Some Thoughts on Gifted ChildrenAuthor(s): F. J. BuddenSource: Mathematics in School, Vol. 10, No. 1 (Jan., 1981), pp. 12-13Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30213603 .

Accessed: 22/04/2014 10:25

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access toMathematics in School.

http://www.jstor.org

This content downloaded from 130.239.116.185 on Tue, 22 Apr 2014 10:25:13 AMAll use subject to JSTOR Terms and Conditions

Some thoughts on

gifted children by F. J. Budden, Royal Grammar School, Newcastle upon Tyne

We need to make a subtle distinction: between gifts, talents, skills, . . . "Gifted" normally means endowed naturally with intelligence or other inborn potential; "talented" implies, I feel, that such gifts have been exercised and developed; "skilled" usually refers to a "motor" skill, such as playing tennis, per- forming on the piano, rock-climbing, and so on. In many such cases, there is an artistic element to a greater or less degree, as well as requirements of imagination, inventiveness, originality, etc. Thus skill in tennis goes far beyond the technical mastery of various shots. . . One would speak of a talented scientist, but hardly of a skilled scientist, though one might well use the word skilled of a surgeon. Again, one talks of "bright" children, and here one is referring mostly to how rapidly they can assimi- late knowledge, how readily they can respond to teaching, how receptive to new ideas. Surely the distinctions are ones of usage rather than of intrinsic meaning. And of course, it is not enough to label a person "intelligent", since intelligence is relative to performance in some direction: a person may be highly intel- ligent in one respect, stupid in another: as, for example a mathematician with a First class in his subject may be very stupid when it comes to trying to understand the intricacies whether of economics or of electronics, to say nothing of un- ravelling the meaning of a legal document!

The figures of the top 1/2% and the top 3% of the population have been quoted in previous papers. To get some idea of the magnitudes involved, suppose that each year group of secondary school pupils in Great Britain contains about 800 000 children (a rough guess, but I hope not too far out). At the 1% level, we should be thinking of no fewer than 8 000 gifted mathe- maticians per year, which is about two-thirds of the number who enter the National Mathematics Competition, though those entrants cover three or four years' age range. To put it another way, it would mean the top c.40 boys and girls of the upper sixth form in the Newcastle area each year. While we may also note that the entries to Hons. Maths. courses in British Universities during each year must be something of the order of 1 000, that is, of the order of the top 0.12%. Thus 1/2% is rather larger than it appears at first sight, while 3% is very wide!

It is probably true to say that mathematics is one of the disciplines where there is the largest range of ability. Second only, perhaps, to music, where the range is from Mozart down to the tone-deaf - a truly astronomical ratio. I would claim that in mathematics, the gap also deserves to be called astronom- ical. My own school is highly selective, taking something like the top 10% of the ability range. Mathematics (before '0' level) is taught in five sets in the lower school. The ability gap between the top set, which streaks ahead, lapping everything up eagerly and with seemingly slight effort, and the bottom set, struggling with the greatest difficulty, committing to paper a high output of sheer rubbish, is enormous. This, be it noted, is within a selected sample. Yet even within a maths teaching set there can be, and more often than not is, a very large range of ability, even though these pupils have been mathematically graded. Of course, selection at the youngest end of the school is bound to be subject to inaccuracies, and of course it depends on many factors other than mathematical ability. But suppose for the moment that selection were by mathematical ability

alone, and also that it were perfectly efficient; and suppose that those with a score above 70 were selected, those in the shaded region in Figure 1 being rejected. The resulting distribution of the grammar school intake should then look like Figure 2. However, allowing for the factors mentioned above (inefficiency of the selection process, selection depending on so many other criteria), one would certainly expect a markedly skew distri- bution as in Figure 3. The curious fact however is that in practice, the ability distribution obstinately refuses to appear anything but like Figure 4! This is a paradox which cannot be explained as the working of the central limit theorem.

I believe that the astronomical range of mathematical ability referred to reflects a near-astronomical range of inborn talent. Certain native gifts are there from birth, not instilled by en- vironment. Such are perfect pitch, an "eye for a ball", draughts- manship, green fingers, a "way" with animals, to take a few perhaps not very well chosen examples. True, such gifts may be sharpened, and indeed perfected, by training. But if A has the gift and B has not, then however much training B is given, however many hours he spends on it, however inspired his teachers, however great his motivation, B will never approach the heights of which A is capable. It is tempting to wonder whether, if one took a random adolescent "from the streets", what are the crafts and skills that he could acquire in the best possible circumstances, that is, given the best teachers, every possible incentive and motivation amounting if necessary to extravagent bribery. In which fields of activity could he be turned into an expert? I suspect one could make a very good marksman of him, but only a tolerable batsman. But for all the conceivable rewards that could be offered, he could never in a lifetime be made into a degree mathematician! Or so I suspect. Has any research been done on this, because it would seem to me to be more worth while than some of the nonsense that nowadays passes for research. Are people aware, I wonder, that there are "green fingers" in all sorts of human activities! - I know a man who can repair TV sets and doesn't even know Ohm's law; and another with the same skill who was a complete

Fig. 1

50 70 100

Fig. 2

50 70 100

12

This content downloaded from 130.239.116.185 on Tue, 22 Apr 2014 10:25:13 AMAll use subject to JSTOR Terms and Conditions

failure at school, and indeed at everything not connected with electronics!

While I do not accept that it is as difficult as some of the writers have made out to spot the mathematical high flier, I do wonder whether there is a very great failure rate in our spotting, and whether this country is littered with "full many a flower ... born to blush unseen and waste its sweetness on the desert air". Do we have hundreds of mathematicians manques simply because they live in the wilds of Cornwall, Northumberland, Perthshire, or wherever, and so have never had the opportunity to show their talent; or, if this talent has been apparent to themselves, has never been observed by others and thereby given the opportunity to flourish? How shall we ever know? It is rather like claiming efficiency for the Post Office on the grounds that large percentages of letters are delivered by the next day, etc., etc., but forgetting about how many letters are lost, for which figures do not exist, and indeed for which figures would be unobtainable since in any case the addressees are not usually aware of the letter having been sent and so cannot report the loss!

I never cease to be amazed when I hear of three year olds learning to play the violin (of all instruments, surely one of the hardest, and the most discouraging and excruciating in its initial stages!) by the Suzuki method. How it is ever possible, I ask myself, to teach at such a young age a technique so complicated, requiring such co-ordination, such concentration, such intel- ligence . . . even the fingers are not equally spaced on the fingerboard! How do you ever get a three-year-old (or even a six-year-old!) who cannot yet read and write, add or subtract, to space his tiny fingers thus

O O O O orO O O O rather than O O O O 1234 1234 1234

Yes, I know you do it by getting him to listen to what comes out to make sure it sounds right, but surely that itself is a complicated enough action for one of such tender years: to distinguish tones from semitones; to know whether the note he is playing is sharp or flat; then to know how to correct it ... the mind boggles! And yet! People answer me by pointing to other very complicated human activities which most people acquire, if not painlessly, without too many tears - learning to speak, to analyse the complications and subtleties of the sounds of language; and then later to read and to write: what appalling complications there have to be grappled with if one pauses to contemplate them! Again, one hears of babies successfully learning to swim; only one who has learned as an adult can appreciate the problems of co-ordination of movements of feet, hands and breathing that have to be overcome in this process; yet animals swim with no instruction! And again, to take a completely different example: campanology. I have never rung bells, having only looked into the theory of it. Although a

Fig. 3

50 100

Fig. 4

50 100

mathematician, it does seem to me awfully complicated, and I am sure I should always be making mistakes both of omission and of commission were I to embark on the practice of camp- anology. Yet, I am told, for all its complexity, successful bell- ringers have been drawn from the ranks of illiterate peasants.

What I am driving at is this: I have described above some extremely complicated human activities, each with intellec- tual ingredients, which can be mastered at a very early age, and without being especially endowed intellectually. Why is it that performance in mathematics lags so lamentably behind that in other activities? The paradox is that those people who sit beside me in choirs and orchestras can perform hair-raising acts of musical acrobatics and of musical intelligence ("better put the second finger on the F sharp so that I can get across to the G-string with the fourth finger in the next bar to play the C sharp when it changes into D flat, and the music moves into B flat minor. . . .!) and yet these same people, and other of similar intelligence cannot do the simplest piece of mathe- matics - I don't mean mere arithmetic: I mean they will not be able to see the difference between (a-b)2 and a2 -b2.

Could it be possible, I ask myself, that there is a Suzuki method around the corner waiting to be applied to the teaching of Calculus, of Group theory, of Probability . . . to infants? This may seem far-fetched; critics may say "but of course, you cannot succeed in teaching a piece of mathematics until the pupils see the need for it." This objection impresses me, till I think of the infant violinists: how can they ever see the pur- pose of struggling to master the intricacies of the violin, even though they are doing it with Mother! They cannot possibly perceive the rich rewards of many happy hours of chamber music playing which lies ahead: motivation appears to be absent. Nor can the babes possibly "see the point of" floating in water and propelling themselves laboriously along, though they may enjoy the sensation, just as the young violinists may enjoy the sensation of the sound (excruciating though it be). Alas, there is no compensating sensation, no instant gratification to be derived from the premature pursuit of mathematics!

Graham Hoare made a sound analysis in a letter circulated to the subcommittee of the qualities needed for the pursuit of mathematics (see Mathematics in School, May 1980, pp 33-34). He was also right to point to the importance of the need for the love and enthusiasm for the subject, and how this derives from the influence and inspiration of a good teacher. How many "flowers born to blush unseen" are there for the lack of an inspiring teacher? The importance of what have been called "non-intellective" factors cannot be over-estimated: character, persistence, determination must all be allied to intellectual gifts for successful mathematical development. A highly gifted student will never flourish if her (or his!) middle name is Inertia! But talents can develop not only by continuous application*, by the contact with an inspired mind, by the existence of some motivation which may or may not be external to, or additional to, a love of the subject for its own sake, but I also believe, finally, by both competition as well as co-operation from fellow students. I do believe this latter component to be important, and have seen it work so often - a group of sixth formers working together, the mutual stimulus acting as leaven to the bread.

In conclusion, I should like to reiterate in the strongest possible terms a point which I made in a previous paper and which I make whenever the opportunity arises, which alas it does frequently: this is to refute the heresy which one hears so often now from egalitarians and others who should know better: "... after all, the brighter children can look after themselves". This carries the implication that if they need any real teaching at all, only the most cursory instruction is required. This is dangerous nonsense, and it cannot be said too often or too emphatically that the brightest children or students need the best teaching that can possibly be provided for them.

*I believe the saying that Genius is 90% perspiration and 10% inspiration to be nothing but nonsense, if only because of the use of the word "genius". But for all but the genius, the formula is about right for the successful pursuit of most disciplines.

13

This content downloaded from 130.239.116.185 on Tue, 22 Apr 2014 10:25:13 AMAll use subject to JSTOR Terms and Conditions