Interpreting teaching materials and embodying mathematical thoughts in preparing lessons
In order to effectively penetrate mathematical thinking methods in teaching, we must first conduct a comprehensive analysis and research on the teaching materials, construct a strategic plan from a strategic perspective, sort out and excavate the main lines and veins of the teaching materials from the overall situation, establish the connection between knowledge, summarize and refine the characteristics of their knowledge, effectively presuppose, connect the preceding with the following, and combine teaching with learning.
For example, in the lesson "The Kingdom of Fractions and the Kingdom of Decimals" in the fifth grade of Beijing Normal University Edition, students' cognitive foundation is excavated, and the default scores and decimals are exchanged, and then the unknown is transformed into the known. The size of the number itself is constant, but it can be compared intuitively, which also lays the foundation for the subsequent study and permeates the "transformed" mathematics. Transforming thinking is an important strategy to solve mathematical problems. Students will experience the process of generating mathematical knowledge such as guessing, reasoning and research, which is a common thinking method in our mathematical thinking.
Digging up teaching materials and infiltrating mathematical thoughts into teaching objectives
When we want to combine the cognitive theory and logical system of teaching knowledge with the knowledge goal in the teaching material as the carrier, and try to dig it out in the teaching process, the key consideration is to let students experience the formation process of knowledge intuitively and combine it with the mathematical thought contained in the teaching conclusion. In the teaching process, we try to dig deep into the mathematical thoughts hidden in the teaching materials in the well-designed classroom teaching process, and use this as a teaching means to fully display the students' thinking process, which will help students understand, master and apply the essence of mathematical thoughts and find its breakthrough.
For example, in the first volume of "Optimization" in grade four, we set the goal as "let students choose the best pancake method in comparison and reflect the idea of operation", expecting students to get various results through mathematical analysis and operation in class, and finally put forward comprehensive and reasonable arrangements to achieve the best results; For another example, the lesson Countdown, the second volume of the fifth grade, aims at "experiencing the discovery process of countdown, understanding the significance of countdown from multiple angles, and infiltrating inductive thinking", expecting students to guide them to study several simple, individual and special situations in the process of summing up the significance of countdown, so as to sum up the general laws and nature and improve the inductive thinking method.
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Infiltrate mathematical culture in teaching
Pay attention to understanding the cultural background of mathematics.
The current textbooks, combined with the teaching content, have been in a lively, interesting and easy-to-understand form since the first grade, with the title "Do you know?" According to the topic, introduce some stories about mathematicians, interesting things about mathematics, discoveries of mathematics, knowledge about the history of mathematics, etc. Through the presentation of these colorful contents, students can understand that the generation and development of mathematical knowledge stems from the needs of human life, enrich their mathematical culture, experience the role of mathematics in the history of human development, and stimulate their interest in learning mathematics.
For example, when learning "quadrilateral", introduce the relevant historical materials of "Tangram" to students, especially the composition of Tangram given by the ancients, so that students can feel the beauty of geometric composition and the wisdom of our ancestors. For another example, when learning "hours, minutes and seconds", the textbook presents an ancient timing tool-lettering. Students know that although we know the time from the clock today, we have gone through a long process of exploration, experienced the difficulties of exploration and the wisdom of our ancestors, and encouraged students to love the culture of the motherland and learn from their ancestors. In teaching, if we can make full use of the cultural characteristics of new mathematics textbooks and let students truly understand the cultural value of mathematics, we can stimulate students' interest in learning and arouse their enthusiasm for learning, so as to really like mathematics from the heart.
Pay attention to highlighting the cultural attributes of mathematics classroom.
Mathematics classroom teaching is to explore the rich cultural resources contained in mathematics, realize the harmonious unity of its scientific value and humanistic value, and promote the sustainable development of students' emotions, attitudes and values. For example, in the "area of a circle" class, when a student proposed to convert a circle into a rectangle and try to calculate the circumference of a circle, the teacher asked the students to cooperate in the experiment in groups. As we all know, a student raised his hand and put forward his own opinion: it is impossible for a circle to be transformed into a rectangle, because it is a curved figure and the sides of a rectangle are straight. There was an instant silence in the classroom, and the students looked at the teacher in an orderly way, waiting for the teacher's ruling. The teacher said slowly, "It's true. On the surface, it is impossible for a circle to become a rectangle.
But through the unremitting efforts of ancient mathematicians, it has been successfully transformed. Do you want to know? "The students all replied," Yes! "The teacher demonstrated the simulation experiment of the courseware, and after the students operated with teaching AIDS, they successfully obtained the area formula of the circle. When the class was over, the students answered very fruitfully that they were eager to learn. At this time, the teacher said meaningfully, "Of course it's easy, because you stand on the shoulders of giants. "However, in the past long years, in order to study and solve this problem, people encountered many difficulties and hardships, spent a lot of energy and time, and condensed the wisdom of many mathematicians. I hope students can explore, guess and practice independently like mathematicians, and make their own contribution to mathematics ... "When the teacher explained this passage, no one did not listen carefully. Mathematical culture should not be sought from outside mathematics. The most intrinsic cultural feature of mathematics should be mathematics itself, which should embody the personality of mathematics and the charm of mathematical thinking. If students really feel the happiness of thinking in math class, and because of the optimization of thinking quality and the improvement of thinking ability, the essential strength of learning individuals has also been reflected, then the cultural tension of mathematics will really stand out.
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Organic Infiltration of Transforming Thinking
1. Improve the consciousness and feasibility of infiltration.
Different from concepts, rules, formulas and other knowledge, the transformed thinking method is not clearly written in textbooks. It is implicit in the mathematical knowledge system, but not systematically implied in all chapters of the textbook. This is an intangible knowledge. As a teacher, we should first renew our ideas, integrate the transformed thinking method into every lesson preparation link, study the teaching materials in depth, and strive to dig out all the factors that can penetrate the transformed thinking method in the teaching materials. For each knowledge point related to the transformed thinking method, we should consider how to infiltrate the reduced thinking method, including how to infiltrate and to what extent. In the teaching of conversion thinking method, we should pay attention to organic integration and natural infiltration, and consciously inspire students to understand the conversion thinking method contained in mathematical knowledge.
2. Emphasize the refinement and guidance of methods.
Solving problems is not only the main way for students to learn mathematics, but also an important means for teachers to teach. Therefore, teachers should pay attention to: first, when designing problems, they should pay attention to the thinking method of containing reduction; Second, in the process of knowledge formation, we should reveal the thinking method of transformation; Third, when teaching examples, we should highlight the method of regression thinking; Fourthly, the thinking method of reduction should be used in problem-solving training; Fifth, we should sum up the inductive thinking method while summing up the knowledge. Sixth, when guiding students to solve problems, we should let them discover the process of the generation, application and development of methods from problem-solving skills, and extract the thinking method of conversion from it to understand the essence of conversion methods.
3. Repeated reappearance and gradual infiltration
Mathematical knowledge is gradually deepened, which leads to the hierarchy of mathematical thinking methods reflected in various stages of knowledge development. When we solve problems, there will be many transformations, sometimes in different directions. Therefore, for the application of transformation method, we should pay attention to its reappearance in different stages of knowledge and students' exploration of the gradual formation of transformation method in different stages to inspire students' thinking. Strengthen the understanding of transforming thinking method. Because the transformed thinking method is gradually formed in the process of enlightening students' thinking, special emphasis should be placed on "reflection" after solving problems in teaching. The transformation method refined in this process is easier for students to understand and accept.
Organic infiltration of the idea of combining numbers with shapes
1. Infiltrate the thinking method of combining numbers and shapes in concept teaching.
In primary school mathematics teaching, the research objects include numbers and shapes. "Number" and "shape" are two main lines, which run through the whole primary and secondary school mathematics textbooks and are one of the basic contents of primary school mathematics teaching. The mutual transformation and combination of "number" and "shape" is not only an important idea in mathematics, but also an important method to solve problems. The combination of numbers and shapes is particularly important in the teaching of mathematical concepts in primary schools.
Case: 24-hour timing method
Teacher: It's evening 12, and most people are sleeping. At noon 12, the hour hand walked a circle, and it was only half a day. It's night 12, and the clockwise has gone twice, which is one day! What do you know through computer demonstration?
Health 1: There are 24 hours in a day. Health 2: A day is a day and a night. Health 3: The hour hand turns twice a day. Health 4: When the hour hand goes to the second lap, 12 should be added to all scales. 1 pm, which is 13 pm according to the 24-hour clock.
Teacher: From zero to noon, the number 12 on the clock face has been used up, just half a day. If we continue to count down, it should be 13, and 13 is what we call L this afternoon.
Summary: This timing method from 0: 00 to 24: 00 is called 24-hour timing method.
"24-hour timing method" is a difficult point in primary school mathematics teaching. Starting from the age characteristics of the third-grade students, in the process of understanding the 24-hour timing method, the teacher chose to use information technology to make the rotation of the minute hand and the hour hand match the alternating picture of night, day, month and day, and extend the movement synchronously with the line segment timing, so that the curve becomes straight and the display process vividly demonstrates the incomprehensible content. Help students to establish the concept of 1 day =24 hours by changing the shape from curve to straight line. Experience 1 day, including day and night, and know that 12 at night is the end of the previous day and the beginning of a new day. Turn clockwise twice to get 1 day, and 1 day is 24: 00. I realized that the 24-hour timing method started from the second lap of clockwise movement and added 12 to the numbers on the clock face.
Second, in the process of solving problems, the combination of numbers and shapes is infiltrated.
The practical problem teaching centered on "solving problems" pays more attention to providing students with problem-solving materials with certain practical significance and interest from their existing knowledge, experience and life background, creating challenging and open problem situations for students, and satisfying their thirst for knowledge and exploration.
Case: A car from City A to City B, due to the slippery road in rainy days, its speed is reduced by 20%. As a result, 1 hour was delayed. How many hours was it originally planned to arrive?
Teachers inspire and guide students to use the painting strategies learned in grade four. The area of a rectangle represents the distance between A and B, and the length and width represent the speed and time respectively. Draw the following picture:
Observing the above figures, students can quickly understand that ① and ③ are equal in area, ③ the length of the figure is the original planned speed 1, the width is 1 hour, and ③ the area is 1× 1. According to the fact that the area of figure ③ is equal to the area of figure ①, the length of figure ① is 65438.
In this way, the abstract application problems are put into intuitive graphics. Under the guidance of intuitive graphics, students can fully understand the relationship between quantity and quantity, find out the basic skills of each copy according to the total number and number of copies, and find out the total number according to the total number and number of copies. Communicate the relationship between graphs, tables and concrete quantities, and improve students' comparative, analytical and comprehensive abilities through the training of combining numbers and shapes.