In daily study, there are often many math problems with multiple solutions, which are easily overlooked in practice or examination. This requires us to carefully examine the problem, awaken our own life experience, scrutinize it carefully, and fully and correctly understand the meaning of the problem. Otherwise, it is easy to ignore other answers and make a mistake of generalizing.
I once heard an Olympic math teacher say: learning math is like a fish like a net; Knowing how to solve a problem is like catching a fish and mastering the method to solve the problem, just like having a net; So the difference between "learning math well" and "learning math well" lies in whether you have a fish or a net. Mathematics, a thoughtful course, is very logical, so it always gives people the illusion. Geometry in mathematics is very interesting. Each number is interdependent, but it also has its own advantages. Such as a circle. The formula for calculating the area of a circle is S=∏r? Because of different radii, we often make some mistakes. For example, "A pizza with a radius of 9 cm and a pizza with a radius of 6 cm are equal to a pizza with a radius of 15 cm". Proposition, this topic first confuses everyone and gives people an illusion. Using the formula of circular area skillfully makes people have a wrong balance. In fact, a pizza with a radius of 9 cm and a pizza with a radius of 6 cm are not equal to a pizza with a radius of 15 cm, because the area of a pizza with a radius of 9 cm and a pizza with a radius of 6 cm is S=∏r? =9? ∏+6? ∏= 1 17∏, the area of a pizza with a radius of 15cm is S=∏r? = 15? ∏=225∏, so a pizza with a radius of 9 cm and a pizza with a radius of 6 cm are not equal to a pizza with a radius of 15 cm. Mathematics is like a mountain peak, soaring into the sky. I felt relaxed at first, but the higher I climbed, the steeper the peak became, which made people feel scared. At this time, only those who really like mathematics will have the courage to continue climbing. Therefore, people who stand at the peak of mathematics all like mathematics from the heart. Remember, people standing at the foot of the peak can't see the summit.
Mr. Qian Xuesen, the pioneer of thinking science in China, believes that human thinking can be divided into three types: abstract (logical) thinking, visual thinking and spiritual (epiphany) thinking. It is suggested that thinking in images should be regarded as the breakthrough of thinking science research. What is thinking in images? The so-called thinking in images is to use the images accumulated in the mind to think. Representation is the image of those object phenomena that we have perceived before and reproduced in our minds. Thinking in images has the characteristics of indirectness and generality. Like abstract thinking, thinking in images is an advanced form of cognition-rational cognition. Why should we cultivate students' thinking ability in images? According to the latest achievements of modern scientific research, the left and right hemispheres of the human brain have different functions. The left hemisphere is the language center, in charge of language and abstract thinking, while the right hemisphere is in charge of the comprehensive activities of image thinking materials such as music and painting. Only by matching, complementing and promoting each other can individuals develop harmoniously. From the characteristics of children's thinking, primary school students' thinking has gradually changed from concrete thinking in images to abstract logical thinking, but at this time, logical thinking is preliminary and still has concrete images to a great extent. Therefore, cultivating students' thinking ability in images is not only their own needs, but also their need to learn abstract mathematics knowledge. So how to cultivate students' thinking ability in images in primary school mathematics teaching? First, fully perceive, enrich appearances, and accumulate materials for cultivating thinking in images. Children can keenly perceive vivid images with rich colors, tones and sounds, and are good at using image colors and sounds to trigger thinking. Image is the cell of image thinking, and image thinking depends on image thinking. To develop students' thinking in images, we must lay a good foundation and enrich the accumulation of image materials. 1. Hands-on operation enriches the appearance of hands-on operation, allowing students to participate in learning with various senses and observe things from various angles. For example, to teach the concept of remainder, let the students divide the sticks first: (1) How many sticks are left in every two of the nine sticks? (2) 13, distributed to 5 people on average. How many sticks can each student get? How much is left? After the operation, guide the students to express the operation process in words and talk about how to divide the sticks, thus forming an image. Then let the students close their eyes and think about how to divide the following questions. (1) There are 7 biscuits. Each biscuit is divided into 3 pieces, which can be distributed to several people. How many pieces are left? ② Pencils 12, distributed to five people on average. How many pencils can each person divide, and how many pencils are left? In this way, students can think in operation and operate in thinking, and understand that dividend is the total number, divisor and quotient are the number of shares to be divided and each share, and the remainder is not enough, and the remainder is less than divisor. Correct and clear representations are formed in the mind, and correct thinking has a solid foundation. 2. Intuitive demonstration to enrich the appearance. Pupils' unintentional attention plays an important role, and the emergence of any new things will arouse students' interest in actively participating in the learning process. In the process of teaching, organize teaching with pictures, teaching AIDS or audio-visual means, visualize abstract knowledge and let students fully perceive the materials they have learned. Only with quantitative perceptual materials can they leave a clear image in their minds. For example, in the teaching of "cuboid cognition", teachers can first show cuboid objects familiar to students in their daily lives, such as matchboxes, chalk boxes, bricks and so on. These objects are cuboids. Then ask the students to list their own rectangular objects (bookcases, wooden cases, thick books, pencil boxes, etc.). ), through the feeling of the object, have a preliminary perceptual understanding of what kind of object a cuboid is. On this basis, teachers guide students to read books while observing the model, and understand the characteristics of the cuboid from different positions and directions, such as the equal area of six faces and opposite faces, the length of twelve sides and parallel sides. We can know the length, width and height of a cuboid by observing the length of a vertex and three sides intersecting with the vertex. Through the flat, lateral and vertical forms of the model, it is shown that the length, width and height are relatively fixed, and the knowledge is "alive", which enables students to establish a clear and profound representation in the process of learning with their mouths and brains, and provides conditions for the rationalization of thinking. The introduction of audio-visual teaching means into the classroom can turn the static into the dynamic and turn the near into the far. With its colorful and flexible teaching form, it provides students with demonstrations reflecting their thinking and thinking process, which can fully mobilize their psychological factors and achieve good results. For example, when teaching "Subtraction Application Problem of Finding Another Addendum", students can vividly understand the relationship between the total and the part through the slide presentation, that is, the total-part = another part. In teaching, we should use a variety of teaching methods to make students fully perceive, establish clear mathematical representations in their minds, and accumulate materials for improving students' mathematical imagination. Second, guide imagination and develop thinking in images. Modern cognitive psychology believes that images can not only be stored, but also the stored image traces (information) can be processed and reorganized to form new images, that is, imaginary images, which is also an important way of image thinking. Therefore, teachers should be good at creating problem scenarios in classroom teaching, such as graphic scenarios and language scenarios, to stimulate students' desire to participate in inquiry and give full play to their rich imagination. For example, after teaching trapezoidal knowledge, students can be guided to imagine: "What shape will the trapezoid become when one base of the trapezoid is gradually shortened to 0?"? When the short bottom of the ladder extends to be equal to the other bottom, what shape does it become? " With the help of representation, seemingly unrelated triangles, parallelograms and trapezoid can be organically combined. You can also memorize the area formulas of triangles and parallelograms according to the trapezoid area formula: 1 S[, trapezoid] =-(a+b) H2 1. When a = 0, it becomes a triangle, and the area formula is: S =-AH2; when A = b, it becomes a parallelogram, and the area formula is: S = AH. Different types of mathematical graphics provide the representation materials of brain thinking in images, arouse the enthusiasm and initiative of brain thinking, improve his thinking ability in images, promote the coordinated development of individual's left and right brains, and make people smarter. For example, illustrations designed with specific plots of applied problems in textbooks broaden the world of students' thinking in images and enhance their will to study hard. For example, examples and review questions in textbooks use colorful colors and various small animals, plants, rivers, mountains and rivers, modern airplanes, cars, ships, satellites, buildings, ancient cultural relics and books to express the quantitative relationship ... These are not only conducive to understanding the quantitative relationship, but also play an important role in the development of students' thinking ability in images and the improvement of their aesthetic ability. Besides, the application problem teaching, because the application problem is a combination of science, art and mathematics, the prototype of the application problem is more complex and abstract, and it is difficult for students to form a clear representation after they get into their heads. If we use the method of combining numbers and shapes to draw line segments, we can help students to establish a correct representation and make clear the hidden and complicated quantitative relationship. For example, "Xiao Liang has 18 yuan in its savings box, and Xiaohua's savings is 5/6 of that of Xiao Liang, and Xiao Xin's savings is 2/3 of that of Xiaohua. How much did Xiao Xin save? " It is often difficult for students to determine the unit "1". In teaching, students can be guided to draw the following line chart to analyze the quantitative relationship: According to the line chart, students can quickly list the formula: 18× 5/6× 2/3- 10 (yuan). Therefore, the line diagram is semi-abstract and semi-concrete, which can not only abandon the specific plot of the application problem, but also vividly reveal the relationship between conditions and articles and problems. The application of line graph and the combination of numbers and shapes here better stimulate students' creative imagination, which not only develops students' thinking in images, but also realizes the complementarity of thinking in images and abstract thinking.
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