Coronavirus and Math Education – Partially Commissioned Collections
The virus that has hit us is driving rapid educational reform. especially at the higher levels of education. On this topic, you can write a longer essay, there will certainly be a stream of doctoral dissertations on the methodology of distance learning. From a certain point of view, this is a return to the roots and to the forgotten habits of self-study. So it was, for example, in the Kremenets secondary school (in Kremenets, now in Ukraine, which existed in 1805-31, vegetated until 1914 and experienced its heyday in 1922-1939). The students studied there on their own - only after they had learned did the teachers come in with corrections, final clarifications, help in difficult places, etc. e. When I became a student, they also said that we should acquire knowledge ourselves, that only order and send classes to the university. But back then it was just a theory...
In the spring of 2020, I am not the only one who discovered that lessons (including lectures, exercises, etc.) can be very effectively conducted remotely (Google Meet, Microsoft Teams, etc.), at the cost of a lot of work on the part of the teacher and just a desire "get an education" on the other hand; but also with some comfort: I sit at home, in my chair, and in traditional lectures, students also often did something else. The effect of such training can be even better than with the traditional, dating back to the Middle Ages, class-lesson system. What will be left of him when the virus goes to hell? I think… quite a lot. But we will see.
Today I will talk about partially ordered sets. It's simple. Since a binary relation in a non-empty set X is called a partial order relation when there exists
(Tadeusz Kotarbinski, 1886-1981, philosopher,
President of the Polish Academy of Sciences in 1957–1962).
- Reflexive, i.e. for each ∈ there is ",
- Passerby, i.e. if ", and ", then ",
- Semi-asymmetric, ie («∧«)→ =
A string is a set with the following property: for any two elements, this set is either "or y". Antichain is...
Stop, stop! Can any of this be understood? Of course it is. But has any of the Readers (knowing otherwise) already understood what is here?
I do not think! And this is the canon of teaching mathematics. Also at school. First, a decent, strict definition, and then, those who did not fall asleep from boredom will definitely understand something. This method was imposed by the "great" teachers of mathematics. He must be careful and strict. It is true that this is how it should be in the end. Mathematics must be an exact science (see also: ).
I must confess that at the university where I work after retiring from the University of Warsaw, I also taught for so many years. Only in it was the notorious bucket of cold water (let it stay that way: there was a need for a bucket!). Suddenly, high abstraction became light and pleasant. Set attention: easy does not mean easy. The light boxer also has a hard time.
I smile at my memories. I was taught the basics of mathematics by the then dean of the department, a first-class mathematician who had just arrived from a long stay in the United States, which at that time was something extraordinary in itself. I think she was a little snobbish when she forgot Polish a little. She abused the old Polish "what", "therefore", "azalea" and coined the term: "semi-asymmetric relationship". I love using it, it's really accurate. I like. But I do not require this from students. This is commonly referred to as "low antisymmetry". Ten beautiful ones.
A long time ago, because in the seventies (of the last century) there was a great, joyful reform of the teaching of mathematics. This coincided with the beginning of the short period of the reign of Eduard Gierek - a certain opening of our country to the world. “Children can also be taught higher mathematics,” exclaimed the Great Teachers. A summary of the university lecture "Fundamentals of Mathematics" was compiled for children. This was a trend not only in Poland, but throughout Europe. Solving the equation was not enough, every detail had to be explained. In order not to be unfounded, each of the Readers can solve the system of equations:
but students had to justify each step, refer to relevant statements, etc. This was a classic excess of form over content. It's easy for me to criticize now. I, too, once was a supporter of this approach. It's exciting... for young people who are passionate about mathematics. This, of course, was (and, for the sake of attention, I).
But enough digression, let's get down to business: a lecture that was "theoretically" intended for sophomores of the Polytechnic and would have been dry as coconut flakes if not for her. I'm exaggerating a little...
Good morning for you. Today's topic is partial cleaning. No, this is not a hint of careless cleaning. The best comparison would be to consider which is better: tomato soup or cream cake. The answer is clear: depending on what. For dessert - cookies, and for a nutritious dish: soup.
In mathematics, we deal with numbers. They are ordered: they are greater and less, but of two different numbers, one is always less, which means that the other is greater. They are arranged in order, like letters in the alphabet. In the class journal, the order can be as follows: Adamchik, Baginskaya, Khoinitsky, Derkovsky, Elget, Filipov, Gzhechnik, Kholnitsky (they are friends and classmates from my class!). We also have no doubt that Matusyak "Matushelyansky" Matushevsky "Matisyak. The symbol for "double inequality" has the meaning "befores".
In my travel club, we try to make the lists alphabetical, but by name, for example, Alina Wrońska "Warvara Kaczarska", Cesar Bouschitz, etc. In official records, the order would be reversed. Mathematicians refer to alphabetical order as lexicographic (a lexicon is more or less like a dictionary). On the other hand, such an order, in which in a name consisting of two parts (Michal Shurek, Alina Wronska, Stanislav Smazhinsky) we first look at the second part, is an anti-lexicographic order for mathematicians. Long titles, but very simple content.
1. Linear order: stations and stops on the Habovka-Zakopane railway from Podhale, built in 1899 (I leave the decoding of the abbreviations to the reader).
All such orders are called line orders. We order in turn: first, second, third. Everything is in order, from the first point to the last. It doesn't always make sense. After all, we arrange books in the library not like this, but in sections. Only inside the department we arrange linearly (usually alphabetically).
2. Linear order: when starting the car engine, we perform actions in a consistent order.
With larger projects, especially in team work, we no longer have a linear order. Let's look at fig. 3. We want to build a small hotel. We already have money (cell 0). We draw up permits, collect materials, start construction, and at the same time conduct an advertising campaign, look for employees, and so on and so forth. When we reach "10", the first guests can check in (an example from the stories of Mr. Dombrowski and their small hotel in the suburbs of Krakow). We have nonlinear order – some things can happen in parallel.
In economics, you will learn about the concept of the critical path. This is the set of actions that must be performed sequentially (and this is called a chain in math, more on that in a moment), and which take the most time. Reducing construction time is a reorganization of the critical path. But more about this in other lectures (I remind you that I am reading a “university lecture”). We focus on mathematics.
Diagrams like Figure 3 are called Hasse diagrams (Helmut Hasse, German mathematician, 1898–1979). Every complex effort must be planned in this way. We see sequences of actions: 1-5-8-10, 2-6-8, 3-6, 4-7-9-10. Mathematicians call them strings. The whole idea consists of four chains. In contrast, activity groups 1-2-3-4, 5-6-7, and 8-9 are antichains. Here's what they're called. The fact is that in a particular group, none of the actions depends on the previous one.
4. This is also a Hasse diagram.
let's go to picture 4. What's impressive? But it could be a metro map in some city! Underground railroads are always grouped in lines - they do not pass from one to another. Lines are separate lines. In the city of Fig. 4 is bake line (remember that bake it is written "boldem" - in Polish it is called half-thick).
In this diagram (Fig. 4) there is a short yellow ABF, a six-station ACFPS, a green ADGL, a blue DGMRT, and the longest red one. The mathematician will say: this Hasse diagram has bake chains. It's on the red line seven station: AEINRUW. What about antichains? There are they seven. The reader has already noticed that I double-underlined the word seven.
Antichain this is such a set of stations that it is impossible to get from one of them to another without a transfer. When we "understand" a little, we will see the following antichains: A, BCLTV, DE, FGHJ, KMN, PU, SR. Please check, for example, it is not possible to travel from any of the BCLTV stations to another BCTLV without a transfer, more precisely: without having to return to the station shown below. How many antichains are there? Seven. What size is the largest one? Bake (again in bold).
You can imagine, students, that the coincidence of these numbers is not accidental. This. This was discovered and proven (i.e. always so) in 1950 by Robert Palmer Dilworth (1914–1993, American mathematician). The number of rows needed to cover the entire set is equal to the size of the largest antichain, and vice versa: the number of antichains is equal to the length of the longest antichain. This is always the case in a partially ordered set, i.e. one that can be visualized. Hassego diagram. This is not quite a strict and correct definition. This is what mathematicians call a "working definition". This is somewhat different from the "working definition". This is a hint on how to understand partially ordered sets. This is an important part of any training: see how it works.
The English abbreviation is - this word sounds beautiful in Slavic languages, a bit like a thistle. Note that the thistle is also branched.
Very nice, but who needs it? You, dear students, need it to pass the exam, and this is probably a good enough reason for studying it. I'm listening, what questions? I'm listening, gentleman from under the window. Oh, the question is, will this ever be useful to the Lord in your life? Maybe not, but for someone smarter than you, for sure ... Maybe for critical path analysis in a complex economic project?
I am writing this text in the middle of June, the elections of the rector are going on at the University of Warsaw. I have read several comments from Internet users. There is a surprising amount of hatred (or “hatred”) towards “educated people”. Someone bluntly wrote that people with a university education know less than those with a university education. Of course, I will not enter into the discussion. I'm just sad that the established opinion in the Polish People's Republic is returning that everything can be done with a hammer and a chisel. I return to mathematics.
Dillworth's theorem has several interesting uses. One of them is known as the marriage theorem.fig. 6).
There is a group of women (rather girls) and a slightly larger group of men. Every girl thinks something like this: "I could marry this one, for another, but never in my life for a third." And so on, everyone has their own preferences. We draw a diagram, leading to each of them an arrow from the guy whom he does not reject as a candidate for the altar. Q: Can couples be matched so that each finds a husband she accepts?
Philip Hall's theorem, says that this can be done - under certain conditions, which I will not discuss here (then at the next lecture, students, please). Note, however, that male satisfaction is not mentioned here at all. As you know, it is women who choose us, and not vice versa, as it seems to us (I remind you that I am an author, not an author).
Some serious math. How does Hall's theorem follow from Dilworth? It's very simple. Let's look again at figure 6. The chains there are very short: they have a length of 2 (running in the direction). A set of little men is an anti-chain (precisely because the arrows are only towards). Thus, you can cover the whole collection with as many anti-chains as there are men. So every woman will have an arrow. And that means she can seem like the guy she accepts!!!
Wait, someone asks, is that all? Is it all app? Hormones will somehow get along and why math? Firstly, this is not the whole application, but only one of a large series. Let's look at one of them. Let (Fig. 6) mean not representatives of the better sex, but rather prosaic buyers, and these are brands, for example, cars, washing machines, weight loss products, travel agency offers, etc. Each buyer has brands that he accepts and rejects. Can something be done to sell something to everyone and how? This is where not only the jokes end, but also the knowledge of the author of the article on this topic. All I know is that the analysis is based on quite complex mathematics.
Teaching mathematics in school is teaching algorithms. This is an important part of learning. But slowly we are moving towards learning not so much mathematics as the mathematical method. Today's lecture was just about this: we are talking about abstract mental constructions, we are thinking about everyday life. We are talking about chains and antichains in sets with inverse, transitive and other relations that we use in the seller-buyer models. The computer will do all the calculations for us. He will not create mathematical models yet. We still win with our thinking. Anyway, hopefully as long as possible!
