home / Self-development/ Application of arithmetic in human life. Application of mathematical knowledge by preschoolers in everyday life and games

Application of arithmetic in human life. Application of mathematical knowledge by preschoolers in everyday life and games

Mathematics in human life

Have you ever heard the following expression: mathematics is a country without borders? This phrase about mathematics has very good reasons. Mathematics occupies a special place in human life. We have become so close to it that we simply don’t notice it.

But our life begins with mathematics. The child has just been born, and the first numbers in his life are already heard: height, weight. The baby is growing up, cannot pronounce the word “mathematics”, but is already doing it, solving small problems of counting toys and cubes. And parents don’t forget about tasks. When preparing food for a child, weighing him, they have to use mathematics. After all, you need to solve a basic problem: how much food you need to prepare for your baby, taking into account his weight.

At school mathematical problems there are many and their complexity is growing every year. They don't just teach the child certain actions. Mathematical problems develop thinking, logic, and a set of skills: the ability to group objects, reveal patterns, determine connections between phenomena, and make decisions. Doing mathematics and solving mathematical problems develops a person, makes him more purposeful, active, and independent.

And after school, mathematics will be very useful. While studying at university, at work and at home, you need to constantly solve problems related to mathematics. What is the probability successful completion 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 do you need to buy to insulate your house? How to correctly calculate whether a girl or a boy will be born? And this is where mathematics comes to the rescue. It follows a person everywhere, helps him solve practical problems, and makes his life much more convenient.

The world and life itself are changing rapidly. It includes new technologies. Only mathematics and problem solving in the traditional sense remain true. Mathematical laws have been tested and systematized, so a person can rely on it at important moments and solve any problem. The math won't let you down.

National action plan for 2012-2016 for the development of functional literacy of schoolchildrenEmphasizes core competencies such as reading, math, and science literacy.

What is the purpose of mathematics education?

Preparation for university.

Preparation for a future profession.

Intellectual development.

Formation of worldview.

Orientation in the surrounding world.

Exercise of the brain.

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

Mathematics is found and used in Everyday life Therefore, every person needs certain math skills. In life we have to count, for example, money. We constantly use, often without noticing it, knowledge about quantities characterizing lengths, areas, volumes, time intervals, speeds and much more. All this came to us in arithmetic and geometry lessons and was useful for orientation in the world around us.

Mathematical knowledge and skills are necessary in almost all professions, primarily, of course, in those related to natural sciences, technology and economics. But there is no doubt the need to use mathematical knowledge and mathematical thinking for a doctor, linguist, historian, and it is difficult to end 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 laws 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 ethical principles of human society. The study of it is designed to cultivate in a person intellectual honesty, objectivity, the desire to comprehend the truth,it also cultivates the ability for aesthetic perception of the world, the beauty of intellectual achievements .

“Mathematics must be taught later because 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, math puzzles develops logical thinking and reaction speed. It is not without reason that they say that mathematics is mental gymnastics.

Mathematics teacher of KSU "Kokpektinskaya Secondary School" Germash E.A.

Back forward

Attention! Slide previews are for informational purposes only and may not represent all the features of the presentation. If you are interested this work, please download the full version.

Participants: 7th grade students.

Goals:

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

Tasks:

study bibliographic sources on this topic;
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.

Progress of the event

Teacher's opening remarks:

1 slide Topic: “Mathematics in human life”

2 slide Fundamental question: Does a person need mathematics?

3 slide Problematic issues:

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

Student performance:

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

Why ships
Don't run aground
And they follow the course
Through fog and snowstorm?
Because because,
Will you notice?
Helps captains
Mathematics!

So that as a doctor, a sailor
Or become a pilot.
First of all we must
Know mathematics.
And there are no professions in the world
Will you notice?
Wherever you need it
Mathematics!

4 slide How and when did mathematics originate?

When we are talking about something very simple and understandable, we often say: “The matter is as clear as two and two are four!”

But before they figured out that two and two equal four, people had to study for many, many thousands of years.

Of course, this teaching did not take place at a desk. Man gradually learned to live: to build houses, find his 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 on arithmetic and geometry were known to the ancient Greeks two years ago. more than a thousand years ago.

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

With each passing decade, mathematics became more and more people need it more.

5 slide Pythagoras

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

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 proven by the Greek mathematician Pythagoras, after whom it was named.

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

6 slide

At the end of the nineteenth century, various speculations were made about the existence of human-like inhabitants of Mars. As a joke, although not entirely without reason, 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 hid a complex problem book under her pillow at night (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 wishes of her parents to Germany, to the university, and came to a famous professor. He didn’t want to take her and, in order to get rid of it, he gave several tasks that they had composed themselves, saying that if she decided, he would take her to him.

Even professors could not solve these problems. The girl decided in twenty minutes.

Sofya Kovalevskaya graduated from 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 calculate a jet plane, a ship, or a new power plant.
For a physicist, mathematics reveals the laws of the atomic nucleus, and for a sailor it shows the path of a ship in the ocean.
In short, mathematics can do everything or almost everything where something needs to be calculated.

But everything begins with mathematics.

The child has just been born, and the first numbers in his life are already heard: height, weight.
The baby is growing up, cannot pronounce the word “mathematics”, but is already doing it, solving small problems of counting toys and cubes.
And parents don’t forget about mathematics and problems. When preparing food for a child, weighing him, they have to use mathematics.
After all, you need to decide elementary tasks: how much food should be prepared for the baby, taking into account his weight.

Slide 9

1 example

You stand at the checkout and pay for the goods. You bought food for 432 rubles, and you have 500 rubles in money in 100 ruble bills. And they give you 40 rubles in change, although they should give you 68 rubles. This means 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 need 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 can you get 100 with one action and five units?

Slide 13

111 - 11 = 100

Slide 14

Where can you do without mathematics?

Here are the builders, building a house. We need to calculate how much cement, how many bricks. Height, width. Make a project.
The dressmaker is about to sew a dress. Measures a person and makes a pattern. Does she need math? Maybe…
The store counts the goods received and the revenue.
The bank counts money, dealing with huge amounts and interest.
Even in music, in poetry, you have to count - rhythm, meter, eighth notes, quarter notes, iambs, trochees.
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 crew needs to make a tunnel, but the distance to cut through the mountain is unknown. 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 crew is unable to build a road around the mountain. Therefore, they undertook a little trick: at the entrance to the not yet cut tunnel they placed a person - (A) and at the exit point too - (B), on the side of the mountain they placed a third person - (C), a triangle ABC was formed. Person A draws a straight line through point C, and person B also lays a straight line through point C. Having drawn straight lines and placed two more people on them at a certain distance - (D,E) So CD =AC, A CB = EU.Corner ACB =ECD by the property of vertical angles, therefore triangle DEC is equal to triangle ABC. Now the team connects points D and E with a segment on the ground. The workers just have to measure the distance from E to D, which will be equal to the required distance from A to B.

16 slide

Do modern people need knowledge of mathematics?

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

However, just a few decades ago there were many problems that were practically impossible to solve, although mathematicians knew how to solve them. It happened that dozens of people worked for several years to solve one 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 work hundreds of thousands of times faster than humans.

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

Pacheva Alina

Project Manager:

Filkova Larisa Nikolaevna

Institution:

MKOU "Secondary School No. 27" Nalchik

In this research work in mathematics on the topic "Mathematics in ordinary life" The author studies the branches of human activity and professions where mathematics is found, proves its necessity, and also finds out whether a person needs mathematics in ordinary, everyday life?

In the presented research project in mathematics on the topic “Mathematics in ordinary everyday life”, the statements of great people about mathematics are studied, the need for mathematics is proved not only in certain professions, but also in everyday (ordinary) life.

In the mathematics research paper “Mathematics in Ordinary Everyday Life,” the student plans to introduce schoolchildren to the results of her research in order to develop interest in this subject, expand knowledge in mathematics and broaden their horizons.

Introduction
1. Mathematics in everyday life.
2. Mathematics in professions.
3. Why is mathematics needed in different areas of life?
3.1. Why is mathematics needed?
3.2. Why does a child need mathematics?
3.3. Why do humanists need mathematics?
4. Statements of great people about mathematics.
Conclusion

Introduction

One day I had a question: What is mathematics for?, Why do we learn various equations and theorems? We only use mathematics in the store when buying groceries. Why do we study mathematics with kindergarten? “And I tried to find out the importance of this subject.

I think my topic is research work in mathematics "Mathematics in everyday life" is relevant .

Purpose of the research work: study where mathematics is found in life and prove its necessity. Find out whether a person needs mathematics in everyday life?

Tasks:

Study types of activities (professions) where a person cannot do without mathematics;
Answer the questions: Why is mathematics needed in everyday life? And What can mathematics give to each individual?;
Study the statements of great people about mathematics.

Hypothesis: mathematics in our lives is necessary not only in certain professions, but also in everyday (ordinary) life.

Mathematics in everyday life and work

Mathematics- a set of sciences that study quantities, quantitative relationships, and spatial forms.

Many famous mathematicians say that the main thing in mathematics is to teach a person to think, sometimes setting him very difficult tasks. " Mathematics develops logical thinking, the ability to independently solve problems, the ability to quickly grasp the essence and find the most suitable and simple approach to a life problem.“- the adults tell us. Mathematics is closely related to our daily life.

Mathematics occurs in our lives at almost every step and it is not so gray and boring, but colorful and cheerful...

Thanks to mathematics, we solve many questions in everyday life. Few people thought that mathematics surrounds us from the first days of life. Any child, even one who has not studied arithmetic, has encountered numbers. At the clinic he finds out his weight, height, and also knows his age. And more than once a day he will be faced with various tasks of counting toys in the room or candies to treat his friends.

Mathematics and daily routine. For example, our daily routine is a routine, nothing more than determining time and planning it throughout the day using simple mathematical calculations.

Lessons at school– this is also the distribution of time between studying different subjects and relaxing during breaks. After school, we need to have time to have lunch, go to extra classes, do homework, have dinner, rest and go to bed in order to get a good night's sleep and start a new day with new strength and in a good mood. And this is how we keep track of time all day by the clock and learn to distribute it correctly so as not to be late and not to arrive earlier than necessary.

At school we study mathematics from the first grade until graduation, then we are taught mathematics at the university. Every year the course expands and becomes more in-depth, with more and more subjects related to mathematics.

IN high school we have algebra and geometry replacing arithmetic. Our horizons are expanding. We can understand, see what previously seemed unclear to us. Mathematical Sciences develop our thinking, teach us to think.

With age we are solving more and more problems: How much food do you need to buy to last for a week? How much do you need to earn to save for a dacha and trips abroad? How much paint should you buy to paint the walls in your bedroom?

Without knowledge of mathematics all modern life would be impossible. We wouldn't have good houses because builders have to be able to measure, count and build. Our clothes would be very rough, since they need to be well tailored, and for this, everything must be measured precisely. There wouldn't be any railways, no ships, no planes, no big industry.

There would be no radio, television, cinema, telephone and thousands of other things that are part of our civilization. Using mathematics, measuring " how much?», « how long?"are a vital part of the world in which we live.

Thanks to mathematics, computing appeared calculating machines. Computer technology has evolved from simple abacus, adding machines, and slide rules to microcalculators and computers. Now computing machines used in all industries National economy: in statistics, trade, automated control of plants and factories. Machines not only count, they can translate from one language to another, they can compose music, play chess.

Home repairs. If we are going to do home renovations, then we definitely cannot do without mathematics. We will need to do a lot of calculations. The accuracy of which will determine whether we will have smooth walls and ceilings, as well as whether we will have enough wallpaper to cover the room and tiles to lay on the floor in the bathroom.

Thus , I can say that we need mathematics everywhere, and there is no area of life where we could do without it.

Mathematics in professions

There is not a single profession in the world where mathematics is not found. And the students’ opinion that mathematics is not useful to us is incorrect. In any profession, a person needs mathematics. Even a person whose work is not related to mathematics needs it.

After all, you need to know mathematics so that you are not shortchanged when giving you a salary or pension. Mathematics also teaches you to solve any problem with several solutions. Thanks to this, a person develops his extraordinary thinking.

There are many clear examples of professions where mathematics is required:

Accountant
Engineer
Seller, programmer and many others...

Accountant.
In the accounting profession, mathematics is simply necessary. The accountant calculates wages, benefits, vacation pay, taxes, insurance premiums, etc.

Salesman.
In the profession of a seller, mathematics is needed in order to count money, received products and goods, the number of remaining products and goods, etc.

Despite the fact that your future profession does not involve a connection with mathematical formulas and calculations, no one knows what you will do in the future. For example, go become an entrepreneur and open your own business.

Such a job change will require you to master new skills in organizing and running a business, including accounting, and without mathematical methods of forecasting, modeling, analysis and calculations there is no way to achieve success.

Why is mathematics needed in different areas of life?

Why is mathematics needed?

What does it give to a person, what abilities and skills does it develop?

First of all, this fundamental science develops our mental abilities - analysis, deduction, and the ability to predict. Math knowledge improves abstract thinking, enhance his speed, teach him to abstract, concentrate and trains his memory.

If we specify what mathematics gives us, the result of getting to know it can be represented by the following list of skills:

communication;
analysis of complex situations, acceptance optimal solutions, regardless of the complexity of the situation;
searching and finding a pattern;
development of logic, reasoning, generalization, competent formulation of thoughts and logical conclusions;
speed of decision making;
planning and keeping in mind a complex step-by-step sequence;
logical construction of complex operations and storing them in memory.

The listed skills are acquired not only as a result of solving problems in various branches of mathematics (algebra, geometry, trigonometry, probability theory, statistics, etc.), but also in the process of using such mathematical and logical methods as puzzles, exact sciences or Mind games, which load your brains and “force” you to look for non-standard solutions and analyze.

Why does a child need mathematics?

Mathematics is essential for children's development. In addition to the fact that it develops the child’s mind, it lays the foundation for rational thinking and intellectual development even at the stage of schooling.

Mathematics, forming logic, trains our mind, which allows us to compare different concepts, analyze them sensibly and comprehend them. A person with a “mess in the head” is more susceptible to delusions, both in thoughts and in reasoning. In other words, knowledge of mathematics will not allow you to be deceived, as millions of people who entrusted their deposits to financial pyramids were deceived.

Mathematics is not just formulas and calculations, it is the logic and order that follow from its rules and functions! Mathematical knowledge allows a person to reason correctly, formulate his thoughts, hold complex sequences in his head and build relationships between them.

Why do humanists need mathematics?

Many humanists believe that they don’t need mathematics, forgetting that mathematical thinking will help in any profession not related to the exact sciences. You don’t need to go far, remember lawyers: they build their defense in court like chess players, coming up with cunning and extraordinary solutions, using legislative framework and logical order of actions.

There is no point in specially studying an in-depth course in mathematics. To obtain the necessary basic knowledge, school and primary education are sufficient. university education, in which general education subjects are compulsory for everyone, both for techies and for humanities students. Studying multidirectional subjects harmoniously complements a person’s knowledge, which will be useful not only in a future profession, but also in everyday life.

Sayings of great people about mathematics

Mathematics is the language in which the book of nature is written. (G. Galileo)
Mathematics is the queen of sciences, arithmetic is the queen of mathematics. (K.F. Gauss)
Anyone who studies mathematics from childhood develops attention, trains their brain, their will, and develops perseverance and perseverance in achieving goals. (A. Markushevich)
“Numbers rule the world,” said the Pythagoreans. But numbers make it possible for a person to control the world, and the entire course of development of science and technology of our days convinces us of this. (A. Dorodnitsyn)
Sooner or later, every correct mathematical idea finds application in one thing or another. (A.N. Krylov)
If you want to participate in a big life, then fill your head with mathematics while you have the opportunity. She will then provide you with great assistance in all your work. (M.I. Kalinin)
Have you not noticed that he who is capable of mathematics is sophisticated in all sciences in nature? (Plato)
It would be good if the state itself required this knowledge and if persons occupying the highest government positions were taught to study mathematics and, when necessary, turn to it. (Plato)
Mathematical sciences from the very ancient times attracted special attention, now they have received even more interest in their influence on art and industry. (P.L. Chebyshev)
Mathematics is the best and even the only introduction to the study of nature. (D.I. Pisarev)
Astronomy (as a science) has existed since it was combined with mathematics. (A.I. Herzen)
Flight is mathematics. (V. Chkalov)
Inspiration is needed in geometry no less than in poetry. (A.S. Pushkin)
Geometry is full of adventure because behind every problem lies an adventure of thought. Solving a problem means experiencing an adventure. (V. Proizvolov)
Mathematics has its own beauty, just like painting and poetry. (N.E. Zhukovsky)
Chemistry is the right hand of physics, mathematics is its eye. (M.V. Lomonosov)
Mathematics must then be taught so that it puts the mind in order. (M.V. Lomonosov)
I love mathematics not only because it has applications in technology, but also because it is beautiful. (R. Peter)
Everything that was previously in the sciences: hydraulics, aerometry, optics and others was dark, doubtful and unreliable, mathematics made clear, true and obvious. (M.V. Lomonosov)
Anyone who wants to study chemistry further must also be proficient in mathematics. (M.V. Lomonosov)
A physicist without mathematics is blind. (M.V. Lomonosov)
A mathematician who is not to some extent a poet will never be a real mathematician. (K. Weierstrass)
Mathematics is the language that all exact sciences speak. (N.I. Lobachevsky)
Only with algebra does rigorous mathematical teaching begin. (N.I. Lobachevsky)
No matter how well a machine works, it can solve all the tasks required of it, but it will never come up with a single one. (A. Einstein)
It is mathematics that gives the most reliable rules: whoever follows them is not in danger of deceiving the senses. (L. Euler)
Numbers (numbers) do not rule the world, but they show how the world is controlled. (I. Goethe)
A close, deep study of nature is the source of the most fruitful discoveries of mathematics." (J. Fourier)
...It would be easier to stop the Sun, it would be easier to move the Earth, than to reduce the sum of the angles in a triangle, reduce the parallels to convergence and move the perpendiculars to the straight line to diverge. (V.F. Kagan)
Counting and calculations are the basis of order in the head. (Pestalozzi)
If you want to learn to swim, then boldly enter the water, and if you want to learn how to solve problems, then solve them. (D. Polya)
To digest knowledge, you need to absorb it with appetite. (A. Franz)
The subject of mathematics is so serious that no opportunity should be missed to make it more entertaining. (B. Pascal)

Conclusion

Today we do not know any areas of human activity where mathematics is not needed. Without it, not a single new discovery can be made, not a single invention works, not a single enterprise or state functions, therefore, the range of everything where mathematics is needed is quite wide.

When we start studying this discipline at school, we do not know whether we will make a discovery in physics, computer science, astronomy or another science. Or maybe we’ll be an engineer or an architect, an aircraft designer or a pharmacist, i.e. a specialist in the profession where mathematics will be needed specifically for us.

It is possible that we will be a housewife, a makeup artist, or a famous fashion designer who needs to make drawings for costume patterns. Or fate will test us in the profession of a programmer, a lawyer, the captain of an ocean-going vessel or the leader of a geological expedition, since all of these are areas where mathematics is simply required.

While working on the project, we became convinced that everyone should know and study this greatest of all sciences, without which one cannot imagine one’s life, since mathematics is a kind of travel ticket, without which it is impossible to hit the road. It develops logical thinking, determination, imagination, and the ability to find a way out of any situation.

Mathematics makes you think, helps humanity discover and use the laws of nature, and has always been a powerful engine of science and technology.

I was convinced that mathematics is simply necessary in life, everyday life and professions. In this regard, I decided to introduce as many students as possible to the results of my research in order to develop interest in this subject, expand their knowledge of mathematics and their horizons in general.

Hypothesis put forward the fact that mathematics is necessary in our lives not only in certain professions, but also in everyday life has been confirmed.

Bibliography

1. Aksenova M.D. - Encyclopedia for children. T. 11. Mathematics / Chief editor. M.D. Aksenova - M. Avanta, 1998.
2. Glazer G.I. "History of mathematics at school"
3. Sergeev I.S. "Apply Math"
4. Spivak A.V. Mathematical holiday. 4.1 - M.: Bureau Quantum, 2000 (Supplement to the magazine “Quantum”, No. 2/2000).
5. Shalaeva G.P. Everything about everything. Popular encyclopedia for children. Moscow “Slovo” 1997, 1999.

We all studied mathematics at school. Moreover, many of those who consider themselves “humanists” because of their passion for literature and languages recall logarithms and quadratic equations like a bad dream. Each of us has more than once asked the question, “Can this ever be useful to me in life?” and, most likely, did not receive an intelligible answer even from his algebra teacher. Jordan Ellengberg, an American professor of mathematics at the University of Wisconsin-Madison, takes it upon himself to say: “As much as he can!”

Errors of airplanes and soldiers' feet

Ellenberg begins his book with a story about the outstanding 20th-century mathematician Abraham Wald, who was forced to emigrate from Austria to the United States in the late 1930s due to the persecution of Jews by the Nazis. During World War II, Wald worked with leading American statisticians to solve secret military problems in the Statistical Research Group (SRG). The military command turned to the SRG with the task of finding a way to minimize the losses of American bombers.

Damage to aircraft returning from the combat zone was unevenly distributed - most of the holes were on the fuselage, a smaller part on the engine. The military came to the conclusion that it was necessary to strengthen the most vulnerable parts of the aircraft with armor. The only question was how much armor should be used on the affected areas so as not to overload the aircraft with iron and at the same time effectively strengthen it.

Wald's answer was unexpected. Naturally, he did not dispute that aircraft require additional protection. But at the same time, he proposed making fortifications not where there are the most holes, but where there are none - that is, on the engines. There is only one reason why there was less damage in these zones: in the event of a direct hit to the engine, the plane simply did not return from the battle. A similar thing happened with the wounded in a military hospital: nurses more often saw those wounded in the legs rather than in the chest. And the point is not that the soldiers did not receive chest wounds, it’s just that, as a rule, few survived after them.

Ellenberg focuses on this story with Wald to make the reader understand what the mathematical way of thinking is. Being a mathematician is not just about solving numerical problems and deriving algebraic formulas. Being a mathematician means thinking outside the box, asking the right questions, and most importantly, questioning assumptions that lead to false conclusions.

A mathematician always asks the following questions: “What assumptions are you making? Are these assumptions justified? Sometimes this causes irritation. However, this approach can be very productive.

Apply math to where it hurts

On school lessons algebra, few people think about it. We study a long list of rules and formulas, from the entire array of which we then use only the skills of performing simple arithmetic operations(in fact, far from it, many people don’t even realize how deeply mathematics is woven into the fabric of our thinking). So, if your ideas about mathematics are limited only to the school course, congratulations, you know almost nothing about this subject! There are such fundamental sections of this science as probability theory, mathematical analysis, coding theory, and statistics. (Scary already? I admit, I’m a little too). After all, we are talking about areas of pure mathematics that seem inaccessible to the common man.

Ellenberg hastens to assure us that at the heart of this abstract complex language lies nothing more than common sense, supported by fundamental methods and theorems. And “the real mental work required in mathematics is not much different from the way we think about solving simple everyday problems.” The professor came to this conclusion while working on mathematical research that was so far from real life, that he does not seek to introduce us to them. The further this work progressed, the more clearly he realized that mathematical laws went far beyond the discussions within the university community.

“Knowledge of mathematics is a kind of X-ray glasses, allowing us to see the structure of the world hidden under a disorderly, chaotic surface. Mathematics is the science of not making mistakes, and mathematical forms and methods have been forged over many centuries of hard work and discussion.”

Unlike his predecessor Wald, who was not interested in the applied possibilities of mathematics, Ellenberg aims to talk about the use of mathematical concepts in politics, medicine, economics, religion, the Internet and even everyday affairs. Here we are dealing with simple and profound facts that form part of the mathematical universe.

When is the best time to arrive at the airport so as not to waste your time and not be late? How do you live in a world where Google, Facebook, and even major retail chains know more about you than your own parents? Should you trust polls? public opinion? What about the results of testing new drugs? What can we learn about the existence (or absence) of God using the laws of mathematics? How are they created statistical research, telling us that in certain geographical areas Is the risk of developing cancer higher than in others? What loopholes exist for candidates in the democratic election procedure? What, after all, does one have to do to cheat the system (legally, of course) and win millions of dollars in the lottery? And so on and so forth.

The examples given in the book clearly show how belief in mindless numbers, unverified facts and dubious statistics, disseminated through numerous communication channels, forces people to come to ridiculous conclusions and complicate their lives. A detailed analysis of each case based on mathematical analysis really helps to take a critical look at the flow of information that bombards our heads every day through statements by politicians and public figures, online advertising and the media.

Mathematics is not just for geniuses

Of particular interest is the author's discussion of the ideas that have taken root in the public consciousness that all mathematicians are mad, obsessed geniuses who choose scientific escapism as the main idea of life. This image is widely replicated in popular culture, take, for example, the story of John Nash’s schizophrenia and hallucinations, around which the plot of the film “A Beautiful Mind” is built, or the entire spectrum of mental disorders of Max Cohen in the film “Pi.”

“In real life,” writes Ellenberg, “mathematicians are ordinary people, no crazier than anyone else. In fact, we do not often go into solitude to fight lonely battles in harsh abstract worlds. Mathematics strengthens the mind rather than strains it to the limit.”

It is also a mistake to think that mathematics rests only on geniuses, and everyone else, whose achievements seem less outstanding, is welcome to this area scientific knowledge closed. Meanwhile, this is the opinion of many students who drop out of universities in the middle of their studies, disillusioned not with mathematics itself, but with the fact that they fail to become the best. Ellenberg regrets this because he believes that mathematics is a collective activity involving thousands of minds around the world, and the discoveries of each of them serve a single purpose. Don't underestimate their contribution.

Mark Twain said it very well: “It takes a thousand people to invent the telegraph, or the steam engine, or the phonograph, or the telephone, or anything else equally important, and we attribute the invention to the last of them and forget about the rest.”

Make decisions based on large quantity possible options, use formal logic when assessing events, not to give in to proposals that promise us impossible prospects, to remember that the incredible happens in the presence of a large number of chances - all this means doing mathematics in everyday life. And we have been doing this since childhood - more precisely, those of us who support a good relationship with common sense.

There is a point of view in society according to which all people in matters of intellectual knowledge have a tendency either to the mathematical pole or to the humanitarian pole. A child goes to school, gets A's in literature, but doesn't do well in math. “It’s okay,” the parents say, “he’s a humanitarian.” The opposite situation often occurs.

But how fair is this? Is mathematics objectively more difficult to master than the humanities? Are human abilities genetically determined or the result of upbringing?

During the study Mathematicians turned out to be smarter than humanists It turned out that if a student passes exams well in exact disciplines, in most cases he also copes well with 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 passes all exams well, this speaks of his responsibility, not his abilities. Many people can operate easily abstract concepts and study languages, but they find mathematics very difficult. In addition, other studies show that there is no connection between mastering mathematical and humanities disciplines at the level of brain activity. These are completely different cognitive abilities.

Physiological basis of intellectual abilities

As part of the study Origins of the brain networks for advanced mathematics in expert mathematicians Scientists recorded the brain activity of mathematicians and other people while performing 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 others 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 children's ability to perform simple algebraic operations is the key to further mathematical success. After all, in early age, even before any upbringing, parts of a person’s brain develop differently. Some people's mathematical areas are better developed, while others have them 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 learn sines and cosines.

The same can be said about humanists. The speed at which a child masters a language and the ability to grasp the basic laws of grammar make it possible 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 the brain.

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

Many other factors influencing 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, which is why his natural talent will not develop.
What we talk about as a physiological tendency may actually be the result of the early educational activities of parents.

As Swiss psychologist and philosopher Jean Piaget notes: Cognition, the development of both language and mathematical cognitive abilities occurs during the preoperational period (2–7 years). It is then that the child’s physiological predisposition to certain activities may appear.

This period in brain development is the most important, since the creation of neural connections occurs 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 zones that are most often involved begin to actively develop.

At this stage, brain development directly depends on human activity and the repetition of certain practices.

The study of twins also sheds light on the formation of human abilities. Their set of genes is approximately the same, and therefore differences in intellectual abilities are most likely due to 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 actually becomes similar under relatively identical external conditions.

Scientists from the University of California at Santa Barbara came to approximately the same conclusion. The high heritability of educational achievement reflects many genetically influenced traits, not just intelligence. External environment the conditions for the implementation of the biological basis matter and play a role.

conclusions

Whether a person becomes a humanist or a mathematician depends on biological factor and heredity, which predetermine the development of his brain. However, the manifestation of this factor is strongly influenced by activities 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 he somehow uses different areas of the brain, stimulating their development.

In practice, this means the following: parents should not force their child to do activities for which he has no particular attraction and in which he is not very successful. We must try to find talent and promote its development.