Under the background of the new curriculum reform, the mathematics curriculum standard puts forward the requirements of "four foundations": cultivating and developing students' geometric intuition, which has become a problem worthy of attention in mathematics education. Cultivate students' geometric intuition, guide students to learn to analyze problems and solve simple practical problems by using graph strategy, and gradually rise to the point where they can draw directly by reasonable conversion between mathematical language and symbolic language. I think intuition is a kind of perception, an insightful setting, and its essence. Geometric intuition mainly embodies two points: first, you can see the relationship between different things at a glance; The second is to see the essence through the phenomenon. Mathematics is an abstract generalization of objective phenomena and gradually formed. It is a science that studies quantitative relations and spatial forms.
First, the intuitive teaching value of geometry
Mathematics Curriculum Standard for Full-time Compulsory Education (20 1 1 Edition) puts forward for the first time that we should pay attention to cultivating students' geometric intuition in compulsory education, which highlights the position and role of geometric intuition in students' mathematics learning and its teaching value. With the proposal of mathematics curriculum standards to cultivate and develop students' geometric intuition ability, geometric intuition has become a concern in mathematics education, which is far beyond the research significance of geometric graphics itself in content, significance and methods.
Mathematical knowledge is abstract, and what is most needed in learning mathematics is abstract thinking and reasoning ability. Therefore, in the process of thinking, it is helpful to express problems with intuitive graphics and symbols, describe the process of thinking, and show and solidify invisible abstract thinking. First, it helps to make complex and abstract problems concise and vivid. Second, it helps to explore the problem-solving ideas and predict the results. Third, it helps students to understand mathematics intuitively. It can be said that attaching importance to the cultivation of geometric intuitive ability from an early age is of great help to the study of mathematical knowledge in the future.
Second, the teaching methods of cultivating geometric intuitive ability
In order to cultivate students' geometric intuitive ability in primary school mathematics, we should start with intuitive teaching, guide students to learn the meaning of analyzing problems by using graph strategy, and solve simple practical problems, and gradually rise to the point where direct drawing can be transformed into mathematical language and symbolic language. In the process of solving mathematical problems, we should gradually infiltrate the idea of combining numbers with shapes, realize the transformation of numbers and shapes, and the relationship between shapes and numbers, so that the cultivation of geometric intuitive can run through the whole primary school mathematics learning process.
(1) Attach importance to intuitive perception and highlight the teaching of painting strategies.
The problem-solving strategy is mainly used to solve practical problems related to area calculation by drawing intuitive schematic diagrams. When teaching the problem of area calculation, the key is to let students think of the benefits of drawing, draw correctly, analyze with drawing, and experience drawing to solve the problem. First of all, students can be presented with examples of pure words, and when faced with more complicated mathematical problems, students can be guided to think about arranging conditions and problems through drawing. Then, students are encouraged to try sketching, so that their thinking can focus on drawing to express the meaning of the problem, and through the communication between teachers and students, the sketch map can be further improved, so that students can feel that drawing can clearly understand the meaning of the problem. Then analyze the quantitative relationship with the help of schematic diagram, and make clear what to seek first and then what to seek. After solving the problem, we should combine formulas and charts to explain the idea of the problem. Finally, reflect on the whole process of solving the problem, highlight the important role of schematic diagram in solving this math problem, and feel the value of drawing strategy. There are some changes in the topics of "try it" and "think about it" compared with the examples. After solving these problems, students should be guided to think: "Can these problems be solved accurately without drawing? What should I pay attention to when painting? " Deepen students' intuitive experience of the value of applying painting strategies. For example, during the May 13th holiday, I smiled and read a story book. On the first day, I read 65,438+0/3 pages of the book. The ratio of pages read in Grade Two and Grade Three is 4∶3, and Grade Two reads more pages than Grade Three 16. How many pages does this story book have?
The first day, the second day and the third day.
Read 16 pages more than grade three.
(2) Pay attention to the sensible transformation between intuitive graphics and mathematical symbols.
"The meaning of direct ratio", after the students know the meaning of direct ratio, the teaching materials are arranged to understand the direct ratio images, and the intuitive images are used to help students further understand the changing law of direct ratio, so as to make appropriate preparations for future study. In teaching, according to the data in the table of example 1, students are first guided to draw a figure representing the positive proportion by "tracing points". In the process of tracking points, guide students to compare the tracked points with the data in the table, so that students can understand the actual meaning of each point on the image, that is, each point represents a set of corresponding values of distance and time. Then, through observation, students can find that these points are exactly on a straight line, clearly understand the characteristics of the direct ratio image, and further understand the changing law of the simultaneous expansion or contraction of two quantities with the help of intuitive images, and understand the significance of direct ratio. After drawing the image, let the students judge the driving distance and time according to the image, further understand the actual meaning of any point on the image, and initially experience the practical application of the scale image. Through the transformation between the positive proportion image and the positive proportion relationship, we can deepen our understanding of the meaning of positive proportion and lay a preliminary foundation for further learning function knowledge in the future.
For another example, when teaching "solving practical problems with hypothetical strategies", students can be prompted to draw a schematic diagram according to their own assumptions, and analyze the changes in the number of ships after the hypothesis and the reasons for this change according to the schematic diagram drawn, so as to guide students to make timely adjustments according to the changes in the number, calculate the number of each ship, and finally conduct inspection. This problem-solving process involves the transformation of direct view and formula. With the help of intuitive diagrams, students abstract the idea of solving problems: hypothesis-comparison-adjustment-test. In the teaching of cultivating students' geometric intuitive ability, students can be guided to learn to describe and analyze mathematical problems step by step through the transformation of intuitive images and mathematical symbols.
(3) Pay attention to the combination of numbers and shapes.
1. Understand and analyze quantitative relations with the help of line graphs.
Line graph is a tool to help understand the visualization and visualization of quantitative relations. With the help of the relationship between quantity and quantity in the title of line graph, it is not only helpful to clarify the relationship between quantity and quantity, but also to further help students analyze the relationship between quantity and quantity and broaden their thinking of solving problems. In teaching, we can solve the problem of sum and difference by using line graphs and mathematical analysis, realize the superiority of problem-solving strategies through the combination of numbers and shapes, and obtain mathematical thinking methods such as substitution, hypothesis and transformation, so as to enjoy the joy of success in the process of solving problems independently, establish self-confidence and stimulate students' interest in learning. For example, some problems are difficult to understand in words, and it is difficult for solvers to clarify the quantitative relationship in their minds. Example: "The railway from Tianjin to Jinan is 360 kilometers long. An express train leaves from Tianjin, at the same time, a slow train leaves from Jinan. The two cars are driving in opposite directions. Four hours to meet, the average speed of the express train is 68 kilometers per hour, and the average speed of the local train is how many kilometers per hour? "
2. Help the number with form, and make the problem intuitive.
Learning to recognize numbers, starting from the lower grades, learning to add, subtract, multiply and divide, understanding fractions in the middle grades, and understanding negative numbers in the upper grades are all based on specific things or numbers. Students abstract concrete figures, calculations, etc. according to their existing life experience. It realizes the help of graphics to shape, and makes the problem visual and intuitive.
3. Analyze the quantitative relationship with graphs.
The combination of numbers and shapes, with the help of simple illustrations made by figures, symbols and words, can promote the coordinated development of students' thinking in images and abstract thinking, communicate the connection between mathematical knowledge, and highlight the most essential characteristics from the complex quantitative relationship. It is an important feature of primary school mathematics textbooks and a common method to solve problems. In order to better understand the topic, teachers should encourage students to use intuitive graphics around the problem to help them understand and concretize a topic that they can't start with. Under the guidance of the teacher, let students understand the mathematical idea of "combination of numbers and shapes", make full use of the intuition and concreteness of graphics, find out the quantitative relationship and find the breakthrough to solve the problem. Painting is not only to solve problems, but more importantly, it is necessary to build maps and maps to make children's thinking more accurate.
"Try it" in the section "Solving practical problems with transformation strategies" in the sixth grade (Volume II): the numerator of several fractions is 1, and the denominator is 2, 4, 8, 16, 32, 64 respectively. It is necessary to calculate the sum of these scores. In order to inspire students to use the reduction strategy and cultivate their initial geometric intuitive ability, the textbook presents an intuitive view, in which 1 is represented by a big square, each score is represented by the relevant part in the square, and the colored part in the whole figure represents the sum of these addends. At the same time, the textbook also reminds students to "look at the picture and think about what formula can be converted into."
In practical teaching, teaching can be divided into three levels, and students' geometric intuitive ability can be cultivated through the process of solving problems. The first level: guide the transformation of graphics. After putting forward the formula, students will generally use the total score method to calculate. At this time, teachers can encourage students to think of other methods. According to the orthographic diagram, we should first understand the meaning of each part of the orthographic diagram and each score, and then inspire students to convert it into 1- 1/64 for calculation. The second level: expand appropriately and highlight intuition. Teachers expand the formula into1/2+1/4+1/8+…+1/256, and students generally convert the formula into 1- 1/256 by drawing a vertical diagram. At this time, teachers should guide students to realize that the perfect combination of numbers and shapes can help us transform complex formulas into simple formulas for calculation. The third level: think deeply and strengthen intuition. Teachers can inspire students to observe the characteristics of the denominator: the denominator is 2, 2 times, 2 times, 2 times, 2 times ... Divide the square into 2 parts on the intuitive diagram and take 1 part; Divide the remaining graphics into two parts, and take 1 ... among them, the final divided graphics are equal to the remaining graphics. With the help of intuition, as long as the size of the remaining graph is subtracted from the unit "1", it is the result of the required solution. While using transformation strategy to solve problems, students' initial geometric intuition ability is cultivated skillfully with the help of geometric intuition.
(4) Integrating the intuitive teaching of geometry into the teaching.
In teaching, teachers can arrange geometric intuitive teaching according to the teaching content. Using intuition to solve mathematical problems is helpful to clarify the thinking of solving problems and help students find, analyze and solve problems; It is also helpful to prove the correctness of the conclusion. For example, when teaching "average" in the third grade, you can intuitively understand the method of "doing more and making up less" and understand the meaning of average. For example, the average score of Xiao Ming's first three math exams is 93, and the score of the fourth math exam is 3 points higher than the average score of the fourth math exam. What was Xiao Ming's score in the fourth math exam? When organizing teaching, teachers can draw line segments according to the meaning of average, help students learn to solve some complicated average problems, and highlight the role of intuition in solving mathematical problems.
For another example, in the class of "the sum of the internal angles of a triangle", let students practice their own operations through mathematical activities such as measuring, cutting, spelling and folding, find the law, and actively deduce the conclusion that "the sum of the internal angles of a triangle is 180". First, students can think of more ways to measure three angles with a protractor and add them together. The measurement results are all around 180. The teacher then guides the students to pay attention to the straight angle characteristics of 180, and then carries out two activities for the students to experience by themselves.
Operation 1: spell it out.
Operation 2: 10% discount
In this teaching process, students guess the conclusion through measurement, and then verify the conclusion through cutting, spelling and folding. Through the comparative analysis of various senses and independent exploration, it is concluded that not only the knowledge of the sum of the internal angles of the triangle is obtained, but also the students learn how to explore the unknown way of thinking from the known and cultivate their spirit of active exploration.
(5) Attach importance to spatial imagination and cultivate students' creative thinking.
In the teaching of understanding the length, width and height of a rectangle, teachers can guide students to observe the frame of a cuboid first, and then have a group discussion. Then, ask the students to remove one of the edges. Can you work out its size? Continue to remove the edge. Question: What edges should be kept at least for you to guess its size? Students communicate while imagining. Finally, the students leave three edges that intersect at one point. Can you remove one of the edges? Look at the three sides on the left, then imagine and draw the size of this cuboid. Finally, the students all think that the edge can't be removed. At this time, the teacher guides the students to know that these three sides are the length, width and height of a cuboid. In this activity, after observation, operation, imagination and communication, the teacher let the students not only know the length, width and height of the cuboid, but also understand that the size of the cuboid is determined by the length, width and height of the cuboid, so that students can cultivate geometric intuitive ability in the process of spatial thinking. So as to improve students' creative thinking ability.
Three. conclusion
To sum up, in teaching, teachers can arrange geometric intuitive teaching according to the teaching content. Intuitive teaching of geometry is inseparable from reasoning and induction. When using intuition to solve mathematical problems, reasoning helps to clarify the thinking of solving problems, find problems and solve problems; It is also beneficial to prove the correctness of the conclusion, and the intuitive teaching of geometry runs through the whole process of mathematics learning in primary schools.