How I reprogrammed my brain to begin to understand math

Excuse me, education reformers - we still need brewing and repetition

I was a capricious child growing up on the lyrical side of life, and I treated mathematics and science as if they were symptoms of a plague. And because it is strange that I have become a person who daily deals with triple integrals, Fourier transforms and, the pearl of mathematics, the Euler equation. It is hard to believe that from a Mathophobe I turned into a professor of applied sciences.

One day, one of my students asked how I did it - how I changed my brain. I wanted to answer - damn it, with difficulty! I still failed the math and physics exams in elementary, middle and high schools. I enrolled in the mathematics math class after serving in the army at 26 years old. At an exhibition of examples of neuroplasticity in adults, I would be the first copy.

Studying mathematics and the exact sciences in adulthood opened the door for me to the technical sciences. But these severe adult changes in the brain opened up my view from the inside on the neuroplasticity associated with adult learning. Fortunately, my doctoral degree in systems engineering, during which I learned the exact sciences, technology, engineering and mathematics (STEM - Science, Technology, Engineering, Math), and my subsequent study on human thinking, helped me understand recent breakthroughs in neurology and cognitive psychology associated with learning.

In the years following my doctoral degree, thousands of students went through my class — they were raised in elementary and middle schools with the belief that understanding mathematics through active discussion is a talisman of education. If you can explain what you have learned, to others — say, drawing a picture — then you probably really understood it.

An example of this technique, “focused on understanding,” and the object of imitation was Japan. But the end of the story often disappears from the discussion: in Japan, they also invented the “Kumon” teaching method, which is based on memorizing, repeating and cramming to achieve an excellent mastery of the material. This intensive after-school education program is preferred by thousands of parents in Japan and around the world, complementing co-education of children with more practice, repetitions, and a clever bog system developed wisely, in order to provide them with an excellent mastery of the material.

In the United States, concentration on understanding sometimes replaces, rather than complements, older teaching methods, which, as scientists confirm, work with the natural processes of the brain, which studies such complex things as mathematics and exact sciences.

The latest wave of math education reform involves the Common Core — an attempt to assign hard general standards across the US, although critics say these standards do not match the achievements of other more advanced countries. Externally, the standards have some perspective. It is assumed that in mathematics students should have equal opportunities in conceptual understanding, practical and procedural skills.

The devil, as usual, in the details of the implementation. In today's educational climate, memorization and repetition of STEM disciplines, as opposed to learning language and music, are often regarded as unworthy activities that waste students and teachers. Many teachers have long believed that understanding concepts in STEM disciplines has the highest priority. Of course, it is easier for teachers to engage pupils in the discussion of mathematical topics (and this process, with the right guidance, can greatly help in understanding the tasks) than build on setting marks for homework. As a result, although procedural skills and fluency must be taught in the same doses as conceptual understanding, this is often not the case.

The problem with concentrating only on understanding is that students who comprehend mathematics and exact sciences can often pick up basic concepts about an important idea, but its understanding quickly eludes without fixing it through practice and repetition. Worse, students often feel that they understand something at a time when it is not. Such an approach can often bring only the illusion of understanding. As one of the poor students recently told me, “I don’t understand why I did so poorly with the task. I understood everything in the class. ” It seemed to him that he understood everything, and it is possible that it was so, but he did not use what was understood in practice so that it would be fixed in the brain. He has not developed procedural proficiency or ability to apply knowledge.

There is an interesting connection between learning sports discipline and teaching math and science. When you learn to strike with a golf club, you bring the perfection to perfection by practicing for several years. Your body knows what to do, just when you think about it - you do not need to remember all the components of a complex stroke for hitting the ball.

Similarly, when you understand why you do something in mathematics, you do not need to explain to yourself the same thing every time. You do not need to carry 25 balls with you, spread them out in 5 rows in 5 columns on the table to make sure that 5 x 5 = 25. At some point you just know it. You remember that when you multiply the same numbers in varying degrees, you can simply add degrees (10 ⁴ x 10 ⁵= 10 ⁹ ). Using this procedure often and in different cases, you will find that you understand why and how it works. A better understanding of the topic comes from the creation of a meaningful pattern in the brain.



I learned all this about mathematics and about the learning process itself not in the classroom, but in the course of the course of my life, like a person who read Madeleine Lengl and Dostoevsky in childhood, who studied languages ​​in one of the world's leading language institutions, and then abruptly changed her path and who became a professor of technical sciences.

As a young girl, eager to learn languages, and did not have the necessary money and skills, I could not afford to pay for college. So after school I went to the army. I liked to learn languages ​​at school, and it seemed that the army was just the place where a person could get money for learning languages, visiting the highly valued language institute of the Ministry of Defense - a place where learning languages ​​was turned into a science. I chose Russian because it was very different from English, but it was not so difficult to study it all my life and eventually reach the level of a 4-year-old child. In addition, the Iron Curtain attracted me - could I use the knowledge of Russian to look behind it?

After the army, I became a translator on Soviet trawlers in the Bering Sea. It was interesting and exciting to work for Russians - but it was also a seemingly embellished work of a migrant. During the fishing season, you go to the sea, earn good money, periodically get drunk, and then return to the port at the end of the season and hope that you will be hired again next year. For the Russian-speaking person there was practically only one alternative to this - working for the NSA. My army contacts were pushing me towards this, but I didn’t have a soul for it.

I began to realize that although the knowledge of another language is good, it was a skill with disabilities and potential. Because of my ability to incline words in Russian, my house was not besieged. Unless I was willing to tolerate motion sickness and periodic malnutrition on smelly trawlers in the middle of the Bering Sea. I could not help but recall the engineers from West Point, with whom I worked in the army. Their mathematical approach to solving problems was clearly useful for the real world — more useful than my failures with mathematics.

So, at the age of 26, when leaving the army and assessing opportunities, I suddenly thought: if I want to do something new, why not try something that would open up a whole new world of perspectives? Technical sciences, for example? And that meant that I had to learn a new language - a number language.

With my poor understanding of the simplest mathematics, after the army I took up algebra and trigonometry in the course for the laggards. Trying to reprogram the brain sometimes seemed like a silly idea — especially when I looked at the faces of my younger classmates. But in my case, and I did learn Russian at a mature age, I hoped that some aspects of learning a language could be applied in the study of mathematics and the exact sciences.

Studying Russian, I tried not only to understand something, but also to achieve fluency in it. Fluency in such a broad subject as language requires such a degree of familiarity that can be developed only by repetitive and differing work with different areas. My classmates, who studied the language, concentrated on a simple understanding, and I tried to achieve inner fluency with words and the structure of the language. It was not enough for me that the word “understand” means “to understand”. I practiced with the verb, constantly used it in different tenses, in sentences, and then I understood not only where it can be used, but also where it is not necessary to use it. I practiced over quick retrieval of these aspects and variants from memory. Through practice you can understand and translate tens and hundreds of words from another language. But if you do not have fluency, then when someone quickly spits out a bunch of words to you, as in a normal conversation, you have no idea what this person is saying, although technically you seem to understand all the words and structure. And you, of course, cannot speak fast enough for native speakers so that they enjoy listening to you.

This approach, focusing on fluency rather than simple understanding, led me to first place in class. I didn’t understand it then, but this approach gave me an intuitive understanding of the basics of learning and developing expert skills - chunking.

Bursting was first proposed in the revolutionary work of Herbert Simon in the analysis of chess. Different mental analogues of chess patterns served as pieces. Neuroscientists have gradually come to the realization that experts, for example, in chess, are such, since they can store thousands of pieces of knowledge in long-term memory. Chess masters can recall tens of thousands of different chess patterns. In any field, an expert can recall one or several well-connected pieces of nerve routines for analyzing and reacting to a new situation. This level of real understanding and the ability to use this understanding in new situations is acquired only from familiarity with the subject, obtained from repetitions, memorization and practice.

The study of chess masters, ambulance doctors and fighter pilots showed that in stressful situations, a conscious analysis of the situation gives way to rapid subconscious data processing when experts turn to a deeply integrated set of mental patterns - pieces. At some point, a conscious understanding of why you are doing what you are doing, only starts to slow you down and interrupts the flow, which leads to making the worst decisions. I was right, intuitively feeling the connection between learning a new language and mathematics. The daily and continuous study of the Russian language excited and strengthened the neural circuits in my brain, and I gradually began to tie together Slavic pieces that could be easily recalled from memory. Alternating studying, practicing in such a way that I knew not only when you could use the word,

I started studying mathematics and science in adulthood with the same strategy. I looked at the equation - for a simple example, take Newton's second law, F = ma. I practiced sensing the meaning of each letter: “f”, that is, force, it was a push, “m”, mass, heavy resistance to pushing, “a” was a joyful sensation of acceleration. (In the case of the Russian language, I also practiced the pronunciation of Cyrillic letters). I remembered the equation, wore it in my head and played with it. If m and a are large, then what will happen to f in the equation? If f is large and a is small, what will m be? How do units of measure fit on both sides? Playing with the equation - how to link the verb with other words. I began to comprehend that the vague outlines of the equation resembled a metaphorical poem in which all sorts of beautiful symbolic representations existed.

Over time, the professors of mathematics and exact sciences told me that building pieces of experience well recorded in memory through practice and repetition was vital for success. Understanding does not lead to fluency. Fluency leads to understanding. In general, I believe that the real understanding of a complex topic comes exclusively from fluency.

Invading a new area for me, becoming an electrical engineer, and, as a result, a professor of engineering, I left the Russian language behind. But 25 years after the last time I raised a glass on the Soviet trawlers, my family and I decided to take a trip to Transsib across Russia. And although I was looking forward to the long-sought journey, I was also worried. All this time I practically did not speak Russian. What if I forgot everything? What did all those years of achieving fluency give me?

Of course, when I first went to the train, I found that I speak Russian at the level of a two-year-old child. I was looking for words, my inclinations and conjugations were confused, and the previously almost perfect accent sounded awful. But the basis did not disappear, and gradually my Russian improved. Even rudimentary knowledge was enough for daily needs. Soon the guides began to approach me for help in translating for other passengers. Arriving in Moscow, we got into a taxi. The driver, as I later understood, tried to deceive us by going the other way and stuck in a traffic jam, believing that foreigners who were not knowledgeable would calmly endure the counter for an extra hour. Suddenly, Russian words that I have not used for decades have flown out of my mouth. Consciously, I did not even remember that I knew them.

Fluency, when it was needed, was at hand - and helped us out. Fluency allows understanding to build in consciousness, and emerge as needed.

Looking at the lack of people specializing in the exact sciences and mathematics in our country, and our current teaching techniques, and remembering my own way, with today's my knowledge of the brain, I understand that we can achieve more. As parents and teachers, we can use simple methods to deepen understanding and turn it into a useful and flexible tool.

I discovered that having basic and deeply learned fluency in math and exact sciences — and not just “understanding,” is extremely important. It opens the way to the most interesting activities in life. Looking back into the past, I understand that it was not necessary for me to blindly follow my original inclinations and passions. The same “runaway” part of me, who adored literature and language, as a result, fell in love with mathematics and exact sciences - and as a result, transformed and enriched my life.