First, the basic concept of solving problems.
(1) Instrumentality and application of mathematics.
(2) The requirements of informationization, digitalization and market economy.
The instrumentality and application of mathematics are accompanied by the emergence and development of mathematics. Since the mid-20th century, with the rapid development of information technology and market economy, mathematics and its application have been greatly developed and penetrated into various scientific fields. Students must learn mathematics and its application to adapt to the development of society.
Second, the teaching objectives of solving problems
The standard clearly puts forward the following four objectives for Scheme 2:
1. Initially learn to ask and understand questions from the perspective of mathematics, and can comprehensively use the knowledge and skills learned to solve problems and cultivate application awareness.
2. Form some basic problem-solving strategies, experience the diversity of problem-solving strategies, and cultivate practical ability and innovative spirit.
3. Learn to cooperate with others and communicate the process and results of thinking with others.
4. Initially form the consciousness of evaluation and reflection.
Third, the comparison between solving problems and applying problems
1. The difference between solving problems and applying problems.
The concept of application problem: according to the practical problems in daily life and production, some known and unknown quantities and their relationships are described by words, languages and graphics, and the unknowns in mathematical problems are solved by four operations, which is called application problem.
The traditional teaching subject of application problems is closed, which provides students with well-sorted known conditions and problems, deprives students of the opportunity to collect and sort out information from real life and form mathematical problems, and makes students feel that they are only doing problems, not solving practical problems; The presentation form is monotonous, and almost all of them are expressed in words. When analyzing the quantitative relationship, the thinking of adults is used instead of that of students, which makes the thinking narrow and single. When solving problems, we can't make full use of students' existing experience, just arrange a lot of imitation exercises and strengthen the problem-solving ideas in textbooks with repeated exercises.
Now the theme of solving problems is closer to the reality of students' lives, and real life scenes are presented in the form of pictures, dialogues and tables. Some topics are still open. Students are required to collect and integrate information from real scenes, put forward their own mathematical problems, and then use their own strategies to solve these problems, which is conducive to cultivating students' ability to solve practical problems.
2. Similarities and differences between problem solving and application problems in content and structure.
In the curriculum standards, problem solving is not an independent content field, but is put forward independently in the teaching objectives. Therefore, the experimental textbook does not arrange the problem-solving into units independently, but studies the application of various knowledge.
In general textbooks, application questions are arranged as an independent content.
Application problem teaching classifies application problems, focuses on one kind of problems, and emphasizes speed and skills; Problem-solving emphasizes the concrete analysis of specific problems, in other words, how to use the learned knowledge to solve problems in new situations, making them more challenging, and one problem may follow.
(1) Teaching content arrangement
The arrangement of the teaching content of application problems is independent, and application problems are put forward independently in the teaching objectives. Because the complicated classification system of applied problems arranges all kinds of questions into corresponding quantitative relations, the problem-solving process of students becomes a simplified problem-solving process.
The arrangement of problem-solving teaching content is decentralized. Rich in content, informative, diverse in questions and unique in answers. Students are required to have independent opinions and creativity, so that students can develop their mathematical thinking ability and learn mathematical ideas and methods.
(2) Presentation of the problem
Application questions are presented in the form of words, and the form is relatively simple. It is not attractive to students, especially junior students, who feel bored and lack interest.
And solving problems is open. There are mainly pure graphs, semi-words, semi-graphs and pure words. Information content pays attention to all aspects of daily life, is closer to students' reality, and information tends to be diversified and open. Let students actively solve problems through exploration and practice, which can better stimulate students' interest and enthusiasm for exploration.
(3) students' training objectives
In the teaching of practical problems, comprehensive method and analytical method are often used to help students analyze. Teaching center is the only operational relationship between quantitative relations, aiming at finding "problem solving methods".
However, problem-solving teaching has no ready-made types and solutions, and students need to explore and practice in the form of individuals or groups, which is novel and challenging. Problem-solving teaching is conducive to cultivating students' innovative spirit, practical ability and cooperative spirit.
3. The experimental teaching material has made a new breakthrough in the arrangement of "problem solving":
First, it is closely integrated with computing teaching. For example, the first volume of the third grade, multiplication formula, first presents a situation map closely related to students' lives. The theme picture shows three children drawing. Each of them has a box of colored pens. This raises a mathematical question: given that a box of colored pens is 12, how many are there in three boxes of colored pens? The elf asked: How to count a * * * and how many branches are there?
In teaching, teachers let students estimate how many branches there are, then let each student try independently, then exchange their own algorithms in groups, and finally report the representative algorithms of each group to the whole class and discuss the solutions to the problems. This design allows students to feel and understand the practical significance of multiplication through activities. Its advantages are:
(1), which can arouse students' enthusiasm in learning calculation. Because learning calculation not only learned mathematical knowledge, but also solved practical problems in life.
(2) It is beneficial to explore the calculation method. Because the real situation is familiar to students, it can awaken their life experience and activate their existing knowledge.
(3) It can cultivate students' consciousness of applying mathematics. Because students often study mathematics in real life, they can feel that there is a lot of mathematical information in real life, and mathematics has a wide range of applications in real life.
Second, it is closely integrated with the development of mathematical thinking. The process for students to solve simple practical problems is also the process for them to use life experience to explain relevant mathematical information and describe simple phenomena in real life with concrete figures. For example, the textbook "Solving Problems", the first volume of the fifth grade, presents a picture of professional dairy farmers, reflecting the milk production of 220 a week. 5 kg is not only related to three cows, but also related to the time of milk production in one week (7 days), which helps students feel the extensive connection between mathematics and life. Second, it is entirely up to students to choose the calculation method of solving problems. The textbook only encourages students to think independently and solve problems actively through the dialogue between two students. Encourage students to think in many directions and experience the diversity of problem-solving strategies, but we can't ask every student to master a variety of problem-solving methods, which will cause unnecessary burden. And it is necessary to adopt a semi-supporting and semi-releasing way, so that students can actively participate in the process of solving problems. This helps students learn to analyze the quantitative relationship from the perspective of quantity and describe the thinking of solving problems. Third, some known quantities in the example are not directly stated, and the condition of "7 days" in the question is hidden by the word "last week", so it is difficult for students to analyze the meaning of the question, which is realistic, interesting and challenging for students. This helps students to choose the correct calculation method to solve problems and develop mathematical thinking by analyzing the relationship between quantity and quantity.
The third is to combine with practical activities. For example, the practical activity "Mathematics Wide Angle" on page p 1 12 in the first volume of Grade Three, according to students' age characteristics and students' actual life, designed some situations of mathematical practical activities, paid attention to infiltrating students' mathematical ideas of permutation and combination, and initially cultivated students' consciousness of thinking about problems in an orderly and comprehensive way, which is also the requirement put forward by the standard: "In the process of solving problems, students can be simple." . This part of the content is more active and operational, and students can practice and study in groups. Students can further experience the application process of mathematical knowledge in activities, find out the number of simple things arranged and combined by listing and connecting according to practical problems, and feel that some of them are related to order, while others have nothing to do with order. In teaching, we should try our best to avoid arranging and combining terms, and there is no need to explain them to students. To experience the fun of learning and using mathematics. The purpose of this arrangement is to enable students to comprehensively apply what they have learned and flexibly solve some practical problems, so that students can initially gain some experience in mathematics activities, understand the simple application of mathematics in daily life, initially learn to cooperate and communicate with others, and gain positive mathematics learning emotions.
Fourthly, in order to better understand the problem-solving in primary school mathematics, the following points are put forward:
Everything can be replaced, only thinking can't.
Error is a problem, and solving this problem is progress.
Everyone will encounter all kinds of problems in life, and solving problems has become an indispensable part of life.
"China measures the success of education by teaching students with problems, and the United States measures the success of education by teaching students with problems." It can be seen that under the guidance of different educational concepts, the cultivation of students' "problem consciousness" has a completely different position in teaching.
The previous application problem teaching and today's problem-solving teaching are not only different in name. Significant changes have taken place from the presentation of teaching materials to teaching objectives, from teachers' teaching methods to students' learning methods.
After the combination of application problem teaching and calculation teaching, the teaching content is more abundant, and the diversity of problem-solving strategies and the organic integration of calculation and estimation also pose new challenges to teaching.
For traditional things, we should learn to sublate and choose.
Inquiry learning needs teachers to guide students to learn to think.
Students are used to accepting ready-made knowledge, finding standard answers, practicing hard and getting high marks. However, it ignores the rational use of learning methods such as independent inquiry and group cooperation, which inhibits the development of its own problem-solving ability.
There are many mathematical problems, ideas and methods in real life. We should make use of these materials in our teaching. Only when problems are closely combined with students' real life can mathematics be lively, energetic and valuable, and students' interest in learning and solving mathematical problems can be stimulated.
Changing the name of "application problem" to "problem solving" is mainly to make primary school mathematics educators not be bound by the original application problem, and change it into a new expression, which can better reflect the goal that primary school mathematics curriculum should pursue, so they should solve the problem. It is usually called problem solving internationally, which means solving problems. According to our country's thinking habit or expression habit, we define it as solving problems.
5. How to teach primary school math problem solving?
How to teach primary school mathematics to solve problems has become a problem worthy of discussion. With the development of social informatization, the application of mathematics is also deepening and expanding. We should pay more attention to learning mathematics and solving problems in real situations. The teaching strategies to solve the problems are designed as follows:
1, create situations and collect information
When teachers start classes, they can create vivid and interesting teaching situations with the help of theme maps or teaching courseware, and connect abstract mathematics knowledge with real life. The information in the theme map or teaching courseware provides clues for students' thinking in a certain sense. After the students report, the teacher instructs the students to sort out the collected information and find out the problems that need to be solved. Observation and report can also provide a cognitive basis for solving problems, stimulate students' desire for knowledge, glow students' subjective consciousness, and create an atmosphere for students to explore and solve problems independently.
2, group cooperation, explore the problem
When students are clear about the problems to be solved, they should leave enough space and time for students, so that each student can use the existing knowledge and experience to independently find ways, methods and strategies to solve the problems, or they can discuss and communicate with each other in groups to form a preliminary plan. In this process, teachers should participate in the group to obtain information in time, and conduct appropriate guidance and norms.
3. Exchange evaluation and solve problems.
Communication evaluation is a key link in the organic combination of teacher-led and student-centered. The main duty of teachers is to organize students to communicate effectively in mathematics, activate students' thinking and broaden their thinking. After clearing the train of thought, let the students choose their own algorithm. When students have their own ideas, let them further summarize and sort out the algorithm through group communication. Finally, through collective communication, the algorithm is clear.
4. Consolidate methods and expand thinking.
Students have mastered the method, but also continue to practice and deepen their understanding in application. In this link, arrange some basic questions for students to answer with the knowledge they have mastered, so as to consolidate the application. Some expanding exercises are also arranged to enable students to use their existing knowledge flexibly and solve problems from different angles, thus expanding their thinking and cultivating their application consciousness.
Six, some suggestions in the process of implementing problem-solving teaching:
1. Attach importance to students' information collection.
Judging from the steps to solve the problem, collecting information is the first step to solve the problem. In the lower grades, questions are presented in the form of pictures, tables and dialogues. With the increase of grade, the number of pure text questions gradually increases. In practical teaching, the most effective method for middle and lower grade students is to guide students to learn to look at pictures and collect necessary information from them. Teachers should pay attention to three situations: first, the information in the question is scattered, and students should be instructed to look at the pictures many times and try to find the information they can know; Second, when the information in the question is hidden, it is easy to be ignored. At this time, students should be guided to look at the pictures carefully. Third, there is a large amount of information, and students should be guided to collect relevant information according to the questions.
2. Guide students to ask questions
The ability to ask questions is more important than solving them. The requirements of asking questions and solving problems are different, but one key point is that they have something in common, that is, they should be able to combine the relevant information provided in the questions. Only by recognizing the relationship between information can we put forward reasonable mathematical problems. However, in actual teaching, teachers lack such awareness. Sometimes, teachers have the same consciousness and give students opportunities, but students can't bring them up or ask the same questions. Therefore, we should create an atmosphere for students to ask questions boldly and guide them to learn to ask questions. Encouraging students to ask questions is actually to awaken students' impulse to explore and cultivate their courage to question.
3. Cultivate students' cooperation and communication.
Cooperation and communication is an important way for students to learn mathematics. In the process of solving problems, teachers should let students have the need of cooperation and communication. Teachers should organize students to cooperate and communicate according to the actual situation of students' problem solving, when some students' problem solving ideas are unclear, or when students put forward different problem solving methods, especially innovative methods. When students cooperate and communicate, teachers should pay attention to students with learning difficulties, on the one hand, encourage them to actively communicate with their peers and express their ideas; On the other hand, let other students take the initiative to care about them and help them explore ways to solve problems. So as to deepen the understanding of the problem itself and the method of solving it, and contribute to the formation of problem-solving strategies.
4. Pay attention to students' evaluation and reflection.
In the teaching process, teachers should not only properly evaluate students' ideas and pay attention to motivating students, but also organize positive and effective evaluations among groups, students and teachers and students. By evaluating others' problem-solving process, let students form their own clear views on the problem. At the same time, teachers should also guide students to review and reflect in the process of solving problems. On the one hand, in the process of solving problems, I have a correct analysis of my problem-solving activities. When encountering difficulties, you can face them squarely and not give up easily; In a smooth situation, can maintain a cautious attitude, good at finding problems that they ignore. On the other hand, after the process of solving the problem, we should also review the process of analyzing and thinking about the problem completely, and reflect on whether our results are reasonable or not, and whether there are other ways to solve the problem. So as to accumulate experience in solving problems and gradually internalize them into mature problem-solving strategies.
In teaching, teachers should first let students solve basic and routine math problems, then encourage students to solve challenging and unconventional problems such as open problems, and guide students to explore solutions in the teaching process. Problem-solving teaching is an important content and goal of new curriculum mathematics teaching. "A good beginning is half the battle" Let's start from the lower grades, pay attention to the cultivation of students' problem-solving ability, let students learn mathematics well in solving problems, and finally achieve the goal of improving students' problem-solving ability and knowledge and skills.
Cultivating students' problem-solving ability is an important direction of primary school mathematics teaching reform experiment and the basic requirement of new curriculum standards. It is a brand-new teaching mode. Therefore, we must seriously study the "problem-solving" strategy in practical teaching.
The problem is the heart of mathematics. Teachers should study the relevant strategies of "problem solving" in teaching, cultivate students' autonomy, creativity and problem-solving ability through "problem solving", promote students' all-round development, provide students with more opportunities to show their talents, and cultivate students' innovative consciousness and spirit.
Cultivating students' "problem solving" ability is the basic requirement of the new curriculum standard, and it is also an important direction of primary school mathematics teaching reform experiment. In the new curriculum, "problem-centered learning" is a new classroom teaching model. In the past, teachers thought that doing problems meant solving problems, while the new curriculum emphasized that students should be guided to participate in exploration and thinking by designing real, complex and challenging open problem situations, so that students can learn by solving a series of problems. The process of "solving problems" is a kind of "rediscovery" and "re-creation" for students. Therefore, in practical teaching, teachers should seriously study the strategy of "solving problems" and cultivate students' innovative spirit.