I think some people have more natural talent at maths than others. But I believe that good teaching, effort and hard work means that less naturally talented people can often catch up. So I think that Cortina and mathanxiety are right that effort is the major factor, but I also think that some are more naturally gifted than others.
I also agree that working on a subject like maths eventually changes your mind and helps you think logically. It trains and develops your mind and you get better at it the more you do, just like in music.
I also think that Cortina is right about the quality of "doggedness" that is needed to do maths. You need persistence, you cannot give up at the first hurdle, you have to perservere until you see the next step you have to make. You start with a clean sheet and have to fill in teh blanks. The more practice you get at it, the more dogged you become and the more confident you get and the less failure you expect. Eventually you realise that nothing is as impossible as it at first seems.
This is a very interesting article and argues for the role of effort
On reading Hereditary genius, by his cousin, Francis Galton, Charles Darwin wrote: ´You have made a convert of an opponent in one sense, for I have always maintained that, excepting fools, men did not differ much in intellect, only in zeal and hard work; and I still think [this] is an eminently important difference.' Thomas Edison, too, believed that genius was ninety-nine per cent perspiration and one per cent inspiration and he ´often work[ed] as many as 112 hours a week' (McAuliffe 1995). Isaac Newton, asked how he made his remarkable discoveries, replied: ´I keep the subject constantly before me and wait until the first dawnings open little by little into the full light' (Andrade 1956).
Hardy told of a visit he made to Ramanujan ´when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to me rather a dull one, and that I hoped it was not a bad omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways"' (Newman 1956). It was said of Ramanujan that every number was his friend and he had plainly thought about and stored away many interesting facts about most of the lower integers. At the age of 10 or 12 he could recite the values of pi and the square root of two to any number of decimal places. Because mathematics was his only interest as a boy, he had failed his scholarship examinations in India. Could it be that Ramanujan's exceptional achievements resulted, not from exceptional endowments but, rather, from the fact that, like Newton, he had kept the one subject of his interest constantly before him since his childhood?
In 1960, at the Center for Advanced Study in the Behavioral Sciences in Palo Alto, Dutch psychologist Adrian de Groot, an expert on the psychology of chess and a chess master himself, simultaneously played and defeated 20 chess duffers like myself. He was not allowed even to see one chess board presided over by two of the Center Fellows who thought themselves to be relatively accomplished players. Well into the game, after they announced the next move they had decided on, de Groot pointed out that their proposed move was impossible; although they had the chess pieces arrayed before them while he had only his mental image to rely on, they got it wrong while he got it right. De Groot himself had played-and been easily beaten-by the future grand master, Bobby Fisher, when Fisher was a boy of twelve. De Groot was careful to point out, however, that even by that early age Fisher had played many thousands of chess games and had derived from this experience a vast armamentarium of chess positions and strategy.
The late Richard Feynman frequently disconcerted physicist colleagues by interrupting their explanations of new findings, to which they had devoted weeks or months of work, and quickly scrawling on a blackboard a more general result of which theirs was just a special case. Was this lightening-like calculation or was Feynman able to draw upon a ´storehouse of previously worked-out-and unpublished-knowledge'? Feynman's biographer, James Gleick, describes a 1960s Caltech seminar at which astrophysicist Willy Fowler proposed that the recently discovered quasars were supermassive stars.
´Feynman immediately rose, astonishingly, to say that such objects would be gravitationally unstable. Furthermore, he said that the instability followed from general relativity. The claim required a calculation of the subtle countervailing effects of stellar forces and relativistic gravity. Fowler thought he was talking through his hat. A colleague later discovered that Feynman had done a hundred pages of work on the problem years before. The Chicago astrophysicist Subrahmanyan Chandrasekhar independently produced Feynman's result---it was part of the work for which he won a Nobel Prize twenty years later. Feynman himself never bothered to publish. Someone with a new idea always risked finding, as one colleague said, "that Feynman had signed the guest book and already left."'
(Gleick 1992)
K.A. Ericsson (1990; Ericsson and Charness 1994; see also the chapter by Lehmann and Ericsson in this volume) has shown that remarkable feats of memory can be achieved by apparently unremarkable people after extensive practice. He has also shown, as the above examples attest, that most examples of exceptional performance, including those by people known as geniuses, are preceded by years of intense and single-minded application and practice.
Ericsson and his colleagues have amassed a truly impressive body of evidence in support of their view that it is deliberate and intensive practice-rather than differences in native ability-that separates elite performers from the rest of us. With hundreds of hours of guided practice spaced over weeks or months, ordinary college students can learn to increase their digit span-the number of digits correctly repeated after hearing them read only once at a rate of one per second-by 10 times. There are techniques of calculation with which, after extensive practice, one can accomplish feats of mental arithmetic impossible for the untrained mind. The conditioning and practice of elite athletes changes their muscle strength, aerobic capacity, the speed of their reflexes, the size of their hearts, and even the relative proportions of fast and slow-twitch muscle fibers, and it is these practice-produced effects rather than just native ability that is responsible for extraordinary athletic performance. The celebrated violinist at last night's concert almost certainly practices more intensely and consistently than the members of the orchestra's violin section. Elite performers tend to do less well as they get older but many of them also tend to practice less intensely as they age.
Ericsson believes not only that genius and exceptional performance generally depends upon intensive years of practice but, moreover, that most of us, given the same teachers and similar preparation, could do as well as these elite performers do. Ericsson and Charness (1994, p.744) are willing to acknowledge that genetic differences in temperament and ´preferred activity level' may determine which of us go for the gold but, curiously, they cling to the assumption that individual genetic differences in both physical and mental capacities are not important, perhaps nonexistent. This would require us to believe that most children could acquire perfect pitch and the ability to reproduce compositions after a single hearing if only we listened to music as long and as intently as Blind Tom and Leslie Lemke did, or draw from memory a construction site after a brief glance, as Sack's savant Steven did. We should have to suppose, as Lewontin seems to imply, that almost any of our children could become world-class athletes, given the right training and the appropriate temperament. We must also accept the proposition that little Gauss's ability to correct his father's arithmetic at three and confound his school master at ten resulted, not from extraordinary mental hardware, but from mental software acquired through self-directed practice in an intellectually unstimulating environment.
Those of us who have studied MZ twins reared apart from one another find these assumptions, which are the concept of radical environmentalism in different clothes, incredible. We cannot believe that MZA twins correlate .75 in IQ merely because, in their separate environments, their similarities in temperament led them to indulge in very similar amounts of practice on very similar topics. One set of Bouchard's MZA triplets each were on their high school's wrestling team before they ever knew of each other's existence. I think this was because they shared a configuration of genetic traits, physical and mental, that made them interested in-and good at-this particular sport. More generally, I think that one reason, although not the only reason, that most elite performers engage in the dedicated pursuit of excellence in their specialty is that they are naturally good at it from the start so that their early efforts are rewarded by early success.
I think we must agree with Ericsson, however, that works of genius tend to be the product of minds enriched by years of concentrated effort. Isaac Newton often became so caught up in cerebration that he would forget to eat or sleep. Edwin Land, inventor of the instant Polaroid camera and of a sophisticated computational theory of color vision, sometimes worked at his desk for 36 hours or more, unaware of the passage of time until he felt faint on standing up. Similar stories were told of Edison. It does not follow, however, that these were ordinary minds to begin with.
Edison, Feynman, Land, and Newton all from their boyhood had intense curiosity, an enthusiasm or zeal for discovery and understanding. Each of them was able to take seriously hypotheses that others thought to be implausible (or had not thought about at all). All four possessed a kind of intellectual arrogance that permitted them to essay prodigious tasks, to undertake to solve problems that most of their contemporaries believed to be impossible. And each of them had quite extraordinary powers of concentration. Even Darwin, plagued as he was by physical miseries that would have invalided most men, somehow mustered mental energy enough to pursue the painstaking researches that yielded the thousands of facts with which he built his theory and defended it against so many critics.
cogprints.org/611/1/genius.html