Hera Sanju # Why do so many Chinese students not study math

There was once a joke that a Chinese elementary school student could skillfully tell you that 3+4=7, and a French elementary school student would tell you that 3+4=4+3 even though he didn't know that 3+4=whatever, because the set of integers (the original joke was the natural numbers) form an abelian group under the addition operation. This joke illustrates at least one point, that is, the Chinese and Western understanding of mathematics is different, the Chinese are from the arithmetic point of view to understand mathematics, emphasizing the calculation, so the goal of basic education in Chinese mathematics is to train problem solvers, while Europe and the United States is less emphasis on specific calculations, but more emphasis on formal logic, that is, the completeness of the subject axiomatic system and the rigor of logical reasoning from axioms to theorems and other conclusions .

This can be seen from the history of pi in general. Circumference has been discovered in many countries around the world, and the work that the Chinese have been doing since the discovery of circumference is to calculate the specific value of circumference, while the work that Europe has been doing is to prove that circumference is an irrational number. This was proved in ancient Greece, and the original geometry has a method of proof, and such a transcendental number as pi is irrational Aristotle recognized it, but could not prove it, so it was conjecture, and this proof took more than 2000 years until Euler's time. So we say that the reason why the value of circumference calculated by Zuchonsky led the world is because European mathematicians spent more time on proving that circumference is an irrational number, and cared relatively little about the specific circumference value. This is the difference between the Eastern and Western understanding of mathematics. The Eastern understanding of "mathematics" is the science of numbers, that is, arithmetic, and the Western understanding of "mathematics" is mathematics.

Sabine and Chinese mathematics brother Qiu Chengtong do a show, during which asked Qiu Chengtong an arithmetic problem (about addition, subtraction, multiplication and division), Qiu thought for a while, said frankly "we mathematicians do not know much about addition, subtraction, multiplication and division.

"These examples all illustrate that mathematics and arithmetic are actually different.

In fact, the difference between Chinese and Westerners' understanding of "mathematics" can be seen from the English translation of mathematics as mathematics. Mathematics, as the name implies, is the study of numbers, while mathematics in European languages does not have the meaning of mathematics, that is, arithmetic, in it.

mathematics: Etymologically, mathematics means ‘something learned’. Its ultimate source was the Greek verb manthánein ‘learn’, which came from the same Indo- European base (*men-, *mon-, *mn- ‘think’) as produced English memory and mind. Its stem form math- served as a basis of a noun máthēma ‘science’, whose derived adjective mathēmatikós passed via Latin mathēmaticus and Old French mathematique into English as mathematic, now superseded by the contemporary mathematical . Mathematics probably comes from French les mathématiques, a rendering of the Latin plural noun mathēmatica.

This English passage shows that mathematics, which comes from the ancient Greek word for "learning," is rooted in the Indo-European family of languages and means memory and thought, or the basis of learning. Therefore, mathematics translates to mathematics, reflecting the Chinese understanding of "mathematics" (that is, the study of numbers, arithmetic), but mathematics means much more than mathematics. Because it refers more to thought, that is, formalized thought, mathematics is the basis of learning. In the Chinese context, mathematics is more about memorization, just like in elementary school when you can do calculations by memorizing the 9-9 multiplication table, in high school most teachers still make students memorize mathematical formulas and theorems to facilitate problem solving. In Chinese traditional culture, mathematics or arithmetic is a cold discipline, and the traditional Chinese eminent studies are poetry, calligraphy, ritual, music, and spring and autumn, which is far less than the eminent studies of mathematics, which is the foundation of learning in Europe. The fact that mathematics has such a status in China today is due to the influence of Western learning in the late Qing Dynasty and afterwards.

In terms of the history of mathematical development, although all peoples in the world have developed similar operations of addition, subtraction, multiplication and division, as well as similar concepts of simple geometry, the use of abstract thinking methods to extract the common, essential characteristics from the many concrete perceptual miscellaneous, and to synthesize and unify, refining the object of thinking with general significance, that is, the concept, and based on this to form a complete system of axioms, and use Only the ancient Greeks understood mathematics as mathematics (i.e., the basis of learning), and the representative work is "Geometry". (That is why before his death, Chen Shoushin wanted to return to the homeland of mathematics, ancient Greece)

By logic, I mean formal logic, not logic in the colloquial sense. The first step in adopting formal logic in mathematics is to abstract and refine the most basic concepts of the discipline, and since the definition of concepts is generally axiomatic nowadays, the axioms are given along with the concepts.

For a discipline, the axiomatic system needs to be complete, which means that one more or one less axiomatic system will not work. A discipline, except for the axioms do not need to prove, other than the axioms of the things need to be derived from the axioms of the proof, the proof process follows Aristotle's trilogy, so as to form the whole complete discipline system. The logic of this process is called formal logic.

Then, using this formal logic as the standard, we can look back at the mathematics textbooks of primary and secondary schools, and even universities, to see if they conform to the formal logic? Obviously not, for example, secondary school to learn plane geometry, is Euclidean geometry, the axiomatic system to the full? Is there a proof of the axioms to theorems? Even if there is, is the proof complete? Obviously these are not.

Even to the university, draw a table, two vertical lines, and say that this is called determinant, this can be the definition? The so-called definition of Aristotle's formal logic said very clearly, is the genus plus species difference, draw a diagram to say that this is so and so, this is not called a definition.

In fact, in primary and secondary schools, and even universities, many people learn mathematics is basically not how to use the textbook. Why? Because the textbook test is not good. So we use more reference books and reference books of exercises, and then cope with the exam. The examination test to learn, not to learn the test.

Textbook logic has not been very strict, but at least it is also repeatedly argued by experts, and the logic of those reference books is completely crotch, but these reference books study a variety of types of topics, summed up the laws of questions and answer techniques, used to cope with the exam is really good. For example, the five-year college entrance examination three years simulation. So I said the basic education of mathematics does not focus on logic, the goal is to train the doer.

The first law of formal logic, the law of identity, requires that the meaning expressed by a concept cannot change during the argument, and if the meaning expressed by a language changes with the context, then it does not conform to the law of identity.

If a language's meaning does not change according to the context, it needs to be fixed by changes in the form of the language itself, for example, Indo-European languages have changes in form and number, tense and morphology, and subordinate clauses. For example, the Indo-European languages have changes in form and number, tense and tense, and subordinate clauses, which make it easier to ensure the meaningfulness of the ideology.

To take an example from Chinese, a white horse is not a horse. The word "non-" means "not" in Gongsun Long's case. But in the gatekeeper's place, non is the meaning of not belonging, white horse does not belong to the horse, then there is a problem. Because the word non does not distinguish between the semantics of "not" and "not belonging", it creates ambiguity between the two, and logically violates the same law, making white horses not horses a so-called sophistry.

Let's take another example from German, I don't love you.

ich liebe nicht dich

ich liebe dich nicht

Translated into Chinese, the first sentence I love is not you, the negation is you, the second sentence is difficult to translate into Chinese, nicht is not, placed at the end of the sentence, the negation is "I love you" full sentence.

This is why we say that German is rigorous.

The more context-dependent a language's ideology is, the less suitable it is for logical expression, and the less context-dependent a language's ideology is, the more suitable it is for logic, and the most suitable language for logical expression is a formal language, fully formalized and not dependent on any context.

Conversely, of course, the more context-dependent a language's ideology is, the better it is for literary expression, and Chinese is the ideal literary language. But in mathematics, science, and other fields that require logic, Chinese is a bit more difficult to use.

In fact, it involves the question of whether or not to do exercises to learn mathematics. In fact, in addition to mastering the logic of mathematics, the exercises after the class should definitely still be done, but that is, just the exercises in the textbook will be done, and the purpose of consolidating knowledge will be achieved.

Generally excellent textbooks have exercises that are selected by the author and are also coordinated with the knowledge of the textbook. Therefore, they have a principle of doing the problems in a precise way, not in a large number. But in Chinese primary and secondary schools, you will definitely not be able to cope with the exam by doing the exercises in the textbook.

Why is this? The reason is that in China, it is through the solution of the problem to select candidates for further education. 1978, the college entrance examination in the examination for more than 40 years, but the test knowledge is still the same content, can come out of the topic has long been out. Then the proposer can only dig in and keep changing new topics to open the gap between students, resulting in more and more difficult topics, but also more and more biased, more and more strange.

The end result is that not many students and teachers use textbooks, but rather use a variety of problem sets and exercises to study, completely deviating from the original intention of doing some selected topics to consolidate knowledge. And the result is that a large number of problem makers are cultivated, which is farther and farther from MATHEMATICS which focuses on systematic logic.

So we are going back to the original question, who is high and who is low in Asian, especially Chinese and European and American mathematics level, then from the perspective of problem makers, Chinese students are certainly generally stronger than European and American students, even if the students eliminated in the college entrance examination are stronger than European and American college students.

Because Europe and the United States college students do the exercises is the extent of their after-school exercises, while Chinese students can be said to be in a large number of exercises, including strange questions in the partial questions in the body of a hundred battles. Not to mention Europe and the United States, in fact, even domestic, such as Shandong, Hebei students, problem solving ability and calculation ability is also far better than the students in Shanghai and Beijing. (Of course, in the field of scientific research, the advantage of Shanghai and Beijing students is particularly obvious).

But looking back, in the field of mathematics, the model of exam-oriented education makes students farther and farther away from systematic mathematics and formal logic, and farther and farther away from creativity, so many students who stay in Europe and the United States will find that although they exceed most of the European and American students in terms of computational ability and problem solving ability, there are always many people in Europe and America who are In the field of mathematics, in the thinking and logic, in the ability to build the system, in the ability to prove the conjecture, you can not catch up even if you try hard.

I don't know if the students of science and technology have found a problem. It is that most Chinese textbooks follow an inductive method rather than deductive logic. Take linear algebra as an example, first on determinant calculation, then on matrix calculation, then on linear expressions and linear systems of equations, then on eigenvalues eigenvectors, and finally on linear spaces. Step by step from concrete to general, from image to abstract.

However, it is very obvious that the definition of matrix and determinant is not strict, and the logic is not strict. In fact, logic only deductive logic, there is no inductive logic. Because you can not exhaust all cases.

And many foreign textbooks are written backwards, as far as linear algebra is concerned, it starts with an axiomatic definition of linear space. The linear space is equipped on the domain, its axiomatic definition, that is, the linear space is first an Abelian addition group, and then define its multiplication with the domain of the number of operations to meet the four axioms, so that the definition of linear space is perfectly given out, and then axiomatically define the linear mapping, and define the multiplication of the mapping.

Then the matrix is induced by the one-to-one correspondence between the linear mapping and the matrix, and the multiplication of the matrix and the multiplication of the linear mapping also correspond, and finally the axiomatic definition of the determinant is given, and the determinant is a mapping, from the matrix to R, and satisfies several axioms.

For example, addition, inverse exchange, and unitary elements. This way of writing the textbook is typical of the way formal logic is written, characterized by starting from the axioms, from abstract to concrete, from general to general, very tight logic. The way of thinking is very clear, of course, there is another advantage is that we will find the so-called infinite series is composed of functions of linear space, such functions are also vectors, while the derivative and differentiation is also a linear mapping, the same we can construct linear space through linear mapping, then the same can be constructed through the derivative and differentiation operations of two linear space, so that the linear algebra and analytics unified. So is the common use of induction in Chinese domestic textbooks a reflection of the fact that they are better at computation than formal logic?