Functional thinking is a kind of thinking method that considers correspondence, movement change and dependence, describes another state definitely from one state, and transitions from research state to research change process. The essence of function thought lies in establishing and studying the corresponding relationship between variables. Model thinking is a mathematical thinking method to solve practical problems by constructing corresponding mathematical models for the problems to be solved.

Function thought emphasizes "infiltration" in primary schools, which makes students feel "the importance of seeking the same in change and grasping the law". Primary schools are not required to learn "formal" function definitions.

The following two basic principles should be grasped in permeating the thought of function in primary school mathematics teaching:

(1) Only by creating the process of "change" can we feel the function thought.

(2) Stimulate students' nature of "inquiry" and grasp "invariability" in "change".

1. Explore the law-a preliminary understanding of "mode".

"Exploring the law" is actually to cultivate students' "patterned" thinking, and discovering the law is to discover a "pattern". For example, in the second volume of Senior One: the law in the table of hundreds of digits, we can not only explore the arrangement law of digits (horizontal, vertical and oblique), but also further explore the law of two adjacent digits in each row, two adjacent digits in each column, and even the law between four adjacent digits in every two rows and two columns. These laws contain various patterns of change. For example, the study of the meaning of positive and negative proportions in the sixth volume of the sixth grade is a concentrated exploration of the "way" of change. In the study of this content, many different changing laws are presented in the form of tables.

2. Basic quantitative relationship, graphic position and transformation-the experience of "relationship".

A function is like a bridge, establishing a "relationship" between two sets.

① One-to-one correspondence runs through primary school mathematics textbooks. For example, when we recognize the number 1- 10, we can show it. The number of objects corresponds to the bitmap one by one, which is a bridge between concrete objects and abstract numbers.

② In primary school, students are exposed to "two affirmations or multiple affirmations of one", that is, binary functions and multivariate functions. For example, "the problem of volume" comes from an exercise in the textbook. Cut a rectangular iron sheet with a length of 30 cm and a width of 25 cm from the four corners to make a square with a side length of 5 cm. How much iron is used in this box? What is its volume? "This problem is just a simple calculation problem. Of course, students' concept of space has also been developed in the process of solving problems. However, if the provision of "cutting a square with a side length of 5 cm" in the original question is changed to guess and verify that "when cutting a square with a side length of several centimeters, the volume of the iron box is the largest", the problem will change from static to dynamic. With the help of such a process of movement and change, students begin to be instilled with functional ideas.

Primary school textbooks provide students with many intuitive experiences about the "relationship" between various materials and forms. The experience of "relationship" makes students have a preliminary understanding of the dependence between variables, which is the essence of the concept of function.

3. Letters represent numbers, images, tables, etc. -the feeling and initial application of various mathematical languages.

Because functions reflect the relationship between variables, they must be represented by symbols other than numbers. There are four commonly used methods: language description, tables, images and analytical formulas. For example, when teaching the laws of addition and multiplication, letters appear to represent various operational laws, so that students can initially feel that letters can represent numbers in the general sense. Another example is the derivation of the cuboid volume formula in the fifth grade. The formula for calculating the cuboid volume in the textbook is summed up by filling in a table after unit of volume's cuboids are put together.

4. Provide more opportunities for students to solve problems by using functional ideas.

Learning function should be organically combined with understanding, feeling and using function to solve problems. Students should be guided to think about the application of functions, especially the application of functions in daily life and other disciplines. For example, students can be provided with electrocardiogram to let them know the functional relationship between time and heartbeat frequency.

Second, the model thought.

Primary school math textbooks are full of models. The process of primary school students learning mathematics knowledge is actually the process of understanding and mastering a series of mathematical models. In primary school mathematics teaching, paying attention to the idea of infiltration modeling and helping primary school students to establish and master relevant mathematical models will help students grasp the essence of mathematics.

How to infiltrate model thought into classroom teaching in primary school mathematics teaching?

1) Make more use of physical models.

In primary school mathematics, students have to touch all kinds of numbers: natural numbers, fractions and decimals, which are abstractions of realistic models. Therefore, there should be some physical models in time in teaching, such as sticks used in lower grade teaching: one by one, one by one. In this way, when students first came into contact with mathematics, they gradually got the concepts of one and ten through their own intuition and hands-on. These intuitive models are very important for students to learn and understand mathematics knowledge, but they are not fully reflected in our textbooks and teaching, which requires our teachers to realize their importance and dig up corresponding materials.

Second, choose a suitable mathematical model, so that students can gradually feel the model idea.

In normal teaching, there are many mathematical models that can be used in a class, but if they are not misused purposefully, it may lead to classroom confusion, students' inattention, or they don't know much about the important and difficult points of this class, which requires teachers to choose appropriate mathematical models according to students' age characteristics, knowledge distribution and personality characteristics, and prepare lessons in advance. For example, in the teaching of lower grades, we can adopt more intuitive and hands-on models. After students have some experience in mathematics learning, we can gradually adopt some abstract models such as chart models and line models, which will not only give students a certain sense of accomplishment, but also help to cultivate students' model thinking.

Third, pay more attention to students' learning process.

Mathematics teaching is not only to teach students knowledge, but more importantly, to teach students to learn to find problems and then solve them with mathematical thinking methods. Therefore, in primary school mathematics teaching, we should pay attention to students' learning process, let students learn how to solve mathematical problems while drawing mathematical conclusions through some intuitive models and abstract models, and cultivate their diligent and fearless quality, so as to lay the foundation for students' lifelong learning and success.