Simply put, the mathematical cognitive structure is the mathematical knowledge structure obtained in students' minds, which is only a mathematical knowledge structure after students' subjective transformation. It is the product of the interaction between mathematical knowledge structure and students' psychological structure, and its content includes mathematical knowledge and its organization and characteristics in students' minds.

Second, the difference between mathematical cognitive structure and mathematical knowledge structure

Mathematical cognitive structure and mathematical knowledge structure are two different concepts, which have close internal relations and strict differences. The relationship between them is mainly reflected in that students' mathematical cognitive structure is transformed from the mathematical knowledge structure in textbooks, which is the material and objective basis for the formation of mathematical cognitive structure. The difference between the two is mainly manifested in the following aspects:

The concept of length has different connotations. Mathematical knowledge structure is a mathematical knowledge system composed of mathematical concepts and propositions, which reflects the achievements of human understanding of the world's quantitative relations and spatial forms in the simplest and most general way, and is an objective reflection of scientific truth. Mathematical cognitive structure is a kind of mathematical knowledge structure that has been subjectively transformed by students. It is the result of high integration of mathematical knowledge structure and children's psychological structure. Its content not only reflects the objectivity of mathematical knowledge, but also reflects the subjectivity of cognitive subject.

2. Information is expressed in different ways. Both mathematical knowledge structure and mathematical cognitive structure express information, but they have obvious differences in the way of information expression. The mathematical knowledge structure in the textbook expresses in detail the information about the quantitative relationship of the world and the cognitive achievements of the spatial form with words and symbols. It is a mathematical knowledge system with strict logic and relatively perfect structure. In this system, the logical starting point of knowledge, the expression form of knowledge, and the relationship between the content before and after. There is a clear and concrete expression in its carrier-mathematics textbook. The mathematical cognitive structure in students' minds mainly expresses information in a semantic way, and usually stores information in their minds in an intuitive way. This expression shows that "cognitive structure has integrated knowledge representation and personal intelligence activities"

3. The structure is constructed in different ways. Mathematics has a high degree of abstraction and strict logic. Mathematics, as the content of primary school curriculum, has been treated by the teaching methods of textbook writers, but its content is still a relatively strict logical system, with coherent and orderly content and relatively perfect overall structure. However, there is no strict logical order between the mathematical cognitive structure and content in students' minds. It is neither a well-organized linear structure nor an orderly network structure. Once the mathematical knowledge structure is internalized by students into cognitive structure, the logical order and level of its contents are often diluted, and different contents show a trend of mutual integration, and its internal structure is not as clear as that of mathematical textbooks.

4. The structural integrity is different. The mathematical knowledge structure in the textbook is relatively systematic, complete and seamless in content, and the structure itself covers all its components. For example, the knowledge structure of "the meaning and nature of fractions" includes the meaning and unit of fractions, the relationship between fractions and division, the classification of fractions, the reciprocity of false fractions and fractions, the basic nature of fractions, reduction and general fractions, etc. These contents constitute a complete and seamless unit knowledge structure. However, the cognitive structure of mathematics is often incomplete because of learners' mistakes in acceptance and understanding and forgetting after learning. For example, after the knowledge structure of "the meaning and nature of fractions" is transformed into students' mathematical cognitive structure, they may not be very clear about every content, and some contents may be vague or even completely forgotten. So it may be an incomplete mathematical knowledge structure for learners. This shows that students' mathematical cognitive structure is transformed from the knowledge structure of textbooks, but not necessarily what is written in textbooks and what the teacher says can be accepted and preserved intact, and there may often be some gaps in its content.

5. The scientific content is different. The content in the knowledge structure of mathematics textbooks is a scientific truth that has been strictly demonstrated by logic and tested by practice. It can correctly reflect the quantitative relationship of the objective world and the universal law of spatial forms, and there are usually no mistakes. The content in mathematical cognitive structure is not necessarily scientific, because it is the product of the combination of mathematical knowledge structure and students' psychological structure, and it is a mathematical knowledge structure that has been subjectively transformed by students. Its content may be correct, it may be wrong, and it is more likely that it is partly correct and partly wrong. Obviously, the scientific nature of mathematical knowledge in students' minds needs to be tested. We can't equate the scientific degree of students' mathematical cognitive structure with the scientific degree of knowledge structure of mathematics textbooks, thus covering up some misunderstandings that students may have in the learning process.

(1) The cognitive structure transfer theory is based on Ausubel's meaningful proverb learning theory (namely assimilation theory).

Cognitive structure is the knowledge structure in students' minds. Broadly speaking, it is all the contents and organizations in students' minds; In a narrow sense, it is the ideological content and organizational form of students in a certain subject field.

Ausubel believes that the essence of "teaching for transfer" is to shape students' good cognitive structure. We can ensure students to form a good cognitive structure from three aspects: teaching technology, teaching material content and teaching material presentation, so as to facilitate migration. Design Advance Organizer Advance Organizer is a kind of guiding learning material presented to students before they learn new materials. It summarizes the relationship between the new materials to be learned and the original knowledge in the cognitive structure in popular language, and provides a cognitive framework for the learning of new knowledge. The advance organizer can be a law, a concept or a general explanatory text, or a visual model.