Here is what some of the great mathematicians, after having done work considered the very peak of human thought, think about the factors in their success:

Grothendieck, Fields Medalist 1966: “Since then I’ve had the chance, in the world of mathematics that bid me welcome, to meet quite a number of people, both among my “elders” and among young people in my general age group, who were much more brilliant, much more “gifted” than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle — while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things that I had to learn (so I was assured), things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates, almost by sleight of hand, the most forbidding subjects.”

Gauss: “If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries.”

The reason why I give credence to these remarks is that, while both Grothendieck and Gauss were considered amazing geniuses by their contemporaries, neither was known for being modest. (Grothendieck said: “In the history of mathematics, I have produced the greatest number of new ideas”, and Gauss was famous for putting down other mathematicians.) This, together with the fact that even at graduate schools in the US which attract the best and the brightest of students, the drop-out in computer science is over 50%, should suggest that other factors play a larger role in determining success or failure. In my opinion, a rather large reason for failure is the following, rather fragile, learning psychology.

In the current environment, everyone wants to be smart, or at any rate, appear smart. This severely interferes with learning, naturally: students who consider being smart important become more conservative in the length and hardness of problems they attempt, which is a reasonable risk-averse way of preserving their image. This approach works for undergraduates, especially under the diseased quarter system since the material covered is relatively shallow and easy. However, once one starts graduate studies and begins to think about problems where it is not even clear if a solution is possible, the habit of following the risk-averse strategy just doesn’t cut it.

Students not used to prolonged thinking on a single problem start off well. However, soon they find motivation and inspiration leaving them, and they start dreading working on the problem as failure would lead them to question something they (by now) crucially identify with: “smartness”. Procrastination kicks in, and soon the student is busy in a diverse set of academic (but non-research!) activities to hide the reality of not working, like writing complicated scripts to automate their soon-to-be-coming publication phase, optimizing their daily vitamin B12 intake, getting heavily involved with political and religious movements and so on. Few students are able to critically introspect, which is reasonable since society has informed them that smartness is what matters, and if they are unable to solve the problem quickly, the logical conclusion is that they are not smart. In this world-view, it is hard to even consider the suggestion that smartness matters fairly little in such matters and most fall prey to heavy depression. Some do manage to climb out: Feynman, physics Nobel Prize 1964, had developed a reputation for being an extremely smart guy at Los Alamos. He paid for this afterwards as an assistant professor at Cornell, where for the first two years he was paralyzed by this fear, and unable to do any worthwhile work. During this time, he received an invitation to join the prestigious Institute for Advanced Studies (where Einstein was one of the members) but refused since he felt useless as a researcher. Fortunately for science, later a positive reaction set in for  him and he was able to overcome his fear (and later ended up writing  books with titles “What Do You Care What Other People Think’”).

Instead of intelligence, persistence is the crucial parameter for success in graduate school:

Gowers, Fields Medalist 1998: “To illustrate with an extreme example, Andrew Wiles, who (at the age of over 40) proved Fermat’s Last Theorem … and thereby solved the worlds most famous unsolved mathematical problem is undoubtedly very clever, but he is not a genius in my sense. How, you might ask, could he possibly have done what he did without some sort of mysterious extra brainpower? The answer is that, remarkable though his achievement was, it is not so remarkable as to defy explanation. I do not know precisely what enabled him to succeed, but he would have needed a great deal of courage, determination, and patience, a wide knowledge of some very difficult work done by others, the good fortune to be in the right mathematical area at the right time, and an exceptional strategic ability.

This last quality is, ultimately, more important than freakish mental speed: the most profound contributions to mathematics are often made by tortoises rather than hares. As mathematicians develop, they learn various tricks of the trade, partly from the work of other mathematicians and partly as a result of many hours spent thinking about mathematics. What determines whether they can use their expertise to solve notorious problems is, in large measure, a matter of careful planning: attempting problems that are likely to be fruitful, knowing when to give up a line of thought (a difficult judgment to make), being able to sketch broad outlines of arguments before, just occasionally, managing to fill in the details. This demands a level of maturity, which is by no means incompatible with genius, but which does not always accompany it.” [Excerpted
from  the excellent book "A Short Introduction to Mathematics"].

Though not directly related to research, the phenomenon that is Judit Polgar provides another fascinating insight into the reasons behind spectacular success in intellectual activities:

“Forty years ago, Laszlo Polgar, a Hungarian psychologist, conducted an epistolary courtship with a Ukrainian foreign language teacher named Klara. His letters to her weren’t filled with reflections on her cherubic beauty or vows of eternal love. Instead, they detailed a pedagogical experiment he was bent on carrying out with his future progeny. After studying the biographies of hundreds of great intellectuals, he had identified a common theme — early and intensive specialization in a particular subject. Laszlo [sic] believed he could turn any healthy child into a prodigy. He had already published a book on the subject, Bring Up Genius!, and he needed a wife willing to jump on board.” [Psychology Today]

The result were three sisters: Susan, Sofia, and Judit. Judit is by far the best female chess player in history, and ranked in the top-10 chess players in the world. Susan is the next(!) best female chess player in history. Sofia has a record-breaking performance in Italy  that has become known as the “Sac of Rome”.

“Anders Ericsson is only vaguely familiar with the Polgars, but he has spent over 20 years building evidence in support of Laszlo’s theory of genius. Ericsson, a professor of psychology at Florida State University, argues that ‘extended deliberate practice’ is the true, if banal, key to success. ‘Nothing shows that innate factors are a necessary prerequisite for expert level mastery in most fields,’ he says … His interviews with 78 German pianists and violinists revealed that by age 20, the best had spent an estimated 10,000 hours practicing, on average 5,000 hours more than a less accomplished group. Unless you’re dealing with a cosmic anomaly like Mozart, he argues, an enormous amount of hard work is what makes a prodigy’s performance look so effortless. ‘My father believes that innate talent is nothing, that [success] is 99 percent hard work,’ Susan says. ‘I agree with him.‘ “

The effect of psychology on learning is illustrated nicely in an interesting recent experiment: A group of researchers led by Carol Dweck of Columbia University went to a very competitive school’s 5th grade class, and randomly split it into two groups. Both groups were given the same easy puzzles to solve, and the performance of each child noted. Both groups scored well. After the exam, the first group was told ‘you must have really worked hard’, while the second group of children were rewarded by saying  ‘you must be smart at this‘. For the second round, both groups were given the same choice: either take another easy exam, or a much harder exam.  Here’s the punchline: over 90% of students in the first group chose the harder exam, while the majority of children in the second group chose the easier exam. In the third round, everyone had to do the harder exam:

Dweck:  “When we praise children for their intelligence, we tell them that this is the name of the game: Look smart, don’t risk making mistakes … [In the third round, children in first group] got very involved, willing to try every solution to the puzzles … Many of them remarked, unprovoked “This is my favorite test” [while for the students in second group] you could see the strain. They were sweating and miserable.”

The NYMag article ends with the following sage advice, on which I’ll also end: “The brain is ultimately just a muscle. Make it stronger by working it out.”

Editor’s Note: The views expressed in this article are solely those of the author and do not necessarily reflect the views of STEP.

Specifically, working hard for extended periods of time on things mainly as means to some other end – in this case admission into graduate school – can seriously harm the long-term ability to enjoy research work. This begs two questions: what are the harmful effects of such hard work, and how do they limit one’s academic career.

Consider the set of courses (together with the grades attained) that a recent good computer science graduate of LUMS took during his four years: Logic and Critical Reasoning (A+), Composition and Writing (A+), Islamic Studies (A+), Microeconomics I (A+), Microeconomic II (A+), Macroeconomics I (A+), Macroeconomics II (A+), Principles of Finance (A+), Development Economics (A+), Principles of Economics (A+). Intro to Political Science (A), Human Behavior (A), Communication Skills (A), Pakistan Studies (A-), Domestic Violence (A-), Gothic Fiction (A-), Philosophy of Education (B). And this does not include around twenty-five computer-science and four mathematics courses. What is one to make of this academic record?

A decade ago, my reaction on seeing this transcript would be first of awe, and then an unabashed admiration for the person and the effort involved. Today, my response is a feeling of distant exhaustion, followed by a pang of sadness for such mindless waste. I now explain my reasons.

Given such a wide variety of courses, it is hard to believe that any normal undergraduate student would like and enjoy more than half of those courses. Which, of course, is perfectly fine: one goal of a liberal arts education is to provide a wide variety of different areas to develop a good breadth of knowledge etc. But there is no reason to do so well in all these courses (other than the already-stated external motivation). For each ‘A+’ grade, one would have to spend more than a hundred hours reading material which is not interesting, spend more than thirty hours attending classes which are not exciting, spend long evenings grinding out empty essays instead of spending time with friends and family, spend long hours sweating and straining to do well in exams which themselves meaning nothing, spend hours and hours peeking around to see where ones lies ‘relative to the mean’, and so on. And this continuously for a period of at least four years. And this is precisely what is harmful: it demands an amount of will power and self-control which exerts a terrible toll to the psychological health of the individual in the long run:

“The complete control of impulse by will … is not really desirable. A life governed by purposes and desires, to the exclusion of impulse, is a tiring life; it exhausts vitality, and leaves a man, in the end, indifferent to the very purposes he has been trying to achieve.” – Bertrand Russell

This phenomenon, of tiring out of will-power due to excessive strain on it, also applies to graduate study. Graduate students in most universities have to pass a comprehensive ‘qualifying exam’, consisting of topics from all major sub-areas of the field. This requires many months of dedicated study; even a year or more for some, as the consequence of failing – being dropped from the programme – is rather dire. In fact, many aspects of this qualifying exam, and the preparation for it, mirror the grade-chasing, GRE acing, thoughtless racing mind-set of an ambitious undergraduate. The course exams, GRE, the graduate qualifying exam – all these test the student for certain specific qualities, which are often antithetical to a fulfilling life dedicated to research in a meaningful way. This is articulately described in the wonderful book “Disciplined Minds” by Jeff Schmidt:

“However, the student who loves the subject and is not alienated from it is profoundly affected by the qualification process. The process of preparing for the qualifying test, because of the kind of questions on the test and the way they must be answered, tragically alienates this student from his or her own field of interest. The test emphasizes quick recall, memorized tricks, work on problem fragments, work under time pressure, endurance, quantitative results, comfort with confinement to details, … and it de-emphasizes physical insight, qualitative discussion, exploration, curiosity, creativity, history, philosophy and so on”

In discussing this with several top undergraduates over the past few years, I find that almost all are fully conscious of the ‘alienated’ effort they are putting in to get high grades. But they all insist that it is a temporary phase, and that they will ‘really really’ start working when they finally get into graduate school. Far be it from me to belittle their youthful optimism, but invariably Nietzsche’s warning – ‘he who fights with monsters should look to it that he himself does not become a monster.’ – springs to mind. Schmidt puts it elegantly:

“Students should not cavalierly label their alienated labor as merely temporary, as instrumental to get the degree but easily reversed afterwards. Performing intense alienated labor for an extended period of time changes the student. It dampens his creativity and curiosity, clouds his memory of his original interests and ideas and weakens his resolve to pursue them, while getting used to doing protracted, disciplined labor on assigned problems. It is empty rhetoric to tell the student who has gone through the qualification process that he is free now to pursue in his career his original goals, for he is now a different person …”

A physical analogy for this situation is that of a spring. It can be stretched occasionally for very heavy loads, or often for light loads. But if stretched too much for too long, it loses its elasticity, and cannot return to its original shape. And remains distorted and disfigured. And so it is with students.

Schmidt writes for American graduate students, but these harmful effects are amplified several times over for those doing their undergraduate studies in Pakistan, for two reasons. One is that for the lack of a social safety net, the consequences are far more severe. Many students will identify with his description:

“The prospect of failing the qualifying test frightens the student, even the student who is the best at answering the kind of questions used on the test. The student is frightened because his desired future as a professional in his field of interest is at stake. But he is also frightened because society does not guarantee his material security…It seems possible for the individual, if suddenly of no value to employers, to go overnight from a job to walking the streets, from being somebody to being nobody, from living in the suburbs to living on skid row, left to suffer and struggle for survival among the desperate at the bottom of society. It doesn’t matter that such individual downfall is very unlikely; by simply featuring the possibility, the system announces the fundamental insecurity of the individual. This insecurity unrelentingly haunts the student studying for the qualifying test. The student sees professional training as his chance for a secure future, with status and non-alienating work, his chance for a life free from the threat of a nightmarish trip to the bottom of the heap …”

The second reason goes as follows. In general, there are so many students applying for graduate school with great grades, high GRE scores, that for graduate admission committees to distinguish between them becomes almost impossible. For example, historically the GRE scores of Chinese students are so high that many departments are forced to have different criteria for them compared with other applicants. Here, those doing their undergraduate studies in the US have a strong edge which the ones in Pakistan do not: a recommendation from a famous professor.

Randy Pausch, in his famous “Last Lecture”, is not hesitant to state that he got rejected from CMU, and got admitted only because his undergraduate advisor was friends with the chair at CMU. This offers a way out if one is an undergraduate at a place where there are well-known academics. A strong recommendation letter from a well-known academic can go a long way towards securing admission into a great graduate school. So instead of having the pressure to ace a large variety of courses, one just has to, in the final year, decide a possible area of specialization, do well in those courses, perhaps do the final-year project with the professor. Even if one’s general grades and scores are not through the roof, a strong recommendation from a good academic can have an overwhelming effect – more so than high
grades in courses or high GRE scores.

This, in itself, is a good reason for any department to have a strong emphasis on research. If it has faculty known for their research, this provides good students with an alternate way to get graduate admissions, without the enthusiasm-destroying effect of the mindless rat-race for grades and GRE scores.

For the current students, it is a classic catch-22 situation: either one does well enough in course-work to get admission into graduate school, but that process damages the ability to perform well, in a meaningful way, in graduate school. Or one devotes the time to meaningful pursuit of individual interests, but then with the not-so-great grades risks getting ‘stranded’. I think it is an important question which deserves more thought than it has currently received. I suffer to see bright energetic students go through all this, but for my part, I see no way out of this for them.