Some of the same math teachers never seem to have any problems, as if everything is under control, so nothing will change except getting old; Others often have various problems, so they think, consult, read and study, so that they can gradually raise some key and important questions and gradually become excellent teachers who can think and accumulate. The pre-service teachers I know belong to the latter kind of high school math teachers.

The "seamless" tendency of 002 concept teaching

Mr. Li Qiuming, a famous special-grade teacher and a full-time senior teacher in the middle school affiliated to Fudan University, thinks that it is particularly important to strengthen the understanding of mathematics itself if we want to do a good job in mathematics teaching, and advocates that as a careful teacher, we should read and sort out the doubts in the textbooks of all grades, go deep into mathematics research and find out the past lives with knowledge development. For the particularly important concept teaching, Dr. Zhang Jianyue of People's Education Publishing House once summed it up as "one definition, three attentions and several points", which does not talk about the background of the concept and does not experience the generalization process of the concept. It only lists the concept elements and matters needing attention in a logical sense, ignoring the mathematical thinking method reflected by the concept, which makes it difficult for students to reach a substantive understanding of the concept and form corresponding psychological significance. Without process teaching, the relationship between concepts can't be understood and the connection can't be established. As a result, students' mathematical cognitive structure lacks integrity, and their functional indicators such as usability, distinguishability and stability will be greatly reduced. In short, the above teaching has not left corresponding "thinking traces" and "psychological traces" for students. The author will call it "seamless" teaching. On the surface, the reason is that teachers are eager for quick success and instant benefit due to class time limitation and score pressure, because the relative process teaching is "speaking and imitating a lot of exercises". However, the author believes that fundamentally, as Mr. Li Qiuming said, the lack of teachers' understanding of concepts is the key reason why they can't "slow transition, brief introduction, generalize and refine, and stop at details", which enlightens us that a deep understanding of concepts is another focus of teachers' professional development.

Materialist dialectics holds that the whole and the part are dialectical unity. On the one hand, the whole is in a dominant position, commanding the local part, having some functions that it does not have, and some of them cannot be separated from the whole, which requires us to establish a global concept, base ourselves on the whole, make overall plans and achieve the optimal goal. On the other hand, the whole is composed of parts, some of which restrict the whole, and the functions and changes of key parts even play a decisive role in the whole function. It requires us to attach importance to the role and do a good job of the role.

Each part and the interrelation between each part presents a whole structure.

Let's enlarge the perspective of thinking, from thinking about solving problems to thinking about elementary mathematics knowledge and even mathematics. Of course, some ideas may still stay at the level of "feeling and experience", and more readers who are interested in them need to supplement, improve and enhance them.

In the new round of curriculum reform, the concept of "structure" has been widely valued. Interpretation of Mathematics Curriculum Standards points out: "Mathematical knowledge has a certain structure, which forms a unique logical order of mathematical knowledge", and "all mathematical knowledge can only be incorporated into its cognitive structure through students' own' re-creation' activities, and it can be called effective and useful knowledge"; Professor Cao of Beijing Normal University pointed out that "only by acquiring structured knowledge can students form a profound and true understanding of knowledge"; Professor Ye Lan of East China Normal University presided over 15 "New Basic Education" research, and summarized the basic characteristics of its "two-stage distance teaching strategy as" learning structure and practice structure ".

These expositions enlighten us that on the basis of learning all chapters of mathematics well, we should always find the relationship between them through thinking, so as to construct the corresponding mathematical knowledge structure and our own cognitive structure to understand these knowledge. From the perspective of knowledge application, we can call it "structural method" more popularly: the mathematical structure that simulates the structure of a certain kind of thing is the "mathematical model" that simulates this kind of thing, and learning mathematics is learning various mathematical models.

For a long time in history, philosophers and mathematicians have tried to unify mathematics. Of course, mathematics at that time was much simpler than today. In Pythagoras' time, there were only arithmetic and geometry. Today's mathematics is a forest of countless leafy trees. It has more than ten main branches: algebra, geometry, number theory, function theory, probability theory, operational research, calculation method, mathematical logic, graph theory and differentiation.

Pythagoras first tried to unify mathematics with natural numbers, but this attempt ended in failure due to the discovery of irrational numbers. For a long time to come, people hope to unify mathematics with Euclidean geometry. Finally, it was found that even geometry was not unified (so-called non-Euclidean geometry appeared), and people's hopes were dashed. Leibniz, Frege and Russell all hope to unify mathematics and logic. Huge, complex and rich mathematics is simplified into popular, intuitive and easy-to-understand logic. From this, an extremely unpopular, extremely complex and difficult-to-see theory and reducible axiom are derived. Intuitionistic Brouwer and formalistic Hilbert hoped that mathematics would be unified with arithmetic. The result is not even unified in arithmetic-this is the inference of Godel's theorem. After all these attempts to unify mathematics failed, mathematics.

Bourbaki, a group of French structuralist mathematicians, has created extremely valuable research results: the whole mathematics can be summarized into three parent structures: algebraic structure, sequential structure and topological structure (later generations supplemented the measurement structure), and all kinds of mathematical knowledge are nothing more than one of these three structures, or their substructures, or their synthesis, and put forward that "mathematics is to study structures". Topological structure is a mathematical structure used to describe spatial properties such as continuity, separability, proximity and boundary. Engels said, "Mathematics is a science that studies the spatial form and quantitative relationship of matter and its motion." In fact, these structures proposed by Bourbaki School are only the reflection of the relationships and forms in the real world in our minds. Algebraic structure: one operation comes from quantitative relationship; The order structure comes from one concept of time after another; Topological structure-continuity-comes from space experience.

The real number systems we have studied include addition, subtraction, multiplication and division, which are two interrelated algebraic structures; It has a size and is a sequence structure; Its continuity (for example, only real numbers can fill the whole number axis, and rational numbers do not have this feature) reflects the topological structure.