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This article comes from the official account of Wechat: ID:fanpu2019, author: Jiang Shusheng

How to give a definition in mathematics tutorials is often worth studying. A good definition should reveal the nature of the concept at the "what" level, not at the "how" level.

The mathematical problems discussed in this paper are mainly related to mathematics education.

For the understanding of a mathematical concept, intuition, definition and expression are all needed, but they have different functions.

In the elementary course of primary school mathematics (specifically, the understanding of natural numbers), these three aspects are mixed together, both intuitive (counting by fingers, actually a lot of experiments have to be done). And to learn the method of numeration (and then can be calculated), and finally to form the concept of natural numbers. In this process, it is inevitable that there will be inappropriate practices, even detours and mistakes, but if the concept of natural number is finally formed, some shortcomings and mistakes in the learning process are beyond reproach. Just like when a child learns to walk, he will inevitably fall and crawl, stumble, and even get hurt, but as long as he finally learns to walk.

However, in recent years, some teaching methods that think they are clever teach children to learn to count and calculate from a very early age, ignoring or even neglecting intuition. As a result, children may win prizes in quick calculation competitions, but they cannot consciously apply mathematics to solve problems in life, let alone cultivate innovative ability. It's actually just a kind of vanity.

In the mathematics curriculum of middle school, the above three aspects are gradually separated, and the teaching method is significantly different from that of primary school.

First of all, let's look at the concept of irrational numbers. It is basically said in most textbooks in the early years and some textbooks today: first of all, examples are given to illustrate the existence of irrational numbers, that is, some "numbers" are not equal to the ratio of two integers. The most common is the length of the diagonal of a square with a side length of 1 (some textbooks give its irrational proof). Recognizing the existence of irrational numbers, we can further form the concept of real numbers, that is, the totality of rational numbers and irrational numbers. As for the expression of irrational numbers as infinite non-cyclic decimals, many textbooks do not talk about it, or only give specific examples for students to understand. Although this argument does not give a definition of real numbers, it is suitable for most students. In fact, most people have never seen the definition of real numbers in their lives, but this does not prevent them from using real numbers in their work, because the rigor of mathematics is guaranteed by mathematicians, and ordinary people can use them boldly.

However, if a student asks "what is an irrational number", it is precisely not satisfied with intuition, hoping to fundamentally understand the concept of real number, how should the teacher answer? Such a student is one in a thousand, and so is the middle school teacher who can answer such a question. The question is only whether one in a thousand students can meet a teacher in a thousand.

Some teachers will reply: "irrational numbers are infinite non-recurring decimals", which can be seen in some textbooks or extra-curricular books. However, "infinitely acyclic decimal" is only an expression of irrational numbers, not a definition. Philosophically, any definition must be aimed at an objective object, otherwise it may fall into a logical trap. A typical example is "the set of all sets". If this "definition" is introduced, the whole mathematical system will collapse. First of all, we need to understand that real number is an objective existence, and then we can talk about its expression.

There are at least two valid definitions of real numbers, one is separated by Dedekin, and the other is defined by basic narration. The two definitions are equivalent to each other, but their styles are very different, with the former having a strong geometric flavor and the latter having a strong algebraic flavor. From the perspective of number theory, real numbers are the localization of integers in the "Archimedes bit". If you want to understand the essence of real numbers, it is best to understand both definitions (it would be better if you can understand them from the perspective of number theory). However, these two definitions are not simple, and after the definition, we have to establish a variety of operations, size relations, limits and so on. For ordinary middle school students and even college students, the difficulty is quite high. Therefore, it is wise not to introduce the definition of real number in middle school mathematics course and university higher mathematics course.

However, it is very unwise to use "infinitely irrational decimal" as the definition of irrational number in middle school or university mathematics course. instead of making students understand, it will make many students think that they understand. As stated in [4]:

"I'm not afraid that I don't understand, but I'm afraid I still think I know it."

Let's take a look at plane geometry. There are many definitions in geometry textbooks, but these definitions are not "primitive". The original concepts such as point, straight line, plane and so on are only intuitively undefined, but they are defined by the axiom system. In modern language, geometric objects can be defined as a system of sets that satisfy some conditions (axioms). Insisting on defining straight lines, planes, etc., will not lead to good results. Fortunately, we have not heard of such a textbook.

However, in the current middle school mathematics textbooks, there are serious defects in the definition of many geometric concepts, such as taking intuition as a definition, or semantic ambiguity (see [2] for details).

Let's go back to the concept of real numbers. What is worth mentioning is the intuition of the number axis. Understanding real numbers as points on the number axis is an effective way for most students to understand real numbers (including irrational numbers). With the example of irrational numbers and the intuition of the number axis, the concept of real numbers can be effectively taught to ordinary students. In other words, geometric intuition is an effective way to understand real numbers, which is indispensable for middle school students.

There are some more difficult definitions for most students, such as probability. For such concepts, it is often wise to talk about intuition rather than definition. However, it is often necessary to give the expression and further give the "operation" (such as calculation) method. In this way, students can use these concepts to do innovative work, although they may not fully understand a concept in the end. In addition, it is also possible to improve the understanding of concepts through application.

In short, if students can understand, direct definition is the most effective for the establishment of mathematical concepts; and if most students can not understand, at least should not talk about false definitions, or deceive students.

There are also definition problems in college mathematics courses.

Let's take a look at the calculus tutorial first. If you look for any calculus (or mathematical analysis) textbook, you will see that the definition of integral (Riemann integral) is not simple. In the course of mathematical analysis, the integral of unary function is defined as an extraordinary limit, and Dabu and others are used to judge its existence, which is quite complex and difficult to solve. In calculus courses for non-math majors, this part is only simplified (in fact, it is Jerry-built), and its complexity remains basically the same, so it may not be easier to understand than mathematical analysis textbooks; on the other hand, there is no assignment for these contents, let alone exams (including graduate entrance examinations), which is a waste of time and gives students a headache.

By the way, the concept of integral in various versions of middle school textbooks is also written in this way, which is of course a bigger headache for middle school students, and even many middle school teachers do not understand it.

Having studied the theory of real variable function, we know that Riemannian product of unary function is equivalent to continuous almost everywhere. Intuitively speaking, it is not far from continuous function. In fact, the application of Riemann integral is mainly for continuous functions, at most piecewise continuous functions. For ordinary students, Riemann integral is actually only a definition of area, not to mention that this is not a general definition, for example, the area of an area surrounded by a general Jordan simple closed curve. It cannot be defined by Riemann integrals (it was known in Cantor's time that the curve might have a non-zero area). So, it takes so much time and effort to learn the Riemannian integral, just to learn the definition of the area of a particular case. However, most people have the intuition of area and do not need the definition of area. If you are concerned about the definition of area, you can look at Lebesgue integral or a more general definition, such as the definition of dimension and measure in dynamical systems. Therefore, in order to understand the concept of integral, at least for most students, it is better to limit it to the integral of continuous function.

If the integral of a continuous function is defined as a "directed area", it is easy to understand and does not take much effort. Specifically, for the continuous function f (x) on the closed interval [a, b], the graph surrounded by the straight line xfanta _ (x) and the curve y _ (f) (x) has an area, and the area above the line YQ _ 0 is regarded as positive and the area below is regarded as negative, so the total area obtained is called directed area. The directed area given by f (x) on [a, b] is called its integral, which is marked as

From this definition, it is not difficult to prove the Newton-Leibniz formula.

Some other basic properties of integral, such as (r, s are real numbers), partial integration method, commutative method and the integral of some elementary functions, can be easily proved by Newton-Leibniz formula (some can even be used as exercises). Using the auxiliary software made by Mr. Zhang Jingzhong, one class is enough to explain the basic concept of integral and Newton-Leibniz formula, which has exceeded the requirements of middle school curriculum standards. As for the original idea of Riemann integral, which is divided into vertical bars to make area and then take the limit, we can talk about it intuitively, it doesn't need to spend a lot of class hours, in fact, only a small number of students will pay attention.

Let's take a look at the linear algebra tutorial. "Vector" is one of the most important basic concepts. In many textbooks seen so far (some of which are early), vectors are defined as ordered arrays. Such a definition is not only difficult to understand (far from the definition in analytic geometry), but also the operation of vectors has to be defined separately. Generally speaking, you don't know what a vector is until you have learned a lot.

This definition is obviously flawed and does not reveal the nature of the vector. To put it in detail, an ordered array is the expression of a vector in a given coordinate system, at the "how" level, while a good definition should be at the "what" level.

From the "what" level, the vector is the element of the vector space, and it is meaningless to discuss the vector without the vector space. The operation of vectors involves multiple vectors and the relationship between them. So, to understand what a vector is, in the final analysis, you have to understand vector space.

However, there is no vector space in many linear algebra textbooks at all. Even if there are, many teachers do not talk about it. The common reason is that vector space is too "abstract" for students to understand. Then, many concepts and theorems based on vector space, of course, can not be talked about.

In fact, the concept of vector space is not very "abstract". Some foreign undergraduate algebra textbooks talk about group theory first and then linear algebra, which is obviously more "abstract" than linear algebra or higher algebra textbooks in China. On the other hand, middle school students in our country now have to spend a lot of time learning to gather, but there is no use in textbooks (except doing exercises). If you have some understanding of the cleverness of the concept of vector space, you will at least find the set useful. Therefore, it is not difficult for at least some students to understand vector space. For students with difficulties, it requires the patience of educators, for example, the following ways can be taken to teach.

Note that students learn about plane vectors and space vectors in analytic geometry and know some physical applications. In elementary mathematics and physics textbooks, we will generally talk about the intuition of vectors, that is, "quantities with both size and direction", and better textbooks will point out that this is only a kind of intuition, not both size and direction is a vector (such as electric current). Students can have a correct understanding of the vector through the physical meaning, although there is no concept of vector space. Then, from these intuitive concepts of the vector to the general vector space, in essence, the dimension can be unlimited. Therefore, we can first review the plane vector and space vector in analytic geometry, including their intuitive meaning and physical application, then systematically review and organize the operation of the vector, and then review and organize the expression of the vector in the Cartesian coordinate system. Then an example is given to illustrate that high-dimensional vectors also have mathematical and physical meanings. Thus leading to the general vector space, it is not very "abstract" and difficult to understand. Of course, this will take a little more time, but it is beneficial to the later study.

You might retort, "I don't find it confusing to define a vector as an ordered array, and it's easy and convenient." All right, let's move on. You are not going to stay at the vector level, so you need to understand the tensor. According to the "ordered array" approach, many textbooks define a tensor as follows: an n-dimensional r-order tensor consists of a set of numbers, where each footer takes an integer from 1 to n, so there is a total number of nr. If an n × n-matrix (aij) is used to change the coordinate system, each needs to make a coordinate transformation with the function of aij as the coefficient (the specific transformation formula is very complex and related to the conformal and inverse variability of the tensor. Do you still find this definition not difficult to understand and convenient?

If you understand the vector space, then you only need to go one step further to understand the tensor. For example, the tensor product of two vector spaces V and W over the field K is defined as the space composed of all K-bilinear functions on their dual space, which is isomorphic to, also isomorphic to. With tensor product, it is easy to define a tensor (see [3]). It is obviously much easier and much simpler to understand this directly from the "what" level.

It is also worth pointing out that generally speaking, we cannot say whether the definition is right or wrong (Yuri Zarhin once said helplessly, "Well,every definition is correct"), but only the pros and cons of the definition. A good definition can reveal objective existence or natural laws, enlighten thinking and guide meaningful research directions. In extreme cases, even a good definition solves the problem. Unfortunately, many definitions are flawed. Some textbooks take intuition as a definition, there is no scientific rigor to speak of, some are quite difficult to understand, or semantic ambiguity, or almost synonymous repetition (see [2]), these are misleading children. Some definitions are rigorous, but they have no background, are unnatural (there are artificial conditions), and in extreme cases even the defined things do not exist at all. Although some theorems can be deduced from this definition and papers can be written and published, it has no contribution to science and will not be applied, it is just a logic game. There is another kind of situation, although the defined object is objective and worthy of study, the conditions of the definition are complex or difficult to understand (as mentioned above, expression is used as the definition), which is especially unfavorable to beginners. Some of them can also lead to prejudice or psychological disorders.

As can be seen from the above, how to give a definition in mathematics tutorials is often worth studying. This is a subject of "educational mathematics" mentioned by Mr. Zhang Jingzhong (see [6]).

reference

Jiang Shusheng: talking about the particularity of mathematics education-- and how to deal with the relationship between mathematics and pedagogy. Mathematics Bulletin No. 4, 2008

[2] Jiang Shusheng: how bad is the current unified mathematics textbook for middle school (2016)

Li Kezheng: the Foundation of Abstract Algebra, Postgraduate Mathematics Series 6. Tsinghua / Springer Press (2007)

[4] Li Kezheng: the demand for the mathematical quality of labourers in modern society (2019)

[5] the reason: mathematicians win the world. Back to Pu net (2019)

[6] Zhang Jingzhong: talking about Educational Mathematics (2021)

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