Problems of modern writing of mathematical texts

In a recent article comrade KvanTTT raised the question 3r3179. :
3r3182. Can you explain what you do not like in modern writing (mathematical positions and) formulas and how can it be improved? 3r3185. 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 3r33258. excessively [/s] detailed response.
I must say, the material is holivarny. Mostly too emotional. Very controversial. Too personal - often based on their own experience, not rich, albeit diverse. Fasting applies to school and university 3-333258. texts [/s] Textbooks: the "professional" literature has its own specifics, its own audience 3r3197. . There is no solution to the problem in the current realities. At the same time, a part of "my" observations long before me were expressed by
such authorities
like whip and hamming; a little less popular guys even wrote down the instruction " How to read maths ".
3r330. So, in my opinion, The main complaints are not so much about writing formulas, but rather about submitting material 3r3-395. . 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.
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. “The constant number a is called the limit of variants 3r3-33140. Problems of modern writing of mathematical texts 3r3142. if for every positive
3r3355. 3r3142. however small it may be, there is a number N such that all values ​​are 3r3-33140. 3r3142. whose number n> N satisfies the equality
3r361. 3r3142. . ”3r33254.  
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 the limit of a 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?
3r3196. False solution to the problem 3r3197. : 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 the original author solved it, why it is so important to kill a professor-watch for it - it falls.

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.
Pro live examples recently wrote 3r3179.

3r33939. 3r3394. I taught statistics and financial analysis

3r3196. I taught statistics and financial analysis for humanitarian people. It was quite difficult to see 30-90 pairs of empty eyes. I myself was beginning to stir up their hopeless misunderstanding of indexes, indicators and formulas. But, of course, the smart guys were, and one day I heard a guy explain something to fellow students: “Yes, you catch the essence! You came to the club and you think that all the girls are there, like Angelina Jolie. You go, and there the first leg is short, the second one has a short haircut, the third one has a fifth size, the fourth one has a zero one, the fifth one has a boyfriend, etc. And neither is Jolie, but you can collect her out of them. But in general, these are young girls with whom you can have a good time. And so far as they are far from ideal, determines the quality of your party. This is the essence of the dispersion - deviations of the heap tsiferok from the most important tsiferki ". It was beautiful, lively and fun. I took over the experience and a week later we had a projector with interesting presentations and examples, and the audience did not stupidly write down the chalk on the blackboard under the boo-boo-boo, and looked for examples. It was the best session in 2 years. 3r3197.
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 3r33253. Remember school? And the discriminant formula? And how is it proven /derived? One way: purely algebraic . Take the equation 3r3-3140. 3r3116. 3r3142. “Multiply each part by
3r3119. 3r3142. and add
3r3122. 3r3142. "( [i] Why exactly on these values?
), A few more transformations - and ready. After the discriminant, the 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 3-333140. 3r3142. . Any. ”And a series of examples begins with complication.
3r33140. 3r3141. 3r3142. 3r3143.
A lot of examples that are organically 3r3197. lead to solving the equation in 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 always visible clear sequence. 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 leaps from example to example, students were forced to jump over the abyss. How many people have failed and still fail during 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 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 in pairs /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 formalism - and really understood what was going on.
I do not know whether it is possible to massively use the [i] approach. general 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 set up 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 3r3178. Hamming
3r3182. I could learn which methods were effective and which were not. Attending meetings, I have already studied why some works are memorized, 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 must 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 that, 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 lectures [i] crowded with information r3r3197. …
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, aha). 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. 3r3196. And what if so? And if this condition clause is not met? What is needed to complement our concept to the semigroup? 3r3197. 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, an article about a compilation of TeXa in pdf was recently skipped in the CI process. 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 can you improve the flow of material in mathematical texts 3r325353.
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 that “they are old, they cannot be stored”. 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 an office that developed the graphic core of a 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 diagrams about algebra /geometry necessary for developing the same CAD systems 3r3258. (if there are such projects - call) [/s] .
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 ​​"3r33258. Product 3r-?359. The text should be convenient for 3r-?358. Client 3r-?659. Reader", and the forces of 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.
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