Figure 1 Wu Wenjun
I would like to talk about the modernization of mathematics education today, for such a big, important and complex problem, it is not a person who casually says and counts, but should be fully discussed and brainstormed to produce a consensus.
When I say this modernization, I actually mean mechanization. Of course, this is also my personal subjective opinion, and there are many different views that can be seen.
The so-called mathematics education is relative to the study of mathematics, which are two very different categories. As a mathematical research, you have to pay attention to innovation, but as a mathematical education, you can't innovate casually, and it is best to avoid casual innovation, which is different in terms of impact: for example, mathematical research, if I write a big paper, I can win the Fields Medal, and what impact will this have on society and the economy? This can be said to be impossible to talk about, for mathematics itself, what big impact can you say this big paper can have? It's also hard to say. But mathematics education is different, and this influence can be great- if it is done well, it can make the future appearance of mathematics completely different; if it is not done well, it will cause many disasters.
Nor is this catastrophe a casual one. In fact, there have been such disasters. We experienced a cultural revolution from the mid-60s to the mid-70s. At about the same time, abroad, probably in France and the United States, there was also a "Cultural Revolution" in the field of mathematics. At that time, the so-called "new mathematics" was promoted in mathematics, that is, the mathematics that was considered "modern" at that time. This "modern" mathematics put the most active research at that time, such as topology, into some courses in primary and secondary schools, into some topological concepts, and many other reforms. In general, I say that their "new mathematics" movement or "modernization" in the "mathematical modernization movement" is extremely non-mechanized mathematics. As you may know, this modern new mathematical movement has caused a disaster in mathematics education. I think we were fortunate that it was the Cultural Revolution at that time, so we didn't keep up with this "Cultural Revolution" abroad. When our Cultural Revolution was over, their "Cultural Revolution" also ended in failure. Otherwise, if we follow them, we will also cause a disaster.
But in addition, we also have some aspects that cause some bad consequences. Of course, since I haven't had much exposure to schooling in decades, that may not be the case. It seems that there was a time when we in middle school either abolished or cut down analytical geometry to a very small extent. One consequence, then, is that when you study calculus in college, you have to spend a few months to complete this analytic geometry. I don't know if that's the case, but I said it was also close to a disaster, certainly not so bad. Therefore, the impact of mathematics education is very large, and it cannot be casually come. It's not the same as mathematical research, where I can go on a rampage in order to innovate. But mathematics education as an education you must be careful, very careful, can not be random. If you don't do it right, it will have great consequences.
I was completely disconnected from math education for decades, but to be honest, I've always been interested in math education. As for the implementation of mechanization, in the past, the conditions were not met, but now the conditions are quite mature, and the conditions in the main aspects are relatively mature. As I have just said, if we don't do it well, there will be such a danger as new mathematics. But on the other hand, there are dangers to the right as well as dangers to the left. I say that the danger of new mathematics is a danger arising from the introduction of extreme de-mechanization. But if you assume that mechanization is pursued, if you do not implement it properly, it can also create many dangers. So for this place, the right and left can have a very bad impact.
I will give you an example, to engage in so-called mechanization, of course, it is necessary to use computers, then computers are implemented in primary and secondary schools, which is of course very important. Computers started with dolls, and I'm very much in favor of it. But suppose that if you don't learn properly, you won't even be able to learn addition, subtraction, multiplication, and division, and when the students in elementary school learn to add, subtract, multiply, and divide, you say I have a calculator, I enter a number, and then I enter a number, and one plus and one multiplication will come out immediately. If you want to learn it like this, then the student will not only add, subtract, multiply and divide, but also the whole mathematics, science and technology in the future will not be able to learn. I said it was a very dangerous one, and that this danger probably wouldn't come. But there is another danger that can cause problems in the understanding of computers in various ways. Improper use can also cause a variety of harmful consequences. So, even though it is impossible for me to get involved, I would like to take this opportunity to share my views on this issue, because it is more important. Of course, there may be many errors in my words, especially those that do not correspond to the current teaching situation. But I think that even though I am not in his position, I still have to seek political affairs now. Let me now turn to my personal views.
Figure 2 Computer
The question of modernization in universities, or, in my opinion, the question of mechanization, is too early to talk about, and it may be more appropriate to talk about it in the 21st century. So I want to talk mainly about the modernization of mathematics in the scope of primary and secondary schools, or, in my opinion, the so-called mechanization of mathematics.
Before liberation, I taught middle school for many years, about 5 years, the middle school I taught was only junior high school and primary school junior high school, I only had half a year as if I had substituted classes in a high school, but the real high school class was not taught. I think there is a very different content between the content of primary and secondary schools and the teaching content of mathematics and the teaching content of universities. It is the content of the teaching of the university, which is basically, in my current words, non-mechanized, generally non-mechanical, and Western, the content of Western mathematics after the 17th century.
But for the mathematical content of primary and secondary schools, I would like to borrow a phrase called Storock (both engaged in differential geometry and the history of mathematics), who wrote a book called "A Brief History of Mathematics", which talks about this mathematics outside the West, which he named Eastern mathematics. He said that the mathematics in primary and secondary schools is very different from the mathematical flavor of university mathematics and mathematical research that we are now engaged in mathematics, and he calls it Oriental mathematics. It is named Oriental Mathematics, indicating that the mathematics in primary and secondary schools is derived from Eastern mathematics.
I said that this East should mainly refer to China. I say that he is very correct in saying that the teaching content of mathematics in the whole primary and secondary schools is basically Oriental, that is, Chinese, or mechanized. The content of university education is Western, generally non-mechanized; but most of the mathematical content of primary and secondary schools is Mostly, basically, and mainly Oriental, which is Eastern mathematics, mechanized mathematics.
So, let's examine now what kind of oriental, mechanized. First, let's look at primary school, in primary school, we first have to talk about four operations, addition, subtraction, multiplication and division, and so on. Secondly, there may be proportions, some I don't know, because I don't know the specifics, others, maybe some statistical common sense, I had to learn abacus for a while before liberation. That's generally the case, and there may be a lot of changes now. However, your basic content cannot be canceled, which is the most fundamental. In addition, before the liberation to the sixth grade of primary school, there was almost a whole year, engaged in the so-called four problems, I also taught junior high school, to junior high school, the first year of the first grade a whole year and then talk about four problems, that is, for this four problems, before the liberation of my time, students spent almost two full years to study, now I do not know the situation.
So what are the four conundrums? Let me give you a typical example, which is the same cage of chickens and rabbits. This reasoning process, it can be said that logical reasoning is very strict, thinking is very clever, if I want to say, this is calculated with some strange tricks, strange tricks.
Fig. 3 Chickens and rabbits in the same cage
Then in the two years of the sixth grade of primary school and the first grade of junior high school, you have to learn a lot of four problems and learn many strange tricks. If you learn more of these strange tricks, they do play a certain role in logical reasoning, thinking ability, etc., but what use will you learn so much? Of course, the more strange tricks you learn, the greater the skill, but in fact, in the future, you can use, I don't think you will encounter, the chance of encountering is minimal.
Probably not now, I don't know when it started, the past junior high school began to learn algebra from the second grade of junior high school, using algebra to deal with problems like this, four difficult problems became very easy. For many of the four problems of chicken and rabbit in the same cage, if you use the algebraic method to do it, it will become very easy. More importantly, although these four problems have created many strange tricks, you can't run far, you can't take off, you can't talk about it, you can't talk about it. But if you introduce algebraic methods, these things become unnecessary and bland. You can do it, and everyone can do it, and it doesn't take a genius to come up with many tricks to do it, and he can take off, not only can he run far and he can take off. So the four puzzles are replaced by algebra, which is completely correct, which is very important for mathematical education. I say you can't spend too much time on tricks, and it's wrong to consume too much time. But I'm not saying to completely rule out, you can not spend a full two years, but spend a few months, two or three months on these problems, and then talk about the algebraic method, make a comparison between these problems done with strange tricks and done with algebraic methods, I think this should be more effective, I don't know what the status quo is. I am very opposed to that kind of practice, you see he seems to have learned a lot of skills - this one is a clever, that one is a clever, in fact, he can't do anything big, you want to rise or take off at all.
This is what I was talking about in elementary school. This place I need to say, I said that the content in the elementary school is mainly based on four operations, and then further into the algebraic method, then these contents are all mechanized, this mechanized content can be easily learned in the elementary school, the main relationship is because it is mechanized, do not need to use clever ideas like the four problems, but can be carried out more mechanically. The point is that the methods of addition, subtraction, multiplication, and division are based on the numerical representation of the so-called bit-value system in the bit-value system. This bit value system is a big leap forward for the carry system, not a simple improvement. We know that the decimal and binary positions, and the two-entry 111 in the machine, the same 1 position is different, it represents a different meaning, assuming that if it is turned into a decimal, the rightmost 1 is 2^0, the middle 1 is 2^1, the left 1 is 2^2, and the same 1 represents a different meaning. The same symbol has different values due to different positions, which is called the bit value system. This bit value system is much broader than the simple position system, and it is generally not well understood for this point.
Figure 4
I would simply like to say that the carry system is only available in China. I'm not saying that there are no carry-ons in foreign countries, for example, the Maya peoples in Latin America have a 20-seat system, Egypt has a decimal system, and Babylon has a 60-seat system. But this position system is incomplete and incomplete. Only China's decimal system is complete, and we not only have a complete decimal system, but also the so-called bit value system, which is a very critical step, which is generally not easy to understand. So now I'm going to introduce a passage from Laplace, which goes something like this: "Now there's a very clever way to represent all the numbers in 10 letters and give them an absolute position value. "This is the bit value, that is, for each position, the three 1 positions are different, and different positions give it different bit values," this clever method came from India. ”
The last sentence is clearly wrong. The mathematical expression in India is simply incredible. The West tends to think that all mathematics from the East comes from India, and we leave it alone. He said, "Express any number in 10 numbers, each of which gives it a different bit value due to a different position." This clever approach came from India. The idea is very delicate and important, it seems so simple, and it is precisely because it is so simple that we cannot know its merits enough. But it is precisely because of its extraordinary simplicity and its extreme convenience that this method can be communicated to all calculations, and it can affect all calculations, so that our arithmetic system can be placed in the first place of useful inventions. As for how difficult it was to invent such a method, it can be seen from the fact that there are two of the greatest figures in history and antiquity, two geniuses, one archimedes and the other Apollonius, and the fact that two great geniuses like this cannot discover such an important thing shows that the discovery of such a method is by no means an easy task. This is a quote from Laplace.
Figure 5 Archimedes
I quote this because there may be many people who don't pay much attention to the bit value system and don't know much about the importance of the bit value system, so I'll explain it. I said that it was precisely because of the invention of the complete position system in ancient China and the creation of the bit value system that various operations such as addition, subtraction, multiplication, division, square, and so on were possible. In Greece, there was no prescription, no prescription, as evidenced by the book. How about addition, subtraction, multiplication and division, I don't know, because there is no book to prove it, I am not good to say casually.
It is precisely because of this bit value system that we can make arithmetic very simple, so that the operation and calculation become very simple, and can stand the test of history. I think it's at least two thousand years, or I think it's at least a few thousand years. Until now, can you cancel these things in primary and secondary schools? That's why I say it's completely up to the test of history.
Note: This article is transferred from the public account "Hele Mathematics"