How mathematical graphs manifest themselves in human life. Mathematics in everyday life. Mathematics in everyday life and work

In society, there is a point of view according to which all people in matters of intellectual knowledge have a tendency either to the mathematical pole or to the humanitarian one. The child goes to school, gets an A in literature, but mathematics is not given to him in any way. “Nothing,” parents say, “he is a humanitarian with us.” The reverse situation is also often encountered.

But how fair is that? Is mathematics objectively more difficult to master than the humanities? Are human abilities inherent genetically or are they the result of upbringing?

During the study Mathematicians were smarter than the humanities it turned out that if a student does well in exams in the exact disciplines, in most cases he also successfully copes with the humanities. And students in liberal arts schools fail not only in mathematics, but also in languages.

Does this mean that mathematical disciplines are more difficult? No.

If a person does well in all exams, this speaks of his responsibility, and not of his abilities. Many people can easily operate with abstract concepts and learn languages, but mathematics is very difficult for them. In addition, other studies show that there is no connection between the development of mathematical and humanitarian disciplines at the level of brain activity. These are completely different cognitive abilities.

The physiological basis of intellectual abilities

Research Origins of the brain networks for advanced mathematics in expert mathematicians scientists recorded the brain activity of mathematicians and other people during the performance of various tasks. As a result, they came to the following conclusion.

When performing mathematical operations, a person activates special areas of the brain that are not associated with language abilities.

It turns out that the difference between mathematical and humanitarian knowledge lies at the physiological level. There are zones responsible for mathematical thinking, and there are zones for linguistic thinking. It cannot be said that any of them is more perfect.

Nature and nurture

In the study mentioned above, scientists also concluded that the ability of children to perform simple algebraic operations is the key to further mathematical success. After all, at an early age, even before any education, parts of the brain develop differently in a person. Someone's mathematical zones are better developed, while someone's are worse.

Since the same neural network is involved in both elementary and more complex tasks, it is possible to predict a child's future talent even before it manifests itself. The kid quickly understood why 1 + 1 = 2? Then in the future it will be relatively easy for him to get sines and cosines.

The same can be said about the humanities. The speed with which a child learns a language, the ability to capture the basic laws of grammar, allow us to assess how good he will be in comprehending the humanities, since early success in this area indicates the potential of the corresponding area of ​​\u200b\u200bthe brain.

It can be assumed that physiological characteristics predetermine our cognitive abilities. However, this is not the case, and here's why:

• Many other factors that influence the manifestation of talent are not taken into account. For example, a person may have the makings of a mathematician at the physiological level, but at the same time there is absolutely no interest in this discipline, because of which his natural talent will not be developed.
• What we speak of as a physiological tendency may in fact be the result of early parenting.

As the Swiss psychologist and philosopher Jean Piaget notes Cognition, the development of both linguistic and mathematical cognitive abilities occurs in the preoperative period (2–7 years). It is then that the physiological predisposition of the child to a certain activity can manifest itself.

This period in the development of the brain is the most important, since the creation of neural connections proceeds according to the principle of the frequency of their use. About the features of brain development from conception to adolescence. That is, after 2-3 years, those of its zones that are most often involved begin to actively develop.

At this stage, the development of the brain directly depends on the activity of a person and the repetition of any practices.

The study of twins also sheds light on the formation of human abilities. The set of genes they have is approximately the same, and therefore the differences in intellectual abilities are likely to be driven by external factors.

Such studies conducted by Russian scientists in the 90s Where do smart kids come from?, showed that from the age of two, the intelligence of twins really becomes similar in relatively identical external conditions.

Approximately the same conclusion was made by scientists from the University of California at Santa Barbara. The high heritability of educational achievement reflects many genetically influenced traits, not just intelligence. The external environment matters and plays the role of a condition for the implementation of the biological basis.

conclusions

Whether a person becomes a humanist or a mathematician depends on the biological factor and heredity that determine the development of his brain. However, the manifestation of this factor is strongly influenced by activity in childhood. We are talking about the period when a person has not yet directly begun to study the disciplines themselves, but in the process of playing and communicating with parents somehow involves different areas of the brain, stimulating their development.

In practice, this means the following: parents should not impose on the child an activity for which he does not have a special attraction and in which he is not very successful. You need to try to find talent and contribute to its development.

Many people often ask why do you need math?. Often the very fact that this discipline is included in the mandatory curriculum of universities and schools, puts people at a loss. This bewilderment is expressed in the following: Like, why should I, a person whose future (or current) profession will not be connected with making calculations and applying mathematical methods, know mathematics?

How can this be useful to me in life? Thus, a large number of people do not see any sense for themselves in mastering this science, even on an elementary basis. But I'm sure the math, more precisely mathematical thinking skills needed by everyone and everyone. In this article I will explain why I am so sure of this. First, I will tell you why this discipline, as scientific knowledge and method, is needed in general and where its place is in the system of all natural sciences and how it is applied in practice.

If you already know this, but still wonder why you personally need to study this discipline, then go straight to the second part of the article. There I will talk about what personal qualities helps to develop mathematics, and what will you lose if you refuse to master this subject, at least at a basic level.

The place of mathematics in the system of sciences

Mathematics is a fundamental science, whose methods are actively used in many natural disciplines, such as physics, chemistry and even biology. By itself, this area of ​​knowledge operates with abstract relationships and relationships, that is, with such entities that in themselves are not something material.

But nevertheless, as soon as mathematics enters the field of any science about the world, it is immediately embodied in the description, modeling and prediction of quite specific and real natural processes. Here she finds flesh and blood, coming out from under the cover of idealized formulas and calculations cut off from life.

Mathematics is a tool for understanding the world

It is an exact science that does not tolerate arbitrariness in interpretation and various speculations. This is the embodiment of order and rigid logic. It helps to understand the world around us, to learn more about its laws, since these laws are subject to the same order that reigns in mathematics!

We can successfully translate the language spoken by nature into the language of mathematics and understand the structure of the relationships of a phenomenon. And, after we formalize these connections, we can build models, predict the future states of the phenomena that these models describe, only on paper or inside the memory of computers!

Einstein, in response to a question where his laboratory was located, smiled and pointed to a pencil and paper sheet.

His formulas for the theory of relativity became milestone on the way to understanding the universe in which we live. And this happened before man began to explore space and only then experimentally confirmed the correctness of the equations of the great scientist!

Application in modeling and forecasting

Thanks to the application of mathematics, we do not need to conduct costly and life-threatening experiments before realizing any complex project, for example, in space exploration. We can calculate in advance the parameters of the orbit of a spacecraft launched from the earth to deliver astronauts to the orbital station. Mathematical calculations will make it possible not to risk people's lives, but to estimate in advance all the parameters necessary for launching a rocket, ensuring a safe flight.

Of course, a model is a model because it cannot take into account all possible variables, which is why disasters happen, but it still provides fairly reliable forecasts.

You can see the embodiment of mathematical calculation everywhere: in the car you drive, in the computer or portable device from which you are currently reading this article. All buildings, buildings do not collapse under their own weight due to the fact that all the data necessary for the construction were calculated in advance using formulas.

Medicine and healthcare also exist thanks to mathematics, which is used, firstly, in the design of medical devices, and secondly, in the analysis of data on the effectiveness of a particular treatment.

Even the weather forecast is not complete without the use of mathematical models.

In short, thanks to mathematics, we have all the technologies available to us today, do not expose our lives to senseless danger, build cities, explore space and develop culture! Without her, the world would be very different.

Why do people need mathematics? What skills does she develop?

So, we found out that mathematics is one of the most important achievements culture and civilization. Without it, the development of technology and the knowledge of nature would be unthinkable things! Well, you say, let's say this exact science is really extremely important for humanity as a whole, but why do I need it personally? What will she give me?

Mathematics develops mental abilities

Mathematics allows you to develop some important mental qualities, which I wrote about in an article about the development of intelligence (). These are analytical, deductive (ability to generalize), critical, prognostic (ability to predict, think several steps ahead) abilities.

It also improves the ability abstract thinking(after all, this is an abstract science), the ability to concentrate, trains memory and enhances the speed of thinking. That's how much you get! But at the same time, you or your children can lose a lot if you do not pay due attention to this subject.

Speaking in more detail and operating with specific skills, then mathematics will help a person develop the following intellectual abilities

• The ability to generalize. Consider a particular event as a manifestation of a general order. The ability to find the role of the particular in the general.
• Ability to analyze difficult life situations, the ability to make the right decision of problems and be determined in the face of difficult choices.
• Ability to find patterns.
• Ability to think and reason logically, competently and clearly formulate thoughts, draw correct logical conclusions.
• Ability to think quickly and make decisions.
• Planning ahead skill, the ability to keep several consecutive steps in mind.
• Conceptual and abstract thinking skills: the ability to consistently and logically build complex concepts or operations and keep them in mind.

An important point: I have already received a number of questions from readers, so I want to clarify something here right away. The above qualities are developed not only by solving problems from different areas of mathematics: trigonometry, probability theory, etc. You do not have to find dusty school textbooks in these subjects at all if you want to improve these abilities.

Here I am talking not only about mathematics as a specific science, but rather about all those areas of knowledge where the mathematical method is applied and precision, order and logic prevail. So for the development of some qualities of intelligence, studying the exact sciences, solving logic puzzles, and even some are suitable.

Take what is closer and more interesting to you, there is no need to force yourself to study boring textbooks, the main thing is that the head works, so that the tasks require you to search for non-trivial solutions and the accuracy of the analysis. I write about it right away so that it will be clear what I'm talking about.

Mathematics is essential for the development of a child!

Mathematics is especially important for the development of a child! It sets the standards for correct, rational thinking for the rest of your life! Gives a huge boost to mental development.

I don’t even know what other school subject is capable of so raising the mental level of a growing individual and serving as such a good help for intellectual development later, already in adulthood. I do not mean mathematics only as a subject, algebra or arithmetic, I am talking about the application of mathematical methods in general, including physics, geometry, computer science, etc.

Mathematics Organizes, Organizes, and Optimizes Your Thinking

I will begin this paragraph with the famous saying of Lomonosov, the great scientist who achieved success both in the natural sciences and in the field of the humanities - the rarest case of a universal mind. He said: “Mathematics only then needs to be taught, that it puts the mind in order.”

Mathematics trains, such mental qualities that form the framework and skeleton of all your thinking! This is, first of all, logical ability. This is all that organizes all your thoughts into a coherent system of concepts and ideas and the connections between them.

Mathematics itself is the epitome of natural order, and it is not surprising that it orders your mind. And without this notorious logic in the head, a person is not able to draw correct logical conclusions, compare concepts of various kinds, he loses the ability for sound analysis and reasoning. What can lead to the phenomenon of "porridge in the head", confusion in thoughts and reasoning, indistinctness of the argument.

It is easy to mislead such a person, which actually usually happens, since he is not able to identify a clear violation of logic in the statements of all schemers and charlatans (Already the second paid experience with financial pyramids in our country suggests that a huge part of people believe that they don't need math. Knowledge of mathematics does not allow you to deceive!

So it's not only calculations and formulas, it's primarily logic and order! It is a set of rules and functions that make your thinking consistent and logical. This is reflected in your ability to reason, formulate thoughts, hold complex concepts in your head and build intricate relationships.

Why do humanities need mathematics?

Which will certainly come in handy for you, even if you are going to succeed on the basis of some humanitarian discipline, since logic, systemic thinking skills and the ability to formulate complex theories are very necessary there too. Without this, it will not become science, but verbiage.

I have heard of brilliant lawyers who, in addition to legal education received, in addition, physical and mathematical. This helped them, like good chess players, build complex combinations of defense options in court, or invent clever ways to interact with the legislative framework and come up with all sorts of ingenious and non-trivial solutions.

Of course, it is not at all necessary to receive a specialized specialized education in mathematics, even, in my opinion, redundant if you are not going to work in this area. But to master this discipline at the basic level of school education and elementary courses of the university, I believe, everyone should and is capable of.

You should not think that this is not given to you by nature, that your vocation is the humanities and you are not able to teach exact subjects. When someone says that he humanitarian mindset and, therefore, he cannot, in principle, count, read formulas and solve problems, no matter how much he wants to, then know that this is such an elegant attempt to justify the fact of the lack of development of mathematical abilities. Not their absence! But only the fact that these skills, for some reason, have not received proper development.

The human mind is a universal thing designed to solve a variety of problems. Of course, this statement has its limits: everyone, due to the peculiarities of their innate and acquired properties of thinking, has certain inclinations to master different sciences. In addition, specialization most often requires knowledge of one thing: it is difficult to be an excellent mathematician, chemist, lawyer, teacher in one (not all of us are Lomonosovs). There will always be something to choose from.

But everyone can master the basic skills of mathematical thinking! For some, it will simply be more difficult, for others it will be easier. But this is for everyone. And as I said, this is necessary for balanced development of your mind. From what you are interested in, for example, literature or psychology, it does not follow that you do not need mathematics and you are simply not capable of mastering it somehow by nature!

One does not exclude the other, but, on the contrary, harmoniously complements. "Humanitarian mentality" in the context of the impossibility of mastering the exact sciences is just one big nonsense and an attempt to justify the reluctance to master those skills that are given with more difficulty than others.

Why is mathematics important in life and work?

Math is useful in business. But maybe the profession that you consider as your future calling will not be related to calculations, formulas, computer science or analytics. Or you don't use it in your current job.

But still, this does not mean that it will always be so. Perhaps you want to change your profession. Or you get so bored with hired work that you decide to start your own business (and this happens quite often). The organization of an independent enterprise always requires calculations, forecasting and analysis. You, as the head of a new business, will need to have the appropriate skills, not everything is possible to delegate to hired employees, their work in any case needs to be controlled.

Without support in the form of mathematical methods of forecasting, modeling and analysis (at least at a primitive level, depending on what kind of business you have), it is difficult to achieve success in organizing your own business. Based on personal statistics, I can say that, as a rule, graduates of technical and mathematical universities achieve the greatest success in business.

It's not just a matter of knowing special techniques calculations, because it is never too late to master it if necessary. The key is in a certain organization of the mind. Business is a highly ordered system, the construction of which requires from its creator certain intellectual skills, structured thinking, the ability to generalize and derive relationships. The study of the exact sciences, as you know, develops these skills.

Conclusion

Mathematics and others exact sciences are very important both for the development of mankind as a whole, and for the intellectual improvement of a particular individual. Of course, a balanced mental development of a person implies the development of not only exact subjects, but also humanitarian disciplines. Reading quality literature, for example, is also essential if you want to develop.

But, this alone is not enough. I would like to supplement the wording of the well-known statement: "if you want to become smart, you need to read a lot", adding to this: "- and doing mathematics." Otherwise, the effect of just reading books will be like a body without a skeleton or a building without a frame. It's hard for one without the other.

That is why many humanists, no matter how well they understand their subject area, suffer from confusion of thinking and lack of sober judgment, and many avid mathematicians and techies become isolated in the world of abstract formulas and calculations, losing touch with the real world.

The golden rule is everything is good in moderation, the destiny of a harmoniously developed mind, universality at the most basic level! All together and books and mathematics! This is not a sermon for the glory of amateurism, no, in your specialization you must be a professional and a narrow specialist, an expert in your field. But as for your basic erudition and knowledge, there should be a little bit of everything.

I believe that the idea of ​​school education and teaching in the primary courses of universities meets this principle of universality (only an idea, I don’t presume to discuss how this is implemented in practice). I would be extremely negative about strengthening the specialization of primary and secondary education, believing that the growing individual should be given as much as possible from different areas, and when he receives it, let him choose what is closer to him!

Mathematics in human life

Have you ever heard such an expression: mathematics is a country without borders? This phrase about mathematics has a very good reason. Mathematics occupies a special place in human life. We are so close to it that we simply do not notice it.

But our life begins with mathematics. The child has just been born, and the first figures in his life are already being heard: height, weight. The kid grows up, cannot pronounce the word "mathematics", but is already doing it, solving small problems of counting toys, cubes. And parents do not forget about the tasks. When preparing food for a child, weighing him, they have to use mathematics. After all, you need to solve an elementary problem: how much food you need to cook for the baby, given his weight.

There are many mathematical problems at school and their complexity grows every year. They do not just teach the child certain actions. Mathematical tasks develop thinking, logic, a set of skills: the ability to group objects, reveal patterns, determine relationships between phenomena, make decisions. Mathematics, solving mathematical problems develops a personality, makes it more purposeful, active, independent.

And after school, mathematics is very useful. While studying at the university, at work and at home, you need to constantly solve problems related to mathematics. What is the probability of passing the exam? How much money do you need to earn to buy an apartment? What is the surface area of ​​the walls of your house, and how much brick should you buy to insulate your house? How to calculate correctly so that a girl or a boy is born? And this is where math comes in. It follows a person everywhere, helps him solve practical problems, makes his life much more convenient.

The world and life itself is rapidly changing. It includes new technologies. Only mathematics and problem solving in the traditional sense do not change themselves. Mathematical laws have been verified and systematized, so a person in important points can rely on her to solve any problem. Math won't let you down.

National Action Plan 2012-2016 for Development functional literacy schoolchildrenpays special attention to such basic competencies as literacy in reading, mathematics, and science.

What is the purpose of mathematics education?

University preparation.

Preparing for future profession.

Intellectual development.

Formation of the worldview.

Orientation in the environment.

Physical education of the brain.

Here are some motivations regarding the importance of mathematical education for the individual.

Mathematics is found and used in everyday life therefore, certain mathematical skills are needed for every person. We have to count, for example, money in life. We constantly use, often without noticing it, knowledge about the quantities that characterize the extent, area, volume, time intervals, speeds, and much more. All this came to us in the lessons of arithmetic and geometry and came in handy for orientation in the world around us.

Mathematical knowledge and skills are necessary in almost all professions, first of all, of course, in those related to the natural sciences, technology and economics. But there is no doubt that it is necessary to use mathematical knowledge and mathematical thinking to a doctor, linguist, historian, and it is difficult to cut off this list, mathematical education is so important for professional activity Nowadays. Hence,mathematics and mathematical education are needed to prepare for a future profession . This requires knowledge of algebra, mathematical analysis, probability theory and statistics.

Philosophical comprehension of the world, its general patterns and basic scientific concepts is also not possible without mathematics. And that's whymathematics is necessary for the formation of a worldview .

Mathematics should contribute to the development of the ethical principles of human society. Its study is designed to educate in a person intellectual honesty, objectivity, the desire to comprehend the truth,it also brings up the ability to aesthetic perception of the world, the beauty of intellectual achievements .

“Mathematics already then needs to be taught, that it puts the mind in order,” - M.V. Lomonosov. Not only arms, legs, body require training, but alsothe human brain needs exercise . Solving problems, puzzles, mathematical puzzles develops logical thinking, speed reaction. No wonder they say that mathematics is the gymnastics of the mind.

Mathematics teacher of KSU "Kokpekty secondary school" Germash E.A.

Back forward

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Members: 7th grade students.

Goals:

• educational: the formation of a sustainable interest in mathematics;
• educational: the formation of such personality traits as cognitive activity.
• developing: development creativity students (imagination, observation, memory), monologue speech, the ability to identify cause-and-effect relationships, the development of logical thinking.

Tasks:

• study bibliographic sources on the topic;
• to introduce the history of the emergence and development of mathematics
• identify areas of application of mathematical knowledge.

Products: computer presentation.

Necessary equipment: projector, screen, computer.

Event progress

Introductory speech of the teacher:

1 slide Topic: "Mathematics in human life"

2 slide Fundamental question: Does a person need mathematics?

3 slide Problem questions:

• How and when did mathematics originate?
• What professions need mathematics?
• What mathematicians do you know?
• Do you need knowledge of mathematics modern man?

Student presentation:

To drive ships
To fly into the sky
There's a lot to know
And at the same time, and at the same time,
You notice,
Very important science
Mathematics!

Why ships
Don't run aground
And they're on course
Through fog and blizzard?
Because because,
You notice,
Helps captains
Mathematics!

So that a doctor, a sailor
Or become a pilot.
We must first of all
Know mathematics.
And there are no professions in the world
You notice,
Wherever you need
Mathematics!

4 slide How and when did mathematics originate?

When it comes to something very simple, understandable, we often say: "The matter is clear, like two times two - four!".

But before you think of the fact that twice two is four, people had to study for many, many thousands of years.

Of course, this teaching did not go behind the desk. Man gradually learned to live: build dwellings, find a way on long trips, cultivate the land.

Because even in the most distant times, when people lived in caves and dressed in animal skins, they could not do without counting and measuring.

Many rules from school textbooks of arithmetic and geometry were known to the ancient Greeks more than two thousand years ago.

Other ancient peoples - Egyptians, Babylonians, Chinese, peoples of India - in the third millennium before our era had knowledge of geometry and arithmetic, which some students of the fifth or sixth grade lack.

With every decade, mathematics became more and more necessary for people.

5 slide Pythagoras

The great scientist Pythagoras was born around 570 BC. on the island of Samos. Pythagoras' father was Mnesarchus, a gem-carver.

Pythagorean theorem- one of the fundamental theorems of Euclidean geometry, establishing the relationship between the sides of a right triangle. It is believed that it was proved by the Greek mathematician Pythagoras, after whom it is named.

The theorem goes like this: IN right triangle the square of the hypotenuse is equal to the sum of the squares of the legs .

6 slide

In the late nineteenth century, various suggestions were made about the existence of human-like inhabitants of Mars. Jokingly, though not entirely unreasonably, it was decided to send a signal to the inhabitants of Mars in the form of the Pythagorean theorem. It is not known how to do this; but it is obvious to everyone that the mathematical fact expressed by the Pythagorean theorem takes place everywhere and therefore inhabitants of another world similar to us must understand such a signal.

7 slide

Sofia Kovalevskaya

A girl from a noble family loved mathematics and even at night hid a difficult problem book under her pillow (her parents did not approve of her hobbies).

At that time, it was not customary for women to go to college, but she went against the will of her parents to Germany, to the university, and came to a famous professor. He did not want to take her and, in order to get rid of it, he gave several tasks that he himself had compiled, saying that if she decides, she will take her to her.

These problems could not be solved even by professors. The girl decided in twenty minutes.

Sofia Kovalevskaya graduated from the university and became a world-famous mathematician

8 slide

What can mathematics do?

• It helps the astronomer determine the paths of distant stars.
• An engineer uses mathematics to design a jet plane, a ship, or a new power plant.
• To a physicist, mathematics reveals the laws of the atomic nucleus, and to a sailor it shows the path of a ship in the ocean.
• In a word, mathematics can do everything or almost everything where something needs to be calculated.

But everything starts with mathematics.

• The child has just been born, and the first figures in his life are already being heard: height, weight.
• The kid grows up, cannot pronounce the word "mathematics", but is already engaged in it, solves small problems of counting toys, cubes.
• And parents do not forget about mathematics and tasks. When preparing food for a child, weighing him, they have to use mathematics.
• After all, you need to solve elementary tasks: how much food you need to cook for the baby, given his weight.

9 slide

1 example

You stand at the checkout and pay for the goods. You bought food for 432 rubles, and you have 500 rubles in banknotes of 100 rubles. And they give you change of 40 rubles, although they should give you 68 rubles. So you were shortchanged by 28 rubles !!!

10 slide

2 example

I need to be at the dacha at 15.40. I spend 1.40 hours on the road. Today I have to go to the store. When should I leave? How much time can I spend in the store?

11 slide

12 slide

Solve the problem.

How to get 100 with one action and five units?

13 slide

• 111 - 11 = 100

14 slide

Where can you do without mathematics?

• The builders are building a house. It is necessary to calculate how much cement, how many bricks. Height, width. Compose the project.
• Here the dressmaker is going to sew a dress. Measures a person, makes a pattern. Does she need math? Maybe…
• The store considers the received goods, revenue.
• The bank counts money, dealing with huge amounts, with interest.
• Even in music, in poetry, you have to count - rhythm, size, eighths, quarters, iambs, choreas.
• What can we say about such complex sciences as space (rockets, satellites), computer technology, television, radio! Of course, none of this would have been invented without calculations, without mathematics.
• That is, mathematics is our whole life?

15 slide

The task of applying the sign of equality of triangles to measure the distance between two inaccessible objects .

Condition: The road-laying team has to make a tunnel, but the distance to be cut through the mountain is not known. What should the team do to find out this distance, if the distance from A to C and from B to C is known (Fig. 1)?

Picture 1

Solution: The brigade cannot make a road around the mountain. Therefore, they undertook a little trick: at the entrance to the not yet cut tunnel they put a person - (A) and at the exit too - (B), they put a third person on the side of the mountain - (C), a triangle ABC was formed. Person A draws a straight line through point C, and person B also draws a straight line through point C. After drawing straight lines and placing two more people on them at a certain distance - (D,e) So CD=AC, A SW = EU.Corner ACB=ECD by the property of vertical angles, so triangle DEC is equal to triangle ABC. Now the brigade connects points D and E with a segment on the ground. It remains for the workers to measure the distance from E to D, which will be equal to the desired distance from A to B.

16 slide

Is knowledge of mathematics necessary for a modern person?

The world and life itself is rapidly changing. It includes new technologies. Only mathematics and problem solving in the traditional sense do not change themselves. Mathematical laws are checked and systematized, so a person can rely on it at important moments, solve any problem. Math won't let you down.

But every year we have more and more wonderful machines: complex machine tools, various automatic machines. In order to work well on such machines, you need a lot of knowledge. Now mathematics is needed not only by a scientist or engineer, but also by a foreman and a factory worker.

However, even a few decades ago, there were many such problems that were almost impossible to solve, although mathematicians knew how to solve them. It happened that dozens of people worked for several years to solve a single problem. The calculations were slow. The main "tools" of a mathematician were the same as in the days of the ancient Greeks - his own head and a blank sheet of paper with a pencil.

And now mathematics has a new powerful assistant, which is called an electronic computer. Existing high-speed computers operate hundreds of thousands of times faster than humans.

Mathematics has never been so comprehensive and such a science needed by people as it is today. It is difficult to talk about what mathematics will be like tomorrow. It is developing so rapidly now, new discoveries are made in it so often that it is probably useless to guess what will be. One thing is certain: tomorrow mathematics will become even more powerful, even more important and more needed by people than today.

FROM THE POINT OF VIEW OF SCIENCE

What was required
to prove: chenye
explain why
modern man
can't do without
mathematics

Text: Elena Kiseleva

The Pythagoreans claimed that numbers rule the world, and Alexander Suvorov called mathematics "the gymnastics of the mind." Now interest in this science is gradually reviving - Yandex together with high school Economics on March 14 conducts an all-Russian test in mathematics. T&P spoke to five famous mathematicians to understand why formulas and equations are needed in everyday life, why mathematics is an interesting and creative subject, and what the humanist loses by brushing aside this science.

"He who does not know mathematics cannot even reveal his ignorance"

Sergei Lando

Doctor of Physical and Mathematical Sciences, Dean of the Faculty of Mathematics, National Research University Higher School of Economics

As my teacher Vladimir Igorevich Arnold said, “the main goal of mathematical education should be to develop the ability to mathematically investigate phenomena real world". The essence of mathematics is the study of general patterns that describe the qualitative nature of the world around us - the change of seasons, the location of the planets, climate change, fluctuations in exchange rates or oil prices, the development of natural grammars or the principles of constructing artificial languages. Mathematicians have developed and developed a variety of methods - computational, algebraic, geometric, the method of evidence-based reasoning, logical inference. In some cases, these methods are so developed that they allow one to achieve a deep understanding of the existing patterns, in others this understanding is a matter of the distant future. Knowledge of regularities allows not only to explain past events, but also to predict future ones.

A person who has never encountered mathematical reasoning experiences serious difficulties in distinguishing a fact from its interpretation, true statements from false ones, and understanding what consequences follow from this or that statement. A person who is unable to estimate the order of numerical values ​​can be easily manipulated by unscrupulous economists and politicians. As Roger Bacon wrote in 1267, “He who does not know mathematics cannot learn any other science, and cannot even discover his ignorance, and therefore does not seek a cure for it.”

In our time, this approach is common - I do not understand mathematics, physics, chemistry, biology, ..., so I'd better go to study something humanitarian. That is, from the very beginning of his independent life, a person agrees to his own inferiority, to the deliberate absence of some, moreover, valuable quality. It doesn't work for the humanities. And I would like people to go to the humanities with a pronounced interest in what they want to do, in the study of man and his activities. IN natural sciences and in mathematics such interest is present, in my opinion, more often. People master them and subsequently engage in them because of an internal need that does not at all negate others, including humanitarian interests.

"A person who has never met with mathematical reasoning has serious difficulties in distinguishing fact from its interpretation and true statements from false"

Have you ever tried to describe the beauty of a painting to a person who has never seen it? This is not an entirely unsolvable task - if your interlocutor has sufficient experience in visiting art galleries, is well acquainted with many masterpieces of world painting. If the listener does not have such an experience, there is no hope that he will receive positive emotions from the description. The ability to perceive the beauty of mathematics also requires constant - or at least regular - work. It can be developed in young children by talking to them about math before school. It is not uncommon for this beauty to be revealed to a student unexpectedly. Initially, school mathematical competitions were aimed at achieving this result: through the prism of beautiful problems and beautiful solutions, to show a small part of the spectrum of beautiful ideas, arouse interest and encourage them to go further.

In order not to remain unfounded and to give a concrete idea of mathematical beauty, I will tell you the following fact: if another, smaller plan is superimposed on the plan of Moscow, then in Moscow there will definitely be a place that will be depicted on two plans by two points lying one above the other - a needle that pierces two plans at these points will indicate one and the same place in the city. Do you understand why this happens? This statement is the beginning of a large and branched mathematical theory and is used in a huge number of applications. It remains true in a much more general situation - for example, if the second plan of Moscow is distorted or crumpled.

"The humanities are entering an era of high precision"

Ivan Arzhantsev

Doctor of Physics and Mathematics, Dean of the Faculty of Computer Science, National Research University Higher School of Economics and Yandex

Why is math needed? Lomonosov's phrase that "mathematics should be taught later, that it puts the mind in order" perfectly reflects the essence of the matter. Rumors of eccentric scientists are greatly exaggerated. People who understand mathematics are valued not only because they have special knowledge, but rather because they can think and analyze.

If physicists, chemists, biologists need laboratories, installations, consumables, then mathematics is always with you. You go, for example, on a train, take a piece of paper and a pen, or just close your eyes and work on solving some problem. There is no less beauty in mathematics than in art. If the work in mathematics is heavy and confusing, most likely the author either took on the “wrong” problem, or the solution still needs to be worked on. Proving a theorem is like assembling a puzzle. You twist this way and that the existing fragments, known facts and methods of proof, and when all of a sudden everything came together - this is beauty!

Mathematics itself needs applications. They not only guarantee its right to exist, but also provide an environment that generates new purely mathematical problems. In addition to applications in the natural sciences - physics, chemistry, biology - mathematics is increasingly used in economics, social and humanities. Mathematical results play a special role in the IT world. Technological breakthroughs are often based on fundamentally new algorithms and theorems, sometimes from very abstract areas of mathematics.

In March 2014, the HSE and Yandex Faculty of Computer Science was opened. We have students who are interested in mathematics and programming. It is they who, after some time, will be able to apply an arsenal of mathematical methods to the problems of information retrieval and computer vision, automatic word processing and bioinformatics, the development of software packages and the creation of Internet services. One of the areas of Computer Science is the "new mathematics" for working with big data. What can be achieved here is on the verge of fantasy.

There is a feeling that right now the humanities are entering the “epoch of precision”. This is not only about the ability to build ever more accurate mathematical models of various processes and calculate these models on super-powerful computers. New technologies allow capturing and storing accurate information about a variety of real events. The only question is what to do with this information: a person or even a scientific team will not be able to analyze the collected piles of data for many years. The idea of ​​modern data analysis is that computer systems and the algorithms implemented on them themselves work with the received arrays of information and give the user only the final result - the statistics of interest to him and certain patterns discovered. This allows not only to confirm or refute hypotheses from the humanitarian sphere with mathematical rigor, but also to detect dependencies that were unknown to specialists. Mathematically savvy humanists are needed here - they can set a task, explain what kind of data it is planned to collect and what kind of characteristics we will be interested in.

Recently, Yandex decided to hold an all-Russian test for everyone who loves mathematics or, perhaps, would like to fall in love, but somehow it didn’t work out: schoolchildren, mothers, fathers, grandfathers and grandmothers. The tasks are simple, according to the basic school curriculum - however, for a successful solution, you need to be careful. Training tasks are already open on the site - you can test your strength.

The test will take place on March 14, the day of Pi. You can participate in the test not only online - in Moscow, tasks can be solved at the HSE, which has become a partner of the project. The project was supported by universities in many regions of Russia: Yekaterinburg, Novosibirsk, Kazan and others. I highly recommend to free an hour from Saturday and join - especially for those who are afraid of mathematics. After the test, university teachers will analyze the tasks together with the project participants.

“Ignorance of mathematics threatens with porridge in the head”

Alexey Savvateev

Doctor of Physical and Mathematical Sciences, Expert of the Department of Theoretical and Applied Developments of Yandex, scientific adviser Social Analysis Laboratories at Dmitry Pozharsky University

There are political debates going on right now. It would seem, what does mathematics have to do with it? But at detailed study situation, it is clear that the humanities, who are not familiar with the basics of mathematics, do not have an opinion in their heads, but “porridge”. They can't focus on anything, jumping from one argument (messy and often contradictory) to another. And this is from each of the warring parties. A person who understands mathematics has order in his head, everything is in its place. He is working out his position, you can't fool him on the chaff. So, ignorance of mathematics threatens with porridge in the head.

Humanitarians need to be taught beautiful mathematics - pictures, pictures and pictures again. They should immediately make you work with your head: think, compare, compare and draw conclusions. Not just to contemplate beautiful mathematical constructions, but to be their active co-organizer, to see the purpose for which this or that is done, to understand simple logical transitions.

Then, at the next stage, you can already move on to abstract concepts and terms - oddly enough, they are better given to the humanities than hardened and stubborn techies! It is quite possible to solve various Diophantine equations, talk about complex numbers, about number systems (rings, fields) and how they help in solving problems. It is quite accessible already at the early stages of comprehension of mathematics, the analysis of problems for the construction of what can and cannot be built with a compass and a ruler. In general, I would advise any humanist to master the book by Courant and Robbins "What is Mathematics".

What is the beauty of mathematics? Find a geometric proof of the Pythagorean theorem - and you will understand!

“This science teaches us that seemingly unsolvable problems can be solved”

Dmitry Vetrov

PhD in Physics and Mathematics, Head of the Department of Big Data and Information Retrieval, Faculty of Computer Science, HSE

per century information technologies voices are heard more and more often that any serious knowledge of mathematics is specific and the average person does not really need it. Many people think that the skills of dividing, say, a seven-digit number by a four-digit one or adding two fractions are already superfluous - because all this can be done on a smartphone. What can we say about the ability to solve logarithmic inequalities, algebraic and differential equations, problems of three-dimensional geometry (stereometry) and other "higher mathematics"?

There are two fundamental arguments against this position. Firstly, we need not so trivial mathematics already in everyday life. For example, to solve a simple task(“In one of the dynamically developing African countries, inflation is 32 million percent per year. The question is: how many percent do prices increase per day in this country?” The answer will surprise many with its smallness), you need the ability to compose and solve equations with logarithms. Without this, it is impossible to calculate at what percentage it is necessary to put 5 thousand dollars in the bank into the account of the born son, so that in 18 years he can receive 25 thousand, or what is better: to invest our savings so that they will increase by 6% every six months, by 13% every year, or 27% every two years? Perhaps even more burning (as of March 2015) topic: in what currency is it better to keep your savings - in rubles (banks promise a high interest rate), in dollars or in euros? Intuition suggests the correct answer (a little bit in all currencies), but in order to choose the right proportions, you need to have the simplest knowledge of risk theory. In meetings at work, we often have to choose which of the various points of view to support, which group of supporters to join. Game theory allows you to do this with the greatest benefit for yourself. It also gives an answer to the question of what salary can be safely demanded if competitors are trying to lure you to their place, not being afraid to sell cheap or lose a profitable offer.

On the day, inflation in this African country was "only" 4%. Despite the seeming smallness, by the end of the year the country ran out of paper for printing more and more new banknotes. These are the insidious properties of the exponent.

Secondly, mathematics (even school) teaches you to think logically. To understand what follows from what and what does not (for example, from the fact that we have herring in the refrigerator, it follows that we also have fish there. But it does not follow at all from the fact that we have fish in the refrigerator (although and maybe) that we have a herring there). Almost the only school discipline that teaches reasoning is geometry. But this is just the tip of the iceberg. In fact, there are more subtle patterns around us that go beyond the usual logic, which operates with the concepts of "truth" - "falsehood" - "unknown". For example, it has been scientifically established that there is a negative correlation (stochastic dependence) between the length of a person’s hair and his height, that is, if we take a random citizen of Russia and tell us that he has short hair, we can say with high probability that his height is taller. average. Now you will know that I have short hair. Does this give you any additional information about my height? The correct answer is no.

It is known about me that I am a man, but not about a random resident of the Russian Federation. The correlation between height and hair length exists as long as the gender of the person is unknown (women, on average, are shorter than men and have longer hair). Once the gender is known, the relationship between hair length and height disappears. This phenomenon, known as conditional independence, underlies the whole theory of probabilistic graphical models, which are actively used in the tasks of analyzing texts, images, videos, social networks, etc.

And there are a lot of such examples (for example, is there a relationship between the price of tomatoes in the supermarket and their quality, or is there a relationship between jeans of a certain brand and my attractiveness in the eyes of girls; is it worth taking a paid diagnostic test that gives the correct answer in 90% of cases to determine the disease, found in one in ten thousand people). To understand when relationships exist, and when they are false correlations generated by unaccounted for factors, you need to have an understanding of the basics of probability theory and Bayes' theorem. Well, or at least developed common sense and a solid four in geometry.

Not worth it. Even if the test gives a positive result, in 999 cases out of a thousand, it will be a false alarm. The above, of course, does not apply to a situation where there are good reasons to assume the presence of a rare disease, for example, its symptoms.

Modern mathematics covers a much wider range of issues that go far beyond the everyday. The major search engines, thanks to which you may be reading this article, are crammed with mathematical models that allow you to adjust the parameters for issuing the result of our search query for a particular user. In other words, for the same request, Google will give me some links, and others for you, simply because we logged into the browser under different gmail accounts. Sophisticated mathematical models are used by investment funds that manage our savings; online stores that recommend certain products to us; traffic lights, which reduce the likelihood of traffic jams on the streets due to the constant adjustment of their mode; and many other technologies around us.

The most active application is found by mathematical methods in modern natural sciences and the humanities. The processing of data from the Large Hadron Collider gave rise to a whole branch of mathematics, the so-called. big data analysis. Biologists use complex mathematical methods to reconstruct the evolutionary tree from the remnants of the genomes and organs of extinct individuals. Chemists search for promising polymers for future synthesis using mathematical modeling algorithms. Art historians use mathematics to determine the authors of anonymous literary works and artistic canvases. Finally, it is impossible not to note the revolution in the field of mathematical methods of machine learning, which is taking place before our eyes. With the advent and successful application of deep neural networks (deep neural networks), humanity has begun to rapidly approach the creation of artificial intelligence. Even now, a high school student who knows how to program can independently build a new neural network model that can solve the next task (for example, synthesizing music, understanding images, etc.), which was previously considered subject only to human intelligence.

Mathematics constantly teaches us that seemingly unsolvable problems can be solved if we move to a new level of thinking. We divide four by seven, subtract nine from two, operate on irrational numbers, we are faced with the fact that one equation can have many solutions, although each time we have to overcome some pattern gap. The study of mathematics helps to understand that many truths that we used to consider absolute are actually relative, and much of what seemed to us to be of a different nature is in fact special cases of the same phenomenon only from a different angle of view. Such effects are observed not only in mathematics and its applications, but also, for example, in politics.