First, focus on cultivating students' ability to perceive and integrate mathematical information.
The arrangement of "solving problems" in the new textbook (People's Education Edition) runs through all parts of mathematics curriculum. Some are presented in the form of unit arrangement, and some run through the content of "calculation teaching". The presentation forms are also diversified, including tables, pictures, situational dialogues, and combination of pictures and texts. The arrangement of examples is to present a realistic situation with mathematical problems, some give one condition and problem, and the other condition needs to be found in the situation. Some have given mathematical materials symbolized by mathematics, and some situation diagrams also contain a variety of information to solve problems. Students often can't understand these mathematical information well and find mathematical problems. Therefore, cultivating students' ability to perceive information and discover mathematical problems is the premise of problem-solving teaching. The author believes that teachers should pay attention to the following points in their usual teaching:
1. The problem-solving situation diagram in the guidance textbook. For example, the problem-solving example 3 on page 4 of the second volume of senior two is to solve the practical problems related to "two-step calculation", and it is also the first time to use the teaching of adding and subtracting mixed operations to solve problems. The textbook presents a picture of a child watching a puppet show (as shown below), which hides a lot of mathematical information. In teaching, teachers can guide students to observe the scene map and let them say, "What are the friends doing in the picture?" "What mathematical information do you find from the picture, so that students can fully perceive the mathematical information in the situational picture, and then let students think," What mathematical problem do you want us to solve in the picture? "From guiding students to integrate information to discovering mathematical problems. In the classroom, teachers should be good at guiding students to explore different presentation forms in the situation diagram of teaching materials, and guide students to observe, collect information and express it in their own language.
2. Cultivate students' ability to perceive and integrate information in the usual practice. In teaching, teachers should not only guide the situation map in class, but also strengthen the training of students' ability to perceive and integrate information in the problem-solving exercises after class. If it does not appear in the form of pure text during the design exercise, you can design some graphic combinations or situational dialogues to hide information; Or just give information for students to ask questions and other forms to cultivate students' ability to observe and capture information.
Second, attach importance to the analysis of quantitative relations and establish mathematical models.
It is pointed out in the Curriculum Standard of Primary Mathematics that "students should be allowed to experience the process of abstracting quantitative relations from practical problems and using what they have learned to solve problems". The author believes that in the process of problem-solving teaching, teachers should pay attention to strengthening students' analysis of quantitative relations and let students establish mathematical models from the perspective of analyzing quantitative relations.
[Case] The second volume of the first day of the first day, "Finding one number is more than the other" (as shown in the figure below).
Teacher: Please look at this picture and tell me what you know.
Health: I know Xiaoxue 12 flowers, Xiao Lei 8 flowers and Xiaohua 9 flowers. (The courseware shows Xiaoxue and the flowers in Xiao Lei)
Teacher: What math questions do you ask?
Health 1: How many flowers did Xiaoxue and Xiao Lei receive?
Health 2: How many more flowers does Xiaoxue have than Xiao Lei?
Teacher: Who can answer the question of 1
Health: 12+8=20 (flowers)
Teacher: Tell me what you think.
Health: Xiaoxue and Xiao Lei sent flowers together.
Teacher: You speak very well. Can you answer the question raised by student 2? Please put a disc instead of a red flower first, and then answer continuously.
(Student hands-on operation)
Teacher: How do you calculate it?
Health: 12-8=4
Teacher: Is this classmate's algorithm correct?
Health: Yes.
Teacher: Please put it on the blackboard and say what 12, 8 and 4 in the formula mean respectively.
Sheng (pointing to the picture) 12 indicates how many flowers Xiaoxue has, 8 indicates how many flowers there are, and 4 indicates that Xiaoxue has more flowers.
Teacher: It's amazing that you can explain the meaning of each number in the formula with operation! Who can say why subtraction is used?
Health: Xiaoxue has 12 flowers and Xiao Lei has 8 flowers. Light snow minus Xiao Lei means more light snow than Xiao Lei.
……
In this teaching process, teachers should not only guide students to understand the meaning of each number in the formula in specific operations, but also firmly grasp the important question of "why subtraction should be used". In the process of thinking about this problem, students will naturally think of "the number of flowers in Xiaoxue-the number of flowers in Xiao Lei = the number of flowers in Xiaoxue more than Xiao Lei". This process is actually a process of analyzing quantitative relations. In the process of this analysis, the mathematical model of "large number-decimal number = difference" is actively constructed in students' minds by combining specific operations. Although this model is not abstract, students have formed representations in the process of analysis and operation. It should be emphasized that teachers should guide students to construct from the context of the topic itself, rather than summarizing the abstract quantitative relationship.
Many quantitative relations are both mathematical and life-oriented. For example, when we pay for something, we know that "the unit price multiplied by the quantity is the total price", and we must know the time and speed of the trip. This quantitative relationship is familiar to students in their daily life. Students can make abstract generalizations on the basis of full experience and directly apply them to solving problems. Some mathematical models can be constructed with the help of students' existing mathematical knowledge. For example, Example 3 on page 29 of the second volume of Grade Two is a simple practical problem related to "average score". Although it is the first time to solve problems by division, students already have the knowledge and experience of division, and it is easy to establish that "how many people are needed in each group" and "how many groups can be divided into" are all calculated by division.
Third, pay attention to the guidance of analytical methods and master problem-solving strategies.
Educational psychologists believe that the process of problem solving is to feel the existence of the problem, sort out all aspects of the problem and form various solutions. Students must have certain problem-solving strategies when choosing solutions, so teachers should guide students to learn scientific and reasonable analysis methods and master some problem-solving strategies.
In the curriculum reform, teachers often wonder: Can the traditional analysis method of applied problems still be used in problem-solving teaching? The author believes that the curriculum reform is not to abandon the traditional teaching methods, but to inherit the essence of traditional teaching. Even the traditional analytical method should be paid attention to in problem-solving teaching, just how to use it well. For example, the method of drawing line segments to guide students to solve problems appeared in the second volume of senior two (as shown on the right).
In teaching, teachers can ask students to put a disk on the table first, then simulate the model of line segment diagram with this intuitive method, and then abstract the line segment diagram. Only in teaching, teachers should not spend a lot of time on how to draw a line segment diagram, but should focus on guiding students to analyze the quantitative relationship according to the drawn line segment diagram and find a solution.
In addition to drawing line segments, other traditional methods can be used to guide problem solving. For example, the list method is mentioned in the solution of the first volume of the fourth grade of the Soviet Education Edition. The author thinks this analysis method is simple and clear, which is convenient for students to observe and compare. It can also be used to solve Example 2 on Page 4 of Volume II of Grade 4 of People's Education Press:
Example: Frozen received 987 people in three days. According to this calculation, how many people are expected to receive in six days? In teaching, teachers can help students make a table to make the quantitative relationship in the questions clearer.
From the table, students can clearly see that six days is twice as many as three days, so the number of people who receive six days is twice as many as those who receive three days. Divide six by three and multiply it by 987 to get it. At the beginning of teaching, the teacher instructs the students to list, and then the students can make their own lists slowly.
Of course, teachers can also use other scientific and reasonable analytical methods to guide students to solve problems, such as enumeration, induction, hypothesis strategy, transformation strategy, analysis, synthesis and so on. In my opinion, students' learning methods are more important than solving several problems.
Fourth, pay attention to the diversification of problem-solving methods and improve the level of thinking.
Because each student has different knowledge, experience and life accumulation, everyone has their own understanding of the problem in the process of solving problems, and on this basis, they form their own problem-solving strategies. The diversified experience of problem-solving strategies is an effective way to stimulate the flexibility and broadness of students' thinking, develop their thinking ability and cultivate their innovative spirit. Therefore, teachers should give students enough time and space to think independently and encourage students to solve problems from different angles and in different ways. For example, the problem of planting trees in the second volume of the fourth grade of the People's Education Edition, Example 3 "The outermost layer of the Go board can hold 19 pieces on each side. How many pieces can you put on the outermost layer? " On the basis of guiding students to communicate and discuss, when teachers ask students to report their different ideas, students may have different ways to solve problems: ① direct counting; ② According to the method of upper and lower parts 19 and left and right parts 17, that is,19× 2+17× 2 = 72; ③ Press each side 18, that is,18× 4 = 72; ④ Count 19 pieces on each side and then subtract 4 pieces from the board feet, that is, 19×4-4=72:⑤ Subtract the outermost pieces from the total pieces placed on the whole board to get the outermost pieces, that is, 19× 17× 65438. Teachers should praise and encourage students' different methods to protect students' enthusiasm for solving problems independently. At the same time, through the comparison of various algorithms, we should guide students to learn and absorb better methods, ideas and strategies to solve problems and gradually improve their thinking level.
Five, pay attention to contact with real life, cultivate application consciousness.
It is an important goal of problem-solving teaching to use mathematical knowledge to solve problems in life and cultivate students' application consciousness. In teaching, on the one hand, teachers should pay attention to creating life situations to attract students' interest and arouse their life experience; On the other hand, using mathematics knowledge to solve problems in life, communicating the relationship between mathematics and life, realizing the application value of mathematics in life, enhancing students' awareness of applying mathematics, and constantly improving students' ability to solve problems. For example, teachers can design common practical problems related to tree planting after learning and practicing, such as sawing wood, ringing bells and setting up stations. These problems are closely related to life, which makes students feel that there is mathematics everywhere in life.
Problem-solving teaching is a new subject of curriculum reform. In teaching practice, we must implement the requirements of the new curriculum standards and combine the students' cognitive laws and actual situation to carry out teaching, in order to receive effective results.