Problems of modern writing of mathematical texts

    In a recent article, comrade KvanTTT raised the question :
    Can you explain what you do not like in modern writing (mathematical positions and) formulas and how can it be improved?
    I tried to answer in one comment, but the size of the text field did not allow me to finish the calculations. This article is an overly detailed answer.

    I must say, the material holivarny. Mostly too emotional. Very controversial. Too personal - often based on their own experience, not rich, albeit diverse. The post concerns school and university texts of textbooks: the “professional” literature has its own specifics, its own audience . There is no solution to the problem in current realities. At the same time, a part of "my" observations long before me were expressed by such authorities as Knut and Hamming; a little less popular guys even wrote down the instruction " How to read mathematics ".

    So in my opinionthe main complaints are not so much about writing the formulas, but about the presentation of the material . Moreover, the presentation of material at virtually all levels of education, starting with school and ending with advanced science. The beginning of the current situation put Euclid, who said about the absence of the royal road in mathematics. The royal road has not been laid so far. Euclid managed, and we can.

    The first problem - the significance is not shown . Another gift from Euclid: "Give the questioner a penny if he is looking for benefits, not mathematics." The authors begin to introduce definitions, prove theorems and create other mathematics without explaining why it is needed at all. Example: a textbook on mathematical analysis from Fichtenholz . Read the first chapter: "from the school course you know about rational numbers, but the needs of mathematics compel us to introduce material ones ..." and rushed. What needs, what kind of mathematics, what does not suit the rational - yes, the dog knows it. "Obviously."

    Or another example from the same tutorial. "A constant number a is called the limit options$ x = x_n $ if for every positive $ \ varepsilon $ however small it may be, there is a number N such that all values $ x_n $whose number n> N satisfies equality$ | x_n - a |  <\ varepsilon $. "

    Most students do not understand the definition above, but six months later they get used to it. Even by the end of their studies, more students are not aware of why they needed the notion of a limit sequence. Similarly, for functions, integrals, series ... Fichtenholz describes some mathematical objects, sometimes gives particular examples - that's all. Well, yes, now I understand that the limits are needed, for example, for a correct description of the upper / lower amounts when introducing integrals, but before the integrals there are two more semesters!

    Or determinant, defined as an skew-symmetric multilinear function. Are you guys serious? The only adequate response of a first-year student to this definition of "what?" What is the benefit of this definition? I do not argue, there is a benefit, but can any freshman realize it?

    False solution to the problem: background. Manifested in all kinds of conferences. “The problem was posed by Jacob, studied by his disciple Abel, and the disciple of the disciple Cain, and one hundred or five hundred incarnations of Vishna.” What is the essence of the problem, why did the original author solve it, why is it so important to kill a professor-clock for her?

    The next problem - the authors do not pose real problems.

    In principle, it is similar to the previous one. Remember the course of probability theory. What tasks prevail there? "In the basket are 25 black and 10 white balls ...". Casino examples, card, D & D, economic - no, not heard. We will use the most politically correct examples, although probability theory has grown out of the study of dice.

    Live examples recently wrote Free_Mic_RS

    I taught statistics and financial analysis ...
    Я преподавала статистику и фин.анализ у относительно гуманитарных ребят. Это было довольно сложно — видеть 30-90 пар пустых глаз. Меня саму начинало мутить от их беспросветного непонимания индексов, показателей и формул. Но, конечно, сообразительные ребятки были, и вот однажды я услышала, как один парень объяснял что-то сокурсникам: «Да вы уловите суть! Вы пришли в клуб и думаете, что все девушки там, как Анджелина Джоли. Идёте, а там у первой ноги короткие, у второй короткая стрижка, у третьей пятый размер, у четвёртой — нулевой, у пятой есть парень и т.д. И ни одна не Джоли, но из них её собрать можно. Но в целом это молодые девушки, с которыми можно приятно провести время. И вот то, насколько они далеки от идеала, определяет качество вашей вечеринки. В этом суть дисперсии — отклонения кучи циферок от самой главной циферки». Это было прекрасно, живо и весело. Я взяла опыт на вооружение и уже через неделю у нас был проектор с интересными презентациями и примерами, а аудитория не тупо записывала под бу-бу-бу и стук мела по доске, а искала примеры. Это была лучшая сессия за 2 года.

    Math starts with a task. And the dead, one-sided tasks leave the impression that theory-belief only works with them. The intention of the authors is good: to give an example, and then go to the general. Abstracted from the example. But a few "live" examples would make the transition to abstraction much more useful. At least, I firmly believe that the reverse process (the transition from the abstract to the particular) would be much easier.

    Problem: excessive brevity and inconsistency

    Remember the school? And the discriminant formula? And how is it proven / derived? One way: pure algebraic . Take the equation$ ax ^ 2 + bx = -c $"We multiply each part by $ 4a $ and add $ b ^ 2 $"( Why exactly these values? ), A few more transformations - and that's it. After the discriminant, students are given discriminant-for-even-b. And then the formula Vieta. And still full squares. And a bunch of examples. And it is not always explained why we need all these methods.

    Now imagine the situation, the student is told: “today we will learn how to solve equations with$ x ^ 2 $. Any. ”And a series of examples begins with complication.

    $ x ^ 2 = 4 \\ x ^ 2 = 9 \\ x ^ 2 + 2x + 1 = 1 \\ x ^ 2 + 2x + 1 = 4 \\ x ^ 2 + 2x + 1 = 9 \\ 4x ^ 2 + 4x + 1 = 9 \\ 4x ^ 2 + 4x + 5 = $ 13

    A lot of examples that organically lead to the solution of the equation through full squares. Then it is already possible to introduce a discriminant (as a simple algorithm for solving equations, when students get tired to select full squares), and Wyeth with an even discriminant as “know-how”.

    A similar approach is used in textbooks. Alas, not all. And not a clear sequence is visible everywhere. According to rumors, some authors lost sheets of drafts in trams, and then replaced the lost pieces with expressions like “it is easy to show that ...”. As a result, instead of calm jumps from example to example, students were forced to jump over the abyss. How many people have failed and still fail for 10 + 6 years of school / university education?

    Personal example (requested in the original post). In the first year of matan I suffered. Calmly solving examples, he did not at all assimilate the theory. I asked a classmate for help with calculating the length of the curve through the integral. He took a bottle of beer, drew a random curve, straightened with infinitely small segments, singled out one such segment, added it to the triangle dl, dx, dy, and asked: "Do you remember the Pythagorean theorem?" Then everything was easy.

    I asked him: why is this not shown on pairs / in textbooks? He showed a couple of counterexamples, explained why the formalism in matane is needed - and I got flooded. I just read the theorem, singled out the main thing, wrote / solved trivial examples, then I dealt with the formalism - and really understood what was going on.

    I do not know whether it is possible to massively use the approachgeneral review => counterexamples => formalism . I do not know how much and what kind of theory / practice a student needs to recruit before a “breakthrough”, I can hardly imagine how to put pedagogical experiments on this topic, and how much work I will have to invest in research. But the memory of that explanation has been living for 10 years. And after all these years, I try to listeners to first give an overall picture, then show the problems, and then plunge into the details.

    You say my personal feelings may be wrong. Besides them, I have only similar ideas from Hamming :
    ... I could learn which methods were effective and which were not. Attending meetings, I have already studied why some works are remembered, and most are not. A technical person wants to give a very limited technical lecture. As a rule, the audience wants a broad lecture of a general nature and wants a much more general overview and introduction than the speaker wants to give. As a result, many lectures are ineffective. The lecturer calls the topic and suddenly dives into the details. Few people can keep track. You have to draw a big picture to explain why this is important, and then slowly expand the sketch of what has been done. Then more people will say, “Yes, Joe did it,” or “Mary did, I really see what it is about. Yes, Mary gave a really good lecture, I understand what she did. ” As a rule, people give a very limited, safe lecture; this is usually ineffective. In addition, many lecturescrowded with information ...

    Ideas in bulk

    I must say, my experience in teaching is extremely limited. You may have noticed that I limited myself to the school curriculum and mathematical analysis. Alas, these are the areas where I had the opportunity to touch theory with practice. I still do not understand the essence of the determinant in algebra, I do not realize the projective geometry, and only six months ago I began to penetrate with matrices (immediately after practice, yeah). A good illustration of the saying "theory without practice is dead."

    As I was told, in NMU a new concept was always introduced with a dozen questions. And what if so? And if this condition clause is not met? What is needed to complement our concept to the semigroup? Listeners were allowed to play with the subject. Get used to. I think that the experience of NMU should be carefully considered.

    Surely in the higher sections of mathematics, the “first example, then abstraction” approach will not work. So, examples “on a piece of paper” do not help to realize RSA. But the growing time of the program with an increase in the length of the key helps to feel the purely practical aspects.

    There is a fear that "ideal / greenhouse" school textbooks will lead to shock when working with the "tower". It seems like, "hardcore should be raised in his youth."

    It is quite difficult to develop courses, hoping that students already know something. The larger the required base, the greater the likelihood that something from the base of the student is misunderstood.

    They say the peak of the form of mathematicians is 30 years. After 30 it is already possible to load them with writing textbooks, giving the methodologist a special partner.

    Current technologies allow you to write texts with a command using git. On Habré recently jumped an article about the compilation of TeXa in pdf in the process of CI. I am sure that a team of authors with good tools can write much better textbooks.

    In addition to professors, teachers, students and schoolchildren in mathematics there are states. And regulations. And requirements. And certification. All of this affects textbooks, authors, teachers, and the quality of the presentation.

    How to improve the flow of material in math texts

    In the current (Russian) reality - no way. There are enthusiasts, there are professionals, there is no motivation.

    Professors of mathematics have enough of their tasks to write textbooks. Sometimes there is a lack of purely humanitarian skills; they don’t learn to write books at universities. Plus, professional deformation: “obvious” for a professor can be very heavy for students. Math teachers are busy. And papers. And tutoring. I will not say anything about the state. Almost did not come across his representatives, so there is nothing to say. Unless, I will mention the policy of replacing textbooks every three years. After school, I wanted to transfer my textbooks to the library, I was told “they are old, you cannot keep them”. Motivation to write good tutorials does not add such an approach.

    In other words, I don’t expect positive developments from the education system. I hope, of course, but not waiting. What helps out are glimpses of IT and other engineering. At one of the mathematical conferences I received a book on computer graphics from one of the participants. The author worked in the office developing the graphic core of some drawing system, and the material was quite good. The mathematics was not “pure”, applied, but the very fact of the existence of good educational material certainly pleases.

    Another approach: teachers from companies working in universities. Mathematical texts from these guys do not have to wait, the specifics are not the same. Unless, game developers will get together to write a manual on a theory, or they will write graphs about algebra / geometry necessary for developing the same CAD systems (if there are such projects - call) .

    Finally, there are various non-state educational platforms, such as the same Coursera. These guys can do anything, for they work for money, compete, and quickly get feedback. But they have their drawback: the format of data submission is different. They do not write texts directly.

    And what will all come in the future?

    Most interesting. Maybe everything will remain as it is. Maybe there will be a departure from the texts in mathematics. Or maybe the authors will be inspired by the idea of ​​“ product text should be convenient for the client’s reader”, and the pioneers will be able to reverse the tradition. Then in 30-50-100 years, we will have textbooks that are understandable to most readers.

    Upd1. Inserted a photo with the calculation of the length of the curve.

    Upd2. In the comments it is often mentioned that the text is devoted to the problems of teaching, and not to professional mathematics. The reason is simple: most of the “professional” works that I have seen in terms of the presentation of the material are no different from textbooks. At the same time, school / university literature is known to the majority in Habré, and “professional” - to the percentage.

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