First of all, the questions raised
(A) to study the reasons for problem-solving strategies
1, the important position of "problem-solving strategy" in primary school mathematics learning
At present, there are some problems that need to be solved urgently in mathematics education in primary and secondary schools. Mainly in the learning process, problems involving the actual situation, students' hands-on ability, ability to understand and solve problems, innovation ability, ability to overcome difficulties and explore independently, ability to cooperate and communicate, and confidence in solving problems are not satisfactory.
Solving problems is mainly to cultivate thinking ability, not to apply ready-made conclusions. So you don't need much knowledge, what's important is to use it flexibly. The formation of problem-solving strategies can effectively cultivate students' thinking ability. The formation of personalized problem-solving experience is conducive to improving students' problem-solving ability. The value of problem-solving activities is not only to solve a certain kind of problem and get the conclusion of a certain kind of problem, but also to develop in the process of problem-solving, that is, to form corresponding experience, skills and methods on the basis of problem-solving experience, and then to form the ability to solve problems through reflection and refining. It can be said that solving problems is one of the core contents of mathematics education.
2. Solving problems is one of the trends of mathematics curriculum reform.
The Mathematics Curriculum Standard for Full-time Compulsory Education (Experimental Draft) defines the overall goal of mathematics curriculum in compulsory education stage, and further expounds it from four aspects: knowledge and skills, mathematical thinking, problem solving and emotional attitude. The overall goal of solving problems is to "initially learn to use mathematical thinking to observe and analyze the real society, solve problems in daily life and other disciplines, and enhance the awareness of applied mathematics." All these fully reflect that solving problems has become the trend of mathematics curriculum reform, and improving students' ability to solve problems has become the requirement of the times and social development.
(B) The reasons for taking the textbook of Jiangsu Education Edition as an example
The experimental teaching materials under the curriculum reform in China are no longer based on the traditional arithmetic application problems, but on the students' life experience, the quantitative relationship embodied in the operations they have learned and the problem-solving strategies. The textbook of People's Education Edition arranges some problem-solving strategies, such as charts, enumerations, lists, finding rules, starting with simple situations, etc. The problem-solving strategies of the textbook arrangement of Beijing Normal University Edition include drawing, listing, guessing and trying, and finding rules from special cases. However, the textbook of Soviet education edition adopts the principle of combining decentralization with centralization. Since the fourth grade, each volume has arranged an independent unit "problem solving strategy", which is rare in other versions of textbooks. In the past, in primary school mathematics teaching, applied problems were regarded as an important carrier or even the only way to cultivate students' problem-solving ability. In fact, the process of mathematics learning itself should be a problem-solving process. The presentation order of this part of the content in the textbook of Jiangsu Education Edition is mainly: "Example presentation-problem guidance-method presentation-strategy summary-attempt-practice-unit practice".
Textbooks are the carrier of implementing curriculum standards and embodying the spirit of curriculum reform.
It is also the crystallization of the wisdom of many education experts and front-line teachers. This study attempts to put forward some views on the effective teaching of problem-solving strategies through the teaching research of this part of the textbook of Jiangsu Education Edition.
Second, the research status
(A) Summary of domestic research
At home, a large number of scholars and front-line educators have also conducted in-depth investigation and research on problem solving. The research of mathematical problem solving strategies mostly focuses on mathematical application problems. They put forward the definition of related concepts, the classification of strategies and the general steps of solving problems through their own education and teaching practice or observing others, combined with psychological theory. Professor Zhang Dianzhou and Professor Liu Hongkun in our country pointed out in "Problem Solving in Mathematics Education" in their "Mathematics Pedagogy":
The problem is a situational state,
"Problems" in problem solving; It does not include conventional mathematical problems, but refers to unconventional mathematical problems and mathematical application problems; The problem is relative. Chinese scholar Wo Jianzhong (200 1) studied the development of primary school students' mathematical problem-solving strategies. This study holds that in the structure of mathematical problem-solving strategies, the gifted students and the poor students go through roughly the same cognitive steps in solving application problems: reading, analyzing, assuming, calculating and checking. The time spent in the analysis stage is closely related to the result of solving problems, and analysis is an important link in solving application problems. The development of primary school students' mathematical problem-solving strategies embodies the following characteristics: from guessing strategy to trial-and-error strategy to grasping the essence of mathematics. Chinese scholar Li Mingzhen and others believe that the basic strategies to solve mathematical problems are: overall strategy, pattern recognition strategy, transformation strategy, media transition strategy, dialectical thinking strategy and memory strategy. Zou Ming combined with his own teaching practice, in 2007, he emphasized in the article "Reflections on Problem-solving Strategy Unit Teaching" that: ① Go into the situation and get information. (2) processing information,
Form a strategy. ③ Apply and expand, and deepen understanding. (4) timely reflection, improve the strategy. ⑤ Apply what you have learned and feel the value. In 2008, Liu Qin put forward in the article "Strategy not to teach":
① Students' experience is the basis of problem-solving strategies; (2) Timely publishing and collecting strategies gradually formed in the process of solving problems; (3) Review and reflect to improve students' awareness of strategy selection and optimization.
Based on the above situation, the discovery research mainly focuses on a series of studies on related concepts, strategies and steps to solve problems from a theoretical perspective; Some domestic educators have also studied how to improve students' problem-solving ability from their own practice. It is hoped that based on the textbook, through the combination analysis of the "problem-solving strategy" unit in the textbook and the analysis of teaching cases, we will focus on the teaching and learning of "problem-solving strategy", thus promoting the formation of effective teaching of problem-solving strategy.
(B) the definition of the concept
1. Problem-solving strategies usually refer to a series of rules for selecting, organizing, changing or operating background propositions in order to fill gaps in problems. The role of strategy is to reduce the randomness of trial and error, save the time needed to solve problems and improve the probability of answering.
2. The problem-solving strategy is the thinking strategy to solve the problem, which is essentially a cognitive strategy. Cognitive strategy is a special kind of intellectual skills, which points to students' internal activities, that is, students' self. It can be divided into general cognitive strategies and specific cognitive strategies.
① General cognitive strategies include retelling strategy, arrangement strategy and organization strategy. Retelling strategy refers to the repeated memory of learning materials, which reflects a kind of "surface" or surface processing of learning materials; Finishing strategies refer to adding details, explaining meanings, giving examples, summarizing, inferring or linking them with related concepts. Organizational strategies enable them to find out the hierarchical relationship between learning materials and help them remember and understand, such as outlining and drawing structural diagrams.
② Specific cognitive strategies are suitable for guiding the learning process of specific learning contents (such as mathematics and Chinese), such as drawing, list analysis, classification, generalization, transformation, analogy, association, modeling, simplification, discovery of laws, estimation, guessing and testing. The "problem-solving strategies" listed in the textbook "Primary Mathematics" published by Jiangsu Education Publishing House belong to the cognitive strategies in a specific subject direction.
3. Problem-solving strategy is an ideological theory that guides students to analyze and explore problem-solving methods. It helps students acquire an easy-to-understand theory and guides students to explore the direction.
4. Mathematical problem-solving strategy refers to the principles and principles by which we can think about assumptions, choose and adopt solutions and steps in the whole process of solving mathematical problems, which is an overall understanding of mathematical problem-solving methods. Mathematical problem-solving strategy is a universal and highest-level information processing method, which is different from mathematical problem-solving methods and specific skills.
5. Problem-solving strategy is a learning strategy that people usually adopt when facing problem situations. It has a high degree of procedural and corresponding steps. It is a kind of general knowledge of operating procedures, the key for people to solve problems, and one of the standards to distinguish novices from experts.
To teach students to learn to learn, students need to master and consciously use learning strategies; Similarly, in order for students to learn to solve problems, it is necessary to master and consciously use problem-solving strategies. The traditional teaching of problem-solving strategies for application problems is to put forward some effective problem-solving methods for a class of problems. Problem-solving strategy can be regarded as a kind of thinking, which can not be mastered by solving a specific application problem. At the same time, the formation of specific strategies can improve their ability to solve related practical exercises.
Third, the theoretical basis of the research
(A) the basis of educational psychology
Educational psychology has conducted in-depth research on problem-solving strategies and put forward that the cognitive strategies that students should learn are mainly thinking and problem-solving strategies. The internal conditions of cognitive strategy learning include: original knowledge background, students' motivation level and reflective cognitive level. From the existing research on cognitive strategy teaching, the external conditions of cognitive strategy learning involve the following problems: the simultaneous presentation of several examples, the discovery of guiding rules and their application conditions, and the opportunity to practice variants. According to the theory of information processing process, cognitive strategy plays a regulatory role in the whole information processing process, and the purpose of using strategy is to improve the efficiency of information processing. The research shows that the application of strategy can not be separated from the processing information itself. The more knowledge children have in a certain field, the more they can use appropriate processing strategies. The learning of problem-solving strategies is essentially the learning of cognitive strategies. The characteristics of cognitive strategy learning are fully considered in the compilation of the textbook "Problem-solving Strategies" of Jiangsu Education Edition. At the same time, combined with students' motivation and reflective cognitive level, some guiding opinions are given for teachers' teaching design.
(B)' New Curriculum Standard' clearly requires' focus on cultivating students' problem-solving ability''
In the Standard published in 200 1, China has listed problem solving and mathematical thinking as important aspects of the process goal of curriculum three-dimensional integration. It can be seen that the practice and research of solving problems is the necessity of the historical development of mathematics education and plays an important role in primary school mathematics learning.
(3) The arrangement of "problem-solving strategies" in the textbook of Jiangsu Education Edition.
Textbooks are the carrier of curriculum reform and the crystallization of the wisdom of many educators. According to the physiological and psychological characteristics of children's development, the teaching content of this part of problem-solving strategies is arranged as follows:
The first stage:
From the first grade to the third grade, there is no independent unit of "problem-solving strategy" in the primary school mathematics textbook published by Jiangsu Education Publishing House, and a problem-solving strategy is introduced respectively. But there are some basic problem-solving strategies in the textbook, such as Grade Two (Volume Two).
Arranging list method to solve problems in Multiplication Formula and Formula Quotient makes students have a preliminary understanding of this problem-solving strategy. In addition, the "statistics" part of the lower grades uses tabular statistics, which makes full preparations for further study.
The second period:
From the fourth grade (the first volume), every volume of primary school mathematics textbooks published by Jiangsu Education Publishing House has written a unit of "problem-solving strategies", introducing a problem-solving strategy respectively. The fourth grade textbook (Volume I) introduces the strategy of solving practical problems with lists. In the textbook of Grade Four (Volume II), on the basis that students have preliminarily learned the strategy of using lists to solve practical problems, this paper introduces the strategy of using drawing or lists to solve slightly complicated practical problems. The textbook is divided into two sections to arrange this part: the first section focuses on solving practical problems related to area calculation by drawing intuitive schematic diagrams; The second paragraph focuses on solving practical problems related to travel by drawing line segments or lists. On the basis that students have learned to solve problems with lists or drawings, the textbook content of grade five (Volume I) introduces the strategy of "listing one by one" to solve some simple practical problems. The fifth grade (the second volume) introduces the strategy of "reverse reasoning" to solve related practical problems. The sixth grade textbook (the first volume) introduces the substitution and hypothesis strategies for solving simple practical problems, and applies the strategy of drawing lists in the process of solving problems. The sixth grade textbook (the second volume) introduces how to solve related practical problems with transformation strategies on the basis that students have learned the strategies of drawing columns, enumerating, pushing backwards, replacing and assuming. Transformation strategy means that when the subject's contact problem is difficult to start with, it is reduced to another familiar and easy-to-solve problem through transformation, so as to achieve the purpose of solving the problem.
The presentation of this part of the textbook "Problem Solving Strategies" not only pays attention to the internal connection of knowledge between different grades, but also pays attention to the connection of knowledge before and after the arrangement of the same volume. The introduction of knowledge conforms to the spiral upward trend. For example, after learning two-step mixed operation in grade four (volume one), this paper introduces how to solve the application problem of two-step calculation with list method. After learning the three-step mixed operation and multiplication and division in Grade Four (Volume II), this paper introduces the strategy of using it as a graph or list to solve slightly complicated practical problems. In the arrangement of teaching materials, we should choose appropriate practical problems to lead to examples, and then achieve the goal of cultivating students' ability through trying, thinking, doing and practicing.
Fourthly, the teaching research of problem-solving strategies.
(1) Lead-in stage: stimulate students' interest in learning and generate demand for learning problem-solving strategies.
Interest is the best teacher. Teachers should be good at concretizing and visualizing abstract content, making boring content lively and interesting, and let students explore strategies to solve problems happily in practical activities, so as to achieve the goal of "knowing why, knowing why". As a problem to be solved, it is different from simple exercises. It is not a simple mathematical problem, it is complete, closed, with sufficient conditions and unique answers. It often provides students with a situation, which is manifested in the reality of content and is related to students' experience; Or the reality of the problem is an open, ill-structured and simple mathematical problem with strong thinking value. When students face different problem situations, teachers need to guide students, remove the non-mathematical components in the situation, find problems and refine them. At the same time, it makes a preliminary analysis of the problem, that is, it analyzes the scope of the problem, the available materials provided in the situation, Lenovo's previous experience in solving problems, formulates a preliminary problem-solving plan, and chooses corresponding problem-solving strategies.
For example, when teaching the design of problem-solving strategy-transformation, in the introduction stage, the teacher first shows a light bulb diagram and asks, "Can you measure its volume?" Then it leads to the story of how Edison and Aptom measured the volume of light bulbs. Finally, they sum up and write a book on the blackboard. The teacher's first question prompted most students to have cognitive conflicts, effectively mobilize their existing knowledge and experience, and then think nervously, hoping to find a strategy to solve the problem. Through stories, students can further understand the relationship between mathematics and life, and stimulate their interest in learning mathematics and confidence in learning mathematics well. For students, learning problem-solving strategies is not building castles in the air. They have accumulated some knowledge about strategies and preliminary experience in solving problems in their daily lives, but students often pay attention to whether specific problems can be solved and lack due thinking. This design can stimulate students' learning experience and promote their positive thinking.
(2) the new authorization stage
First, pay attention to the process of strategy formation and experience the value of strategy.
"Problem solving" is a process of intellectual activity, which is embodied in the process of teachers guiding students to use mathematical knowledge for thinking activities. It organizes and implements teaching from the following aspects: creating problem situations, finding problems, exploring problems, solving problems, and evaluating processes and results. Its essence is to give full play to students' main role in teaching and let students participate in and experience the process of knowledge and skills from unknown to known. In this process, students' awareness of applied mathematics is improved, their ability of independent inquiry is stimulated and cultivated, and their creative thinking is developed.
Whether strategies can be truly understood, mastered and flexibly used by students requires students to experience, experience and feel in problem-solving activities. In the process of solving problems, students need to go through the process of individual inquiry and cooperative inquiry, and need to implement plans, adjust plans, re-implement plans and solve problems. Teachers should attach importance to students' learning process, give them enough time, create a relaxed environment for them, and let them turn their strategies into their own in the process of using certain strategies in order to gain direct experience.
For example, there is an example in the "Problem-solving Strategy" unit in the first volume of the fifth grade: Uncle Wang forms a rectangular sheepfold with a fence of 18 long 1 meter. How many different ways are there? How to maximize the surrounding area?
In the teaching process, Mr. Zhang Yanping first guides students to "wave a stick". Through calculation, he made it clear that the circumference of a rectangle is 18m, and deduced that the sum of its length and width is 9m. Then, different closure methods are found by grouping operation; Then guide the students to master the specific thinking method of "listing one by one" in the process of filling out the form, and talk about the strategies to solve this problem in the group; Finally, ask the students to calculate the area of each rectangle. By comparison, they realize that in rectangles with equal perimeters, the areas are not necessarily equal, and the closer the values of length and width are, the larger the area is. In this teaching process, students use the methods of operation, list or drawing, which not only initially perceive the function of the strategy of "listing one by one", but also help to list without repetition or omission. At the same time, by analyzing problems from different angles, it embodies the orderliness and thoroughness of strategy and thinking, and effectively trains students' divergent thinking ability and inquiry ability.
Second, organize students to review and reflect and master the methods of strategy acquisition.
Influenced by traditional teaching concepts and methods, a considerable number of teachers pay more attention to the knowledge points in books in mathematics teaching. The task of teaching is to help students put the knowledge in books into their pockets and heads. Their concept of teaching efficiency is to teach students more knowledge in a limited time. Due to the lack of understanding of problem solving, students' problem solving activities, experience and reflection in activities are not paid attention to in the selection and development of teaching content, the organization and implementation of teaching activities and the evaluation of students' learning activities. Obviously, students' learning is more about the acquisition of indirect knowledge than the experience of problem-solving learning activities. The goal of teaching is not to let students acquire a specific strategy, but to let students master the formation process of inquiry strategy in the learning process and use it flexibly in practical problems.
Learning is not only a process of constantly acquiring knowledge and skills, but also a process of accumulating activity experience. When a problem is solved, stop and review: What problem have I solved? What difficulties did you encounter in solving the problem? How did I solve it? What teachers or classmates inspired me? What will I do next time I encounter similar problems? Not how? In teaching, if teachers pay attention to reflection and often guide students to reflect on the above problems, students will naturally form the habit of reflection, which will greatly improve students' comprehensive strategies to solve problems, thus effectively strengthening students' ability to solve problems.
For example, the teaching fragment of "problem-solving strategy-transformation method": when students sum up three transformation methods to solve this problem, teachers guide students to think about the formation process of this strategy: "transformation method". After * * * came to three conversion methods, the following dialogue appeared:
Teacher: Please observe the similarities and differences between these three schemes.
Health 1: all the juice in the second cup returns to the first cup.
Health 2: We always find out two cups of juice now, and then return the second cup of juice to the first cup.
Health 3: The difference is the method, and it is also known now that all three methods are to pour 40 ml of cup B back into cup A, and then find out how much ml is in two cups of juice.
Teacher: No matter whether the students used numbers, tables or formulas just now, it is actually based on the fact that the current two cups of juice are 200 ml, and 40 ml of the second cup is poured back to the first cup, so as to find out the ml number of the original two cups of juice.
Teacher: Please review the similarities between the two problems we just solved.
Health: Playing cards and pouring juice, their similarities are the result of the development of known things. Go back and find the initial state of things according to the changes of things.
Teacher: Yes, this is the strategy of "reverse reasoning" that we learned today.
Review and reflection is a rational reflection on what we have experienced, and this process is also a process in which students select solutions to problems so as to optimize the formation of strategies. When students put forward several methods to solve problems, there is a process of collective communication, comparison and discovery of essential connections. From the above case, we can see the comments and reflections organized by two teachers: "Please observe the similarities and differences of these three solutions". This process of communication and review is the process of promoting students to screen and optimize strategies. Please review the similarities between the two problems we just solved. This problem links the problem of playing cards and juice that has just been solved, which is convenient for students to understand and master the characteristics of such problems, and at the same time consciously cultivate the habit of timely reflection in the teaching process.
(3) Consolidation stage: design layered exercises to consolidate students' formation strategies.
As strategic knowledge, mathematical problem-solving thinking strategies must be transformed from "declarative" to "procedural" in order to guide students' thinking. A more effective transformation method is "variant exercise", that is, by changing the irrelevant conditions of strategy application, students can identify the essential conditions of thinking strategies, thus improving their mastery of strategies. Teachers should carefully design exercises, which require levels and diverse presentation methods. Only in this way can students realize the benefits of using strategies to solve problems in the process of solving problems and cultivate their consciousness of consciously using strategies to solve problems. The design of exercises can be divided into three levels:
One is imitation exercise, that is, presenting the situation of normalization problem, with the aim of consolidating new knowledge; The second is to change the practice and present the situation of summing up the problem. The purpose is to further experience the specific advantages of problem-solving strategies through the change of problems and attach importance to the cultivation of students' analytical ability.
Avoid students copying the problem-solving model of examples; Third, comprehensive exercises, providing relevant information, and cultivating students' ability to choose information flexibly and solve problems. In actual teaching, teachers can appropriately increase the amount of training,
Pay attention to changing the problem situation, often remind students to apply problem-solving strategies, so that students can form strategies in the process of applying strategies.
For example, in the class of "Problem-solving Strategy-List Method", Teacher Chen arranged such exercises:
Teacher: The school is going to buy some teaching and daily necessities, and the video in the store is playing relevant information (scrolling price information on the big screen).
Football: each 56 yuan chair: 3 sets 100 yuan.
Volleyball: each 42 yuan blackboard eraser: 10 20 yuan.
Chalk: 20 boxes of 46 yuan desks: 2 150 yuan.
Mop: One basketball from 39 yuan and one 48 yuan.
Computer: one broom from 24 yuan: three 10 RMB.
Teacher: Please solve the problem according to the above information. (Computer display)
1. How many basketballs can a sports group buy with the money for six footballs?
2. How much does it cost to buy seven desks at school?
3. 124 yuan, how many blackboards can schools buy?
Each class is divided into three brooms, which can be divided into 24 classes. If you pay 4 yuan per class, how many classes can you give?
Teacher: Can each study group solve a problem? Read the questions carefully first and think about what information you need to collect and how to organize it.
Teacher Chen showed this comprehensive exercise after class, which made the training form diverse, novel, distinct, distinct and coherent.
Focus on solving practical problems in life. In the process of inquiry and training, we should pay attention to cultivating students' interest in mathematics learning and how to collect and sort out information according to problems, so as to cultivate students' ability to solve problems. After students have a full understanding and definition of problem-solving strategies, teachers arrange such exercises to strengthen the training of the strategies of list method, so as to deepen students' understanding and mastery of strategies, make their countermeasures slightly deeper and gradually reach the realm of using them freely. Make students deeply understand the magical function of list method, and use it in time in the process of solving problems in the future.
In a word, "Problems are the heart of mathematics". Learning mathematics is inseparable from solving problems, but solving problems is not the purpose. It is for students to deepen their understanding of knowledge, strengthen skills training, improve their strategic awareness of problem solving, improve their thinking ability and problem solving ability, and cultivate their innovative spirit and practical ability. In this way, it is particularly important for students to learn correct thinking methods and problem-solving strategies in the process of solving problems. The above research on the teaching of problem-solving strategies for primary school students aims to reflect on the problems that should be paid attention to in the teaching of problem-solving strategies, provide operable guiding strategies, and promote the formation of problem-solving strategies. It is hoped that through our practice, the effectiveness of problem-solving teaching for primary school students will be gradually improved, so as to achieve the goal of comprehensively improving students' mathematical literacy.