CHAPTER V. ARITHMETIC. Two kinds of counting. Counting is the first step in arithmetic. It may be divided into two kinds, correct and intelligent. To count correctly is to say one, two, three, four, &c., and requires simply an acquaintance with the names of the figures and with the order in which they should be repeated; to count intelligently requires in addition a true appreciation of the numerical difference of each group. 'Correct' counting is not an arithmetical operation. It is a mere verbal exercise; nothing of number enters into it but the names. A child may repeat the names of the figures from one to a thousand, and yet perceive as little reference to different groups of objects as he perceives, and as actually exists, when he repeats the letters of the alphabet. Correct counting is, however, an introduction to intelligent counting, and is, so far, indispensable; but it should not be forgotten that in itself it is of very little arithmetical value-that, in fact, it is but a means to secure a desired end. I am particular in mentioning this, as it is the only proficiency in counting required by the majority of teachers from the children of their junior classes. Intelligent counting. A child should be taught not only to count correctly,' but to know that each name which he repeats represents a number of ones, or units, and that each group differs from the next to it by one, and only one-that nine, for instance, is one more than eight, and one less than ten, and only one. This is taught by means of the ball-frame. One ball is brought over on the first line, two on the next, three on the next, &c., and as they are brought over the master and children repeat together the number of balls. This frame should form a portion of the apparatus in every school. When it has not been procured, books, marbles, pens, or any similar objects easily grouped will answer the purpose; even the pupils themselves in default of all else may be arranged into the groups represented by the different names, and be thus used in place of a ball-frame. Instead of saying one, two, three, &c., they might judiciously be made at first to say one ball, two balls, three balls, &c. They will thus soon come to see that the names are the names of groups, which differ from each other by a certain quantity. Tests of proficiency in counting intelligently. When pupils can repeat the names of the figures, and, in addition, answer such questions as the following, they may be said to count intelligently: If I have 6 marbles, and get 1, how many shall I have ? lose 1, Each consisting of the addition or subtraction of unity. Those involving the addition or subtraction of two or more units are not proper questions, as the object in view is to determine whether the pupils know the precise difference between the groups which are in immediate connection with one another. Intelligent counting embraces two arithmetical laws. Intelligent counting may be said also to embrace a knowledge of the two arithmetical laws deducible from the mere names of the figures. The first law is exemplified in the teens, or tens, in thirteen, fourteen, fifteen, &c.; and the second in the figures that end in ty, as twenty, thirty, forty, &c. The first names are corruptions of the compounds three-ten, four-ten, five-ten, &c., and these again are contractions of the expressions three and ten, four and ten, five and ten. The second are corruptions of two-tens, three-tens, fourtens, &c., and these are contractions of the expressions 'two tens added together,' three tens added together,' 'four tens added together,' &c.; and both are instances of that peculiar species of addition in which one word registers the answer, and indicates the addends employed. When a child says eighteen, or twenty, for instance, he ought to learn from the mere word (1) that the group it represents is formed by the union of two others, and (2) that these others are indicated by the names eight and ten, or, in the case of the word twenty, by ten and ten (two tens). Test questions on these laws. When properly taught these laws, the pupils should be able to answer questions like the following: NOTATION AND NUMERATION. 113 The laws admit of easy application in this way, and the skill and intelligence required are quite within the powers of young children. It is very easy to employ them in forming a system of mental addition, and, by a reference to the fact which they so frequently bring before the children—that ten enters into the composition of all numbers beyond itself—it is easy to explain to all children the ten-group or decimal character of our system. NOTATION AND NUMERATION. Most important rules. Pupils should now be enabled to read and represent numerically by the recognised symbols the numbers whose names they have learned to repeat. In other words, they should begin the study of numeration and notation. These are two of the most important rules in arithmetic. Unless they are fully understood it is quite impossible that there can be any true or rational knowledge of the powers of the numbers made use of in any calculation; and it is found to be the case that intelligent and successful answering upon arithmetic is in exact accordance with an intelligent and a successful comprehension of these elementary rules. And yet I regret exceedingly to say that there are few schools in which they are taught carefully or skilfully. This statement is borne out by the combined testimony of all the inspectors in Great Britain and Ireland. If there is any point upon which there is a perfect agreement among these gentlemen, it is this.1 That they are not known arises from the system of uniting. The system generally practised for teaching these rules is the cause of the great ignorance which the pupils exhibit. It may be called the system of unit-ing, by which each figure gets a name, independent of, and longer than its predecessor. By it such a number as fifty millions, sixty thousand, and five, instead of being grouped in the child's mind as 50, 060, and 005, must be decomposed into the following: 5 units O tens O hundreds O units of thousands 6 tens of thousands O hundreds of thousands O units of millions 1 See all the Reports of the Commissioners of National Education, Ireland, and the following in England: Min. of Council, 1845-6, vol. ii. p. 188; 1846-7, vol. i. p. 134; vol. ii. p. 93; 1847-8, vol. ii. p. 190; I 1856-7, pp. 467 and 631; 1857-8, p. 585; 1859-60, pp. 30 and 193; 18601, pp. 170 and 188; 1861-2, pp. 114 and 130; and also Education Commissioners, 1861, vol. i. p. 259, vol. ii. p. 558. How few children can see that all this is equivalent to the simple expression above! It cannot, therefore, be a matter of surprise that the junior classes are unable to make use of notation at all, and that the senior classes practise it with extreme slowness and difficulty. It is impracticable in a large school. Teaching notation on this system is quite impracticable in a large school. Suppose, for instance, that thirty children are assembled for a half-hour's lesson, and that they are told to put down four hundred billions, fifty millions, sixty thousand and five. Now, leaving out of the question the blunders they make and rectify before they are correct, let us suppose that the slates are ready for inspection. Before the master can decide upon the accuracy or inaccuracy in any one case, he must repeat the whole of the following: 5 units ◊ tens O hundreds O units of thousands 6 tens of thousands O hundreds of thousands O units of millions 5 tens of millions O hundreds of millions O units of billions O tens of billions 4 hundreds of billions. And, to test the accuracy of the whole class, he must go over all this no less than thirty times. When, therefore, we consider the errors which, as a matter of course, he will meet with, and the instructions he will, as a consequence, be called upon to give, we can have no hesitation in saying that he will be unable to go over all within the half-hour. Notation and numeration are, therefore, omitted from the school teaching, for this omission is imperative by the system of instruction adopted. Unit-ing, though a bad system of numeration, is not at all a system of notation. It must, however, be acknowledged that it is a system of numeration, though so bad as to be worthless, but it is not at all a system of notation in the true sense of the word; for, if a child taught thus is required to put down fifty-six thousand, three hundred, and eight, for instance, he is obliged, from having no primary guide, to make the best guess he can at it; and this may probably be : 56,300,8. NOTATION NOT KNOWN. He then begins to unit' or 'numerate' it, and says:— 8 units O tens O hundreds 3 units of thousands. 115 Finding thus that the three is in the wrong place, he blots out a cipher, and begins again : 8 units O tens 3 hundreds, &c., &c. And so he continues blotting out the figures, and repeating the above rhyme, until the whole is right. Many at Practice fail in notation. Is there any wonder, then, that there should be ignorant pupils and despairing teachers! I have met with frequent cases in which pupils who were able to do Practice were quite unable to subtract two numbers of seven or eight figures when dictated to them, and with other cases where, though right, they exhibited more doubtfulness and slowness than they would evince in the solution of a difficult question in the higher commercial rules. Teachers say that the failure is caused by nervousness, but this is not so. Teachers cannot conceive the cause of these failures, and are too prone to attribute them to that nervousness and diffidence which most children feel when under examination, forgetting altogether that the same nervousness, were it really in existence, would produce bad answering with more certainty on the higher and more difficult rules. Real cause. The real cause is to be found in the teacher's own neglect, consequent upon the laborious and impracticable systems adopted. Most of them would rather teach any rules than these, and some of them pretend to look upon them as entirely beneath their notice. The children, also, taking their cue from the masters, refuse very frequently even to learn what is thus so much despised. It is scarcely necessary to say, that if they were thoroughly known they would indeed be beneath the notice of the teachers and advanced classes; but when not known, the more rudimentary they are the greater must be the shame attached to those who are ignorant of them. It will always be found, however, that those who despise these rules on account of their simplicity are merely seeking a cloak for inefficiency. The true system is simple. The true system of teaching numeration and notation is very simple. |