems.

4. Change thinking methods

The so-called "transformation" is to simplify the problem to be solved into another easy problem or solved problem.

5. Thinking methods of equations and functions

Using the thinking method of equation is to transform the problem into solving the equation (group) problem by using the symbolic language of mathematics according to the quantitative relationship between the known quantity in the problem and the unknown quantity in the teaching method.

The so-called functional thinking method is to analyze and study the quantitative relationship in specific problems from the viewpoint of movement and change, and to characterize and study this quantitative relationship through functional forms, so as to solve problems.

6. Holistic thinking method

The whole way of thinking is that when considering a mathematical problem, we don't focus on its regional characteristics, but on the overall structure of the problem. Through comprehensive and profound observation, we can understand the essence of the problem from a macro perspective and treat some independent but closely related quantities as a whole.

Thirdly, the way to infiltrate mathematical thinking methods in teaching.

1. In the process of knowledge generation, timely infiltrate mathematical thinking methods.

The content of mathematics teaching can be roughly divided into two levels: one is called superficial knowledge, which contains basic contents such as concepts, properties, laws, formulas, axioms and theorems; The other is called deep knowledge, which mainly refers to mathematical thoughts and methods. Surface knowledge is the foundation of deep knowledge and has strong maneuverability. Only by studying textbooks and mastering and understanding certain superficial knowledge can students further learn and understand relevant deep knowledge. Mathematical thinking method is based on mathematical knowledge and contained in surface knowledge. It is the essence of mathematics, which supports and guides superficial knowledge. Therefore, in the teaching process of concepts, properties and formulas, teachers should constantly infiltrate relevant mathematical thinking methods, so that students can grasp the surface knowledge and understand the deep knowledge at the same time, thus making students' thinking leap qualitatively. Only talking about concepts, theorems and formulas without paying attention to the teaching of infiltrating mathematical thinking methods will not be conducive to students' real understanding and mastery of what they have learned, so that students' knowledge level will always remain in the primary stage and it is difficult to improve. In the teaching process, students should be guided to actively participate in the process of exploration, discovery and deduction of conclusions, find out the causal relationship among them, understand its relationship with other knowledge, and let students experience the mathematical ideas and methods they have experienced and applied in creative thinking activities.

How to infiltrate mathematical thinking methods into primary school mathematics 1 The concepts, laws, formulas, properties and other knowledge of improving infiltration consciousness are clearly written in textbooks, which are "tangible", while the mathematical thinking method is implicit in the mathematical knowledge system, which is "intangible" and scattered in all chapters of textbooks in an unsystematic way. The teacher talks too much and talks too little, which is very arbitrary. Because of the tight teaching time, it is often squeezed out as a "soft task". 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 mathematical thinking methods, integrate both mastering mathematical knowledge and infiltrating mathematical thinking methods into teaching purposes, and integrate the requirements of teaching mathematical thinking methods into lesson preparation. Secondly, we should study the teaching materials deeply and try our best to find out all kinds of factors that can penetrate mathematical thinking methods in the teaching materials. For each chapter and section, we should consider how the specific content permeates mathematical thinking methods, which mathematical thinking methods permeate, how to permeate, and to what extent. It is necessary to have an overall design and put forward specific teaching requirements at different stages. 2. Grasping the feasibility of infiltrating mathematical thinking method teaching must be realized through specific teaching process. Therefore, we must grasp the opportunity of teaching mathematical thinking methods in the teaching process-concept formation, conclusion derivation, method thinking, thinking exploration, law revelation and other processes. At the same time, we should pay attention to the organic combination and natural infiltration in the teaching of mathematical thinking methods, consciously and imperceptibly inspire students to understand all kinds of mathematical thinking methods contained in mathematical knowledge, and avoid the counterproductive practices such as mechanically copying, generalizing and being divorced from reality. 3. The repetitive mathematical thinking method that pays attention to infiltration is gradually accumulated and formed in the process of inspiring students' thinking. Therefore, in teaching, we should first emphasize "reflection" after solving problems, because the mathematical thinking method refined in this process is easy for students to understand and accept. For example, through the regular comparison of scores and percentages, guide students to sum up the main points of solving such application problems, find the scores corresponding to specific quantities, and let students experience the corresponding ideas and reduction ideas themselves. Secondly, we should pay attention to the long-term nature of infiltration. It should be noted that the infiltration of students' mathematical thinking methods can not see the improvement of students' mathematical ability overnight, but a process. Mathematical thinking methods must be trained step by step and repeatedly, so that students can really understand.

What are the main mathematical thinking methods, elementary number theory and operational calculus rules, graphics, daily mathematical application,

Elementary algebra, geometry, set and corresponding concepts. ..