*Yes, some people are brighter than others but I really believe tha...
The importance of practice (hard work) for being successful:
**A...
need reference for the grothendieck modesty quote
Dropout rates vary across Science, Technology, Engineering and Math...
*"They expected me to be wonderful to offer me a job like this and ...
"*Yes, some people are brighter than others but I really believe th...
Lazlo Polgar believed that **any child had the capacity of being a ...
Estimated amount of ...
The title of the book is Mathematics: A Very Short Introduction
On Being Smart
∗
Nabil H. Mustafa
What is the crucial quality important for succeeding in graduate school? I will provide a few
examples that suggest that: i) The answer is not intelligence – a minimum of intelligence,
such as what everyone reading this article has, is sufficient for succeeding in any graduate
school, ii) it is ... hard work. I apologize for the disappointment.
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 under-
standing 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 i) while both Grothendieck and
Gauss were considered amazing geniuses by their contemporaries, ii) both were not exactly
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 mathemati-
cians). 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.
∗
This article is more a collection of interesting quotations with my commentary. This has the added
benefit of providing references for those interested.
1
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 undergrad-
uates, 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 prob-
lems where its 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 ge-
nius 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 mathemati-
2
cians develop, they learn various tricks of the trade, partly from the work of other math-
ematicians 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 mea-
sure, 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.’
1
Though not directly related to research, the phenomenon that is Judit Polgar provides an-
other 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.’
2
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 vi-
olinists 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
3
:
A group of researchers led by Carol Dweck of Columbia University went to a very competi-
1
Excerpted from the excellent book ‘A Short Introduction to Mathematics’.
2
http://psychologytoday.com/articles/index.php?term=pto-3789.html&fromMod=popular_
parenting
3
See the excellent article: http://nymag.com/news/features/27840/
3
tive 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.’
4
Dropout rates vary across Science, Technology, Engineering and Math (STEM) disciplines. They range from 38 percent among mathematics majors to 59 percent among computer sciences majors.
You can learn more about it in this full report: [STEM Attrition: College Students’
Paths Into and Out of STEM Fields](https://nces.ed.gov/pubs2014/2014001rev.pdf)
You can find the book here:
[](https://books.google.com/books/about/Mathematics_A_Very_Short_Introduction.html?id=DBxSM7TIq48C&printsec=frontcover&source=kp_read_button#v=onepage&q&f=false)
need reference for the grothendieck modesty quote
Agree!
The title of the book is Mathematics: A Very Short Introduction
Estimated amount of time for solitary practice as a function of age for the middle-aged professional violinists (triangles), the best expert violinists (squares), the good expert violinists (empty circles), the least accomplished expert violinists (filled circles) and amateur pianists (diamonds).
Read on here:
- [The Role of Deliberate Practice in the Acquisition of Expert Performance](http://fermatslibrary.com/p/787c0427)
- [Giftedness and evidence for reproducibly superior performance: an account based on the expert performance framework](http://fermatslibrary.com/p/6ac1e26f)
- [The Making of an Expert](https://hbr.org/2007/07/the-making-of-an-expert)
*Yes, some people are brighter than others but I really believe that most people can really get to quite a good level in mathematics if they're prepared to deal with these more psychological issues of how to handle the situation of being stuck.* **- Andrew Wiles, British Mathematician most know for proving Fermat's Last Theorem**
*Genius is one percent inspiration, ninety-nine percent perspiration.* - **Thomas Edison**
The importance of practice (hard work) for being successful:
**At the age of 18 the expert pianists had accumulated 7,606 hr of practice, which is reliably different from the 1,606 hr of practice accumulated by the amateurs.**
Learn more here [The Role of Deliberate Practice in the Acquisition of Expert Performance](http://fermatslibrary.com/p/787c0427)
"*Yes, some people are brighter than others but I really believe that most people can really get to quite a good level in mathematics if they're prepared to deal with these more psychological issues of how to handle the situation of being stuck.*
[...]
*I get manic about it. Once I'm stuck on a problem I just can't think about anything else. It's more difficult. So I just take a little time off and then come back to it.*"
You can watch this interview here, **What does it feel like to do maths?**: [](https://www.youtube.com/watch?v=KaVytLupxmo)
*"They expected me to be wonderful to offer me a job like this and I wasn't wonderful, and therefore I realized a new principle, which was that I'm not responsible for what other people think I am able to do; I don't have to be good because they think I'm going to be good. And somehow or other I could relax about this, and I thought to myself, I haven't done anything important and I'm never going to do anything important. But I used to enjoy physics and mathematical things and because I used to play with it. It was never very important but I used to do things for the fun of it, so I decided I'm going to do things only for the fun of it."*
Feynman during an interview about turning down Princeton, see more here (minute 20:14):
[](http://www.dailymotion.com/video/x24gwgc_richard-feynman-the-pleasure-of-finding-things-out_news)
Lazlo Polgar believed that **any child had the capacity of being a master in any chosen field**. In 1970 he started an experiment with his daughters. The premise, in his words:
"*with a simple premise: that any child has the innate capacity to become a genius in any chosen field, as long as education starts before their third birthday and they begin to specialize at six.*"
His 3 daughters became chess grandmasters. You can learn more about it here: [Lazlo Polgar](https://en.wikipedia.org/wiki/László_Polgár#Education_and_career)