Quoted from The Aims of Education by Alfred North Whitehead:
The most austere of all mental qualities; I mean the sense for style. It is an aesthetic sense, based on admiration for the direct attainment of a foreseen end, simply and without waste. Style in art, style in literature, style in science, style in logic, style in practical execution have fundamentally the same aesthetic qualities, namely, attainment and restraint. The love of a subject in itself and for itself, where it is not the sleepy pleasure of pacing a mental quarter-deck, is the love of style as manifested in that study.
…Style, in its finest sense, is the last acquirement of the educated mind; it is also the most useful. It pervades the whole being. The administrator with a sense for style hates waste; the engineer with a sense for style economises his material; the artisan with a sense for style prefers good work. Style is the ultimate morality of mind.
…Style is the fashioning of power, the restraining of power… with style the end is attained without side issues, without raising undesirable inflammations. With style you attain your end and nothing but your end. With style the effect of your activity is calculable, and foresight is the last gift of gods to men. With style your power is increased, for your mind is not distracted with irrelevancies, and you are more likely to attain your object. Now style is the exclusive privilege of the expert. Whoever heard of the style of an amateur painter, of the style of an amateur poet? Style is always the product of specialist study, the peculiar contribution of specialism to culture.
I have crossposted this quote because it has direct relation to Krutetskii’s concept of “economy of thought” that I discuss in my book. In mathematically able children, this is still not a skill to work “without waste”, but an instinctive striving to think economically and choose between possible approaches to a problem the one which promises the most streamlined and elegant solution.
And here is a quote form Krutetskii’s classical book. It concerns one of his favourite subjects, Sonya L, 8 years old girl.
A distinctive tendency for “economy of thought.” Sonya is notable for a striving to find the most economical ways to solve problems, a striving for clarity and simplicity in a solution.
Although she does not always succeed in finding the most rational solution to a problem, she usually selects the way that leads to the goal most quickly and easily. Therefore many of her solutions are “elegant.” What has been said does not apply to calculations (as was stated above, Sonya is unfamiliar with calculation techniques). Consider a few examples.
Problem: “How much does a fish weigh if its tail weighs 4 kg, its head weighs as much as its tail and half its body, and its body weighs as much as its head and tail together?”
Solution: “Its body is equal in weight to its head and tail. But its head is equal in weight to its tail and half its body, and the tail weighs 4 kg. Then the body weighs as much as 2 tails and half the body — that is, 8 kg and half the body. Then 8 kg is another half of the body, and the whole body is 16 kg.” (We omit the subsequent course of the solution. The problem is actually already solved.)
“The sum of the interior angles of a triangle is 180 degrees. Angle BAC yields 180 degrees either when added to the exterior angle (as supplementary adjacent angles), or when added to angles B and C (as interior angles). Therefore, the exterior angle is equal to the sum of angles B and C.”
The issue of “economy of thought” in young children can be compared with the issue of style in sport. There were times, say, in swimmimg, when young boys and girls routinely won over adults – and it was before steroids coming into use. Interestingly, this happened in long distance swimming, where economy of effort was paramount, and not in short distances, where sheer power and force of adults prevailed.
As every boy in Sibirea, I had to do a bit of cross-country skiing — without much success, I have to say. Aged 15, I knew a 12 years old girl from my school who could beat me at any distance. She liked to tease 18-year old male conscripts on their compulsory 5 km skiing test, by flying past them, effortlessly like a snowflake in the wind. For sweating and short-breath guys, it was an ultimate humilation. For reasons unknown, the little girl had style.
At that point, the reader will not be surprised by other qute from Krutetskii:
The reduced fatiguabiliy in mathematics lessons that characterizes Sonya should also be noted. Not only is she very hard-working and fond of solving problems “on reasoning”: she
tires comparatively seldom during these lessons (excluding long, involved calculations, which she does in her head). Neither the lessons at home nor those with the experimenter were ever ended on her own initiative. Even prolonged lessons (for her age) did not lead to marked fatigue. For experimental purposes we set up a few lessons with her of an hour and a half, without interruption (a 45-minute lesson doubled!). Only at the very end of this period did the little girl of 8 show signs of fatigue (mistakes, slackening of memory). When occupied with other types of work (music, reading, writing), Sonya tires normally.
Should ve be suprised that “style”, “economy of thought” and non-fatigability go hand in hand?
Still, “style” in the sense of Whitehead is not something which can be attained at a very young age in a fully developed form. But it is something which could be seriously, perhaps irreversibly compromised at early stages of education by a student accumulating wrong habits and mannerisms.