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however, she should work out every question, mentally, along with the class. The main advantage of mental arithmetic is, the wonderful manner in which it disciplines some of the most important faculties of the mind, particularly those of attention, abstraction, and reasoning. But to gain these advantages, in any considerable degree, the pupils should distinctly know, that the questions are never to be repeated. They must give their whole attention, while the question is reading, and they must retain the whole in their minds, until they have found the answer, and explained the process by which it was discovered. The books ought to be used thus : The teacher reads, 6. Your brother William gave you nineteen cents, your brother John, ten, and your cousin Mary, two. How many have been given to you, in all ?”
c. (after consideration.) Thirty-one. T. How do
know ? C. Because brother William gave me nineteen, brother John, ten ; now ten and nineteen make twenty-nine ; and cousin Mary gave me two; twenty-nine and two make thirty-one.
T. Very well. Twelve men are to have ninety-six dollars for performing a piece of work. How much is due to each ?
C. Eight dollars.
C. Because, as the twelve men were to have ninetysix for their work, and as there are eight twelves in ninety-six, of course each man would have eight.
I now give a question in a more advanced stage.
T. A cistern has two cocks; the first will fill it in three hours, the second in six hours ; how long would it take both to fill it ?
C. Two hours.
C. Because if the first can fill it in three hours, it will fill one third of it in one hour ; and if the second will fill it in six hours, it will fill one sixth in one hour ; but one third is equal to two sixths ; therefore, both will fill three sixths, or one half, in one hour; or the whole, in two hours.
Some of the exercises in addition, in Colburn's 'First Lessons,' are so easy, though not the less important, that there is some danger of the class allowing their minds to wander, and yet answering correctly. This may be checked, by varying the questions, as follows : Instead of, Nine and four ? Nineteen and four ? Twenty-nine and four ? Thirty-nine and four ? regularly increasing the number of ty, let them be varied, thus : Twenty-nine and four ? Forty-nine and four ? Thirty-nine and four ? Fifty-nine and four ? &c:
It requires some tact, to gain the utmost advantage from mental arithmetic ; but it is easily acquired. The main point is, that the attention of the teacher be kept wide awake. The dull and slow must be allowed time; the bright must not be suffered to monopolize the answers. At the same time, it will not do for the answers to be received, in the order in which the pupils stand in the class; for, in this case, only one child would be occupied at
Each pupil would attend only to his own question; whereas all should be occupied, and should actually solve every question put to the class. The best plan, then, is, for each to hold up a finger, when ready to answer, leaving the teacher to select whose turn it shall be. Thus, every one might have an equal chance. The dull and the bright, however, ought not to be together, but in different classes. In fact, it would be well to have the classes differently arranged, for each separate study. Some are bright at reading, and dull in arithmetic, and vice versa.
To chain the dull to the bright has bad effects on both.
The Pestalozzian plates, at the end of Colburn's book, may or may not be studied, at the option of the teacher. They are explained in the 'Key,' page 141. At all events, they should be clearly understood by the teacher.
Abbreviations in Mental Arithmetic. The following abbreviations have never before been published. They may probably not only be useful to the student, but lead to the invention of others, equally profitable. To multiply by 5.
Take half the number, and multi
ply by 10. We take hall, because multiplying by 10 gives' double of multiplying by 5. Thus, 5 X 64 =
X 10 = 32ty, or 320. When the number is odd, halving leaves a remainder of 1, which, of course, is one 5. Thus, 73 X 5=73 X 10 = 36ty and five, or 365.
Let us next proceed to 15, 20, 25, 30, &c., and afterwards take up the intervening numbers.
Fifteen is 10 and half of 10, therefore, increasing any number a half, and multiplying by 10, is the same as multiplying by 15. Thus, as 64 and half of 64 make 96, 64 x 15 =96ty, or 960. When the number is odd,
, proceed as above, in speaking of 5. Thus, 75 X 15 = 112ty and five, or 1125, and the square of 15 is 22ty and five, or 225.
Twenty being two tens, to multiply by 20, double the number, and inultiply by 10. Thus, 20 X 45 = 90ty, or 900.
Twenty-five is one fourth of 100; therefore, to multiply by 25, take ţ of the number for hundreds : every unit in the remainder is one twenty-five. Thus :
24 x 25 4 x 100 600.
27 X 25 = 27 X 100=675, &c. Fifty is half of 100; therefore, to multiply by 50, take 4 the number for hundreds. Thus, 24 X 50=4 X 100
Thirty is thrice ten ; therefore, to multiply by 30, take thrice the number, and multiply by 10. Thus, 24 X 30 = 72ty, or 720.
Let us now examine the intermediate numbers, which are all done on one principle. Fourteen times any number is 15 times that number, less once the number ; and 13 tiines any number is 15 times the number, less twice the number. Thus, 14 X 24 = 15 X 24, less once 24 ; and 13 X 24 = 15 X 24, less twice 24. Again, 16 X 24 = 15 X 24, more once 24 ; and 17 X 24 =15 X 24, more twice 24. Thus, by connecting two numbers less, and two numbers more, with our 15, 20, 25, 30, &c., we have all the intermediate numbers.
Division is performed by reversing these processes ; that is, multiplying, where division is shown above; and dividing, where multiplication is indicated. Though not so easy as multiplication, some practice in it will be useful.
This system of abbreviations may seem obscure or difficult, perhaps, to those who have never practised mental arithmetic. But nothing is hazarded in the assertion, that, where Colburn's Arithmetic is used, as pointed out above, the class will understand and apply it with ease and rapidity, before they have gone half through that work. The teacher may exemplify the abbreviations for himself, on the slate ; but they should be performed by the school, exclusively in the mind.
It is a matter of the first importance, that the teacher should have a distinct idea of the objects to be gained by the practice of mental arithmetic ; as, otherwise, the main advantages that might result from it will assuredly be lost. Let it be constantly borne in mind, then, by the teacher, that the knowledge of arithmetic is not the chief benefit to be derived from it, but one of secondary importance. It is the mental discipline, the power of abstraction, the habit of attention and of reasoning which it developes, that constitutes its chief value. But all these advantages are lost, if the child is allowed to study the book ; more especially by working out the questions on the slate. They can only be completely attained, by calling on the class to solve each question mentally, merely from hearing it once read, and then to give a clear account of his mental operations. And, so beautifully are the questions arranged, so completely does the knowledge gained in each question, come into requisition in those that follow, that, if the plan of study be commenced right, and strictly followed, the most intricate and difficult questions will give no trouble to the class.
It may, perhaps, be incredible to some, but it is not the less true, that Colburn's book may be gone through, and correct notions be attained of the principles of arithmetic, without the knowledge of a single character. А child, who can neither write nor read, who has never even seen a figure, will probably acquire this knowledge more readily than those who fully understand them. Notwithstanding this, however, as the knowledge of figures is an indispensable part of education, and as its acquisition is much the easiest in early youth, as soon as a child can hold his pencil correctly, and can write the ten characters, he should proceed to the practice of
Written Arithmetic. Notation.—The method of expressing large numbers and of performing large operations, by words, is so inconvenient and tedious, that, from the earliest periods known, characters have been invented to
them more concisely. Almost every ancient nation had a method peculiar to itself ; only three of which, however, are necessary to be known by the American student.
The first is the Greek mode, in which the first nine letters of their alphabet represented the numbers from 1 to 9, and the next nine letters represented the tens from 1 to 9; that is, 10, 20, &c., to 90. The hundreds they represented by the other letters, supplying what were wanting by other marks, or characters; and in this order they proceeded, using the same letters, again, with different marks, to express thousands, tens of thousands, &c. This method of notation is now only used in Greek books, to denote the numbers of chapters, sections, &c.
The second is the Roman method, which is now used by all the European nations, and their American descendants, for distinguishing dates, chapters, and sections of books, &c. This, like the Greek, is derived from the alphabet, but in a different order. The origin of the Roman numerals is sufficiently evident; and, as a knowledge of this origin is not only an aid to its acquisition, but will enable the student to recall it to mind when forgotten, an explanation of it will not be considered out of place here.
The ten fingers present so obvious and convenient a mode of numeration, that every tribe and people, hitherto