I was a wayward child who grew up on literature and treated math and science as if they could catch the plague. Therefore, it is rather strange that as a result I became a person who deals with triple integrals, Fourier transforms, and the pearl of mathematics, the Euler equation every day. It's hard to believe that from a person with a literal inborn phobia towards mathematics, I turned into an engineering professor.
One day one of my students asked me how I did it: how I changed my brain. I wanted to answer: "Damn it, it was extremely difficult!" After all, I couldn't do math and science in elementary, middle school, and high school. In truth, I only started taking math classes after I was fired from the army when I was 26 years old. If there was an example of the potential for flexibility in the adult brain, I would become Model # 1.
Studying mathematics and science as an adult opened the door for me to a world of many possibilities - engineering. Through hard work in adulthood, my brain change has allowed me to see firsthand the neuroplasticity that underlies adult learning. Fortunately, preparing for a doctoral dissertation in systems engineering, linking a huge picture of different STEM disciplines (science, technology, engineering, mathematics), and then for my further research and work, which centered on the structure of human thinking, helped me to realize the latter discoveries in neurosciences and cognitive psychology related to the learning process.
Since I got my PhD, thousands of students have passed through my hands, students in elementary school and high school believed that the sacred talisman of understanding mathematics is active discussion. It is believed that if you can explain to others what you have learned, for example, by drawing a picture, then you understand it.
Japan has become an admirable and emulated example of these active learning methods of “understanding”. However, the downside of this concept is not often talked about: Japan also became the birthplace of the Kumon method of teaching mathematics, which is based on memorization, repetition, cramming and work on how the child masters the material. This intense extracurricular program (and others like it) has been eagerly received by parents in Japan and around the world who supplement their kids' online education with lots of practice, repetition and, yes, sophisticated cramming to give them the freedom to master the subject.
In the United States, the emphasis on understanding sometimes supplants another older method used (and used) by scientists: to study mathematics and science, you need to work with the natural process of the brain.
The latest wave of educational reforms in mathematics is about the compulsory school curriculum: it is an attempt to set strong, uniform standards across America, although critics point out that the standards do not stand up to comparison with the best performing countries. At least superficially, the standards provide a reasonable perspective. They assume that in mathematics, students should have equal conceptual knowledge, fluency in problem solving skills and the ability to apply them.
The catch, of course, lies in getting things done. In the current educational climate, memorization and repetition in STEM disciplines (versus learning a language or music) is often viewed as a demeaning waste of time by both students and teachers. Many teachers have long been taught that conceptual knowledge is key in STEM disciplines. Indeed, it is easier for teachers to engage students in a discussion of a math topic (and if done properly, it develops a better understanding) than it is tedious to evaluate homework done. The implication is that fluency in skills and the ability to apply them must develop in equal measure with conceptual knowledge, and this very often does not happen. The dissemination of conceptual knowledge reigns supreme, especially during precious class times.
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The difficulty in focusing on understanding is that in math and science classes, students can often grasp an important point, but this knowledge can quickly slip away without being established in practice and repetition. To make matters worse, students often think they understand something when, in fact, they don't. By highlighting the importance of understanding, teachers can unwittingly push their students towards failure while children indulge in the illusion of knowledge. As one engineering student recently told me (failing an exam): “I just don't understand how I could get such a bad result. I understood everything when you explained in class. My student might have thought that he understood the topic then, and perhaps he did, but he never put this knowledge into practice to really learn it. He has not developed any decision skill or ability to apply what he thinks he has already understood.
There is an interesting relationship between studying mathematics and science and mastering a sport. When you learn to hit with a golf club, you perfect this movement through constant repetition over several years. Your body knows what to do when you just think about it (the whole block), instead of remembering all the difficult steps it takes to hit the ball.
In the same way, once you understand something about math and science, you don't have to constantly re-explain it to yourself every time you come across a topic. You don't have to carry 25 marbles with you, constantly laying out rows of five pieces in order to understand that 5 × 5 = 25. At some point, you just know it by heart. You remember the idea that you just need to add the exponents (small numbers written on top), when multiplying the same number in different degrees (104 × 105 = 109). If you do this procedure frequently, solving many different types of problems, you will find that you have a very good understanding of both the reasons and the actions behind the procedures. Understanding is expanded by the fact that your brain has built meaning schemes. The constant focus on understanding itself is actually a hindrance.
I learned all this about mathematics and the learning process not in classrooms K-12, but on my own experience, as a child, growing up reading Madeleine Langle and Dostoevsky, who studied language at one of the world's leading language universities, and then suddenly turned into a professor of engineering.
In my youth, with a talent for languages and without enough money or skills, I could not afford to go to college (there was no talk of college loans then). So from high school, I went straight to the army. I loved studying foreign languages back in high school, and the army seemed to me a place where people were paid money to learn foreign languages, even if they were studying at the prestigious Military Institute of Foreign Languages, a place where language learning grew into a science. I chose Russian because it was very different from English, but it was not so difficult that I had to learn it for ages and learn to speak it at the level of a four-year-old. In addition, the Iron Curtain beckoned with its mystery: suddenly I will be able to use my knowledge of the Russian language and take a look,what is behind it?
After serving in the army, I began to translate for the Russians who worked on Soviet trawlers in the Bering Sea. Working for Russians was fun and exciting, plus it was a slightly glamorous kind of job for migrants. You go to the sea during the fishing season, earn decent money, constantly get drunk along the way, then return to the port at the end of the season and hope to be called back to work next year. For a person who spoke Russian, there was only one alternative to employment - to work for the National Security Agency (my friends in the army constantly suggested this option to me, but it was not for me).
I began to understand that knowledge of a foreign language in itself is a useful business, but with limited potential and number of opportunities. No one cut off my phone, no one needed my knowledge of declensions in Russian. Unless I was going to get used to seasickness and occasional malnutrition on fetid trawlers in the middle of the Bering Sea. All the time I remembered the engineers who studied at West Point, with whom I worked in the army. Their mathematical and scientific approach to problem solving was obviously useful in the real world, far more useful than my misadventures with mathematics in my youth would have allowed me to imagine.
Thus, at the age of 26, leaving the army and in search of new opportunities, I realized: if I really want to try something new, why not start with what could open up a whole world of new perspectives for me? Something like engineering? This meant that I would try to learn a completely different language - the language of calculus.
With my poor understanding of even the simplest mathematics, my post-army endeavors began with restorative lessons in algebra and trigonometry. This was well below the zero level of most college students. At times, trying to reprogram my brain seemed to me a ridiculous undertaking, especially when I looked at the young faces of my younger classmates and realized that they had already abandoned their difficult classes in mathematics and natural sciences, and I decided to go straight to meet them. But in my case, in my experience of mastering the Russian language as an adult, I suspected (or simply hoped) that there would be something in the aspects of learning a foreign language that I could use when mastering mathematics and science.
When I was learning Russian, I focused not only on understanding the language, but also on being fluent in it. Free use of the whole system (in this case, the language) requires close acquaintance, which is achieved exclusively through repeated and varied interaction with its elements. Where my classmates were content with a simple understanding of spoken or written Russian, I tried to develop an inner, deep connection with the words and structure of the language. I was not content with just knowing the meaning of the word "understand." I used the verb in practice: I constantly conjugated it in different tenses, used it in sentences, and finally, I understood not only when to use this form of the verb, but also when not to do it. I trained with the challenge of quickly recalling all these aspects and variations. If you are not fluent in the language and someone is chattering quickly at you, as it happens in normal conversation (which always sounds awfully fast when you are learning a foreign language), you have no idea what you are in fact they say, although technically you understand each word individually and the structure of the phrases. Of course, you yourself cannot speak fast enough for native speakers to enjoy listening to you.
With this approach (focusing on fluency instead of just understanding), I got ahead of everyone in the class. I didn't realize it then, but this approach to language learning gave me an intuitive understanding of the fundamental basis of learning and developed competence - the formation of blocks.
Block formation was originally developed in the revolutionary work of Herbert Simon, where he analyzed chess: blocks were seen as various neural equivalents of different chess schemes. Gradually neuroscientists realized that specialists like chess grandmasters achieved this by storing thousands of blocks of knowledge about their area of expertise in long-term memory. Grandmasters, for example, can remember tens of thousands of different chess patterns. Regardless of the discipline, connoisseurs can awaken in their consciousness one or several well-welded, assembled in a block of neural subroutines, with the help of which they analyze and respond when faced with the need to learn new things. The level of true understanding, the ability to use it in new situations appears only with that clarity, level of knowledge,which can only provide repetition, memorization and practice.
As studies carried out among chess players, ambulance doctors and fighter pilots have shown, at moments of greatest stress, a quick subconscious processing comes to replace the conscious analysis of the situation, since all these specialists develop a system of neural subroutines, blocks, at a deep level. At a certain moment, a conscious "understanding" of why you are performing this or that action serves only as an obstacle and results in not the most successful decisions. When I intuitively understood that there was a connection between learning a foreign language and learning mathematics, I was right. Daily long-term practical mastery of Russian charged and strengthened my neural connections, and I gradually began to link together the blocks of language knowledge that could be easily used now. By organizing your learning in "layers" (in other words,practicing in such a way that I not only knew when to use the word, but also when not to use it, or rather another version of it), I actually used the same approach that practitioners in mathematics and science take. While studying mathematics and engineering as an adult, I began to use the same strategy as when studying a foreign language. I looked at equality, to take the most elementary example, at Newton's second law f = ma. I trained in understanding what each letter means: f - gravity - meant pressure, m - body weight - put a kind of resistance on my pressure, and a - an invigorating sensation of acceleration. (The equivalent in learning Russian was to say aloud the letters of the Russian alphabet). I memorized equality so that it settled in my memory,and I could play with him. If m and a were large numbers, how did this affect f when I substituted them in the formula? If f was large and a was small, how did this affect m? How did the parts of the equality fit together? Playing with equality was like verb conjugation. I was beginning to intuitively understand that the blurry outlines of equality were like a poem saturated with metaphors, in which many beautiful symbolic images are hidden. Although at that time I would not call it that, in truth, in order to master mathematics and science well, I had to slowly, day after day, build strong neural "block" routines (like those that I did with the formula f = ma), so that I can easily use information from long-term memory, as I did with the Russian language. If m and a were large numbers, how did this affect f when I plugged them into the formula? If f was large and a was small, how did this affect m? How did the parts of the equality fit together? Playing with equality was like verb conjugation. I was beginning to intuitively understand that the blurred outlines of equality were like a poem saturated with metaphors, in which many beautiful symbolic images are hidden. Although at that time I would not call it that, in truth, in order to master mathematics and science well, I needed to slowly, day after day, build strong neural "block" routines (like those that I did with the formula f = ma), so that I can easily use information from long-term memory, as I did with the Russian language. If m and a were large numbers, how did this affect f when I plugged them into the formula? If f was large and a was small, how did this affect m? How did the parts of the equality fit together? Playing with equality was like verb conjugation. I was beginning to intuitively understand that the blurred outlines of equality were like a poem saturated with metaphors, in which many beautiful symbolic images are hidden. Although at that time I would not call it that, in truth, in order to master mathematics and science well, I had to slowly, day after day, build strong neural "block" routines (like those that I did with the formula f = ma), so that I can easily use information from long-term memory, as I did with the Russian language.when did I substitute them in the formula? If f was large and a was small, how did this affect m? How did the parts of the equality fit together? Playing with equality was like verb conjugation. I was beginning to intuitively understand that the blurred outlines of equality were like a poem saturated with metaphors, in which many beautiful symbolic images are hidden. Although at that time I would not call it that, in truth, in order to master mathematics and science well, I needed to slowly, day after day, build strong neural "block" routines (like those that I did with the formula f = ma), so that I can easily use information from long-term memory, as I did with the Russian language.when did I substitute them in the formula? If f was large and a was small, how did this affect m? How did the parts of the equality fit together? Playing with equality was like verb conjugation. I was beginning to intuitively understand that the blurred outlines of equality were like a poem saturated with metaphors, in which many beautiful symbolic images are hidden. Although at that time I would not call it that, in truth, in order to master mathematics and science well, I needed to slowly, day after day, build strong neural "block" routines (like those that I did with the formula f = ma), so that I can easily use information from long-term memory, as I did with the Russian language.how did this affect m? How did the parts of the equality fit together? Playing with equality was like verb conjugation. I was beginning to intuitively understand that the blurred outlines of equality were like a poem saturated with metaphors, in which many beautiful symbolic images are hidden. Although at that time I would not call it that, in truth, in order to master mathematics and science well, I needed to slowly, day after day, build strong neural "block" routines (like those that I did with the formula f = ma), so that I can easily use information from long-term memory, as I did with the Russian language.how did this affect m? How did the parts of the equality fit together? Playing with equality was like verb conjugation. I was beginning to intuitively understand that the blurred outlines of equality were like a poem saturated with metaphors, in which many beautiful symbolic images are hidden. Although at that time I would not call it that, in truth, in order to master mathematics and science well, I had to slowly, day after day, build strong neural "block" routines (like those that I did with the formula f = ma), so that I can easily use information from long-term memory, as I did with the Russian language.which contains many beautiful symbolic images. Although at that time I would not call it that, in truth, in order to master mathematics and science well, I had to slowly, day after day, build strong neural "block" routines (like those that I did with the formula f = ma), so that I can easily use information from long-term memory, as I did with the Russian language.in which many beautiful symbolic images are hidden. Although at that time I would not call it that, in truth, in order to master mathematics and science well, I had to slowly, day after day, build strong neural "block" routines (like those that I did with the formula f = ma), so that I can easily use information from long-term memory, as I did with the Russian language.as I did with the Russian language.as I did with the Russian language.
At times, math and science teachers told me that building blocks of information deeply embedded in the mind was the absolute foundation of their success. Understanding does not create fluency in knowledge; on the contrary, fluency builds understanding. In fact, I believe that true understanding of a complex subject arises only in the conditions of free mastery of it.
In other words, in teaching the natural sciences and mathematics, it is easy to switch to teaching methods where the emphasis is on understanding, and routine repetition and practice, which are the basis of fluency in the subject, are avoided. I learned Russian not only because I understood it - after all, understanding is not such a difficult task, but it can easily slip away from you. (What does the Russian word for “understand” mean?) I learned Russian, striving for fluency through practice, repetition and cramming, only a type of cramming that stimulated the ability to think flexible and fast. I learned math and science using exactly the same principles. Language, mathematics, natural sciences, like almost all areas of human knowledge, use the same mechanisms of the brain.
When I burst out into a new life, becoming an electrical engineer and then a professor of engineering, I left Russian in the past. But 25 years after the last time I drank on a Soviet trawler, my family and I decided to drive through all of Russia on the Trans-Siberian railway. Despite the fact that I was looking forward to this trip, which I had long dreamed of, I was worried. Over the years, I have hardly spoken at least a word in Russian. What if I completely forgot him? What did all these years of mastering fluency in the language give me?
Of course, when we first boarded the train, I spoke Russian like a two-year-old child. I frantically searched for words, made a mistake in declension and conjugation, my former almost perfect pronunciation turned into a terrible accent. But the foundations were laid, and from day to day my Russian got better and better. But even with a basic level, I was able to cope with the daily tasks during our journey. Soon the guides began to approach me so that I could help translate them for other passengers. Finally we arrived in Moscow and got into a taxi. The driver, as I soon realized, was going to rob us like a sticky man - he drove us in exactly the opposite direction, through traffic jams, expecting that foreigners, who do not understand anything, would tacitly pay for an extra hour at the rate. Suddenly Russian words escaped me,which I have not spoken for decades. I didn't even realize that I knew them.
Somewhere deep in my mind, my fluency in the language remained and came out at the right moment: it quickly got us out of trouble (and helped find another taxi). Fluency allows understanding to become part of consciousness and emerges when you need it.
When I see today how much there is a lack of specialists in natural sciences and mathematics in our country, I observe modern trends in pedagogy, reflecting on my own path, on the knowledge I have gained about the abilities of our brain, I understand that we could achieve much more. As parents and teachers, we can use simple, accessible methods to deepen our understanding, making it useful and flexible. We can push other people and ourselves to study new disciplines that seemed too difficult to us - mathematics, dance, physics, language, chemistry, music - thereby opening completely new worlds for ourselves and others.
As I understood for myself, having a fundamental, deeply rooted free knowledge of mathematics (and not just "understanding") is the basis of everything. It opens doors to many interesting specialties. Looking back, I understand that I shouldn't have blindly followed my inclinations and interests. The part of me that “freely” loved literature and languages was the same one that made me love mathematics and science as a result, it changed and enriched my life.
