The idea that effort rather than "natural" ability is more decisive in developing genius is a long-standing cultural belief of ethnic Chinese people, and they are one demonstration of the truth of that idea. The psychologist who most strongly believes this is K. Anders Ericsson, who has conducted a number of interesting experimental and biographical studies of "expert performance," all of which show a big role for deliberate practice and a vanishingly small role for innate ability. Ericsson's publications make the blood of other psychologists boil, but I think that they express the truth of the matter.

What I find interesting after this last year is the role of social support in encouraging the effort necessary to reach the highest level of ability. It's not just coincidence that many of the best math competitors come from communities in which they have frequent face-to-face contact with other top competitors. Certainly in my state most of the top MATHCOUNTS mathletes come from a school with a large enrollment and a very active math team. The students make friends while learning the material in the school's abundant supply of old tests. They are the team to beat both in our state junior high contest and in MATHCOUNTS. Effort makes genius, but social networks sustain effort.

Indeed, the role of local culture is a decisive one for many. Jeff Boyd's article is another example of the importance of culture, and what parents and teachers can together do to achieve it. It's also the main reason we continue to build the San Diego Math Circle, and why we'll be rolling out a new project in the fall to help local efforts (more details coming in September!)

To those of you in college or beyond who want to pitch in locally: let us know where you are, and we may be able to hook you up with a group of students. We are learning about more and more local efforts that have started in the last couple years.

No fair. I don't know anyone who lives around me who enjoys competitive mathematics.

Then you'll have to teach them to.

Well considering I live next to a university and my dad is a math professor, there are plenty of people who enjoy mathematics (and have participated in competitive mathematics in the past). We managed to pull together a "math aid" class in middle school where we had three lectures from professors but after starting classes at the high school the "real" math (USAMO, proofs, etc.) has become minimal while fast paced competitions have become dominant. The irony of the situation is that despite our lack of training for the "more serious" competitions our school had three USAMO qualifiers this year and we managed to win (team-wise) every contest we attended. With this do you think it's reasonable to ask the school to provide a "math study hall" type class where students who are interested in competitive mathematics (USAMO in particular) can get together and work on problems and even participate in AoPS classes? If so could you make some suggestions about approaching the school district.

-Thanks
Dimitar Popov

DPopov -

I think it's absolutely reasonable to ask for such a class. In fact, it would be beyond the pale for an organized group of students and teachers to demand such a class. I'm betting the band and the basketball team each have their own classes.

Your first step should be to organize - this means finding a group of students that wants such a class, and a teacher who will go to bat for you with the school. If you have a teacher that likes to work with y'all on this sort of stuff, you're set on this already. Otherwise, you'll have a little convincing to do. Probably the best bet in this regard is to present the teacher that needs to be convinced with your already made plan for the year (i.e. it won't be much work for them).

When you are ready to go to the administration, make sure you outline how successful your students already are - point out how few schools in the whole country did as well as you, and that no other school in Texas had 3 USAMO qualifiers.

If you're going to get involved in lots of contests, or start building a math library for students to use various books like AoPS and Zeitz, you'll want to raise some funds. You might ask around a bit on the site of others what has been successful - I know that a number of students and teams have been successful simply sending letters to their community businesses and leaders. You might also read through Jeff Boyd's article for some pointers.

Thanks for the advice! As for at teacher my former middle school Mathcounts coach would be delighted to teach (although she thinks that we are beyond her) or direct such a class and there are plenty of students who are interested in these types of competitions and would enroll in the class if they can find a spot in their schedules(I think we had about 50 take the AMC). Once again, thanks for your advice!

-Dimitar Popov

Your teacher shouldn't be intimidated by many of the students being beyond her -- this is the case in nearly all the top programs (except Exeter of course). Most of the best teachers/math coaches are facilitators - they get the students access to opportunities and materials, and give the students the encouragement to do the rest.

How is the result of the trivia contest evidence for hard work versus "raw talent" being the relevant factor? Could it not be just as well interpreted by saying that the winners dominated because their superior minds simply absorbed more from reading similar books and being in similar environments, than their peers?

I'm not advocating either theory here, but I don't understand how the trivia contest points toward one or the other.

That's certainly possible, but from all I've seen of students, it's not the way I'd bet. Spelling names, for example, is far more likely the result of repeated exposure than a 'superior mind'.

I admit, I have no hard data on this, just my empirical observations of the most successful people I know - they work harder and more intensely than others. This, more than their 'genius' is their success. Perhaps I'm also colored by my personal experience - I'm hardly a genius, but I work pretty hard

I'll also add as a note that it's very hard to quantify or measure 'genius' or natural aptitude, but it's very easy to see effort. Look across the greats throughout time in any pursuit: Einstein, Gauss, Euler, Newton, Feynman, Curie -- go outside math/science -- Yo-yo Ma, Segovia, Michael Jordan, Tiger Woods, Pablo Picasso, Didrickson, Michaelangelo. . .

Who knows what kind of natural talent these people had? But we do have evidence that they worked *very hard* and that they had an early and deep immersion in their passions.

I'll take a hard worker over a genius any day. I've known too many 'prodigies' who haven't yet amounted to anything. But I know a great many non-geniuses who have, through their passion and dedication, become very impressive.

As a person who has gotten more through work than talent, I hope you're right.

However, the upper levels of mathematics and athletics and many other specialized endeavors are already full of hard workers, and many overachievers. It's not at all clear to me that Einstein or the others on your list, were fantastically hard workers relative to their peers. Certainly nothing amazing is likely to happen without strong effort, but academia, the NBA and similar selective endeavors are littered with hard workers who attain mediocre results.

As to reading books obsessively and learning more, it is the learning style of some mathematicians, but not necessarily a requirement. It is amazing how far some go in depth but not breadth of knowledge, and some of the best are not even close to being "well rounded" or well-versed in the literature. Grothendieck told a visitor at IHES "we don't read books here, we write them."
One of the smartest mathematicians I ever met seemed to have read almost nothing after his PhD thesis, but he would absorb anything at a seminar with incredible speed, and seem to know it better than the speaker in a short time. I don't know what kind of hard work would lead to such abilities.
I also read one of the child prodigies, tenured at very young age, mention that he had never had the feeling of difficult labor when doing mathematics, it was sort of easy and natural.

A person who naturally remembers more of the names that they encounter, has a larger spelling database (from the same level of exposure) than a person with ordinary memory.

A person who naturally is more prone to detect patterns and make associations with other information (correlating names with country of origin, recognizing cognate names/words in various languages, getting spelling clues from similar words) has a more efficient spelling database than a person who is less prone to make associations and notice patterns. Again, this is assuming the same level of exposure.

If in addition to such native mental processes you ALSO enlarge the set of exposures, then of course the resources for good spelling are increased.
In my experience, however, voracious reading is less of a predictor of good spelling than the sort of mental habits and capabilities I mentioned above.

In a study of problem-solving (olympiad-type problems in geometry and trigonometry) among mathematicians and others, the top mathematicians actually often knew less than average mathematicians about Euclidean geometry or the relevant formulas in trigonometry, but they pursued more productive paths in attempting solutions, and were able to rediscover relevant facts on the way. They ended up doing significantly better on the problems, even though not having seen the material in years.

The study was by Allan Schoenfeld, whose name I may have just mis-spelled.

That should read "...than a person who is LESS prone to make associations", etc

i.e. a person who does not have an eye/ear/mind for the structure of words is not necessarily likely to develop it by extensive reading, and having such skills is at least as useful for good spelling.
Of course, more reading never hurts, either.

Surely natural talent is relevant - but in my experience, hard work is far, far more important. I know of virtually no people who work hard and are passionate about what they do who are not successful. I know lots of extremely talented people who don't work as hard who are not successful. I'll bet on the hard workers every time.

And from what I have read of the biographies of the Feynmans and von Neumanns and Newtons, etc., they were just as much at the upper end of effort as their results would suggest they might be. These people were passionate about what they did and pursued it with a single-mindedness found in all those who are successful. Perhaps they were 'simply talented' - but that's hard to measure. Effort is not as hard to measure.

I have deep skepticism about any study as to which is important, though. I strongly suspect that any such study will reveal exactly what the researcher is looking for.

Again, I am very sympathetic to deconstructions of popular notions of "talent" and "genius", which are overblown in many respects.

However, the fact that effort is easier to measure (and encourage, train, reward), does not mean that one should fall into the trap that "what we can't easily measure doesn't exist". Einstein's brain is notorious for the unusual proportion of glial cells, although that could not be measured in his lifetime. (Not that the relevance of glia to intelligence is clear, and we don't know how many dummies had a surfeit of glia, but anyway.) The Edward Witten phenomenon is hard to explain on the basis of hard work and threshold-level talent, considering that he started in his mid-20's and was at the top of the field 4-5 years later. He started ten years later than most, after studying history and working as a journalist and political campaigner, in a field (theoretical physics) where most of his competition began in their early or mid-teens and worked hard doing it. As a side interest, he is also one of the top 2-3 mathematicians on the planet.

My initial point was something like your last comment; if you're looking for proof that hard work is worth more than talent, or vice versa, the trivia competition you observed can be cited as "evidence", because it is consistent with ANY presumed mixture of talent and work as the ingredients of success.

With that said, given the choice it is far more productive to hold the work in higher esteem and see every success as evidence of that (the "Asian" attitude). Even if the belief turns out to be false, it encourages behavior that leads to other benefits.

By the way, Schoenfeld's study had nothing to do with work versus talent, or nature/nurture, or any other such controversy. He is a mathematician who became an educational psychologist (at Berkeley, I think), was interested in the psychology of mathematical problem-solving, and included various mathematicians as part of the group in the study.
The stronger abilities of the top mathematicians even though having lesser knowledge, was something observed afterward in reviewing the session transcripts, not something that the study was designed to probe.