The Mathematics Curriculum Standard (Experimental Draft) (hereinafter referred to as the Standard) points out: "Students will initially learn to ask questions from the perspective of mathematics, understand problems, comprehensively apply the knowledge and skills they have learned to solve problems, and develop their sense of application. Form some basic strategies to solve problems, experience the diversity of problem solving, and cultivate practical ability and innovative spirit. " With the implementation of the new curriculum standards, we are more and more aware that teachers are not only aiming at imparting knowledge, but also good at arousing students' awareness of problems, deepening the depth of problems and exploring ways to solve them.
Since the 1990s, many scholars and teachers in China have begun to pay attention to the research on problem solving, but most of these studies are about how to cultivate students' problem solving ability in the middle and senior grades of primary schools or the whole primary school stage, and there is little research on students' problem solving ability in the first period. At the same time, we also found that some teachers did not adapt to the arrangement characteristics of the new textbooks, and weakened the problem-solving teaching in teaching, focusing on the calculation teaching, especially in the first phase of mathematics teaching. We also find that the students in the first stage are not strong in analyzing and solving practical problems, so their follow-up study will be very difficult. Therefore, in actual teaching, the teaching goal of solving problems for students in the standard has not really been realized.
In the new round of curriculum reform practice, our school has accumulated some experience by studying students' problem-solving ability, and also trained a group of backbone teachers who are interested in "solving problems" and have certain professional mathematics knowledge. We choose to study the "problem-solving" ability of the students in the first phase. Guided by the spirit of Standards, based on promoting students' all-round, sustained and harmonious development, we will start with the basic materials and application of "problem-solving" in the first phase of primary school mathematics textbook, emphasizing on improving students' ability to analyze and solve problems from their existing life experience.
The purpose of this topic is to find a way to cultivate students' problem awareness and problem-solving ability from an early age through the combination of mathematics teaching and learning in the first stage of primary school, so that students can learn to analyze and solve problems with mathematical ideas and methods and lay the foundation for their subsequent study. It can also promote teachers to explore educational ways to improve students' application consciousness and comprehensive quality.
Second, the definition of the concept
(1) Core concept: the first learning period, "problem solving" and "problem solving" ability.
(2) the first phase: primary school 1~3 grades. "Problem solving" is a learning activity in which students actively analyze, explore and propose solutions according to certain problem situations with the help of teachers. "Problem-solving" ability refers to finding and discovering problem materials and cultivating the ability to analyze and solve problems.
Third, the theoretical basis
(A) "Standards" pointed out: "Teaching materials provide basic clues for students' learning activities, and are important resources for achieving curriculum objectives and implementing teaching. "Therefore, we believe that teachers should fully tap the basic materials of' problem solving' in textbooks and creatively use these materials in classroom teaching to serve students' learning activities.
(2) The results of psychological research show that the process of solving problems is divided into four stages: asking questions, clarifying questions, putting forward hypotheses and testing hypotheses. Based on this, we try to study the basic procedure of "problem solving" in classroom teaching, and take action to improve students' problem solving ability, so that students can start from their existing life experience, carry out observation, reasoning, communication and other activities, explore some "problem solving" strategies, learn to think about problems from the perspective of mathematics, and then stimulate students' interest in mathematics.
(3) Explanation of cognitive principle: The process of students' constructing mathematical cognitive structure is a process in which new mathematical knowledge interacts with students' original cognitive structure and expands the original cognitive structure or establishes a new mathematical cognitive structure. Therefore, by exploring the factors affecting problem solving, students' visual, auditory and kinesthetic organs can be effectively mobilized to participate in the teaching process, so that students are willing and good at actively thinking, proposing and solving problems.
Fourth, the research objectives
By studying the basic materials and application of the first phase of "solving problems" in mathematics textbooks, combined with real life, we can improve students' problem-solving ability in the first phase of learning, explore the factors that affect problem-solving, cultivate students' initial problem-solving ability, guide students to apply their learned mathematics knowledge to life, solve mathematical problems around them, understand the role of mathematics in real life, and realize the importance of learning mathematics. At the same time, help teachers to further study and master teaching materials, and combine them with students' real life. Explore the methods and ways of "problem solving" in the first phase of primary school mathematics teaching, and form a certain teaching strategy of "problem solving". Form a good scientific research atmosphere in the school, thus improving the teaching quality of the school.
Verb (abbreviation of verb) research method and object
(1) research methods:
1. Literature method: Collect information about "problem solving", learn knowledge about "problem solving", learn from the relevant research results of "problem solving", understand its shortcomings, constantly update educational concepts, and form the basic concept of subject research.
2. Action research method: combined with classroom teaching practice, the first phase improves students' problem-solving ability, explores problem-solving strategies, stimulates students' interest in active inquiry learning, promotes the formation of students' problem-solving consciousness, and conducts confirmatory research in practice.
3. Experience summary method: in the process of practice, experience and practice are continuously refined, and through reflection and revision, practical experience with certain promotion value is formed.
(2) Research object: Students from Grade One, Grade Two and Grade Three in our school.
VI. Main contents and achievements of the research.
(a) the main content of this topic (sub-topic)
Sub-topic 1: Research on the basic material of "problem solving" and its application in the mathematics textbook of primary school last semester
By sorting out and studying the basic materials of "problem solving" in the first issue of mathematics textbooks, the most basic teaching resources are provided for teachers to carry out "problem solving" teaching.
Subtitle 2: The Action Research on Cultivating Students' Problem-solving Strategies in the First Stage
According to the spirit of the Standard, based on whether teachers can respect children's psychological development process in the teaching process, this paper studies how to make good use of these materials reasonably and creatively in classroom teaching, how to solve problems in classroom teaching, so that students can learn to ask questions, understand problems from a mathematical perspective, and form some basic strategies to solve problems.
Subtitle 3: Explore the factors that affect problem solving.
The process of students' constructing mathematical cognitive structure is a process in which new mathematical knowledge interacts with students' original cognitive structure and expands the original cognitive structure or establishes a new mathematical cognitive structure. By exploring the factors that affect problem solving, students' visual, auditory and kinesthetic organs are effectively mobilized to participate in the teaching process, so that students are willing and good at thinking, asking questions and solving problems on their own initiative.
Every semester, the research group can seriously carry out activities around the plan, including practical class research, lesson analysis, open teaching, observation of excellent classes and so on. After more than three years of practice, exploration, revision, re-practice and re-study, our school has made some progress in the research of cultivating students' problem-solving ability.
Understand "What is a problem?"
In order to study our subject, we think we should first make clear the meaning of "problem". Actually, "problem" and "exercise" are not the same thing. The problem we often talk about at ordinary times is actually "sports". The "problem" in solving problems should have the following characteristics: (1) goal. The problem arises from people's needs. Without demand, there would be no problem. The manifestations of this demand are often "doubt" and "want to know" in people's thinking. In other words, to organize teaching with the idea of "problem solving", it is necessary to stimulate students' domestic demand and make them have doubts and eager desire to solve problems. (2) obstacles. That is, the question should be challenging. If a student blurts out the question, it can't be called a "problem" in solving the problem. (3) Relativity. That is, the problem is not absolute, it is closely related to the problem solver. It may be a "problem" for some students, but it is not a "problem" for another student. The "problem" varies from person to person. In our opinion, the problems we design should be: meeting some needs of students, being conducive to the formation of some knowledge, being conducive to the consolidation and evaluation of some knowledge, and being conducive to the training of some ability or skill.
(2) sorting out the basic materials of "solving problems" in textbooks.
On the basis of studying the new curriculum standards carefully, fully tap the basic materials of "problem solving" in the teaching materials and carefully sort them out. Basic information mainly includes the following aspects:
First, there is no separate problem-solving unit in the textbook. As far as the content arrangement of solving practical problems in the field of number and algebra is concerned, some arrangements have been made in the primary stage of calculation teaching, which not only solves practical problems, but also enables students to understand the significance of four calculations in combination with reality. Some are taught separately as examples. The content of solving practical problems mostly starts from the life situation or fairy tale world that students are familiar with, selects events that are directly related to students and can be recognized and understood as basic materials, emphasizes the connection between mathematics and real life, and emphasizes solving practical problems with mathematical knowledge. In this way, the original boring application problems become more vivid, interesting and humanized, which not only mobilizes students' enthusiasm for learning, but also helps students feel the close relationship between mathematics and daily life and appreciate the value of mathematics learning.
Solving practical problems in teaching materials has completely changed the presentation mode of text narration, and the presentation mode of problems is rich and colorful. Almost all questions have scenes, such as physical photos or pictures, cartoons or dialogues. This presentation is very consistent with the interests and cognitive characteristics of students of this age group, and students are willing to solve these problems. In fact, the illustrated and intuitive presentation of the teaching materials is also very enlightening, which is conducive to inspiring students' thinking.
The textbook arranges a certain number of topics in examples, "think and do" and exercises, many of which require students to collect information from situations, sort out and combine information, and propose and solve mathematical problems. Some topics are open, which provides opportunities for cultivating students' mathematical thinking and improving their ability to solve practical problems.
Second, solving practical problems is arranged in all content areas of the textbook. In the field of "practice and comprehensive application", the textbook also arranges some activities to solve practical problems by comprehensively applying the learned knowledge. This can not only stimulate students' interest in learning, make them realize that mathematics has a wide range of applications in the real world, but also help teachers change teaching methods and let students experience a vivid, active and personalized learning process.
Of course, there are also some problems in the textbook that need our serious study. For example, for some practical problems presented in the form of pictures, students must first understand the pictures and correctly collect and sort out the information in the pictures in order to solve the problems. This is very difficult for some students, especially one condition is provided in written form, and the other condition is obtained by collecting information in pictures, which makes it more difficult for students to learn. For another example, the curriculum standard does not explicitly require students to master the common quantitative relations in practical problems, and textbooks often require students to solve problems according to existing knowledge and life experience. To some extent, this weakens students' understanding and application of comparison, analysis and synthesis, which is not conducive to cultivating students' problem-solving ability.
(C) Classroom teaching strategy research and action attempts to penetrate problem-solving strategies.
For the students in the first phase, although there is no independent unit system teaching strategy in the textbook, students have accumulated some knowledge about strategies in their daily life and accumulated previous experience in solving problems, but they have not summarized and improved them. Through research and attempt, in the first stage, we mainly infiltrated various strategies such as collecting information, trying, drawing and listing information to solve problems for students.
1, information collection strategy
Many problems solved by junior students are presented through pictures and dialogues. Therefore, teachers should first cultivate students' strategies of collecting information. After presenting the situation map, students should be instructed to make clear the order of looking at the map and learn to collect corresponding information from specific pictures or dialogues. After continuous exploration, we pay attention to guiding students to adopt the method of "12③ reading questions", in which "12③" is the condition and "③" is the problem. Whether it is the practical problem of drawing, the practical problem of combining pictures and texts, or the practical problem of pure words, students should mark the questions with "① ② ③" after reading them preliminarily to improve their ability of collecting information.
2. Try this strategy.
Trying strategy is a process of "trial and error" of various methods. Different students have different levels of mathematics, so we should fully respect each student's personality differences, allow students to learn mathematics in different ways, and let students use trial strategies to solve problems. For example: "Each boat can take up to eight people. How many boats do 50 students need to rent? " The common practice is to guide students to calculate 50 ÷ 8 = 6 (boats) ... 2 (people), so it is necessary to rent 7 boats. However, this kind of teaching lacks attempts and explorations of various problem-solving strategies. So, you can let students try to explore:
(1) 8× 6 = 48 (people), six boats can accommodate 48 people, and two more people need to rent seven boats.
(2) Eight plus eight places, * * * plus six times for two people, you need to rent seven boats.
(3) Remove 8 people from 50 in turn. After the 6 th trip, there are still 2 people who need to rent 7 boats.
(4) 6× 8 = 48 (people), 8× 8 = 64 (people), six boats can only accommodate 48 people, which is not enough, and eight boats are too many, so 7×8 = 56 (people), so it is more appropriate to rent seven boats.
3. The strategy of painting.
Due to the limitation of age, students in the lower grades of primary school can adopt the strategy of drawing assistance, so that students can spread their thoughts, enlighten their thinking and stimulate their interest in learning mathematics, thus helping them find the key to solving problems.
For example, in the "Number Recognition" unit of Senior One, students are required to count and write the number 1 1 ~ 20. Students can circle the "ten" first, and then add up the rest, which can ensure that the numbers written are correct and help students understand the relationship between "ten" and "one" vividly.
4. List the strategies of the results.
Sometimes, listing the results one by one in the process of solving problems can often get twice the result with half the effort in characterizing problems and finding solutions to them. For another example, there is a question in the math exercise book for Senior Two (Volume II): What three numbers can be spelled with two, five and eight cards? Students either don't know how to think about this problem and can't write; Or random combination, although you can find the answer, but there is no order, so there will be omissions. So I took the opportunity to guide the students to find out the arrangement method: first, determine the number in the hundred places, and then consider how to choose the number in the ten places and one place, so that the students can arrange their answers in an orderly manner to prevent omissions.
Through research, we think that we can start from the following aspects and infiltrate problem-solving strategies in junior high school mathematics classroom.
1. Go deep into the situation, get information and find problems.
Creating situations is the beginning of every classroom teaching, introducing the mathematical knowledge learned from real life. Mathematics comes from life. In classroom teaching, teachers should be good at digging up mathematical materials in life and introducing mathematical knowledge from students' real life, so that students can feel that mathematical knowledge is around and there are mathematical problems everywhere in their lives.
For junior students, the key to obtaining information is to learn to read questions. Problem-solving teaching should start with students of this age learning to read questions. They are a blank sheet of paper. Teachers need to teach children to walk, step by step slowly, teach them how to read questions, and gradually develop good reading habits. Generally speaking, students can be trained to say a complete sentence from the "preview class", and then gradually train students to say two or three sentences. On this basis, students can be guided to try to change the third sentence into a question in combination with specific topics, and gradually become familiar with the quantitative relationship in the topics.
2, suitable for guidance, using strategies, analyzing problems and solving problems.
In teaching, teachers should give students enough time to think and adopt various means and methods to guide students to think seriously, so as to improve their thinking ability. When students come into contact with new knowledge points, they don't know where to start at once, and they are a little at a loss. However, after a careful look at the topic, many people found the connection and difference between the new knowledge and the old knowledge, and immediately smiled. In actual teaching, teachers should introduce relevant problem-solving strategies to students according to specific conditions, and provide them with a "scaffold" to help them find support in the process of solving problems.
Teachers should provide students with discussion time to exchange their exploration and discovery of knowledge. Bernard Shaw, a great British writer, once said, "If you have an apple in your hand and I have an apple in my hand, exchange it with each other, then there is still only one apple in your hand and mine;" But if you have an idea and I have an idea, and exchange these ideas with each other, then everyone will have two ideas. "Indeed, classroom discussion can enable teachers and students to communicate with each other, inspire each other, open their minds, draw inferences from others, promote each other and gain something. Different understandings and viewpoints collide with each other, fully mobilizing students' enthusiasm and initiative in learning, stimulating their interest in learning, inducing their motivation in learning and igniting their desire for knowledge. At the same time, it deepens students' understanding of new knowledge, promotes digestion and absorption, and promotes the consolidation of memory.
(4) Several important factors that affect problem solving.
1, environmental factors. Piaget said: only by requiring children to act on the environment can their cognitive development proceed smoothly; Only by letting children make up fairy tales and adapt to the stimulation in the environment can we ensure the development of children's cognitive structure. It can be seen that the environment plays a vital role in students' cognition, and at the same time, the environment also directly affects the speed and quality of solving problems. Based on this understanding, we have done some exploration and research on environmental factors. We believe that the environmental factors that affect students' problem solving include classroom learning atmosphere, the role of teachers, the background of the problem, that is, the problem situation and so on. We find that (1) a democratic, harmonious and progressive learning atmosphere can stimulate students' enthusiasm for solving problems and make students' various senses play the greatest role. (2) Teachers should play two roles in problem solving. One is the "foreign squad leader", that is, they should guide the whole class to discuss the problems to be solved, observe, ask questions and guide students when they solve problems individually or in groups, and organize the whole class to discuss the solution. Second, as an "assistant" for students to solve problems, the specific behavior is to ask questions, design tasks, ask students to analyze their own mathematical performance, point out the characteristics of mathematics used in solving problems, and help students construct heuristic and control strategies and the knowledge used. (3) The problem situation should meet the following requirements: the problem situation is realistic (students' cognitive and emotional reality), challenging (students are not easy to solve) and open (meeting different needs of different students).
2. Students' consciousness of solving problems. In practice, we find that many students rarely take the initiative to find solutions when facing problems, and often wait for teachers' hints more or less. Make students in a passive position in the process of solving problems. How to make students take the initiative to solve problems, first of all, we should cultivate students' awareness of solving problems. Only when students have keen insight and strong information literacy (the ability to collect, extract and process information) can they become experts in solving problems.
(five) around the topic, to carry out the subject competition.
1, Senior One and Senior Two "quiz". Through this competition, we can stimulate students' interest in learning, enlighten mathematics and scientific thinking, develop students' personality characteristics, cultivate students' exploration spirit and application ability, and improve their comprehensive quality.
2. Organize students to participate in Dongsheng Primary School Students' Mathematical Knowledge Application Competition. Many students in the research group won the first and second prizes in the grade group.
3. Every semester (1 ~ 3 grades) primary school students' "problem solving" ability competition. First, the preliminary contest is held in the class, and then the former 10 students are selected to participate in the competition organized by the school. Students participate in a wide range of competitions and their ability to solve practical problems is further improved.
(6) Promoting teachers' professional development.
There are 27 main members of the mathematics research group in our school. Before participating in this study, 30% teachers have never participated in this study. Therefore, for all mathematics teachers, this study will guide teachers to the study and research of teaching practice and scientific theory. Through study and practice, teachers have mastered the basic teaching and research methods, learned how to write sub-project plans and how to conduct research and analysis. The mathematics teaching and research group carried out activities around the research goal of the subject, and organized a series of discussion activities such as "talking with experts", "problem solving" case writing, practical classroom analysis, and paper writing. Teachers' enthusiasm for research is very high, and their awareness of "solving problems" is gradually enhanced. The teachers of our research group have also promoted the improvement of their professional level in the implementation of the project. Both competitions and thesis writing have gained a lot. For example, when Mr. Wu taught the first-grade course "Knowing Clocks", he was able to combine specific life scenes to make mathematics problems come alive, and won the second prize in the evaluation of high-quality courses for mathematics teachers in Longquan City. The lesson "Understanding Part" taught by teacher Zhu Wenhui can be demonstrated by multimedia courseware, and the problem situation can be carefully created, so that students can find and solve problems independently through observation, analysis, judgment, comparison and discussion, and also won the first prize in the evaluation of high-quality courses for math teachers in Longquan City. Teacher Wen's mistakes are equally wonderful was published in Educational Research. Teacher Liu Xiaochun's How to Cultivate Students' Oral Ability in Low-level Problem-solving Teaching, Teacher Zhan's How to Cultivate Students' Problem-solving Ability in Junior Middle School Classroom Teaching, Leave Problems to Students and Liberate Students' Thinking in Open Questions won the first and second place respectively in the selection of excellent papers on primary school mathematics in Longquan City. Many papers by teachers such as Zhong Chunwei, Liu Rixing and Zeng Qibo have been published or won prizes in provincial and municipal journals. As teachers realize, the core content of "problem solving" is to let students solve problems creatively. Teachers will never replace the problems that students can solve themselves; Teachers never hint at questions that students can think for themselves. Teachers begin to pay attention to teaching methodology in teaching design, and use the strategy of "problem solving" to realize students' autonomous learning and creative learning.
Seven, the form of results:
1, "problem solving" classroom teaching demonstration, writing teaching cases.
2. Write research reports and compile teachers' essays.
Eight, some thoughts after investigation.
(A) In the course of the subject experiment, the main key to affect the experimental effect is the improvement of teachers' subject consciousness. Teachers' teaching behavior directly affects students' learning state and experimental effect. It is mainly manifested in teachers' reasonable grasp of teaching content, careful design of teaching links and rational use of teaching strategies in the process of learning textbooks. Therefore, only by helping teachers to constantly establish the consciousness of "problem solving" and improving the quality of each lesson can we truly realize that the mathematics classroom is centered on the problem raising and solving, and truly reflect that the classroom is returned to the students, and the students are the masters of problem solving.
(2) The formation of students' ability to solve problems is not achieved overnight. It is necessary to solve some practical problems in life, accumulate experience and realize independent construction in practice. Therefore, the foothold lies in whether teachers can respect children's psychological development process, optimize construction conditions, attach importance to the excavation of mathematics learning materials in life, and guide students to construct knowledge independently.
(3) Exploring the teaching method of "solving problems" in the classroom. It is our main task to solve problems effectively in class. After defining some concepts and theories, our main energy should be this.