Regarding the cultivation of students' ability to solve applied problems, the Mathematics Teaching Syllabus for Full-time Primary Schools of Nine-year Compulsory Education (Trial) has not been clearly put forward, but in the teaching purpose, it is said that students should "use what they have learned to solve simple practical problems", which in essence includes cultivating students' ability to solve applied problems, and of course it is still preliminary in the primary school stage. It can be said that cultivating students' ability to solve practical problems is the basic content and important way to enable students to use their own mathematical knowledge to solve simple practical problems. Because application problems reflect the common quantitative relations and various practical problems in the surrounding environment, different mathematical knowledge is needed to solve them. By solving practical problems, students are encouraged to link their mathematics knowledge with real life and some simple scientific and technological knowledge, so that students can not only understand the practical application of mathematics, but also initially cultivate their ability to solve practical problems with mathematical knowledge. In addition, as a tool discipline, mathematics should also take solving practical problems as a key point in teaching. This point is increasingly recognized by mathematics educators in various countries. For example, in the early 1980s, the United States proposed that "solving problems is the focus of school mathematics in the 1980s; In the mathematics curriculum standards for primary and secondary schools drawn up for the 1990s, it is once again emphasized that one of the goals of mathematics education is to make students become "people with the ability to solve mathematical problems" and "people who effectively apply mathematical methods to solve problems". Of course, the significance of cultivating students' ability to solve application problems goes far beyond this. It can also develop students' logical thinking ability and cultivate students' good thinking quality (such as flexibility and creativity) and moral quality. These are the abilities and qualities that citizens with high cultural literacy in modern society must possess.

For a long time, China's primary school mathematics has attached importance to the teaching of applied problems, whether from textbooks or teaching, but there are also many problems, mainly focusing on the teaching of content, ignoring the cultivation of ability. In addition, the selection and arrangement of teaching materials are not reasonable, and the teaching methods are not appropriate, which leads to a large amount of energy consumption and a small collection effect. Therefore, how to improve students' ability to solve application problems and make students' burden lighter is a problem worthy of serious study and discussion.

Second, the reform trend of teaching to solve application problems

In recent years, some mathematics educators and experienced teachers at home and abroad have made some attempts to solve practical problems, especially how to cultivate the ability of practical problems, and gained some useful experience. There are mainly the following development trends.

(A) the content of application questions has a tendency to expand.

The first is to strengthen the combination with practice. It is not limited to the ready-made application problems compiled in textbooks, but collects materials and data from real life and makes some calculations. For example, in the United States, when adding and subtracting, students are required to collect some digital materials by classification, and then make statistics and calculations. When teaching in Britain, giving students a train timetable can not only let them know the departure and arrival time of a certain train, but also make various calculations. Through some practical work, students can know that the concepts and ideas of mathematics exist in people's activities, and can use mathematical knowledge to solve practical problems in life. Some teachers in our country also pay great attention to math problems in real life. For example, a teacher had such a problem: "A workshop cut a piece of iron sheet with a length of 90 decimeters and a width of 60 decimeters into a circular iron sheet with a radius of 10 decimeter. How to make full use of iron sheet? " Results Most students are as follows: 90× 60 ÷ (3.14×102) ≈17; Some students draw pictures (bottom left) and get the answer of 12. Some students also passed the operation (as shown at the bottom right).

The answer is 13. Through discussion, students realize that the last blanking method has high utilization rate, while the first calculation method is divorced from the reality of this iron sheet. Through such questions, students can initially realize that when solving practical problems, they should never mechanically copy the computational knowledge they have learned, but also pay attention to the concrete analysis of practical problems.

Secondly, the problems solved by using mathematical knowledge are not limited to those encountered in real life, but also include some problems that help to cultivate students' ability to use mathematical knowledge to explore. For example, fill in the appropriate numbers in the following χ, so that the sum of the numbers in every two adjacent χ is equal to the number in the middle χ. Ask students not only to write different answers, but also to find out the rules of filling in and whether the numbers at the beginning and end are the same. Because of the wide range of solving problems, many countries do not use the name "application problem", but call it "problem" directly. In Japan, it was originally called "application problem" and now it is called "article problem" to reflect the expansion of its scope.

(2) The difficulty of application problems tends to decrease.

This problem has been solved in most countries. For example, Japan, the United States, Britain and other countries have a wide range of solutions, which are more practical, but less difficult. For example, most articles in Japanese textbooks are calculated in two steps. In a few countries, such as Russia, the original application questions were more difficult and had more steps. Later, the difficulty was reduced or appropriately moved back. Especially after the three-year primary school system was changed to the four-year primary school system, with the extension of the teaching time of arithmetic content, the teaching time of application problems was correspondingly extended, and the difficulty of application problems was further reduced. The "Mathematics Learning Objectives" compiled in Hong Kong stipulates four integer application problems, "the number of operations for each problem shall not exceed two times", and fractions and decimals are limited to solving simple application problems. Many countries or regions take these measures to make the application problem teaching more suitable for the age characteristics of primary school students, which undoubtedly helps to reduce the learning burden of students and better stimulate their interest and enthusiasm in solving application problems. There are many problems in solving practical problems in our country, especially in the entrance examination, which often goes beyond the scope of teaching syllabus and teaching materials, and brings great pressure and burden to teaching. After the implementation of compulsory education in recent years, it is a gratifying phenomenon to emphasize the overall improvement of national quality and begin to pay attention to appropriately reducing the difficulty of application problem teaching.

(3) Pay attention to training students to master the general strategies of solving problems.

This is one of the important conditions to cultivate students' ability to solve practical problems. It is closely related to the teaching purpose and function of application problems. For a long time, whether at home or abroad, it is more or less the ultimate goal of teaching to teach students to solve some kind of application problems in primary school mathematics classroom. From this perspective, teaching students the types of application problems and remembering the conclusions or formulas are the basic knowledge. As a result, students formed the habit of setting formulas, and did not really cultivate their ability to solve problems. In recent years, more and more mathematics educators have realized that the ultimate goal of application problem teaching should be to make students master the general strategies or methods of solving problems through the answers to some representative questions, so as to truly cultivate students' ability to solve simple practical problems. For example, Takeshi Ito of Japan said that in the past, application problems were solved in a formal and mechanical way, and application problems were divided into several categories, and each category had a definite solution. As a result, students can easily solve some problems, but the application problems they have not learned will not be solved. Lidman of the former Soviet Union wrote "Psychological Principles of Mathematics Teaching in Primary and Secondary Schools", which said: "It is one of the basic functions of mathematics teaching to form and develop students' general skills to solve any mathematical problems (including practical problems)". 65438-0988 The Sixth International Conference on Mathematics Education also emphasized teaching students how to use general strategies to solve problems. Some representatives pointed out that the traditional problem-solving teaching method is often that teachers give examples for students to imitate; Teachers not only fail to prepare real problem situations for students, but also fail to teach students general problem-solving strategies, which can neither improve students' problem-solving ability nor improve their enthusiasm for solving problems. Some representatives put forward general strategies to solve mathematical problems: contact, analysis, classification, imagination, selection, planning, prediction, reasoning, testing and evaluation. In the newly drafted "Mathematics Curriculum and Evaluation Criteria for Primary and Secondary Schools" in the United States, the primary criterion of each learning stage is learning and applying problem-solving strategies, but the required level is different, which reflects gradual improvement. At present, most primary school mathematics textbooks in the United States are compiled into the general strategy of solving problems as the formal teaching content. For example, there are the following contents in a fifth grade textbook: drawing, testing, compiling questions with redundant conditions or lacking conditions, solving steps of multi-step questions, estimating numbers, and solving with tables.

In recent years, some mathematics researchers and teachers in China have begun to pay attention to how to teach students the general ideas and methods of solving problems, paying special attention to the quantitative relationship in analyzing problems. Some experimental textbooks also strengthen the training of understanding the meaning of the questions, extracting the conditions of the application questions, supplementing the conditions of the application questions and testing the answers of the application questions. This is of great help to improve students' ability to solve application problems.

(d) Strengthen the equation solution and supplement the arithmetic solution.

Since the modernization of mathematics education in 1960s and 1970s, many countries have added simple equations and sequence equations to solve practical problems in primary mathematics. However, the teaching of solving application problems with equations has great differences in starting period, depth and breadth. For example, the solution of the teaching equation in the former Soviet Union began in the second grade of primary school, and there are two steps to solve the application problem. This involves the relationship between arithmetic solution and equation solution. In recent years, it has gradually become consistent. On the one hand, in many countries or regions, such as Japan, Russia, Hong Kong and so on. The application problem of solving equations in primary school teaching is limited to two or three steps. On the other hand, after the application of arithmetic to solve application problems has a certain foundation, the application problems are gradually solved by equations, so that the two solutions complement each other.

In China, from the early 1980s, primary schools began to add equations to solve practical problems, and there have been different views. More than ten years' practice shows that adding simple equations and sequence equations to solve application problems is really helpful to develop students' abstract thinking, reduce the difficulty of solving application problems, cultivate students' ability to solve problems flexibly, and is conducive to the connection of mathematics in primary and secondary schools. But there are still different methods in actual teaching. Especially in the teaching of fractional division application problems, many teachers regard equation solution as the transition to arithmetic solution, and finally emphasize arithmetic solution and ignore equation solution. This still can't achieve the purpose of reducing the difficulty and reducing the burden on students. In recent years, some reform experiments have emphasized both arithmetic solution and equation solution, which complement each other and have achieved good results. For example, according to the experiment conducted by Shanghai Hongkou Education College in the third issue of "Primary School Mathematics Teachers" (1989), there was no obvious difference between the experimental class and the control class in the first test, and it was followed up to the middle school in the autumn of the following year. Results The results of the experimental class are obviously better than those of the control class, and the students who only learn arithmetic solutions have negative transfer in middle school. According to "Mathematics Teachers in Primary Schools" No.2, 1992, the experimental textbook compiled by the Teaching and Research Section of Wuxi Municipal Education Commission has achieved similar results. The two experimental classes strengthened the connection between arithmetic solution and equation solution, and both of them paid equal attention, while the two control classes still taught problem-solving mode. Results After the unit teaching, there was no significant difference between the experimental class and the control class, but there was a significant difference after the winter vacation. The scores of the experimental class and the control class were 87.3 and 78.7 respectively. However, according to the experiment of a primary school in Beijing, after unit teaching, there are obvious differences between the experimental class and the ordinary class in testing the three-step problem and solving the application problem flexibly.