I’m sorry, education reformers. We still need a rote and a repeat.
I was a naughty kid who grew up on the lyrical side of life, and I treated mathematics and science as if they were symptoms of plague. And that’s why it’s strange that I turned into a person dealing with triple integrals, Fourier transformations and, the pearl of mathematics, the Euler equation. It’s hard to believe that I’ve turned from a mathofob to a professor of applied sciences.
One day, one of my students asked me how I managed to do it – how I changed my brain. I wanted to answer – damn it, hardly! I failed my math and physics exams in primary, secondary and higher schools after all. I enrolled in the math class after serving in the army when I was 26. At an exhibition of examples of neuroplasticity in adults, I would have been the first copy.
Studying math and exact sciences in adulthood opened the door for me to technical sciences. But these severe adult changes in the brain opened up a glimpse of the neuroplasticity associated with adult learning from the inside. Fortunately, my PhD in Systems Design, during which I learned the exact sciences, technology, engineering, and mathematics (STEM – Science, Technology, Engineering, Math), and my subsequent research on human thinking, helped me to understand recent breakthroughs in neurology and cognitive psychology related to learning. Free gre online practice tests on math knowledge
In the years following my doctorate, thousands of primary and secondary school students have passed through my class with the belief that understanding mathematics through active discussion is a learning mascot. If you can explain what you have learned to others – let’s say by drawing a picture – then you have probably really understood it.
Japan is an example of this “understanding-focused” technique and has been emulated. But the end of history is often missing from the discussion: Japan also invented the Kumon teaching method, which is based on memorizing, repeating and rote learning in order for a student to achieve an excellent command of the material. This intensive post-secondary education program is preferred by thousands of parents in Japan and around the world, complementing their children’s collaborative learning with a large number of practices, repetitions, and a cleverly designed rote system to ensure excellent material possession.
In the U.S., a focus on understanding sometimes replaces rather than complements older teaching methods that scientists confirm work with the natural processes of the brain, learning complex things like mathematics and exact sciences.
The latest wave of reform in mathematics education includes the Common Core, an attempt to set stringent common standards across the United States, although critics say these standards do not match those of other, more advanced countries. On the face of it, the standards have some perspective. In mathematics, students are expected to have equal opportunities in conceptual understanding, practical and procedural skills.
The devil, as usual, is in the details of implementation. In today’s educational climate, memorizing and repeating STEM disciplines, as opposed to learning language and music, is often seen as unworthy activities that waste the time of students and teachers. Many teachers have long believed that understanding concepts in STEM disciplines is of the highest priority. Of course, it is easier for teachers to engage students in a discussion of mathematical topics (and this process, with the right guidance, can be very helpful in understanding the tasks) than it is for teachers to take notes for their homework. As a result, while procedural skills and fluency should be taught in the same doses as conceptual understanding, this is often not the case.
The problem with focusing only on understanding is that students in math and science are often able to grasp basic concepts about an important idea, but understanding it quickly slips away without fixing it through practice and repetition. Worse, students often think they understand something at a time when it is not. This approach can often only bring the illusion of understanding. As one failed student recently told me, “I can’t understand why I did such a bad job. I understood everything in class,” he said. He thought he understood everything, and maybe he did, but he didn’t use what he understood in practice to get it into his brain. He didn’t develop the procedural knowledge or the ability to apply it.
There’s an interesting connection between learning sports and learning mathematics and exact sciences. When you learn to hit a golf club, you perfect the punch by practicing it for several years. Your body knows what to do, just when you think about it – you don’t have to think about all the components of a complex swing to hit a ball.
Similarly, when you understand why you are doing something in maths, you don’t have to explain the same thing to yourself every time. You don’t need to carry 25 balls, put them in 5 rows of 5 columns on a 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 to different degrees, you can simply add degrees (104 x 105 = 109). Using this procedure often and in different cases, you will find that you understand why and how it works. Better understanding of the topic comes from creating a meaningful pattern in the brain.
I learned all this about mathematics and the learning process itself not in the classroom, but in the course of my life, as a man who as a child read Madeleine Lengle and Dostoyevsky, studied languages at one of the world’s leading language institutes, and then dramatically changed his path and became a professor of technical sciences.
As a young girl with a passion for languages, and without the right money and skills, I could not afford to pay for college. So I joined the army after school. I loved learning languages at school, and it seemed that the army was exactly the place where a person could get paid for learning languages by attending a highly regarded language institute of the Ministry of Defense – the place where learning languages was turned into 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. Besides, the Iron Curtain attracted me – couldn’t I use my knowledge of Russian to look behind it?
After the army, I became an interpreter on Soviet trawlers in the Bering Sea. Working for Russians was interesting and exciting – but it was also an outwardly embellished work of a migrant. During the fishing season you go to the sea, earn good money, occasionally 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 a Russian-speaking man, there was almost only one alternative to that – working for the NSA. My army contacts pushed me towards it, but I had no soul for it.
I began to realize that while knowing another language is good, it was a skill with disabilities and potential. Because of my ability to bend words in Russian, my house was not besieged. Unless I was ready to tolerate sea sickness and occasional malnutrition on stinking trawlers in the middle of the Bering Sea. I couldn’t help but think of the West Point engineers I worked with in the army. Their mathematical approach to solving problems was clearly useful to the real world – more useful than my failures with mathematics.
So, when I was 26 years old, leaving the army and evaluating the possibilities, I suddenly thought: if I want to do something new, why not try something that would open up a whole new world of perspectives for me? Technical sciences, for example? And that meant that I had to learn a new language – the language of computation.
With my poor understanding of the simplest mathematics, after the army, I took algebra and trigonometry on a course for the retarded. Trying to reprogram my brain sometimes seemed like a stupid idea – especially when I looked at the faces of my younger classmates. But in my case, and I learned Russian as a mature student, I hoped that some aspects of language learning could be applied to the study of mathematics and exact sciences.
While studying Russian, I tried not only to understand something, but also to achieve fluency in it. Fluency in such a vast subject as language requires a degree of familiarity that can only be developed through repeated and different work with different areas. My classmates who studied the language focused on simple understanding, and I tried to achieve inner fluency with the words and structure of the language. It was not enough for me that the word “to understand” meant “to understand”. I practiced with the verb, used it constantly in different tenses, in sentences, and then understood not only where it could be used, but also where it was not needed. I practiced to quickly extract these aspects and options from memory. Through practice, you can understand and translate tens and hundreds of words from another language. But if you have no fluency, then when someone quickly spits out a bunch of words to you, as in a normal conversation, you have no idea what that person is saying, although technically you seem to understand all the words and structure. And you certainly can’t speak fast enough for native speakers to enjoy listening to you.
This approach, focusing on fluency rather than simple understanding, brought me to the first place in the class. I did not understand it at the time, but this approach gave me an intuitive understanding of the basics of learning and expert skills – chunking.
Chunking was first proposed in Herbert Simon’s revolutionary work on the analysis of chess. The pieces served as various mental analogues of chess templates. Neurobiologists gradually came to the understanding that experts, say, in chess, are such, since they can store thousands of pieces of knowledge in long-term memory. Masters in chess can remember tens of thousands of different chess templates. In any area, an expert can remember one or more pieces of nerve subroutines well connected together for analysis and reaction to a new situation. This level of true understanding, and the ability to use this understanding in new situations, is acquired only through acquaintance with the subject, obtained from repetitions, memories and practice.
The study of chess masters, ambulance doctors and fighter pilots has shown that in stressful situations conscious analysis of the situation gives way to rapid subconscious data processing, when experts turn to a deeply integrated set of mental templates – pieces. At some point, the conscious understanding of why you are doing what you are doing only begins to slow you down and interrupts the flow, which leads to making the worst decisions. I was right intuitively aware of the connection between learning a new language and learning mathematics. Daily and continuous learning of the Russian language excited and strengthened the nerve loops in my brain, and I gradually began to bind together Slavic pieces that could easily be called from memory. By alternating my studies, practicing in such a way that I knew not only when I could use a word but also when I didn’t need to use it, or when I needed to use another variant of it, I used the same approaches that I used to study mathematics.
I started studying mathematics and exact sciences as an adult with the same strategy. I looked at the equation – for a simple example, take Newton’s second law, F = ma. I practiced in the sense of the meaning of each letter: “f”, i.e. force is a push, “m”, mass is heavy resistance to push, “a” was a joyful feeling of acceleration. (In the case of Russian, I also practiced pronunciation of Cyrillic letters). I memorized the equation, carried it in my head and played with it. If m and a are big, what happens to f in the equation? If f is big and a is small, what will m be? How do the units of measurement converge on both sides? To play with the equation is how to connect the verb to other words. I began to understand that the vague outlines of the equation resembled a metaphorical poem in which there were all sorts of beautiful symbolic notions. Even though I would not have expressed it that way at that time, I had to slowly and daily build strong nerve subroutines for good study of mathematics and exact sciences.
Over time, professors of Mathematics and Precise Sciences told me that building well-recorded pieces of experience through practice and repetition was vital to success. Understanding does not lead to fluency. Fluency leads to understanding. In general, I believe that real understanding of a complex subject comes only from fluency.
By invading a new field for me, becoming an electrical engineer, and eventually a professor of engineering, I left the Russian language behind. But in 25 years after last time I picked up glass on Soviet trawlers my family and I decided to make a trip along Transsib through all Russia. And though I gladly expected the long desired trip, 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?
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Of course, the first time I got on the train, I found myself speaking Russian on the level of a two-year-old child. I was looking for words, my declensions and conjugations were confused, and my almost perfect accent sounded terrible. But the base did not go anywhere, and gradually my Russian improved. Even the rudimentary knowledge was enough for everyday needs. Soon the guides began to come to me for help in translating for other passengers. When we arrived in Moscow, we took a taxi. The driver, as I understood later, tried to deceive us by going the other way and getting stuck in a traffic jam, believing that unknowing foreigners could easily withstand the extra hour of the counter. Suddenly, the Russian words I hadn’t used for decades flew out of my mouth. Consciously, I didn’t even remember knowing them. Same workbooks for 4th grade.
The fluency, when I needed it, was at hand – and helped us out. The fluency allows the understanding to fit into the mind, and pop up 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 path, with my current knowledge of the brain, I realize that we can achieve more. As parents and teachers, we can use simple methods to deepen our understanding and turn it into a useful and flexible tool.
I have discovered that having basic and profound fluency in mathematics and exact science – rather than simple “understanding” – is essential. It paves the way for the most interesting activities in life. Looking back, I realize that I didn’t have to blindly follow my original inclinations and passions. The same “fugitive” part of me, who adored literature and language, eventually fell in love with mathematics and exact sciences – and eventually transformed and enriched my life.