The Secret Life of the Mind

by Mariano Sigman

‘Do you agree that this is the side of a square whose area is twice the original’s?’ asked Socrates.

  To which the slave answered yes, thus sketching out the basis of the Pythagorean theorem, the quadratic relationship between the sides and the diagonal.

  The dialogue concludes with the slave, who realizes, just by answering questions, the basis of one of the most highly valued theorems in Western culture.

  ‘What do you think, Menon? Did the slave express any opinion that was not his own?’ asked Socrates.

  ‘No,’ replied Menon.

  The psychologist and educator Antonio Battro understood that this dialogue was the seed of an unparalleled experiment into whether there are intuitions that persist over centuries and millennia. I undertook this task with my graduate student Andrea Goldin, a biologist. We posed Socrates’ questions to children, teenagers and adults and found that their responses, 2,500 years after the original dialogue, were almost identical. We are very similar to the Ancient Greeks,* we get the same things right and we make the same mistakes. This shows that there are ways of reasoning that are so deeply entrenched that they travel in time through cultures, changing little.

  It doesn’t matter–here–whether the Socratic dialogue actually took place or not. Perhaps it was merely a mental simulation by Socrates, or by Plato. However, we did show that it is plausible for the dialogue to have happened just as it was written. When faced with the same questions, people respond–millennia later–just as the slave did.

  My motivation for doing this experiment was to investigate the history of human thought and examine the hypothesis that simple mathematical intuitions expressed in fifth-century Athens could be identical to those expressed by twenty-first-century students in South America or elsewhere in the world.

  Andrea’s motivation was different. Her drive was to understand how science can improve education–an ambition I learned by her side–and this led her to ask very different questions in the same experiment: was the dialogue really as effective as presumed? Is answering questions a good way to learn?

  The illusion of discovery

  In our experiment, Andrea proposed, once the dialogue was finished, to show each student a new square of a different colour and different size and ask them to use it to generate a new one with twice the area. It seemed to me that the task was too easy; it couldn’t be exactly the same as what had been taught. So I suggested we test what they’d learned in a more demanding way. Could they extend the rule to new shapes; for example, a triangle? Could they generate a square whose area was half–instead of double–the original square?

  Luckily Andrea stood her ground. As she had supposed, a large number of the participants–almost half, in fact–failed the simpler test. They couldn’t replicate what they believed they had learned. What happened?

  The first key to this mystery has already appeared in this book; the brain, in many cases, has information that it cannot express or evoke explicitly. It is like having a word on the tip of your tongue. So the first possibility is that this information was effectively acquired through the dialogue but not in a way that can be used and expressed.

  An example in daily life can help us to understand the mechanisms in play. Someone is given a lift time after time to the same destination. One day they have to take the wheel to follow the route they have been driven along thousands of times and find out they don’t know where they’re going. This doesn’t mean that they didn’t look at the route, or weren’t paying attention. There is a process of consolidating knowledge that needs practice. This argument is central throughout the problem of learning; it is one thing to assimilate knowledge per se and another to assimilate it in order to be able to express it. A second example is the learning of technical skills, like playing the guitar. We watch the teacher, we see clearly how he or she articulates their fingers in order to make a chord, but, when it is our turn, we are unable to do the same thing.

  The analysis of the Socratic dialogue shows that just as extensive practice is needed to learn procedures (playing an instrument, reading or riding a bike), it is also necessary for conceptual learning. But there is a crucial difference. In learning an instrument we recognize immediately that seeing is not enough to learn. Yet with conceptual learning both the teacher and the student feel that a well-sketched-out argument can be taken in without difficulty. That is an illusion. In order to learn concepts, meticulous practice is required, just as when learning to type.

  Our further exploration of Menon’s dialogue revealed a pedagogical disaster. The Socratic process turns out to be very gratifying for the teacher. The students’ response seems highly successful. But when the class is put to the test, the result is not always so promising. My hypothesis is that this educational process sometimes fails for two reasons: the lack of practice in using the acquired knowledge and the focus of attention, which should not be placed on small fragments of facts that are already known but rather on how to combine them to produce new knowledge. We have already sketched out the first argument and we will explore it in further depth over the next few pages. A concise example of the second can be found in educational practice.

  Beyond demographic, economic and social factors–which are, of course, decisive–there are countries in which mathematical teaching works better than in others. For example, in China students learn more than is expected–based on the GDP and other socioeconomic variables–and in the United States less. What explains this difference?

  In the United States, teachers solving a complex multiplication like 173 × 75 on the blackboard usually ask the children things they already know: ‘How much is 5 × 3?’ And they all, in unison, answer: ‘Fifteen.’ It is gratifying because the entire class gets the right answer. But the trap lies in the fact that the children were not taught the only thing they didn’t know, the path. Why start with 5 × 3 and then do 5 × 7, and not the other way around? How should they combine this information and how do they establish a plan for being able to resolve the other steps in the problem 173 × 75? This is the same error we found in the Socratic dialogue. Menon’s slave would never have drawn the diagonal on his own. The big secret to solving this problem is not in realizing, once the diagonal has been drawn, how to count the four triangles. The key is in how to get the student to come up with the idea that the solution requires thinking of the diagonal. The pedagogical error lies in bringing the student’s attention to fragments of the problem that had already been solved.

  In China, on the other hand, in order to learn to multiply 173 × 75, the teacher asks: ‘How do you think this is solved? Where do we start?’ This, first of all, takes the students out of their comfort zone, inquiring about something they do not know. They have to establish how to break down this complex calculation into a series of steps: First multiply 5 × 3, and write the result, and then multiply 5 × 70, and so on… Secondly, it leads them to make an effort and, eventually, to make mistakes. The two teaching methods coincide in that they are based on questions. But one asks about already known fragments and the other about the path that unites the fragments.

  Learning through scaffolding

  In our investigation into the contemporary responses to Menon’s dialogue we found something odd. Those who followed the dialogue to the letter learned less. On the other hand, those who skipped over some questions learned more. The odd thing is that more teaching–more of the dialogue–favoured less learning. How can we resolve this enigma?

  We found the answer in a research programme carried out by the psychologist and educator Danielle McNamara in order to decipher a text’s legibility. Her project, vastly influential in the worlds of academia and educational practice, shows that the most pertinent variables are not the ones you would expect, such as attention, intelligence and effort. The most decisive, in fact, was what the reader already knew about the subject before starting.

  This led us to a very different reasoning from the one any of us would have naturally sketched out in the classroom; learning doesn’t fail b
ecause of distraction or lack of attention. In fact, students with almost no prior knowledge can follow the dialogue with great concentration, but their attention is focused on each step, on the trees and not the forest; while those students whose knowledge already brings them close to the solution will not need so much concentration to follow the dialogue.

  So Andrea and I sketched out a seemingly paradoxical hypothesis: those who pay more attention learn less. In order to test it out we created a pioneering experiment, the first simultaneous recording of cerebral activity while one person was teaching and another was learning.

  The results were conclusive. Those who learned less activated their prefrontal cortex more, which is to say, they made more effort. To such an extent that, by measuring cerebral activity during the dialogue, we could predict whether a student would later pass the exam.

  Of course, it is not always true that paying more attention means learning less. With equal prior knowledge, those who pay more attention achieve greater results. But in this dialogue–and so many others in school–it turns out that effort is inversely related to prior knowledge. Those with less knowledge follow the dialogue step by step, in detail. Yet those who are able to skip over whole parts can do so because they already know many of the fragments. The path is well learned only when one can follow it, without needing to stop at every step.

  This idea is closely linked to the concept introduced in the 1920s by the great Russian psychologist Lev Vygotsky of the zone of proximal development, which made such an impact on pedagogy. Vygotsky argued that there must be a reasonable distance between what students can do for themselves and what a mentor demands of them. Later in the book we will revisit this idea when looking at how to lessen the gap between teachers and students by having the children themselves act as mentors. But at this point I want to leap through another window that was opened in the minute analysis of the Socratic dialogue: learning, effort and leaving one’s comfort zone.

  Effort and talent

  We intuit that the few people who learn to play the guitar like a rock star such as Prince* do so based on a certain mix of biological and social factors. But to understand how these elements interact with each other and, above all, how to use that knowledge better to learn and teach, we need to divide this general concept into smaller sections.

  The idea that genetic factors determine the maximum skill that each of us can achieve is very deep-seated. In other words, anyone can learn music or football to a certain level, but only a few virtuosos can reach the level of João Gilberto** or Lionel Messi. Great talents are born, not made. They were touched with a magic wand, they have a gift.

  This idea that we all go through a similar educational trajectory, but the ceiling depends on a biological predisposition, was coined and sketched in 1869 by Francis Galton, one of the most versatile and prolific British scientists. The clearest example appears when the predisposition is a body trait. For example, becoming a professional basketball player is much more likely if you are tall. It is hard to become a great tenor without having been born with the proper vocal apparatus.

  Galton’s idea is simple and intuitive but doesn’t coincide with reality. When investigating in detail how great experts learned what they know, and avoiding the temptation to draw general conclusions based more on myth, it turns out that the first two premises of Galton’s argument are wrong. The upper limit of learning is not so genetically based, nor is the path towards that upper limit so independent of genetics. Genetics are involved in both parts, but not decisively in either.

  Ways of learning

  The great neurologist Larry Squire sketched a taxonomy that divides learning into two large categories. Declarative learning is conscious and can be explained in words. A good example of this is learning the rules of a game; once the instructions are learned, they can be taught (declared) to a new player. Nondeclarative learning includes skills and habits that are usually achieved without the learner being aware of the process. These are types of knowledge that would be difficult to make explicit in the form of language, such as by explaining them to someone else.

  The more implicit ways of learning are, in fact, so unconscious that we don’t even recognize that there was something to learn: for example, learning to see. We can easily identify that a face expresses an emotion but we are unable to declare this knowledge in order to make machines that can emulate this process. Our ability to see is innate in most people. So much so that the inverse of this naturalness of the gaze has poetic strength. The Uruguayan author Eduardo Galeano wrote: ‘And the sea was so immense, so brilliant, that the boy was struck dumb by its beauty. And when he finally managed to speak, trembling, stuttering, he asked his father: “Help me see!”’ A similar thing happens when learning to walk or keep our balance. These faculties are so well-incorporated that it seems they’ve always been there, that we never had to learn them.

  These two categories are useful when exploring the vast space of learning. However, it is equally important to understand that they are inevitably abstractions and exaggerations; almost all learning in real life is part declarative and part implicit.

  For example, learning to walk is an implicit and procedural form of learning, it doesn’t require instructions or explanations, and it’s learned slowly and after a lot of practice. Yet there are many aspects which can be consciously controlled. The same thing happens with breathing, which is fundamentally an unconscious process. It would not be sensible to delegate to each of our distracted free wills something that would be fatal if forgotten. But, to a certain point, we can control our breathing consciously, its rhythm, volume, flow. And it is breathing, the bodily function that spans the conscious and the unconscious, that is used as a universal bridge in meditative practices and other exercises to learn to direct one’s consciousness to new places.

  Establishing this bridge between the implicit and the declarative turns out to be, as we will see, a key variable in every form of learning.

  The OK threshold

  A fundamental concept for understanding how much we can improve is called the OK threshold, the level at which everything feels fine. People engaged in learning to type, for example, begin by searching out each letter with their eyes, exerting great effort and concentration. Like Menon’s slave, they pay attention to each step. But later it seems as if their fingers have a life of their own. When we touch-type, our brains are somewhere else, reflecting on the text, talking to someone else or daydreaming. What’s curious is that once we have reached this level of ability, despite typing for hours and hours, we no longer improve. In other words, the learning curve grows until it reaches a value where it stabilizes. Most people reach speeds close to sixty words per minute. But, of course, this value is not the same for everyone; the world record is held by Stella Pajunas, who managed to type at the extraordinary pace of 216 words per minute.

  This example seems to confirm Galton’s argument; he maintained that each of us reaches our own inherent ceiling. Yet by doing methodical, sustained exercises to increase our speed, all of us can improve substantially. What happens is that we stagnate very far from our maximum performance, at a point at which we benefit from what we’ve learned but we do not generate further learning, a comfort zone in which we find a tacit balance between the desire to improve and the effort that would require. This point is the OK threshold.

  The history of human virtue

  What happens in the example of typing speed occurs with almost everything we learn in life. One example that most of us experience is reading. After years of intense effort at school, many of us achieve reading quickly and with little effort. We read more and more books, without increasing our reading speed. Yet if any of us revisited a methodical, sustained process, and devoted time and effort, we could significantly increase our speed without losing comprehension along the way.

  The narrative of learning in each of our life cycles is replicated in the history of culture and sport. In the early twentieth century, the fastest runners
of the times achieved the extraordinary feat of running a marathon in two and a half hours. In the early twenty-first century, this time isn’t enough to qualify for the Olympics. This is not limited to sports, of course. Some compositions by Tchaikovsky were technically so difficult that in his day they were never played. The violinists of the period thought that they were impossible. Today they are still considered challenging but there are many violinists who can play them.

  Why is it that we can now achieve feats that years earlier were impossible? Is it that, as Galton’s hypothesis suggests, our constitution–our genes–changes? Of course not. Human genetics, over those seventy years, has remained essentially the same. Is it because technology has radically changed? Again, the answer is no. Perhaps that would be a valid argument for some disciplines, but a marathon runner with trainers from a hundred years ago–and even barefoot–could today achieve times that were once impossible. Likewise, a contemporary violinist could now play Tchaikovsky’s works with period instruments.

  This deals a fatal blow to Galton’s argument. The limits of human performance are not genetic. Violinists today manage to play those pieces because they can devote more hours to their practice, because the point at which they feel the goal is accomplished has changed, and because they have better training procedures. This is good news; it means that we can build on these examples to attain goals that today are inconceivable.

  Fighting spirit and talent: Galton’s two errors

  When we judge athletes we usually separate their competitive spirit from their talent as if they were two different substrata. There are the Roger Federers of the world–who have talent–and the Rafael Nadals–who are mainly driven by intense competitive spirit to the extent that they leave their bodies and souls on the field. A typical observer views those with innate talent with a distant respect that denotes admiration for a gift, a competitor’s divine privilege. Fighting spirit, on the other hand, feels more human because it is associated with will and the feeling that we could all achieve it. This is Galton’s hunch: the gift, the ceiling of talent, is innate, and fighting spirit, the path to advancement through learning, is available to all of us. Both of these conjectures, however, are wrong.