From many published articles, it can be found that most teachers regard "times" as an expression to describe the relationship between two quantities, and its mathematical core idea is: use one quantity to describe another quantity. "Times" means the comparative relationship between two quantities, the premise and basis of which is the comparison between them, and the key is to divide one quantity into the same parts based on one quantity. [1] "Times" is the first time that students describe the relationship between two quantities, from the ratio of absolute quantities to the relationship between relative quantities. It plays a vital role in the study of "fraction", "percentage", "ratio" and even future functions that middle and high grades will be exposed to. Some teachers think that "secondary" is one of many ways to describe the relationship between two quantities, and it belongs to the same position as other description methods, so it should be linked with many description methods, and the "secondary" method should be accurately included in the knowledge system of expressing the relationship.

At present, from the current research situation, most front-line teachers understand the "times" at the level of "relationship". Liu Jiaxia, a teacher from Beijing Institute of Education, thinks that "time" is a turning point from addition structure to multiplication structure. The mathematical cognitive structure of primary school students is mainly additive structure and multiplication structure, while multiplication structure is an advanced mathematical cognitive structure based on additive structure. Multiplication structure is not a single cognitive multiplication, but a conceptual system. The basic concepts are multiplication and division, as well as multiple, greatest common factor, least common multiple, algorithm and even area, volume, surface area, speed and other related concepts and laws. [2] From the current situation, it is not necessarily a new research direction to further study "times" from the perspective of multiplication structure. We can focus on "knowledge of the times" from the multiplication structure, and we can also radiate from "knowledge of the times" to the multiplication structure.

Second, what should teachers pay attention to in teaching?

(A) the basis of students' learning

Ausubel, a famous American cognitive psychologist, said: "If I had to boil down educational psychology to a principle, I would say that the most important reason that affects learning is what students already know, and we should teach according to their original knowledge." When teaching, front-line teachers can help students establish the connection between old and new knowledge from the students' existing cognition. The concept of "multiple" in the textbook is based on the concepts of "multiple" and "share" in the knowledge of multiplication and division. Through the comparison of two numbers, the word "times" is derived from "shares", which is the basis for students to understand the meaning of "times". [3] To understand the meaning of "multiple" is to let students relate the new knowledge of "multiple" and "multiple" in the sense of multiplication. Teachers should guide students to understand the essence of concepts by using existing knowledge and experience, and establish the connection between old and new concepts to promote the effective transfer of knowledge.

(B) Teaching situation

The textbook of Beijing Normal University takes happy animals as the situation, and guides students to compare the numbers of different animals step by step through a series of questions, so as to understand the concept of "times". The textbook of People's Education Edition allows students to establish the concept of "time" in the process of spelling figures with wooden sticks. In actual teaching, the teacher further transformed it into various interesting situations, some started with guessing games, some used clapping songs to stimulate interest, some put pictures of different colors before discovering the problem of quantity relationship, some compared the quantity relationship of different things, some compared the length relationship of objects, and some created situations that challenged the mathematical problems brought by three cartoon characters, and so on. No matter what kind of problem situation, the purpose is to let students get positive emotional experience in the situation they are interested in, and master the concept of "times" while feeling the charm of mathematics.

(C) from the relationship, understand the meaning of "times"

As mentioned above, most teachers regard "times" as an expression describing the relationship between two quantities. Therefore, teachers often understand the meaning of "times" by studying relationships. For example, Guo Lijun, a teacher from Haidian Teaching and Research Section, first showed three red disks and three yellow disks in her teaching, and asked what the relationship was between the numbers of these two disks. Then add three yellow plates and ask the students to talk about the relationship between three red plates and six yellow plates. Students will first find the relationship of how many, and then the teacher will slowly guide students to transition from the relationship of how many to the relationship of multiples, that is, let students' thinking rise from the addition structure to the multiplication structure.

(D) Pay attention to hands-on operation, and promote the understanding of the "times" by using painting performance.

Teachers usually guide students to draw a picture, compare and so on. To trigger and activate the original cognition in students' minds and make it explicit; Then guide the students to discuss and communicate, and understand the meaning of "times" from the perspective of "times". According to cognitive psychology, the formation of concepts can be summarized into two stages, that is, from complete representation to abstract concepts, and the concrete representation of abstract concepts in the thinking process can be realized. It is the most important link to form a complete representation in these two necessary transformations. However, due to the lack of expressive ability of junior students, it is particularly important for students to express their conceptual representations through painting representations. Picture representation builds a bridge between physical operation and abstract thinking, allowing students to initially establish the representation of "time" in a circle and draw a picture, which is helpful for students to effectively express their conceptual representation.

(E) In constant comparison and abstraction, gradually understand the meaning of "times"

Ushinski said: "Comparison is the basis of all understanding and all thinking. It is through comparison that we understand everything in the world ... "How can students further understand the concept of" times "after they have a preliminary understanding of" times "through drawing and other means? This requires abandoning all kinds of non-essential features in constant comparison and abstraction and grasping "invariance" in change, and this "invariance" is their quantitative characteristics, that is, the essence of "the times". Many teachers will design multiple comparison links, such as: one is unchanged, several are changing, and the multiple will change; One copy is changing, and several copies are changing, but the multiple remains the same. Through comparison, guide students to gradually clarify and grasp the essence of concepts, so as to deepen their cognition and understanding.

Third, new thinking on the understanding of "times"

Although we know the essence of "times" and the basis of students' learning, it is still difficult for students to understand "times". Although students can distinguish who is whose era, they can also use multiplication to calculate, but this does not mean that students really know and understand the "era", and there will still be many difficulties in application. First of all, it is influenced by the children's own cognitive structure level. From addition structure to multiplication structure, students' cognitive structure needs to undergo a certain degree of "qualitative" changes. The study of the times is the first chance of qualitative change. Students have to go through the transition from addition structure to multiplication structure when learning multiplication, and the transition of cognitive structure is the biggest difficulty for students to learn. [4] Secondly, due to the difficulty of knowledge itself, "times" is not a single multiplication structure, it is an extension of the meaning of multiplication, but it also includes the understanding of the meaning of division, which will undoubtedly cause certain difficulties for students who understand the concept of "times" for the first time. Recently, the school has carried out the study of "classroom observation". Through careful observation of teachers and students in one-on-one class, we also find that there are three problems in the process of learning Time.

(A) students do not understand the "relationship"

The definition of the word "relationship" in modern Chinese dictionaries is: the state in which things interact and influence each other. We often use this word in our daily life, which is easy to understand, but the word "relationship" used in the mathematical background is not understood by children. Most students hear the teacher's question "What is their relationship?" Sometimes, if you don't understand, the teacher might as well stop here and give an example to further explain, so that students can continue their later studies better.

(2) Understanding the concept of "times" cannot be observed only.

By combing the literature, we found that many front-line teachers let students directly find the relationship between quantity and quantity by observing static things, such as teaching AIDS set on the blackboard or simple graphics drawn. Teachers put six red plates and three white plates on the blackboard, and students will quickly say that 6 is greater than 3 or 3 is less than 6, but it is hard to think that there are two 3s in 6, or 6 is twice as much as 3, which just shows that it is very difficult for children's cognition to rise from addition structure to multiplication structure. How do students transition naturally? By trying, we found that "hands-on operation" is a feasible method. Prepare some swing learning tools for students before class, such as the simplest small round paper. In the process of placing learning tools, students will naturally have the idea of splitting piles (as shown below). They will put three pieces of white paper in 1 pile and six pieces of red paper in two piles. The "heap" here is "1 serving" or "65440 serving".

In order to let students better understand "times" and see clearly the relationship between quantity, many teachers let students circle once in class, and then directly lead to the concept of "times". The author thinks that this may be a bit too fast, and the understanding of the "times" needs a time to wait. We might as well slow down. When students initially establish the concept of "time", they can start posing through the guidance of teachers. In fact, "1 heap" and the circled "1 serving" can achieve the same effect, and hands-on modeling will make students understand "times"

(C) "Time" is difficult to express

Students can tell who is several times as many as who, and they can also calculate by multiplication, but it is still difficult to describe. Understanding the "times" requires students to understand the meaning of "times" and learn to express "times". When expressing "times", don't rush to give students a standard expression form, but wait. According to the characteristics of students' age, students' understanding ability precedes students' expression ability, so we think it is normal for students to express unclear. When expressing the "times", we can try to express it from multiple angles by painting to deepen our understanding.

"Double Understanding" is not a new course, but as young teachers, we need to have the consciousness of studying it as a new course and the spirit of thinking, researching and innovating on the shoulders of our predecessors.