Theoretical basis: When people encounter a new problem (target problem), they often think of similar problems (source problems) that have been solved in the past, and use the methods and steps to solve the target problem. This problem-solving strategy is called analogical transfer. In some cases, analogy transfer occurs in two different conceptual domains with the same structural characteristics, which is called inter-domain analogy transfer. In other cases, analogy transfer occurs in the same or very close conceptual domain, which is called intra-domain analogy transfer. Many researchers believe that analogy transfer is the main method to solve all new problems; What's more, I think this is the only way to solve new problems. Although this extreme view is questioned by people, it is enough to show the important position of analogy transfer in solving problems.

Practical significance: on the one hand, the process of students acquiring knowledge through learning is actually a process of promoting knowledge transfer; On the other hand, migration is the continuation and consolidation of learning, and it is also the condition for improving and deepening learning. Learning and migration are inseparable. In mathematics teaching, if teachers can effectively use the law of transfer and pay attention to the role of positive transfer, it will not only help to consolidate the knowledge, skills and concepts they have learned, but also help students acquire new knowledge and cultivate their learning ability and exploration and discovery ability in the learning process.

Fact: Immigration is not automatic. The knowledge, skills and concepts you have learned cannot guarantee a positive transfer to the subsequent study, nor can you avoid a negative transfer to the subsequent study. Therefore, how to use the transfer law to guide learning in mathematics teaching has certain exploration value. Whether teachers can scientifically and correctly use the transfer law of learning in teaching and guide students to acquire knowledge plays a positive role in improving classroom teaching efficiency.

Second, our exploration process.

1. Strengthen the connection between old and new knowledge and set up migration scenarios.

Finding the connection between new knowledge and old knowledge, finding the growth point of new knowledge, making knowledge transfer smoothly through teaching and avoiding the influence of negative transfer as much as possible can greatly improve the efficiency of the project and achieve twice the result with half the effort.

We study the textbook carefully and deeply understand the knowledge structure of the textbook and the cognitive structure of students. Find out the connection between knowledge, clarify the children's existing knowledge and ability, find out the growing point of new knowledge, the difference between new knowledge and old knowledge, presuppose the negative transfer that these differences may cause, highlight this piece in the design of lesson plans, and try to eliminate the problems in new teaching.

In our special research, we focused on the knowledge transfer in the following teaching contents:

How to read and write large numbers?

Comparison of the size of numbers.

Multiply three digits by two digits.

(See the lesson plan design and reflection in the materials for details. )

2. Pay attention to the significance of multiplication and division, and transfer the calculation method under specific circumstances.

In the second grade of primary school, children know the meaning of multiplication and division, and learn simple calculations on this basis. Multiplication represents the sum of several additions, and division has two meanings: average score and inclusion. When solving a problem in a specific situation, we should choose the appropriate method (multiplication or division) according to the problem. To make the right choice, we must judge by the meaning of multiplication and division. After children are familiar with the meaning of multiplication and division, they can compare the calculation of multiplication and division that they have not learned.

After learning division in the second grade, children can calculate the division formula in the formula table, but it cannot exceed this range. For example, there are 24 books, which are distributed to two classes on average. How many books are distributed in each class? Children can correctly list formula 24÷2. But children can't get results without memorizing the multiplication formula of 2×( )= 24 when calculating.

In teaching, I found that if we combine the situation of "divide" with the meaning of "divide equally", we can transfer arithmetic and algorithm. I guide students like this:

Teacher: What is this question doing? Student: Separate the books.

Teacher: How to divide it? How to solve it? Student: The average score is solved by division.

Teacher: Divide 24 books into 2 books on average at a time. I don't know the result. Then you can divide 20 of the 24 books first. How to arrange them? 20÷2= 10 (Ben)

Teacher: There are four books left. What about these four books? Health: Divide these four books equally, and 4÷2=2 (copies).

Teacher: Each class is assigned to 10, then to 2, and a * * * is assigned to 10+2= 12. So 24÷2= 12 (Ben).

Inspired by this, the students thought of dividing 12 books first, and then dividing 12 books; Or divide it into 18 books, then divide it into 6 books, and so on.

With this experience, it is not difficult for students to understand the formula (20+4)÷2=20÷2+4÷2 when learning the distribution law in grade four.

Similarly, divisor is the division of two digits, which has not been learned in Book 2 of Grade Three, but it is involved in practice. This problem can be solved by the meaning of "containing" in division. For example, there are 200 exercise books, and each class is divided into 25. Can you divide them into several classes? The core of solving this problem is how many 25s does 200 contain? Formula 200÷25 will not be calculated. Guide students to think about how much 25 is 200, and 25×( )=200. The problem is solved, and the child can initially understand the reciprocal relationship of multiplication and division.

In the third grade, I learned to multiply two or three numbers by one. In a unit test, a question appeared: Xiaoming reads about 125 words a minute, so how many words did he read in 18 minutes? From the meaning of multiplication, the solution of this problem is the addition of 18 125, and the formula is 125× 18. Children can't calculate, because they only need to calculate two or three digits multiplied by one digit, and this is a problem of multiplying three digits by two digits. What do we do? The importance of connecting multiplication. From the meaning of this formula, to calculate the sum of 18 125, first calculate the sum of 10 125, that is,125×10 =1250; Then add eight 125, that is,125× 8 =1000; Finally, add up 10 125 and 8 125, that is 18 125, 1250+ 1000=2250.

This experience of students laid the foundation for the fourth grade to learn multiplication and division.

3. Guide "reflection", cultivate "enlightenment" and transfer mathematical thinking methods.

Bruner pointed out that mastering basic mathematical ideas and methods can make mathematics easier to understand and remember. Understanding basic mathematical ideas and methods is the bright road leading to the "migration road", and internalizing them into students' cognitive structure is the premise for students to have mathematical quality. The study of mathematical thinking methods needs students' understanding more. The teacher's role is to guide students to understand mathematical ideas and methods.

In practice, we find that it is an effective measure to seize the opportunity to give students "reflection" guidance in teaching, so that students can reflect and evaluate the answers to mathematical questions and the inquiry process itself after mathematical inquiry activities, which will improve students' "understanding" and help them master mathematical ideas and methods.

Reflection is the ability to critically examine one's behavior and situation based on self. As one of the basic processes of solving mathematical problems in primary schools, reflection mainly refers to students' investigation and reflection on the answers and thinking process of mathematical problems, analysis and evaluation of whether the selected problem-solving process is the simplest, whether the reasoning is rigorous, and whether the methods can be popularized, so as to understand the basic mathematical ideas and methods and enhance the transfer ability of the learned knowledge. For example, teachers and students can work out "There are 9 cups in the box, 3 cups outside the box, a * * *, how many cups are there?" As a prototype, after operation, observation, analysis, synthesis and generalization, the mathematical model shown in the left figure is obtained: 9+3 = 12, and then students are guided to reflect and ask them to express their thinking process in mathematical language, that is, "See 9, think of 1, divide 3 by 1 and 2,9 plus/kl". When students understand this thinking mode of "add up to ten", they can move towards "eight plus several" and "seven plus several" ... which greatly develops students' cognitive ability in learning mathematics and improves learning efficiency.

For example, in the second and third grades, when solving the problem of renting a car and renting a boat, we all use the list method. List all the schemes in the table, and finally choose the scheme with the highest cost performance. In mathematics, we call this method of solving problems "enumeration". When children master this method, they can use it to solve other problems.

4. Create life situations, apply what you have learned and improve students' migration ability.

Mathematics is a process in which people qualitatively grasp and quantitatively describe the objective world, gradually abstract and generalize, form methods and theories, and widely apply them. The research shows that if the teaching situation is similar to the situation of using knowledge in the future, the knowledge learned by students will be easier to transfer. Therefore, in the teaching process, students should be helped to get as many "realistic" learning opportunities as possible, namely "mathematization of life problems" and "life mathematics problems".

Strengthening the connection between mathematics and life, making "life problems mathematized" and "mathematics problems living" can not only stimulate students' interest in learning mathematics, but also deepen their understanding of knowledge in the process of applying knowledge. Take the simplest example, for example, when learning line segments, rays and straight lines, we take zebra crossings, spotlights and rails as prototypes, abstract mathematical models through understanding these prototypes, and summarize the characteristics of these mathematical models.

For another example, after the preliminary teaching statistics, teachers can ask students to investigate and collect the height information of the whole class, and make statistical tables to find out the average height. Before solving these problems, students must think about what knowledge can be counted and how to get it, so as to turn life problems into math problems. If students can always unify the learning situation and problem-solving situation, what they have learned will be more practical and promote the transfer of mathematical ability on the basis of application.

5. Use variants to master key features and overcome negative migration.

Mathematics curriculum standard (experimental draft) points out: "Mathematics teaching activities must be based on students' cognitive development level and existing knowledge and experience. "Students' existing knowledge and experience have a positive or negative transfer to the learning of new knowledge. Teachers often do this in teaching: promote positive transfer and prevent negative transfer. However, "negative migration" is often impossible to prevent.

Proper arrangement of some counterexamples can help students pay attention to new features that they have not noticed before and understand which features are related or irrelevant to certain concepts. Appropriate counterexamples can be used not only for perceptual learning, but also for conceptual learning. The understanding of when, where and how to use the learned knowledge, that is, knowledge restriction, can be enhanced by using "counterexamples"

In mathematics learning, students are prone to make mistakes in generalization of non-essential attributes, which is the result of negative transfer of non-essential attributes. As an effective way to overcome this negative transfer, counterexamples or analytical questions are often used in teaching to create cognitive conflicts and help students master the essential attributes of mathematical objects. Counterexample teaching, analysis problem teaching and variant problem teaching all belong to the category of variant teaching. The characteristic of counterexample is to change the essential attribute of an object while keeping its non-essential attribute unchanged, and the characteristic of analysis problem is to change its non-essential attribute while keeping its essential attribute unchanged. Arranging variant learning can help students clarify the difference between non-essential attributes and essential attributes that they have not noticed before, and avoid the mistakes of generalization of non-essential attributes as much as possible. The application of variant problems lies in improving the cultivation of transfer ability in problem-solving learning, which is a common method in mathematics teaching.

Third, our harvest.

The teachers in our small research group study hard and reflect constantly. With our efforts, we have gained some gains.

While improving teaching methods, we should give priority to guiding students' learning, how to study, how to attend classes and how to review. Inspire students' thinking and guide them to carry out reasoning or imagination activities, thus promoting the development of migration. In addition, study habits are also an important factor affecting learning transfer. Cultivating students' good study habits can not only improve the development speed of migration, but also improve the efficiency of learning, so that students can use their time regularly, be energetic when studying, and have enough time to engage in other activities, so as to fully develop themselves and cultivate various interests.

Teaching practice shows that cultivating students to learn new knowledge by using the transfer law can not only stimulate students' interest in learning and master knowledge systematically, but also enable students to master the methods of learning mathematics. The concrete manifestations are as follows: first, clear thinking, paying attention to foresight and retrospection in teaching, so that students can clearly understand the characteristics of close connection between mathematical knowledge and learn new knowledge with old knowledge; Secondly, teachers constantly use the transfer theory to carry out overall teaching, which constructs a good cognitive structure for students and makes them easy to learn and master; Thirdly, because of the application of migration, students can always feel relaxed and happy when learning new knowledge but not new, and always keep interested in learning; Fourth, students' self-study ability has been significantly improved. Due to the continuous application of transfer ability, students have a wide range of associations, and their ability to learn new knowledge through old knowledge is getting stronger and stronger. Fifthly, students' thinking ability has been improved, and they can not only master the algorithm well, but also correctly express the process of arithmetic and analysis. In a word, using the new knowledge of transferring to learn mathematics, we can push students to the main position, give play to their initiative, make their knowledge solid and understand thoroughly, and cultivate and improve their mathematical ability.