Counting

“ Such a child may count correctly from the point of view of a grown-up listener, without having any idea of numbers at all. The operation of counting, in fact, can only be intelligently performed by a person who already has some idea what the numbers are; and from this it follows that counting does not give the logical basis of number. ”

“ Counting by means of his fingers, without including the thumbs, the savage begins by dividing into fours instead of fives. Traces of this form of counting are somewhat numerous, especially among the North American aboriginal tribes. Hence the quinary form of counting, however widespread its use may be shown to be, can in no way be claimed as the universal method of any stage of development in the history of mankind.
In the vast majority of cases, the passage from the base to the next succeeding number in any scale, is clearly defined. ”

“ Any such mode of counting, whether involving the use of the fingers or not, is to be regarded simply as an extraneous aid in the expression or comprehension of an idea which the mind cannot grasp, or cannot retain, without assistance. The German student scores his reckoning with chalk marks because he might otherwise forget; while the Andaman Islander counts on his fingers because he has no other method of counting,—or, in other words, of grasping the idea of number. A single illustration may be given which typifies all practical methods of numeration. ”

“ But the one primitive method of counting which seems to have been almost universal throughout all time is the finger method. It is a matter of common experience and observation that every child, when he begins to count, turns instinctively to his fingers; and, with these convenient aids as counters, tallies off the little number he has in mind. This method is at once so natural and obvious that there can be no doubt that it has always been employed by savage tribes, since the first appearance of the human race in remote antiquity. ”

“ Indeed, it is only by experience that we can know that any definite process of counting will give the true cardinal number of some class of entities. It is perfectly possible to imagine a universe in which any act of counting by a being in it annihilated some members of the class counted during the time and only during the time of its continuance. ”

“ The purpose of counting is extremely manifold. Its most frequent and most important application lies in the fact that the amount affords a measure for the effectiveness or the value of the corresponding group, both increasing and decreasing simultaneously. ”

“ As counting is the fundamental operation of arithmetic, and all other arithmetical operations are simply devices for speed by using remembered countings instead of going through the detailed work again each time, the hint is furnished 19 the teacher for the first lessons in arithmetic. This hint has been generally followed out and the child set at work at first upon the counting of objects so much alike that the qualitative difference is not suggested to him. He constructs gradually his tables of addition, subtraction, and multiplication, and fixes them in his memory. ”

“ In illustration and as an example it may be noticed that the eyes of the majority of persons convey the consciousness of numbers precisely, up to a certain point, without any operation of counting. Most people can say at a glance, of any small group of objects, that there are two, three, four, or even seven. With almost all individuals, counting, and counting from the beginning, becomes necessary when there are eight or more objects together. ”

“ This suggests that mistakes in counting cannot be identified without an arithmetic calculus and this is plainly not true. Four types of mistakes in counting can occur:
Case C1. A child counting apples in a bucket says "one apple, two apples, four apples, five apples" and concludes that there are five apples in the bucket when there are in fact only four. In this case it is clear that the child has not learned how to count. The mistake can be identified and corrected by a parent or teacher.
Case C2. A person who has learned to count correctly makes a mistake through inattention. ”

“ I count, but I count, in ordinary life, what I take to be real objects, existent quite apart from my counting. Suppose that I count ships seen from the shore. There, says common sense, are the ships, sailing by themselves, and quite indifferent to whether anybody counts them or not. In advance of the counting, the ships, in so far as they are a real collection, have their number. ”

“ If you once commit the addition table thoroughly, you have it fixed for life; whereas if you do not, you have to make the calculation over again every time, and thus lose a vast amount of labor. I have no doubt that there are some in this class who are in the habit of counting, who have ascertained that seven and eight for instance, make fifteen; by counting up from seven to fifteen, hundreds of times. ”

“ As a rule, however, the system has been none of these, but counting has been done by fives, by tens, or twenties, and this simply because it has been based upon the antique but persistent habit which man has of counting upon his fingers. ”

“ I noticed some of the company prepare their mouths and draw themselves up at the prospect of some agreeable forfeit. "Let us play at counting," said Charlotte. "Now, pay attention: I shall go round the circle from right to left; and each person is to count, one after the other, the number that comes to him, and must count fast; whoever stops or mistakes is to have a box on the ear, and so on, till we have counted a thousand." ”

“ This method of counting, however, is not confined to the counting of ages, but it characterizes all their counting. If you ask a man how many days before a certain festival near at hand he will say ten where we would say but nine. ”

“ That is our ordinary way of counting. The speculative computation is quite different. Each of the things is always that thing plus me. So that supposing the things to be represented by the figures 1, 2, 3, 4, and ourselves by the figure 5, while following the ordinary ciphering, we should count them and ourselves as 1, 2, 3, 4, and 5 ”

“ When the fingers and toes of the person counting are exhausted, he has recourse to those of another person, if he wishes to count further, although he has then passed the limit of numerical phraseology. ”

“ The place occupied, in the intellectual development of man, by finger counting and by the many other artificial methods of reckoning,—pebbles, shells, knots, the abacus, etc.,—seems to be this: The abstract processes of addition, subtraction, multiplication, division, and even counting itself, present to the mind a certain degree of difficulty. To assist in overcoming that difficulty, these artificial aids are called in; and, among savages of a low degree of development, like the Australians, they make counting possible. ”

“ This, of course, is unintelligible on the grouping system of treating number; on the counting system it merely means that we count backwards from 0, just as we might count inches backwards from a point marked 0 on a scale. It should be remembered that the counting is performed with something as unit. If this unit is A, then what we are really considering is −4A; and this means, not that A is multiplied by −4, but that A is multiplied by 4, and the product is taken negatively. ”

“ It is this view of number as generated by counting which has been the chief psychological obstacle to the understanding of infinite numbers. Counting, because it is familiar, is erroneously supposed to be simple, whereas it is in fact a highly complex process, which has no meaning unless the numbers reached in counting have some significance independent of the process by which they are reached. And infinite numbers cannot be reached at all in this way. ”

“ Sometimes people will speak of mere counting as one of the simplest operations in the world; but on occasions, as I shall show, it is far from easy. Sometimes the labour can be diminished by the use of little artifices; sometimes it is practically impossible to make the required enumeration without having a very clear head indeed. An ordinary child, buying twelve postage stamps, will almost instinctively say, when he sees there are four along one side and three along the other, "Four times three are twelve;" while his tiny brother will count them all in rows, "1, 2, 3, 4," etc. ”

“ There can be little doubt that many of us were taught to count very badly, and that we were hampered in our arithmetic throughout life by this defect. Counting should be taught be means of small cubes, which the child can arrange and rearrange in groups. It should have at least over a hundred of these cubes—if possible a thousand; they will be useful as toy bricks, and for innumerable purposes. Our civilization is now wedded to a decimal system of counting, and, to begin with, it will be well to teach the child to count up to ten and to stop there for a time. ”

“ The origin of the quinary mode of counting has been discussed with some fulness in a preceding chapter, and upon that question but little more need be said. It is the first of the natural systems. When the savage has finished his count of the fingers of a single hand, he has reached this natural number base. ”

“ Sometimes a number above ten, but not over twenty, is indicated with the hands only by counting up to ten in the ordinary way, and then opening all the fingers and counting again, until they reach the requisite amount in excess of ten. ”

“ On the whole, the grouping method refers mainly to concrete numbers and the counting method to abstract numbers. If we sort objects into groups of ten, and find that there are five groups of ten with three over, we regard the five and the three as names for the actual sets of groups or of individuals. The three, for instance, are regarded as a whole when we name them three. If, however, we count these three as one, two, three, then the number of times we count is an abstract number. Thus number in the abstract is the number of times that the act of counting is performed in any particular case. ”

“ They know, as soon as they reach 40, just how many there are, and it is hard to keep from counting the rest in a [Pg 531] group. Otherwise they do not feel any difference in counting Cards 8 and 9. Arranging the letters in words does not affect the speed of counting, so far as they can see, for in counting they do not notice the words at all.
When we average the records of all the subjects giving equal weight to each subject, though the number of tests may be different with the different men, we get the following table. Time given in thousandths of seconds. ”

“ Meanwhile, words to serve the purpose of pure counting are all the scarcer because hands and feet supply in themselves an excellent means not only of calculating, but likewise of communicating, a number. ”

“ Otherwise the pupil is likely to limit the meaning of, say, four to four sensibly discrete things and to have difficulty in multiplication and division. Measuring, or counting by insensibly marked off repetitions of a unit, binds each number name to its meaning as —— times whatever 1 is, more surely than mere counting of the units in a collection can, and should reënforce the latter. ”

“ Or if an observer were stationed at a certain point, and 10,000 persons were to pass him in single file without his counting them as they passed, what sort of an estimate would he make of their number? The truth seems to be that our mental conception of number is much more limited than is commonly thought, and that we unconsciously adopt some new unit as a standard of comparison when we wish to render intelligible to our minds any number of considerable magnitude. ”

“ On the other hand, just as soon as these same people find themselves in contact with civilization, and when they acquire standards of value that have to be counted, they adopt with perfect ease higher numerals from other languages, and develop a more or less perfect system of counting.... It must be borne in mind that counting does not become necessary until objects are considered in such generalized form that their individualities are entirely lost sight of. ”

“ In whatever form we meet with it, the very notion of counting involves the highly artificial conception of a number of objects which for some purposes are treated as absolutely alike, whilst yet they can be distinguished. ”

“ The theory that man’s primitive mode of counting was palpable reckoning on his hands, and the proof that many numerals in present use are actually derived from such a state of things, is a great step towards discovering the origin of numerals in general. Can we go farther, and state broadly the mental process by which savage men, having no numerals as yet in their language, came to invent them? ”

“ THE ART OF COUNTING. Ideas of Number derived from experience—State of Arithmetic among uncivilized races—Small extent of Numeral-words among low tribes—Counting by fingers and toes—Hand-numerals show derivation of Verbal reckoning from Gesture-counting—Etymology of Numerals—Quinary, Decimal, and Vigesimal notations of the world derived from counting on fingers and toes—Adoption of foreign Numeral-words—Evidence of development of Arithmetic from a low original level of Culture. ”

“ SCORE (O.E. scor, from sceran, to cut, notch, cf. “shear”) , properly a notch or groove cut in a piece of wood, called a “tally” (q.v.) , as a method of counting; hence an account or reckoning made in this way. Either from a custom of keeping each series of twenty numbers or notches on a separate tally, or of marking the twentieth number by a longer or deeper mark, the word was early used to denote the number twenty; it is still used as a measure of weight, equivalent to 20 ℔, computing the weight of animals sold for slaughtering for food. ”