Popular Science Monthly/Volume 29/June 1886/Counting Unconsciously
OF THE UNIVERSITY OF JENA.
AT first sight the superscription, "counting unconsciously," seems to contain a contradiction. For, whoever counts from one to one hundred, realizes at each number, that he is counting; yet, in truth, there are so many instances where an educated person counts without realizing it, that he would feel utterly lost in this world should this faculty be suddenly taken from him.
Three coins being placed on a table, any one will, on being asked, "How many are there?" answer, after but a glance, "Three." Even when four or five coins are seen but for a moment, the answer as to their number will be correctly given. So quickly is the answer returned that no time can possibly have been taken for counting. Hence, it follows that counting unconsciously is really an every-day occurrence. The objection that this is no longer to be termed counting, is not valid; for if any one can positively state that there are lying before him three, or four, or five objects, he must be able to distinguish numbers; and it is certainly a fact that one who can not count, can also not answer such questions. Children, in order to distinguish three marbles from four, must first add each marble to the other in this way many learn to count before knowing the numerals. From this it follows that, in order to count, a knowledge of the numerals is not a necessity; even untrained deaf-mutes, who can neither read nor write, are capable of counting, without figures, merely by the aid of their fingers.
From the action of a child who has learned the meaning of the numerals, it furthermore follows that it is only by practice, that is by oft-repeated counting of actual objects, that surety is gained in the art of counting small numbers unconsciously. An idiot, or whoever does not practice, can not count three without adding one by one, and will never rise above the lowest plane of mental development.
Now, however, as is well known, no one can tell in a moment how many objects are lying before him, provided the number of these objects is somewhat large—approximates, say, fifty. Some persons can count more rapidly than others; a broker's apprentice will make groups of three, of five, of ten coins, and then add the groups together; the experienced money-broker is able to determine in a few seconds what the amount is, and this, perhaps, without even touching the coins. But he too, as well as every one else, must count attentively as soon as the number of pieces exceeds a certain limit. But what is this limit?
Dase, the well-known calculator, who died in 1861, stated that he could distinguish some thirty objects of a similar nature in a single moment as easily as other people can recognize three or four, and his claim was often verified by tests. The rapidity with which he would name the number of sheep in a herd, of books in a book-case, of window-panes in a large house, was even more remarkable than the accuracy with which he solved mentally the most difficult problems. Not before or after his time has such perfection been attained; but as every one possesses this faculty to a small extent, and as it can be improved by practice, it is not impossible that in future other experts in this line may appear. The only trouble is that so few know how easy it is to practice.
In the first place, one can by a few trials readily gain the conviction that, without practice, not every one can distinguish six and seven objects as easily as three and four.
In order to learn that it is a comparatively easy matter to estimate up to six and seven, and then up to nine, as correctly as from three to five, one need only make a few trials in guessing at an unknown number of matches or pins that are concealed beneath a sheet of paper, and are then exposed to view but for a second.
Great care must be exercised, however, that one does not consciously count in these attempts; nor will it answer to attempt analysis from memory, after the objects are again hidden from view; all this would consume too much time. It is, in fact, necessary to do nothing more than to estimate, but this must be done with the utmost attention.
Whoever has for any length of time tried seriously to guess correctly will be surprised to find that his guesses will soon grow to be generally correct, whereas at the start they were often erroneous. Only when the number of objects seen exceeds nine will mistakes again occur more frequently. However, further practice in estimating greater numbers of small objects will soon cause considerable improvement even here. Many, however, do not succeed in estimating correctly beyond ten, probably because the attention is not sufficiently concentrated at the time, and as it is necessary, at the start at least, that one's whole attention be closely given; only after having attained some degree of proficiency will the exercise of this power no longer prove fatiguing.
In order to practice this kind of counting, dots and small circles were drawn on white paper squares. Some of these dots were arranged symmetrically, others were irregularly placed. These were glanced at for a moment, and proved of considerable aid in acquiring the art. A good deal depends on the arrangement of the dots. A card-player will immediately, and without stopping to count, realize that there are ten hearts on a ten-spot of that suit, but he will not be able to give as correctly the number of hearts or of dots if arranged, for in-stance, in the form of a cross.
Hence it follows that it is not the symmetry of arrangement that facilitates the estimating, but acquaintance with the manner of arrangement used.
It is more difficult to correctly estimate the number of dots arranged in the form of a cross than to determine them if arranged as on cards and in similar ways.
It is more easy to estimate the dots if arranged as on dominoes; the dots must not be too small, and must be made a deep black on a white ground, or the reverse.
The estimation of the number of dots is most difficult if they are grouped in an irregular manner, as, for instance, in the following figures:
Practice, which is naught but patient and correct repetition, will, however, even here make perfect. However, it may be regarded as proved by the case of Dase, before referred to, that practice, however long continued, can not aid beyond a certain limit. It seems that, for the rapid estimation or the unconscious counting of dots placed in unknown symmetrical arrangement, and for objects grouped into irregular forms, twenty is the limit.
Probably already, when the number of the objects exceeds twenty—undoubtedly, when it exceeds thirty—accuracy in estimating can no longer be attained, even after the greatest amount of practice, in which Dase for one certainly was not wanting.
However, this is not to say that more than thirty dots can not, under any circumstances, be simultaneously determined; but in order that this may be done they must be presented in some well-known manner of arrangement, which must, as it were, have been fairly learned by heart. Thus, very skillful card and domino players are able at a glance to take in as many as forty points, in nines, tens, fives, sixes, etc. This they do so rapidly as not to be conscious of any addition. Bat in such cases it is no longer the seeing of single dots, but seeing the pictures they form, which makes the feat possible. As no one on seeing the number 8 will count from one to eight, so no card-player will stop to count on seeing an eight of hearts, for instance. A child, however, not yet familiar with the appearance of cards, will count each heart separately, perhaps even touching each one in turn with his finger.
In order to quickly attain the faculty of counting unconsciously, a book may be used to advantage. If one takes a book, opens the same—the eyes to be kept closed in the mean time and then casts a rapid glance at a part of the page and tries to estimate how many lines are visible, this way of doing, if often repeated and always tried on different pages, will soon conduce to great accuracy in estimating. A small child is not able to estimate even three lines correctly, though looking at them for fully a second.
As the mind develops, it acquires a more simple and rapid process of counting. Something that at first had to be undertaken slowly and with care, perhaps in separate stages, may later on be accomplished much more quickly and without requiring any special effort, or calling for any great amount of attention, in fact almost "mechanically."
One is fully conscious of every perfectly new impression received by the brain; hence the fascination of a novel idea. The more the charm of novelty fades with the recurrence of the same sensation, the less will consciousness be called into play.
Counting from one upward, by constant repetition, finally comes to be done unconsciously, even as the quick movement of the fingers in practicing on the piano gradually becomes almost automatic, though at first this, too, required great care and attention. In all similar cases consciousness is no longer called into play.
An impression that seemed most startling when first received may, if too often repeated, grow to be trivial. The simple work of counting finally comes to be an unconscious action of the nerve-fibers and cells of the brain.
On newly built roads, the trains are run but slowly; the longer such roads have been used, the more rapidly are trains run on them, and stops at way-stations are no longer needed; it is even thus with the trains of thought in the human brain.
And on this rests the practical importance of rapid counting. Whoever can, unconsciously but correctly, count up to twenty or even only up to twelve, has a great advantage over others who can not, without error, distinguish six from seven in this manner. For such a one can turn his consciousness to other matters and greatly increase his knowledge, where another would make but slow progress.
Those movements in man, which take place through some impression received from without and not aided by any conscious act of the brain (as, for instance, the contracting of the pupil when a bright light strikes the eye), are termed reflex actions.
In part these are brought about by arbitrary but oft-repeated motions, inasmuch as such will gradually take place more rapidly and without premeditation.
In this way, through practice, counting from one to five is done unconsciously, and somewhat resembles a reflex action.
If many such simple mental acts (by the repetition of which nothing new is learned, and time only is lost) could be caused to pass off more rapidly—somewhat resembling reflex actions the brain would be left free to turn to other, to higher aims.—Translated for the Popular Science Monthly from Die Gartenlaube.