The biggest reason I wanted to be a mathematician is that the concept of certainty is not as clear in any other field as it is in mathematics. At first, I wanted to be a painter, but the arguments over the concept of excellent painting and lousy painting changed my mind after a while. I was also interested in philosophy at one point in time, but again, I saw that the concepts in philosophy are too obscure, difficult to judge and take years to prove whether they are true or false. I felt that it is not even clear what philosophy is all about. One day I asked our neighbor’s daughter what they were doing at school, and she said they were focusing on proving some stuff. I was puzzled, so she added, “we are trying to figure out what is for what.” There was only one truth in math, and it utterly attracted me. For me, it was breathtakingly good.
With the help of certainty in mathematics, people understand and explore life, the universe, and its rules. In other words, in mathematics, instead of what is right, you realize for what reason it is correct. If you are trying to understand the logic for your creation and the universe, nothing can be more profound or more philosophical than it.
To become a professional mathematician, one has to enjoy mathematics. Because, by definition of professionalism, the individual fully devotes themselves to their job. As humans, we like to forget ourselves. For example, if an average person tries running 40 kilometers, his muscles build up fatigue and often fail. Nobody runs just for the sake of running. But Haile Gebrselassie, who was the best marathon runner, for example, can run 40 kilometers because he takes pleasure in running and, more importantly, he is ecstatic in the process. He continues to run the marathon as if in a state of hypnosis, with rhythm and pace. In other words, he forgets everything else during the marathon.
As a math teacher, I teach 6 hours a day on an average, and when I finish my classes and sit at my desk, I feel that my feet are hurting because, the moment I sit at my desk, I have the opportunity to think about myself, this is when the pain manifests. I do not feel tired while teaching because I love to teach. Here is another example: Most viewers who watch a horror movie in the cinema are sometimes terrified. Initially, everyone knows that it’s just a movie, but they still get afraid of it. The reason is that people are ecstatic while watching the film to do something they like at that moment. We can use the same example for reading books. Sometimes when we come home and pick up an exciting novel, we feel that we would like to continue reading it all night, unable to let it go as we think ecstatic and get lost in the story.If we want to make students love mathematics, we need to find ways to make them delve into it by making it enjoyable.
First of all, as educators, we need to know mathematics well because we cannot teach what we do not know. Everyone is afraid of the unknown. Students should not feel the hours pass by when they are doing math or learning math. It is not impossible to do so. All we have to do is explain math the way it is: in its natural form. If we manage to show the math in its purest form, every student will love it. Since human emotions are taken from nature, everything that humanity has been contemplating for a long time turns beautiful. Mathematics is also fascinating because thousands of intelligent people have been pondering on how to teach mathematics as it has been for 5,000 years. Something we have spent time on for more than 5,000 years cannot be ugly or uninteresting. Of course, this is true for philosophy, art, and history.
Thinking is a fun endeavor. For example, a child who solves Sudoku will have fun. As Sudoku gets harder, it becomes even more fun, and the child is not afraid of it. Because that child is interested in it and is aware that there is no harm in solving Sudoku, the same child often cannot take pleasure in math. The biggest reason for this is the fear of losing or failing. Failure to do math has harmed a lot of people. But, the fear instilled in students that they will lose their future if they fail in math is a massive obstacle for mathematics itself. Unfortunately, most educational systems in the world do not tolerate failures in math. Worse still, the student’s fear of a dark future upon failing math turns mathematics into a nightmare.
Imagine, before we teach anything about math, we tell a student, “We will give you an exam, and, if you fail, your life will be ruined for good,” and then we expect him to be successful.
I think this discourse is the worst thing that can happen to a person. Comparing this to the Sudoku example, one has nothing to lose when solving Sudoku. So he takes pleasure when he solves Sudoku.
A little bit of stress is good and keeps people on their toes. While the university admission tests certainly measure a student’s math skills, the stress level for the takers of these tests should not be excessively high. In many parts of the world, students usually take tests for admission to universities at the age of fifteen through eighteen. In theory, these tests measure how much students have worked in the past and their capacity to work. It is also true that students with high test scores are more successful than others because we observe that successful students have work discipline or find that their capacity to understand is higher than other students. But these measures do not make the university entrance exams legitimate.
Sometimes, millions of students take those tests at the same time. The reason for this is something known as the centralized education system. In other words, an education system where millions of students learn the same things and are tested at the same time. Moreover, it can be defined as an education system that students learn from the same books, the same curricula, the same methods, in very similar buildings, from teachers who have survived the same education system. Even the children’s holidays are the same! In a country like Finland, the central education system works very well because there are not many students, and they are all blond, tall, and speak English. The structure there is homogeneous.However, there isn’t a homogeneous structure in regions where the population is dense, like Turkey or the USA. There are wealthy and poor students, students growing up in towns, and those in metropolises. Naturally, students’ needs, cultures, and capacities are very different. It is not wise to teach children mathematics through a central education system.
Giving everyone the same education does not mean giving equal opportunities. Our schools are officially similar to prisons, with no freedom for students. Where there is no freedom, there will be no creativity.
We , as educators, should take steps to build better quality schools as much as possible. But first, we have to recognize the problem(s). It is only when we realize the problem; something can be done to fix it. The schools to be built should be completely independent of the old education system and still provide high-quality education.
However, it is challenging to achieve this because there will be barriers due to bureaucracy from the state. Unfortunately, what is called the state is a structure against any original idea. The state wants everyone to do the same thing. Since the state does not trust anyone other than itself, it judges students through standardized exams.
In many societies, the most educated people are found in the universities, but they operate under the state. Ruling the universities means that the state does not fully trust these institutions. The state should be open to the ideas of creative individuals. Even if the plan fails, it must be tolerant because the project could be one of many that work did not, but I would make a big difference if it were to be successful.
For example, if an institute that is good in mathematics educates 20 good students every year and does it for 30 years, it will produce 600 good mathematicians. If we assume that these 600 mathematicians had students as well, it means that it has produced a couple of thousand good mathematicians. It would be an ideal plan to make the world a more beautiful place. The so-called state has to allow young people to be educated differently.Because of the central education system, the students are so focused on success that they think they have to solve a problem quickly to be called successful students. For example, sometimes, I ask my students a question. Many of them respond with quick answers; even when I tell them that their response is wrong and that it would take longer to solve the question, they continue to respond with prompt answers. The reason is that the students are convinced that they have to find the answer quicker than others to be considered above the “ordinary” students. However, they have to give it a thought before they start solving it and take time to solve it.
One of the essential things in mathematics is to be convinced that you can convince someone else.
Proofs are no longer being included in the mathematics curriculum. Mathematics without proof is no different than reading a dictionary to learn English. Math books taught in schools are full of mistakes and not designed aesthetics. In my opinion, textbooks are an insult to students’ intelligence because they do not include writing or narrative. Books are just exercises, questions, theorems, and formulae. The students are not aware of this because they have not seen the proper form of a textbook before.
I would not say I wasn’t a very successful student during my student life; I failed many exams and graduated from university, taking seven years. But I never cared about it because I knew what I wanted to do and what I did. A lecture note given by a simple person would never be an obstacle to my future. If I were to be considered successful, I owe my family’s trust in me for not interfering. I wouldn’t be a mathematician if my parents put me under “test stress” and made me feel anxious about the future.
Also, sometimes, families can become the main obstacle to their children’s learning without realizing it. Sometimes, I meet families who want their 5-year-old to be a good mathematician. However, those families should leave their children alone. Abstract intelligence needs to be developed to learn mathematics because mathematics is abstract. Grasping the concept of abstract starts in healthy brains during puberty. Therefore, insisting on pre-adolescent math education is of no use to anyone.
Parents should direct their children to art, music, sports, or literature rather than mathematics. Children should read lots of books, play football or basketball, swim, play the piano, play chess, and even collect random things before learning math. These activities will help with the development of abstract thinking in children. The overwhelming brainpower of a genius cannot be achieved by working as it is a gift of God. Many people have similar levels of intelligence and can improve their level. Education serves to increase the capacity of people with average intelligence.Mathematics, by itself, is a tricky subject. First, it is necessary to make this distinction: you cannot learn mathematics; you need to understand it. Understanding mathematics is like building a pyramid to make a simple analogy. If we want to create a masterpiece, we have to build each layer firmly. Without the first, the second layer of the pyramid cannot be constructed. Pyramid building should not start from the top as well, and this is what makes mathematics difficult, and that is what separates mathematics from other subjects.
For example, if you have a final exam in history and the exam is about World War II, without knowing the history of antiquity or remembering the history of humanity, you can still learn about the history of World War II and get a passing score. But in mathematics, it will not work. A student has to remember the mathematics he learns every year. Without knowing Algebra I, you cannot understand a heavy subject such as Algebra II or Calculus. It is impossible to learn trigonometry without learning the concept of Pi.
The mathematical foundation has to be very solid. To give you a more straightforward example, you need to know what root two plus root three is before you can solve it. If I write a Hungarian theorem on these lines right now, you would ask yourself, “what the hell did I just read?” because to understand, you need to know the language first. To mention an expression, to prove it, you first need to know the meaning of that expression. The mathematical universe is a symbolic language for understanding the world, existence, and movement. Mathematics was not made up because we were bored.
Being a mathematician is also a difficult task because one has to devote quite a few years of life to becoming a true mathematician.
Those who claim it to be easy will be fooling you. You may have heard of mathematicians who devote their entire lives to solving one just problem, but when a journey in mathematics ends, one enjoys it. The pleasure from it cannot be replaced by anything else. If a person is engaged in something for years, life will certainly reward it. Nobody gets pleasure from the riddles solved in 5 to10 minutes. The person who deals with tricky riddles pushes himself seriously, works for days on a puzzle, gets real pleasure when he finishes it. For example, multiplying two 5-digit numbers will not help anyone. Because most people know how to multiply two 5-digit numbers, and when they do, they do not tend to get pleasure out of it. So even knowing the multiplication table does not matter as much. If a person doesn’t know the multiplication table, he will not make a good accountant, at the most.
However, successful people love the challenge, and they want to be taller when they finish something; humanity should go one step further to gain something.
When I was 15, I fell in love with a mathematical phenomenon called proof by induction. I liked it before, but I never thought I would love it that much. If a student aims to be a mathematician, if they like mathematics above all else, they should delve into it. Mathematics cannot be learned without being loved, and more importantly, it cannot be learned while being sloppy.
A person who wants to be a mathematician cannot be confined to his environment, and that is why a good math student often has to spend time in the library. Besides, all good mathematicians know proper English because, in the scientific world, nothing can happen without the knowledge of English, and most of the time, that person is not even held in high esteem if he does not know proper English.
A good mathematician must refer to many books, but a person cannot read a math book like a novel. In other words, starting from the first page and reading each page might not be the best course of action to learn mathematics. Usually, the reader starts in the middle of the book, then goes back to the beginning, if necessary, he goes forward, and goes back to the beginning again, if needed. They should not look immediately at the proof of a theorem he read. If necessary, he must try to deduce the proof of the theorem himself.When learning math, one must write and always have paper and pencil ready and utilize a lot of paper, if needed. Math students even should not use unnecessary things such as erasers and rulers. They shouldn’t be afraid of making a mistake, and if they do, they should just cross it. Apart from these, a good math student should meet other cultures. He should see other kinds of education and other types of human relationships and get a different perspective.
It is not necessary to be born intelligent to be good in mathematics. If a student wants to understand mathematics, he must have a tendency to focus and concentrate. Even an autistic or hyperactive child can be a mathematician as long as they have the ability to concentrate.
Unfortunately, societies have changed a lot in today’s world, and because of the internet and television, almost all children have problems with concentration and focus. Technology is the biggest obstacle today for students to stay focused or concentrate. Students are constantly exposed to external factors in their lives. They follow the news, listen to music, have fun, or send each other something more than a few times each day.
The percentage of students who can stay focused for half an hour and ask, “what am I doing, where I am, what is my goal, why am I here, what should I do, or what should I think” is very low. Today child does not know how to express themselves because they do not know how to understand and interpret what they read. They do not know what to do in schools if you do not ask them any questions.
There were not many distractions in my student years. Most of the time, I enjoyed watching the discussions of guests coming home because I would try to understand who was right and who was wrong. So, as a child, I learned how to defend a thought, what counter-thought is and how to defend it.
When can a person be alone? They can be alone while reading a book, playing the piano, praying, or thinking. Mathematics is an excellent tool for thinking. Today, as teachers, we need to focus on this. For example, during a math instruction that lasts for an hour and a half, the student should be made to think and solve for about an hour. Sometimes, after a 20-minute lecture, I ask children some abstract questions and let them think about it for the rest of the lecture. At the end of the lesson, when I ask if they liked the lesson at the end of the lesson, they usually mention that they liked it very much. They realized that they wanted to use their minds to think and solve. To sum up, listening to a lecture is the hardest job in the world, but it is much easier to think about it.
If the students are constantly looking at the clock and praying for the lecture to end while a teacher is teaching, the teacher should stop. Alternatively, they can ask a question or a riddle, which is slightly above their student’s level but not impossible to solve. This way, the students will be encouraged to think, or sometimes a teacher should just chat and play with her students.
I recently played a game with one of my students. In the beginning, I had $5, and the student had $8. We bet with a fair amount, toss a coin. If it is a head, I win, and the student wins if it is a tail. He gives me a dollar if it is a head, and I give him a dollar if it is a tail. The game goes on until one of us has no money left. Who will win this game, and with what percentage of probability?Logically, if my student has $8 and I have $5, the student is more likely to win. But what exactly is this percentage? What makes the game beautiful here is that we started a series of discussions while playing the game. The game we played was an excellent example of positive discrimination.
For instance, if I had a dollar and the student had a million dollars, I would have no chance of winning against the student. The odds for the student to win would have been very high. If the bet were not a fair one ($1 per toss), I would not be able to win, and the game would be over immediately. However, I wished we could play it for at least 5–10 turns; this game of possibilities can be a positive example for the students. This fact can also be related to politics; it is a game of balance; it is math.
Mathematics is the best version of the adaptation of abstract thought to real life.
Although engineers do not like it, a real mathematician always insists on abstract thinking, and when it is time, the student’s interest can be directed to mathematics with such practices. It is an unnecessary effort for a mathematician to look at whether every question has real-life applications; engineers should take care of this. As a teacher, sometimes I tell students that my problems are of no use to them. However, I show them that the method they will use to solve the problem, their handling of it, and their persistence in solving it is the life lesson itself.
It should not be forgotten that there could have been no technological development without basic sciences such as mathematics. An engineer can make a refrigerator that cools more quickly or a solar-powered car, but it is not technological innovation; it is technological development. If an engineer can charge his mobile phone wirelessly, then it is a technological innovation requiring basic sciences mathematics. Unfortunately, many big science centers have been taken over by engineers, and they perceive every scientific development as technological development. For a long time, the sole purpose of many engineers was to build electric cars.
There should be departments such as mathematics, philosophy, and art in an educational institute. Unfortunately, those working in these fields are not seen as professionals in many societies.
A profession is expected to produce tangible things. For example, a carpenter makes a chair or a table. But, mathematics, philosophy, and art are forms of existence and thought. These basic sciences do nothing, but they are useful for everything. I have not come across a person who says that he will study mathematics or philosophy and earn loads of money. It should be noted that some things require devotion and do not need a specific approach, and you should not always work to get an award. Therefore, society and the state must support basic sciences such as mathematics and philosophy.
Every person has to do something in life. It is sometimes useful to be struggling in life because when you are struggling, you feel, “I exist!”. There is nothing that one cannot understand in this world. Sometimes, we cannot grasp things at all, but that is because we do not think deeply about them. We may not always understand all human relations or commercial relations, but we can still understand mathematics.
Mathematics is a purely mental thing, it is abstract, but it is not something that people make up because they are bored. Mathematics also has aesthetic beauty. A life in which one out of every three numbers was prime would be much easier and tremendously dull. Mathematics is not just about numbers. In mathematics, A’s, B’s, X’s, and Y’s are more natural than numbers. Mathematics is the ability to see a problem, symbolize it, and cast it on paper. Mathematics is not the art of finding the truth but the art of finding out why it is the truth.
In ancient Greece, at the door of a school opened by Plato, it was inscribed, “Let no one ignorant of geometry enter.” In today’s modern schools, I think we should alter it slightly to: “Let no one ignorant of geometry leave.”
*** I had a mathematics professor who made me love mathematics in college. His name is Ali Nesin. He created a beautiful mathematics village in Turkey and was awarded Leelavati Prize in 2018. He has perfect solutions for mathematics education. I decided to combine his solutions and opinions with this long read.
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